Survival and extinction for a contact process with a density-dependent birth rate

To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the d-dimensional integer lattice, with natural birth rate and natural death rate one. The process also depends on a payoff a₁₁ = a modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by times the exponential of a times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when a > 0, in which case the payoff a models mutual cooperation, whereas the birth rate decreases with the local density when a < 0, in which case the payoff a models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case a = 0), we prove that, for all payoffs a, there is a phase transition from extinction to survival in the direction of. Using various block constructions, we also prove that, for all birth rates, there is a phase transition in the direction of a. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on. This underlines the importance of space (local interactions) and stochasticity in our model.