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, and have natural birth rate λ 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, they give birth at rate λ times the exponential of a times the fraction of occupied neighbors. In particular, the birth rate increases with the local density when a>0, in which case individuals have a beneficial effect on each other (cooperation), whereas the birth rate decreases with the local density when a<0, in which case individuals have a detrimental effect on each other (competition). Using standard coupling arguments to compare the process with the basic contact process, i.e., 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 the natural birth rate λ. Using various block constructions among other techniques, we also prove that, for all natural birth rates λ, there is a phase transition in the direction of the payoff 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 λ. This underlines the importance of space (local interactions) and stochasticity in our model.