Existence of traveling wave solutions appearing in game theory

Abstract We study a two-component reaction-diffusion model for a symmetric $$2\times 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> game with two strict Nash equilibria, where local interactions follow the best response dynamics. When the diffusion coefficients are equal, the direction of traveling wave solutions connecting these equilibria is determined solely by the payoffs. In this paper we consider the general case in which the diffusion coefficients may differ. This generalization is mathematically significant, as it leads to richer properties of traveling wave solutions, and conceptually meaningful, as it models situations where different strategies may correspond to different diffusion coefficients. For the resulting two-component system, we prove the existence, non-existence, uniqueness and multiplicity of traveling wave solutions connecting these equilibria, determine the sign of the wave speed, and describe their qualitative shape. Our results show that the direction of propagation depends on both the ratio of the payoffs and the ratio of the diffusion coefficients. Numerical simulations illustrating stability are also presented.