Travelling waves and dominance of ESS's
The authors consider a simple two-strategy evolutionary game in which each strategy is an evolutionarily stable strategy (ESS). Their model differs from the conventional one in that there is also a spatial domain (taken to be the \(x\)-axis) over which the individuals playing the various strategies are distributed. Denote by \(U(x,t)\) resp. \(V(x,t)\) the number density at position \(x\) of the first resp. second strategy. Following earlier work of one of the authors, this gives the dynamical system \[ \partial U/\partial t=UV(\alpha U-\beta V)(U+V)^{-2}+\mu(\partial^ 2 U/\partial x^ 2), \] \[ \partial V/\partial t=-UV(\alpha U-\beta V)(U+V)^{-2}+\nu(\partial^ 2 V/\partial x^ 2). \] If both diffusion rates \(\mu\), \(\nu\) are zero and \(U\), \(V\) are spatially independent, then this system has two asymptotically stable equilibria, viz. (1,0) and (0,1). The population converges to one or the other depending on whether \(U\) is initially greater or less than \(\beta/(\alpha+\beta)\). The interesting case is the one where \(\mu\), \(\nu\) are not zero and the initial spatial distribution plays a role. Specifically, the authors consider the case where for \(x>0\) strategy 1 is played by almost everybody and for \(x<0\) strategy 2 is prevalent. The authors show that there exists a travelling wave solution of the form \[ U(x,t)=u(x- ct),\quad V(x,t)=v(x-ct). \] Such travelling wavefronts can in effect replace one ESS by another. The ``strength'' or ``dominance'' of each ESS which decides the ``winner'' in a precisely defined sense is determined by its pay-offs and by its diffusion rate. Good strategies have large pay-offs and small diffusion rates.
scientific article; zbMATH DE number 62362
scientific article