Evolutionary games on the lattice: multitype contact process with density-dependent birth rates

Interacting particle systems of interest in evolutionary game theory introduced in the probability literature consist of variants of the voter model in which each site is occupied by one player. The goal of this paper is to initiate the study of evolutionary games based more realistically on the multitype contact process in which each site is either empty or occupied by a player following one of two possible competing strategies. Like in the symmetric multitype contact process, players have natural death rate one and natural birth rate $\lambda$. Following the traditional modeling approach of evolutionary game theory, the process also depends on a payoff matrix $A = (a_{ij})$ where $a_{ij}$ represents the payoff a type $i$ player receives from each of its type $j$ neighbors, and the actual birth rate is an increasing function of the payoff. Using various couplings and block constructions, we first prove the existence of a phase transition in the direction of the intra payoff $a_{11}$ or $a_{22}$ while the other three payoffs are fixed. We also look at the behavior near the critical point where all four payoffs are equal to zero, in which case the system reduces to the symmetric multitype contact process. The effects of the intra payoffs $a_{11}$ and $a_{22}$ are studied using various couplings and duality techniques, while the effects of the inter payoffs $a_{12}$ and $a_{21}$ are studied in one dimension using a coupling with the contact process to control the interface between the 1s and the 2s.