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

Abstract

I study a population model in which the reproduction rate λ 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 Dickman (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<λ<λc,1), in which a low density of diffusing particles renders triplets exceedingly rare, and (ii) frequent reproduction (λ>λc,2). For λ between λc,1 and λc,2 there is no active steady state. In the rare-reproduction regime, a mutating λ 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 λ becomes excessive. The global population can nevertheless survive due to the presence of other regions, with reproduction rates that have yet to drift upward.