diff --git "a/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/load_file.txt" "b/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/FNAyT4oBgHgl3EQfrPlH/content/tmp_files/load_file.txt" @@ -0,0 +1,772 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf,len=771 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='00556v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='PE] 2 Jan 2023 Competition of alliances in a cyclically dominant eight-species population Junpyo Parka, Xiaojie Chenb, and Attila Szolnokic,∗∗ aDepartment of Applied Mathematics, Kyung Hee University, Yongin 17104, Republic of Korea bSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China cInstitute of Technical Physics and Materials Science, Centre for Energy Research, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Box 49, H-1525 Budapest, Hungary Abstract In a diverse population, where many species are present, competitors can fight for surviving at individual and collective levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In particular, species, which would beat each other individually, may form a specific alliance that ensures them stable coexistence against the invasion of an external species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our principal goal is to identify those general features of a formation which determine its vitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, we here study a traditional Lotka-Volterra model of eight-species where two four-species cycles can fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Beside these formations, there are other solutions which may emerge when invasion rates are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The complete range of parameters is explored and we find that in most of the cases those alliances prevail which are formed by equally strong members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, there are regions where the symmetry is broken and the system is dominated by a solution formed by seven species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our work also highlights that serious finite-size effects may emerge which prevent observing the valid solution in a small system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Keywords: cyclic dominance, alliances, competition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Introduction If a species is stronger and beats the other one then how can the “weaker” species survive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is a fundamental problem of ecology to explain the amazing biodiversity of life [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A possible expla- nation could be the so-called intransitive or cycli- cally dominated relation among competing species where a third (or more) species beat the preda- tor species, hence establishing a delicate balance among all competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the simplest case, these relations remind the well-known rock-scissors-paper game and this type of interaction can really be observed among animals, plants, or even among bacterias [3, 4, 5, 6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Evidently, a three- member loop can be easily enlarged for more par- ticipants, as it was made in the extended Lotka- Volterra model [10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Motivated by the complexity of such interactions, researchers have studied related models actively over the last decade and several interesting observations have been made [14, 15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' ∗Email addresses: junpyopark@khu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' xiao- jiechen@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' szolnoki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='attila@ek-cer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='hu ∗∗Corresponding author It would be almost impossible to sum all impor- tant observations, such as reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [19, 20, 21, 22, 23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Instead, the reader is referred to specific review papers about the milestones along this research path [26, 27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nevertheless, it is also worth noting here that the mentioned cyclic or intransitive relation among participants should not necessarily be defined by a Lotka-Volterra-type mi- croscopic rule, but could be the result of collective behavior among competing strategies in evolution- ary game models [30, 31, 32, 33, 34, 35, 36, 37, 38] The most exciting question is whether we can predict the direction of evolution based on the food- web that defines the microscopic dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Can we identify general principles which may inform us about the vitality of an alliance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' An early observa- tion was, for instance, that when two three-member cycles fight then the one, where the inner invasion is faster, is more viable [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The picture, how- ever, is more subtle, because a faster general rota- tion does not necessarily provide an advantage if the members of the trio are unequal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the lat- ter case, the triplet becomes vulnerable no matter the average of inner invasion rates is higher than in the rival triplet which is formed by less aggressive, Preprint submitted to Chaos Solitons and Fractals January 3, 2023 but equally strong partners [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Notably, hetero- geneous invasion rates may emerge temporarily or locally [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Or, the extension of the original food-web by adding a reverse link can also change a stable solution [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Furthermore, the number of members could also be a decisive factor because a smaller loop is generally stronger than a large one [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this work we introduce a minimal eight-species model where two four-member loops fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The interaction between these quartets practically establishes a traditional Lotka-Volterra circle of eight species, which can be described by an inva- sion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Not to break the balance between the quartets, we assume that the inner invasion is uni- form and equally strong in both formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The only difference between them is we introduce two short-cuts in one of the loops which generates extra interactions within the related alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our major question is how the stability of alliances change due to this extension and can we identify new solutions which would be hidden otherwise?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It could also be interesting whether the previously established principles, observed for simpler systems formed by trials and duets, remain valid when we apply them to alliances formed by larger groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Alliances formed at different levels To make our observations comparable with pre- vious works [45, 46, 47, 48, 49, 50], we assume that species are distributed on an L × L square lattice where periodic boundary conditions are ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Each node is occupied by one of the species which are denoted by Xi where indexes are from i = 0 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' During an elementary step, we select a neighboring pair nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If they are occupied by the same species or represent species which are neu- tral then nothing happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Otherwise, an invasion happens with a specific probability and predator species invades prey species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The microscopic rules are defined in the following way: XiXi+1 γ−→ XiXi (1) XiXi+2 α−→ XiXi (2) X2X6 β−→ X2X2, X4X0 β−→ X4X4, (3) where i + 1 and i + 2 are calculated in cyclic man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The parameters α, β and γ define the proba- bilities of successful invasions between the involved predator-prey neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' γ α β 1 2 4 0 3 5 6 7 Figure 1: Food-web of competing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the basic model eight species are invade cyclically each other with probabil- ity γ, similar to the extended Lotka-Volterra model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ad- ditionally, we introduce a cyclic inner invasion among odd and separately among even species with probability α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To break the symmetry among the quartets, we also introduce an inner invasion between species “2” and “6” and between species “4” and “0” with probability β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Importantly, we use the color-code of species in later figures where spatial distri- butions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For a deeper insight about the model, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 1 we show the food-web of competing species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our first note is about the biggest loop where all species are involved, according to the extended Lotka-Volterra type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here every species is simultaneously a predator and a prey of another one, hence es- tablishing a possible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The invasion rate is γ for all interactions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Another solution is formed by “1”, “3”, “5”, and “7” species who in- vade each other cyclically with probability α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For later reference, we denote this alliance as A4 1,3,5,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A similar A4 0,2,4,6 quartet is formed among species “0”, “2”, “4” and “6”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Additionally, we introduce an extra chance of invasions among group members here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In particular, species “2” invades species “6” and species “4” invades species “0” with probabil- ity β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, there are no neutral pairs in the mentioned quartet anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As a consequence, the average inner invasion is augmented among the four species, which could be a support to their via- bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, however, there are many other possible solutions in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For example, the new invasions among even-numbered species offer 2 the chance for trials to emerge: A3 0,2,4 or A3 0,2,6 can work as a rock-scissors-paper-type solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Beside the mentioned formations, there are A5 (five-), A6 (six-), or A7 seven-member solutions, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Per- haps, it is not necessary to specify them because the reader can easily construct examples, based on the food-web, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As we noted, there are three parameters in our model and in the following we systematically scan the whole parameter field to identify the dominant solution for each combination of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For this reason, we executed Monte Carlo simulations where a Monte Carlo step (MCS) means that on average every node has a chance to update its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The linear system size varied between L = 100 to L = 3200 and the necessary relaxation steps were between 103 to 3 × 105 MCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' According to the standard protocol, simulations were launched from an initial state where species are distributed ran- domly on the lattice and monitor the ρi concentra- tion of Xi for all species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We, however, would like to stress that this initial state does not always could be a good choice to find the solution which is valid in the large size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is because a small sys- tem size does not necessarily “offer” equal chance for all possible solutions to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Some solution, which involves large typical lengths, would emerge just later, but to that stage of the evolutionary pro- cess other solutions may invade the whole available space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, a more complex solution practi- cally has no chance to emerge at a small system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As a consequence, other solutions may win and an independent run can easily terminates onto a differ- ent solution, no matter we use the same set of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To overcome this uncertainty, we not simply enlarged the system size, but also used alter- native initial states where larger and homogeneous domains of competing species were distributed ran- domly on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Similar approach was used previously in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [51, 52], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, we can create all types of interfaces which could be the foundation to build more sophisticated solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To make our analysis more complete, at the critical values of control parameters we also gener- ated sub-solutions independently in restricted areas of space and after we let them fight directly along the separating interface [53, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This technique can identify the dominant solution unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The details of the above mentioned protocols will be specified in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phase diagrams We first summarize our main findings and later we give further details about the microscopic mech- anisms which are responsible how dominant solu- tions emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' According to our simulations, we can distinguish three main cases which determine the system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The decisive condition is the intensity of interaction between the quartets men- tioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If this interaction is weak, means the value of γ is small, then the A4 {1,3,5,7} solution dominates the whole β − α parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In other words, only the quartet of “1+3+5+7” sur- vive independently of the α and β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For intermediate values of γ, when the invasion flow in the large cycle becomes substantial, a con- ceptually new behavior emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We represent this phenomenon by showing the phase diagram ob- tained at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Figure 2 shows that beside the mentioned A4 {1,3,5,7} quartet, other solutions be- come dominant at certain values of β − α pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When both α and β are small then the “large cir- cle” is the winner, hence all eight species coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For intermediate β values this solution is replaced by A7 {0,1,2,3,4,6,7} where only species “5” is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 A4 {1,3,5,7} A7 {0,1,2,3,4,6,7} all β α Figure 2: Phase diagram on β − α parameter plane obtained at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If α is large enough, the quartet of “1+3+5+7” species always win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If both β and α are small, all species survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, for low α and intermediate β a seven- member solution dominates where species “5” is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Dashed blue line denotes discontinuous, while solid red line marks the positions of continuous phase transition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Qualitatively similar system behavior can be ob- served when γ is large, hence the flow in the ex- 3 ternal loop is intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In other words, the inter- action between the quartets becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3 where we present the com- plete phase diagram on the β − α plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The only difference between the diagrams is the area, where complete eight-species solution dominates, is larger and the seven-member solution is shifted toward larger β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 A4 {1,3,5,7} A7 {0,1,2,3,4,6,7} all β α Figure 3: Phase diagram on β − α parameter plane obtained at γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The diagram is similar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The only difference is the eight-member full solution occu- pies larger area in the parameter field, while seven-member solution is shifted toward higher β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We must stress that the presented phase dia- grams are valid in the large system size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If the system size is small then we may observe that the system can easily terminate onto different so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To illustrate it, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4 we present the survival probabilities at each (α, β) parameter pairs for γ = 1, when the linear system size was L = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' More precisely, we launched the evolution from a random initial state and monitored the fractions of species until N = L×L MCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' After, we recorded the number of surviving species and repeated the whole process 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way we can esti- mate the probability that S species survive for long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4 show these surviving probabilities for all possible S values at x different (α, β) pairs on the parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' These pan- els indicate that different solutions may emerge no matter we use the same values of (α, β) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, the destination is rarely unambiguous, hence the surviving probability equals to 1 only for S = 4 when α is high and β is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nevertheless, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 1 S=8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 Figure 4: Heat maps of the survival probability on β − α parameter plane obtained for γ = 1 by using 100×100 system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Each panel shows the probability to reach a state which contains S species after N = L2 MCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The value of S is indicated on each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The results are averaged over 100 independent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' the contour of areas around “maximum” values are roughly agree with the diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4 illustrated, this system size is insuffi- cient to make reliable conclusions about the proper system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Detecting phase transitions Therefore, to detect the phase transition points more precisely, we need to apply systematic finite- size analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' An example is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 5 where we present the “lack” of species “5” in dependence of α at fixed β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='75 and γ = 1 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When the value of 1 − ρ5 reaches 1 then the system enters onto the mentioned A7 {0,1,2,3,4,6,7} solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Evi- dently, we also checked the portions of other species, because ρ5 = 0 is fulfilled in other solutions, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here the symbols are the average of many indepen- dent runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As an example, for L = 100 we exe- cuted 2000 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Importantly, the average hides the proper system behavior for small system size, because it mixes different destinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For exam- ple, at L = 100, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='15 we never measured ρ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Instead, the majority of indepen- dent runs terminated onto a state where ρ5 = 0 or ρ5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This ambiguity disappears for large system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The plot also highlights that the tran- sition from the eight-member to seven-member so- lution is continuous because species “5” vanishes gradually as we increase α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The transition, how- ever, between A7 {0,1,2,3,4,6,7} and A4 {1,3,5,7} phases is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To illustrate another type of phase transition, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 6 we show an alternative order parameter in dependence of β at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here, we calculate the difference between the portions of 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 all A7 {0,1,2,3,4,6,7} A4 {1,3,5,7} 1 - ρ5 α 100 200 400 800 1600 Figure 5: The absence of species “5”, as an order param- eter, in dependence of α at γ = 1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='75 for different system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The linear sizes are indicated in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The dominant solutions are marked on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The plots are the average of 100-2000 runs depending on the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Lines are just to guide the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' quartets formed by odd- and even-indexed species, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When β is small, all available species coexist, the system is in the “all” phase, hence the mentioned difference is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When this parame- ter becomes 1, then species with even indexes van- ish and the quartet of “1+3+5+7” species becomes dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interestingly, this formation is viable even if the inner invasion, due to the tiny α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='01, is extremely slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The explanation of this surpris- ing behavior is given in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Similarly to the previously discussed Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 5 case, the average of the order parameter could be mis- leading when the system size is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This can be clearly seen for L = 100 and L = 200, where the av- erage is larger than the value for larger system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is because the system can easily be trapped in the A4 {1,3,5,7} state already at small β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The pos- sible destinations, however, are less ambiguous for larger sizes, but the jump in the order parameter is valid signaling a discontinuous phase transition at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Battles of solutions In the following, we analyze the possible mech- anisms which explain the system behavior summa- rized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To get a deeper insight about the dominant processes, which drive the pat- ter formation, it is instructive to use a specific ini- tial state where we divide the available space into two halves and both regions are occupied by one of the main quartets formed by odd- or even-indexed 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='8 all A4 {1,3,5,7} (ρ1+ρ3+ρ5+��7) - (ρ0+ρ2+ρ4+ρ6) β 100 200 400 800 1600 Figure 6: The difference between symmetric quartets in de- pendence of β at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When this order param- eter becomes 1, the system enters to a four-species state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The dominant solutions are marked on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The plots are the average over 20-2000 runs, depending on the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The linear size of squares are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, we can follow how these solu- tions compete, or how new possibilities may emerge due to their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The β − α parameter plane can be divided into four main sectors which fundamentally determine the relation of different solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We first con- sider the low α – low β region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is impor- tant to note that the A4 {0,2,4,6} quartet formed by even-indexed species is not a proper solution here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Instead, we can see the battle of A3 {0,2,4} and A3 {0,2,6} triplets when even-indexed species are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Since species “4” beats species “6”, this fight ends up with the victory of the former solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This domain is marked by “A” on the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When γ is small then there is just a weak interaction between odd- and even-indexed species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, the mentioned domains remain compact and they fight along the separating inter- face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Finally, A4 {1,3,5,7} quartet win this battle and the whole space will be occupied by the domain, marked by “B” on the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If γ is high enough then the evolution changes drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is illustrated on the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Because of high γ, species “7”, who has no predator in the “0+2+4” triplet, can easily en- ter into the A3 {0,2,4} domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The raid of species “7” results in a neutral duo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This A2{2, 7} so- lution is marked by “C” on this panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interest- ingly, however, this solution has limited life, be- cause the intensive interactions of the original quar- tets at the border offer a chance for the complete 5 A B C D Figure 7: Pattern formation in the low α – low β region when we launch the evolution from a prepared state where left (right) side of the space is occupied by even-indexed (odd- indexed) species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the former case only the A3 {0,2,4}triplet survive, shown by “A” on the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For small γ, when the interaction is weak around the main loop, the A4 {1,3,5,7} quartet, marked by “B”, can beat the other solution and gradually invades the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If γ is high enough, species “7” can easily invade the triplet leaving a neutral A2 {2,7} pair behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But this solution, marked by “C”, can- not survive because the interface between the original quar- tets is a birthplace of the complete eight-species solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is marked by “D”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Since γ is high enough, this solution is viable and prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Color codes agree with those we intro- duced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Parameters are: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='05 (left panel), and γ = 1 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' eight-member solution to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This solution is marked by “D” on this panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Because of the high γ value, the general invasion flow is intensive in the largest loop, which makes it strong against A4 {1,3,5,7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Notably, the latter quartet is weak due to small α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The above described mechanism explains the left-down corners of phase diagrams shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2 and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We can face a conceptually different situation in the large α – small β corner of the parame- ter plane, because it provides supporting conditions both for A4 {0,2,4,6} and A4 {1,3,5,7} quartets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' There- fore, both solutions would be viable in the absence of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Importantly, the high α value makes the difference from the previously discussed cases, which generates a fast rotation withing both quar- tets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' There is, however, a crucial difference between these solutions: while A4 {1,3,5,7} is formed by equal partners, the extra inner invasions described by β probability break this delicate balance for the ben- efit of species “0” at the expense of species “6”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way the pattern formed by even-indexed species becomes heterogeneous, including larger domains of “0” species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This makes the whole alliance vulner- able against the attack of the rival well-balanced loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This process is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 8, where we 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0 20000 40000 60000 80000 100000 120000 0 6 2 4 1 3 5 7 ρi time [MCS] Figure 8: The time evolution of species when quartets are fighting at high α, low β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Initially both A4 {0,2,4,6} and A4 {1,3,5,7} solutions evolve independently in separated areas of available space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When separating borders opened, marked by an arrow, symmetric quartet gradually crowd out the rival gang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Color codes are the usual, except white line is replaced by grey one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Parameters are: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, L = 1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' monitor the battle of these quartets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Initially, we allowed both solutions to emerge peacefully in re- stricted areas and the stationary portions of species “0” and “6” change, as we described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Let us stress that in the initial state all species repre- sented equally, which is practically invisible because the unequal stationary distributions of “0+2+4+6” species evolve very fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' After 20000 MCSs, when the separating borders opened, the symmetric so- lution gradually invades the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Notably, the presented simulation was recorded at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1, where the interaction between the quartets is mod- erate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Still, this light communication is capable to reveal the advantage of A4 {1,3,5,7} solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If we apply larger γ values then nothing changes concep- tually, but the victory of A4 {1,3,5,7} becomes faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hence, in agreement with the phase diagrams, the symmetric four-member solutions is always the win- ner in the mentioned corner of the β − α parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The lastly discussed observation also answers one of our original questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Namely, one may argue that the introduction of an additional inner inva- sion within A4 {0,2,4,6} solution results in a faster ro- tation among these species, which could be useful them [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But the weakening consequence of sym- metry breaking is stronger, as it was also the case for three-member loops [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Next, we discuss the small α – large β corner of the parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here, the situation is partly 6 similar to the first discussed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' More precisely, A4 {0,2,4,6} solution is unstable and replaced very soon by A3 {0,2,4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But this triplet is vulnerable, be- cause the large heterogeneity of inner flow, (α ≪ β), results in huge homogeneous domains, as it was al- ready reported in [56] for traditional rock-scissors- paper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Such a large homogeneous domain is also an easy prey of the rival A4 {1,3,5,7} quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This picture is valid for small and medium γ values, where the interaction between the even- and odd- indexed species is not too strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For large γ val- ues, however, the evolution could be more complex, which can be observed more easily from a “random- patch” initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This starting pattern is illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' As we already mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 2, this prepared initial state can offer all possible in- terfaces to be present at the very beginning, which is extremely useful when there are large difference among the invasion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this way, we can ob- serve the valid solution already at smaller system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' We stress, however, that the mentioned state should be reached from all kind of initial states if all available species are present and the system size is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' (a) (b) (c) Figure 9: Alternative initial states from where evolution is launched when L = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Panel (a) shows the traditional starting state where every node is uploaded by a randomly chosen species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Panel (b) shows a state where two competing quartets are generated first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Panel (c) shows a state where larger patches of species are distributed randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The so- lution, which is valid in the large size limit, can be reached from all starting points, but the necessary system size could be largely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Turning back to the large γ, large β, small α re- gion, the above specified prepared initial state can help us to identify the valid A7 {0,1,2,3,4,6,7} solution already at relatively small system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' At first sight, this solution may seem weird or exotic, but its emergence can be understood if we follow how the portion of species change by increasing β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This phe- nomenon is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 10 where we present the stationary fractions of all species when we in- crease the intensity of additional invasions at fixed γ = 1 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' At β = 0 we have a completely symmetric food-web where there are equally strong 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='7 all A7 {0,1,2,3,4,6,7} 0 1 2 3 4 5 6 7 ρi β Figure 10: Stationary fractions of species in dependence of β at γ = 1, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The results are obtained at L = 3200 system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Species are denoted to every lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For clarity we only show the lines without symbols here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The stable phases are shown on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' quartets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Consequently, all species are present at the same level here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When we introduce a non-zero β then not only the fractions of species “0” and “6” start decaying, but simultaneously, the portions of their principal preys, species “1” and “7” start growing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This increment involves the decay of their preys, which are species “2” and “0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This dou- ble stress explains why species “0” suffers the most at small β values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Naturally, the decay of species “6” affects the population of its principal predator, hence the portion of species “5” also a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' One may argue that the decay of species “0” should affect its main predator species “7”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But the predator of the latter, which is species “6”, can- not grow here, hence in sum species “7” should not necessarily decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This difference between the status of species “5” and “7” explains why there are diverse consequences when both of their principal preys, species “6” and “0”, are attacked directly via an inner invasion described by parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Impor- tantly, the direct support of species “4” via the in- tensive “4”→“2” helps the spreading of species “4”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This process is also dangerous for species “5”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' On the other hand, the intensive “2”→“6” process will not simply strengthen species “2”, but also weaken its prey species “3”, which consequence is also en- joyed by species “4”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When the latter species die out, the remaining seven species form a heteroge- neous seven-member Lotka-Volterra loop where the “weakest” species (species “4”, who beats species “6” with probability α) occupies the largest frac- tion of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This effect is an example for the phe- 7 C B F H G A D E I Figure 11: Intermediate stage of the evolution taken from the battlefield when different solutions emerge and fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The domains mark the following solutions: A4 {0,1,3,4} (A), A5 {1,3,4,5,7} (B), A4 {1,3,5,7} (C), A5 {0,2,3,5,7} (D), A5 {0,1,3,4,6} (E), A3 {0,2,4} (F ), A6 {1,2,3,4,5,7} (G), A5 {1,2,4,5,7} (H), and A5 {1,2,3,5,7} (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Finally “C” domain wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Param- eters are α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, γ = 1, and L = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' nomenon observed first by Tainaka in the simplest three-member loop [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Last, we discuss the remaining large α – large β corner of the parameter plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Here the A4 {0,2,4,6} solution is not stable again, hence the remaining A3 {0,2,4} solution fights against A4 {1,3,5,7} quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' When γ is low and the interaction is weak between these groups then the latter formation wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This evolution is similar to the case we reported in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Practically, the same happens when γ is high, but in this case a large set of solu- tions may emerge temporarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Despite of this di- versity, however, the winning alliance remains the symmetric quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' To give an impression about the variety of different candidates, we present an in- termediate snapshot about the “battlefield” taken at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='9, and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Figure 11 shows that in the intermediate stage of the evolution at least nine(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=') solutions emerge locally and fight for space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' But eventually the symmetric A4 {1,3,5,7} so- lution crowds out all other competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Conclusions Which are the most important features of a cycli- cally dominant alliance that determine its viabil- ity against alternative formations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Motivated by this question, we introduced an eight-species model where there are two four-member quartets who in- teract each other, hence forming a complete eight- member loop, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' The original model is com- pletely symmetric where the inner invasion within the quartets are equally strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Additionally, we break the symmetry and introduced an extra inner invasion within one of the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Based on pre- vious observations about three-member loops, one may argue that this faster rotation of alliance mem- bers could be positive to make them stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' From the other viewpoint, the broken symmetry is always detrimental, hence the new opportunity would just weaken the involved quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' It is also worth not- ing that the slight alteration of the original “sym- metric” food-web offers the chance several potential solutions to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Indeed, an armada of candi- dates can be observed at intermediate stage of the evolution, but it is believed that the final pattern is determined by some basic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' According to our findings, keeping the symme- try is vital and the solutions, which maintain a balance among their members, are fitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In the complete parameter space we studied, it can hap- pen that there are more than one solution which possess this attractive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' In this case the general speed of inner invasion could be a decisive factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' If the rotations are equally strong then we could give examples when a short or a larger loop is the winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' For example, a quartet can beat a trio, but a quartet can also beat an octet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, it cannot be made a simple conclusion that a smaller or larger alliance is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Probably, there are two competing effects whose relation determines the ac- tual fitness of a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' On one hand, a short loop may involve relatively large homogeneous domains, which could always be dangerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' On the other hand, too large loop also means that the reaction of the alliance could be delayed, because several inner invasions should happen to produce the predator of the external intruder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Therefore, the sum of these adverse impacts could be case-sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Naturally, our present study is a sterile theoreti- cal model, but we strongly believe that the observa- tions we made could be useful when real systems are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Our another important message is the im- portance of appropriately chosen system size, which 8 is always a central issue when cyclic dominance is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Otherwise, the observations could be di- verse without deeper understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This is why we should always consider the actual size when we want to give predictions about the pattern forma- tion in a finite system driven by intransitive inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' This work was supported by the National Re- search Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' NRF- 2021R1A4A1032924).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' was also supported by a grant from Kyung Hee University in 2022 (KHU- 20220901).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' was supported by the National Natural Science Foundation of China (Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 61976048 and 62036002) and the Fundamental Re- search Funds of the Central Universities of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' was supported by the National Research, De- velopment and Innovation Office (NKFIH) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' K142948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Sigmund, Games of Life: Exploration in Ecology, Evolution and Behavior, Oxford University Press, Ox- ford, UK, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Bascompte, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Sol´e (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' ), Modeling Spatiotemporal Dynamics in Ecology, Springer, New York, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Sinervo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Lively, The rock-paper-scissors game and the evolution of alternative male strategies, Nature 380 (1996) 240–243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Burrows, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hawkins, Modelling patch dy- namics on rocky shores using deterministic cellular au- tomata, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 167 (1998) 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Kerr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Riley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Feldman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Bohan- nan, Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors, Nature 418 (2002) 171– 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Cameron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' White, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Antonovics, Parasite- grass-forb interactions and rock-paper-scissor dynam- ics: predicting the effects of the parasitic plant Rhi- nanthus minor on host plant communities, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ecol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 97 (2009) 1311–1319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ruifrok, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Janzen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Kuijper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rietkerk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Olff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Smit, Cyclical succession in grazed ecosys- tems: The importance of interactions between different- sized herbivores and different-sized predators, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Popul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 101 (2015) 31–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Garde, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ewald, ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Kov´acs, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Schuster, Mod- elling population dynamics in a unicellular social organ- ism community using a minimal model and evolutionary game theory, Open Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 10 (2020) 200206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Liao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Miano, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nguyen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Chao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hasty, Survival of the weakest in non-transitive asymmetric interactions among strains of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' coli, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 11 (2020) 6055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Frachebourg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Krapivsky, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ben-Naim, Spatial organization in cyclic Lotka-Volterra systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 54 (1996) 6186–6200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Esmaeili, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Brown, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, Perturbing cyclic predator-prey systems: How a six-species coars- ening system with nontrivial in-domain dynamics re- sponds to sudden changes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 98 (2018) 062105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Baker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, The effect of habitats and fit- ness on species coexistence in systems with cyclic dom- inance, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 486 (2020) 110084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Bayliss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nepomnyashchy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Volpert, Beyond rock-paper-scissors systems - deterministic models of cyclic ecological systems with more than three species, Physica D 411 (2020) 132585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [14] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Durney, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Case, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Zia, Saddles, arrows, and spirals: Deterministic trajectories in cyclic competition of four species, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 83 (2011) 051108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Cazaubiel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' L¨utz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Arenzon, Collec- tive strategies and cyclic dominance in asymmetric predator-prey spatial games, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 430 (2017) 45–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Choi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Xu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hui, Adaptive cyclically dominating game on co-evolving networks: numerical and analytic results, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' B 90 (2017) 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Roman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Dasgupta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, A theoretical approach to understand spatial organization in complex ecologies, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 403 (2016) 10–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szab´o, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Sznaider, Segregation pro- cess and phase transition in cyclic predator-prey models with even number of species, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 76 (2007) 051921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Park, Correlation between the formation of new com- peting group and spatial scale for biodiversity in the evolutionary dynamics of cyclic competition, Chaos 32 (2022) 081101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Yoshida, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Mizoguchi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hatsugai, Non-Hermitian topology in rock-paper-scissors games, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 12 (2022) 560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Tainaka, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nakagiri, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Yokoi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Sato, Multi- layered model for rock-paper-scissors game: A swarm intelligence sustains biodiversity, Ecological Informatics 66 (2021) 101477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Serrao, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' T¨auber, Stabilizing spiral struc- tures and population diversity in the asymmetric may– leonard model through immigration, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' B 94 (2021) 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Park, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Do, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Jang, Multistability in the cyclic competition system, Chaos 28 (2018) 113110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Nagatani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ichinose, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Tainaka, Heterogeneous network promotes species coexistence: metapopulation model for rock-paper-scissors game, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 8 (2018) 7094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Mobilia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rucklidge, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szczesny, The influence of mobility rate on spiral waves in spatial rock-paper- scissors games, Games 7 (2016) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Mobilia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Jiang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szczesny, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rucklidge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, Cyclic dominance in evolution- ary games: a review, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interface 11 (2014) 20140735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szab´o, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' F´ath, Evolutionary games on graphs, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 446 (2007) 97–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [28] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Dobramysl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Mobilia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' T¨auber, Stochastic population dynamics in spatially extended predator-prey systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 51 (2018) 063001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' de Oliveira, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Bazeia, Pattern for- 9 mations driven by cyclic interactions: A brief review of recent developments, EPL 131 (2020) 68001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [30] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hauert, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' De Monte, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hofbauer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Sigmund, Vol- unteering as Red Queen mechanism for cooperation in public goods game, Science 296 (2002) 1129–1132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Chen, Strategy dependent learning ac- tivity in cyclic dominant systems, Chaos Soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Fract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 138 (2020) 109935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Palombi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Ferriani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Toti, Coevolutionary dy- namics of a variant of the cyclic Lotka-Volterra model with three-agent interactions, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' B 93 (2020) 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Chen, Alliance formation with exclu- sion in the spatial public goods game, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 95 (2017) 052316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [34] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Canova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Arenzon, Risk and interaction aver- sion: Screening mechanisms in the prisoner’s dilemma game, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 172 (2018) 279–292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, Second-order free-riding on anti- social punishment restores the effectiveness of prosocial punishment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' X 7 (2017) 041027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [36] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Tao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Hu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Shi, Risk-preference-driven partici- pate willingness provides alternative routes to solve so- cial dilemma, EPL 135 (2021) 28001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Zhu, Dynami- cally generated cyclic dominance in spatial prisoner’s dilemma games, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 82 (2010) 036110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [38] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Xiao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Chen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, Early exclusion leads to cyclical cooperation in repeated group interac- tions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Interface 19 (2022) 20210755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szab´o, Cyclical interac- tions with alliance specific heterogeneous invasion rates, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 75 (2007) 052102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Blahota, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Blahota, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, Equal partners do better in defensive alliances, EPL 131 (2020) 58002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, Biodiversity in models of cyclic dominance is preserved by heterogeneity in site-specific invasion rates, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 6 (2016) 38608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, Zealots tame oscillations in the spatial rock-paper-scissors game, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 93 (2016) 062307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [43] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Bazeia, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' de Oliveira, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, Breaking unidirectional invasions jeopardizes biodiver- sity in spatial May-Leonard systems, Chaos, Solitons and Fractals 141 (2020) 110356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [44] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' de Oliveira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, Competition among al- liances of different sizes, Chaos, Solitons and Fractals 157 (2022) 111940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Intoy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, Synchronization and extinction in cyclic games with mixed strategies, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 91 (2015) 052135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' L¨utz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Risau-Gusman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Arenzon, Intransitivity and coexistence in four species cyclic games, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 317 (2013) 286–292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [47] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Mir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Stidham, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, Emerging spa- tiotemporal patterns in cyclic predator-prey systems with habitats, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 105 (2022) 054401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [48] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szab´o, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, Phase transitions induced by variation of invasion rates in spatial cyclic predator- prey models with four or six species, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 77 (2008) 011906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [49] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Brown, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Meyer-Ortmanns, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Pleimling, Dy- namically generated hierarchies in games of competi- tion, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 99 (2019) 062116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Park, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Jang, Robust coexistence with alternative competition strategy in the spatial cyclic game of five species, Chaos 29 (2019) 051105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [51] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szab´o, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, Phase diagrams for the spatial public goods game with pool punishment, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 83 (2011) 036101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Chen, Competition and partnership be- tween conformity and payoff-based imitations in social dilemmas, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 20 (2018) 093008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, Reentrant phase transitions and defensive alliances in social dilemmas with informed strategies, EPL 110 (2015) 38003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [54] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Perc, Competition of tolerant strate- gies in the spatial public goods game, New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 18 (2016) 083021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [55] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Szolnoki, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Danku, Dynamic-sensitive coopera- tion in the presence of multiple strategy updating rules, Physica A 511 (2018) 371–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Juul, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Sneppen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Mathiesen, Clonal selection pre- vents tragedy of the commons when neighbors compete in a rock-paper-scissors game, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' E 85 (2012) 061924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' [57] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Tainaka, Paradoxial effect in a three-candidate voter model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' A 176 (1993) 303–306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'} +page_content=' 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAyT4oBgHgl3EQfrPlH/content/2301.00556v1.pdf'}