deForest Russ, Belmonte Andrew
The W. G. Pritchard Laboratories, Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jun;87(6):062138. doi: 10.1103/PhysRevE.87.062138. Epub 2013 Jun 28.
We introduce a nondiffusive spatial coupling term into the replicator equation of evolutionary game theory. The spatial flux is based on motion due to local gradients in the relative fitness of each strategy, providing a game-dependent alternative to diffusive coupling. We study numerically the development of patterns in one dimension (1D) for two-strategy games including the coordination game and the prisoner's dilemma, and in two dimensions (2D) for the rock-paper-scissors game. In 1D we observe modified traveling wave solutions in the presence of diffusion, and asymptotic attracting states under a frozen-strategy assumption without diffusion. In 2D we observe spiral formation and breakup in the frozen-strategy rock-paper-scissors game without diffusion. A change of variables appropriate to replicator dynamics is shown to correctly capture the 1D asymptotic steady state via a nonlinear diffusion equation.
我们将一个非扩散性空间耦合项引入进化博弈论的复制者方程。空间通量基于各策略相对适应度的局部梯度所导致的运动，为扩散耦合提供了一种依赖于博弈的替代方式。我们通过数值研究了一维（1D）情况下两种策略博弈（包括协调博弈和囚徒困境）以及二维（2D）情况下剪刀石头布博弈中模式的发展。在一维情况下，我们观察到在有扩散时修正的行波解，以及在无扩散的固定策略假设下的渐近吸引态。在二维情况下，我们观察到在无扩散的固定策略剪刀石头布博弈中螺旋的形成与瓦解。结果表明，一个适用于复制者动力学的变量变换能够通过一个非线性扩散方程正确地捕捉一维渐近稳态。
