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

TL;DR

This paper investigates a population model with mutating reproduction rates within a triplet annihilation framework, revealing complex survival strategies and boom-bust cycles driven by local reproduction rate fluctuations.

Contribution

It introduces a variant of the triplet annihilation model incorporating mutation in reproduction rate, highlighting new survival dynamics and population fluctuations.

Findings

Two distinct survival regimes identified based on reproduction rate

Mutations cause stochastic boom-and-bust cycles in local populations

Global survival possible despite local extinctions due to rate fluctuations

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.