Abstract

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 prey 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.“Three at play. That’ll be the day!” (a child in Wings of desire [W. Wenders, 1987])

Graphical abstract

中文翻译：

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具有三主体相互作用的循环Lotka–Volterra模型变体的协同进化动力学

摘要

我们研究了具有三主体相互作用的循环Lotka–Volterra模型的变体。受到剪刀石头布多人游戏的启发，该模型描述了一种理想的生态系统，其中三个物种之间的周期性竞争通过合作捕食得以发展。在充分混合的环境中，它的速率方程显示简并的Hopf分支，这是因为涉及两个捕食者加一个猎物的反应具有与涉及两个猎物加一个捕食者的反应相同的速率。我们估计分叉点处的随机噪声的大小，在分叉点处，有限的大小效应会将中性稳定的轨道转变为不规则的发散轨迹。特别是，我们比较了以Fokker-Planck近似得出的灭绝概率的解析预测，基于Gillespie随机算法的数值模拟。然后，我们将相像的分析扩展到异构速率。在一个充分混合的环境中，我们观察到简并的Hopf分叉的连续性，将上述情况概括化。中性稳定性来自不同反应之间的复杂平衡。值得注意的是，在二维晶格上，所有分叉因相互作用的空间局部性而消失。在本文的第二部分中，我们研究了在带有多个代理的补丁的晶格元人口模型中迁移率的影响。我们发现，策略沿旋转螺旋的臂部传播，就像通常在循环优势模型中那样。我们观察到大波长范围内的传播不稳定性。“三个在玩。那将是一天！” （一个有渴望之翼的孩子[W. Wenders，1987]）

图形概要