Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

Abstract

We consider a nonlinear degenerate parabolic equation containing a nonlocal term, where the spatial variable $x$ belongs to $\mathbb{R}^d$,$d \geq 2$. The equation serves as a replicator dynamics model where the setof strategies is $\mathbb{R}^d$ (hence a continuum). In our model the payoff operator (which is the continuous analog of the payoff matrix) isnonsymmetric and, also,evolves with time. We are interested in solutions $u(t, x)$ of our equation which are positive and their integral (with respect to $x$)over the whole space $\mathbb{R}^d$ is $1$, for any $t > 0$. These solutions, being probability densities, can serve as time-evolving mixedstrategies of a player. We show that for our model there is an one-parameter family of self-similar such solutions $u(t, x)$, all approachingthe Dirac delta function $\delta(x)$ as $t \to 0^+$. The present work extends our earlier work [11] which dealt with the case $d=1$.