Population dynamics in the triplet annihilation model with a mutating reproduction rate

Abstract

I study a population model in which the reproduction rate lambda is inherited with mutation, favoring fast reproducers in the short term, but conflicting with a process that eliminates agglomerations of individuals. The model is a variant of the triplet annihilation model introduced several decades ago [R. Dickman, Phys. Rev. B~{\bf 40}, 7005 (1989)] in which organisms ("particles") reproduce and diffuse on a lattice, subject to annihilation when (and only when) occupying three consecutive sites. For diffusion rates below a certain value, the population possesses two "survival strategies": (i) rare reproduction (0 < lambda < lambda_{c,1}), in which a low density of diffusing particles renders triplets exceedingly rare, and (ii) frequent reproduction (lambda > lambda_{c,2}). For lambda between lambda_{c,1} and lambda_{c,2} there is no active steady state. In the rare-reproduction regime, a mutating $\lambda$ leads to stochastic boom-and-bust cycles in which the reproduction rate fluctuates upward in certain regions, only to lead to extinction as the local value of lambda becomes excessive. The global population can nevertheless survive due to the presence of other regions, with reproduction rates that have yet to drift upward.