Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions

We study a variant of the cyclic Lotka-Volterra model with three-agent interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors game, the model describes an ideal ecosystem in which cyclic competition among three species develops through cooperative predation. Its rate equations in a well-mixed environment display a degenerate Hopf bifurcation, occurring as reactions involving two predators plus one prey have the same rate as reactions involving two preys plus one predator. We estimate the magnitude of the stochastic noise at the bifurcation point, where finite size effects turn neutrally stable orbits into erratically diverging trajectories. In particular, we compare analytic predictions for the extinction probability, derived in the Fokker-Planck approximation, with numerical simulations based on the Gillespie stochastic algorithm. We then extend the analysis of the phase portrait to heterogeneous rates. In a well-mixed environment, we observe a continuum of degenerate Hopf bifurcations, generalizing the above one. Neutral stability ensues from a complex equilibrium between different reactions. Remarkably, on a two-dimensional lattice, all bifurcations disappear as a consequence of the spatial locality of the interactions. In the second part of the paper, we investigate the effects of mobility in a lattice metapopulation model with patches hosting several agents. We find that strategies propagate along the arms of rotating spirals, as they usually do in models of cyclic dominance. We observe propagation instabilities in the regime of large wavelengths. We also examine three-agent interactions inducing nonlinear diffusion.Comment: 22 pages, 13 figures. v2: version accepted for publication in EPJ

Competição cíclica e jogos assimétricos de predador-presa

Padrões cíclicos em populações biológicas competitivas têm ganhado destaque considerável em Teoria Evolutiva do Jogos nas últimas décadas, uma vez que eles parecem ser características importantes na manutenção da biodiversidade em sistemas com competição. Em contrapartida, uma parte considerável das populações que interagem competitivamente na natureza, não parece apresentar padrões cíclicos de comportamento, como por exemplo as interações entre predadores e presas (especialmente quando há apenas duas espécies envolvidas). Neste trabalho, investigamos e comparamos modelos em ambos os contextos. Inicialmente, revisamos uma generalização com quatro estratégias do jogo Pedra-Papel-Tesoura, analisando o papel da intransitividade na manutenção da coexistência entre as espécies, tanto em Campo Médio quanto em uma rede espacialmente estendida. Em seguida, consideramos o modelo de Lett et al. [1] em que predadores podem atacar colaborativamente presas isoladas ou agrupadas. As vantagens e desvantagens desses comportamentos dependem de uma série de condições, e a Teoria Evolutiva dos Jogos dispõe de ferramentas úteis para estudar tais sistemas, uma vez que ela se propõe a resolver problemas envolvendo conflitos de interesse tanto em sistemas sociais quanto em biologia evolutiva e ecologia. Consideramos uma versão estocástica espacial do modelo de Lett et al. [1] através da distribuição das populações em uma rede quadrada. Comparamos então os comportamentos evolutivos das densidades populacionais com os resultados previstos na versão do modelo em campo médio, mostrando que na presença de organização espacial surgem comportamentos mais ricos envolvendo novas transições de fase. Mostramos também que a coexistência entre as estratégias coletiva e individual, tanto para predadores quanto presas, está presente também nas simulações em rede, sendo uma fase estável. Além disso, a persistência dessa fase se deve a um mecanismo efetivo de dominância cíclica, similar á generalização do jogo Pedra-Papel-Tesoura com quatro estratégias, revisada na primeira parte do trabalho. Esse resultado demonstra, por uma abordagem não usual, que a intransitividade é um mecanismo robusto de manutenção da diversidade.Cyclic patterns in competitive biological populations have been gaining popularity amongst evolutionary game theorists in the last decades, since they appeared to have an important role on biodiversity maintenance in competitive biological systems. On the other hand, a substantial part of competitive populations in nature does not seem to present any cyclic behavior, as is the case of the majority of the interactions between predators and prey (especially when there are just two species involved). Here we investigate and compare models in both contexts. First we analyze a cyclic competition model, which is a generalized version of the Rock- Paper-Scissors game with four strategies, exploring the role of intransitivity on the maintenance of the species coexistence both in a mean field approach as well as in a spatially extended network. Next we study a predator-prey model in which predators may attack isolated or grouped prey in a cooperative, collective way. Whether gregarious behavior is advantageous to each species depends on several conditions and Game Theory has some useful tools to deal with such a problem, since its main purpose lies in dealing with conflicts of interest, even in the context of Evolutionary Biology and Ecology (the Game Theory branch which covers those topics is called Evolutionary Game Theory). We here extend the Lett et al. [1] model to spatially distributed groups and compare the resulting behavior with their mean field predictions for the coevolving densities of predator and prey strategies. We show that the coexistence phase in which both strategies for each group are present is stable because of an effective, cyclic dominance behavior similar to a generalization of the Rock-Paper-Scissors game with four species presented in the first part of this work, a further example of how ubiquitous this mechanism is

Cyclic Competition and Percolation in Grouping Predator-Prey Populations

We study, within the framework of game theory, the properties of a spatially distributed population of both predators and preys that may hunt or defend themselves either isolatedly or in group. Specifically, we show that the properties of the spatial Lett-Auger-Gaillard model, when different strategies coexist, can be understood through the geometric behavior of clusters involving four effective strategies competing cyclically,without neutral states. Moreover, the existence of strong finite-size effects, a form of the survival of the weakest effect, is related to a percolation crossover. These results may be generic and of relevance to other bimatrix games