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 R d , d ≥ 2. The equation serves as a replicator dynamics model where the set of strategies is 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 R d is 1, for any t > 0. These solutions, being probability densities, can serve as timeevolving 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 δ(x) as t → 0 +. The present work extends our earlier work [11] which dealt with the case d = 1.