# American Institute of Mathematical Sciences

January  2016, 3(1): 51-74. doi: 10.3934/jdg.2016003

## Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

 1 Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece, Greece

Received  November 2015 Revised  January 2016 Published  March 2016

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 set of strategies is $\mathbb{R}^d$ (hence a continuum). In our model the payoff operator (which is the continuous analog of the payoff matrix) is nonsymmetric 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 mixed strategies of a player. We show that for our model there is an one-parameter family of self-similar such solutions $u(t, x)$, all approaching the Dirac delta function $\delta(x)$ as $t \to 0^+$. The present work extends our earlier work [11] which dealt with the case $d=1$.
Citation: Vassilis G. Papanicolaou, Kyriaki Vasilakopoulou. Similarity solutions of a multidimensional replicator dynamics integrodifferential equation. Journal of Dynamics & Games, 2016, 3 (1) : 51-74. doi: 10.3934/jdg.2016003
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