Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

Vassilis G.  Papanicolaou; Kyriaki  Vasilakopoulou

doi:10.3934/jdg.2016003

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2016, Volume 3, Issue 1: 51-74. Doi: 10.3934/jdg.2016003

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Similarity solutions of a multidimensional replicator dynamics integrodifferential equation

Vassilis G. Papanicolaou^1, and
Kyriaki Vasilakopoulou^1,

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

Received: November 2015

Revised: January 2016

Published: January 2016

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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 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 $d=1$.

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Mathematics Subject Classification: Primary: 91A22; Secondary: 91B52.

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References

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