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 $λ$ and natural death rate one. The process also depends on a payoff $a_{11} = a$ 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 $λ$ 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 $λ$. Using various block constructions, we also prove that, for all birth rates $λ$, 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 $λ$. This underlines the importance of space (local interactions) and stochasticity in our model.