具有小选择系数的空间进化博弈
Spatial Evolutionary Games with small selection coefficients
摘要：
在这里，我们将使用Cox，Durrett和Perkins的结果为选民模型扰动$Z^d$，$d\ge 3$的相互作用内核是有限范围，对称的，并具有协方差阵$σ^2I$的空间进化博弈研究。我们考虑的博弈具有${\bf 1} + wG$形式的支付矩阵，其中${\bf 1}$是全1的矩阵，$w$是小的正的。由于我们的群体大小$N=\infty$，我们称我们的选择小而不是弱，这通常意味着$w =O(1/N)$。研究这些博弈的关键是，当动力学在空间和时间上适当地重新缩放时，它们收敛于反应扩散方程（RDE）的解。受Ohtsuki和Nowak和Tarnita等人工作的启发，我们证明了反应项是修改后的博弈矩阵的复制子方程，并且博弈矩阵的修改仅通过相关合并随机游走的两个或三个简单概率的值依赖于相互作用核。两个策略博弈导致了具有立方非线性的RDE，因此我们可以完整地描述相图。三个策略游戏导致一对耦合RDE，但使用我们早期工作的想法，我们能够表明，如果有一个排斥函数的复制方程的修改后的游戏，然后有共存的空间游戏时，选择是小的。这使我们能够证明共存的空间模型中的各种各样的例子，其中的复制方程的修改后的游戏有一个吸引力的平衡与所有组件的积极。利用这一结果，我们能够分析最近在癌症建模中使用的四种进化游戏的行为。
英文摘要：
Here we will use results of Cox, Durrett, and Perkins for voter model perturbations to study spatial evolutionary games on $Z^d$, $d\ge 3$ when the interaction kernel is finite range, symmetric, and has covariance matrix $σ^2I$. The games we consider have payoff matrices of the form ${\bf 1} + wG$ where ${\bf 1}$ is matrix of all 1's and $w$ is small and positive. Since our population size $N=\infty$, we call our selection small rather than weak which usually means $w =O(1/N)$. The key to studying these games is the fact that when the dynamics are suitably rescaled in space and time they convergence to solutions of a reaction diffusion equation (RDE). Inspired by work of Ohtsuki and Nowak and Tarnita et al we show that the reaction term is the replicator equation for a modified game matrix and the modifications of the game matrix depend on the interaction kernel only through the values of two or three simple probabilities for an associated coalescing random walk. Two strategy games lead to an RDE with a cubic nonlinearity, so we can describe the phase diagram completely. Three strategy games lead to a pair of coupled RDE, but using an idea from our earlier work, we are able to show that if there is a repelling function for the replicator equation for the modified game, then there is coexistence in the spatial game when selection is small. This enables us to prove coexistence in the spatial model in a wide variety of examples where the replicator equation of the modified game has an attracting equilibrium with all components positive. Using this result we are able to analyze the behavior of four evolutionary games that have recently been used in cancer modeling.
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