Cooperation under pressure: a phase diagram for altruism

Cooperation and altruistic behavior are enhanced in spatially structured
populations as compared with mean-field scenarios [Nowak and May, 1992].
In most studies of evolutionary game theory with spatial structure,
however, the size of the population is assumed to be constant. Here I
study a model where the population may vary in response to environmental
pressure p. I discuss two non-trivial effects. First, as increasing p
dilutes the population, cooperation becomes dominant and prevents
extinction for any non-zero efficiency of altruistic acts. Second, in a
large region of the parameter space, the increase of cooperators
overcompensates the decrease of defectors with growing p. Thus the overall
population size rises in response to increasing environmental pressure. I
obtain the phase diagram displaying these effects by stochastic
simulations and confirm it by solving the equations of pair approximation.