Noise-induced stabilization and fixation in fluctuating environment

Immanuel Meyer,Nadav M Shnerb

doi:10.1038/s41598-018-27982-1

Abstract

The dynamics of a two-species community of N competing individuals are considered, with an emphasis on the role of environmental variations that affect coherently the fitness of entire populations. The chance of fixation of a mutant (or invading) population is calculated as a function of its mean relative fitness, the amplitude of fitness variations and their typical duration. We emphasize the distinction between the case of pairwise competition and the case of global competition; in the latter a noise-induced stabilization mechanism yields a higher chance of fixation for a single mutant. This distinction becomes dramatic in the weak selection regime, where the chance of fixation for a single deleterious mutant is an N-independent constant for global competition and decays like (ln N)−1 in the pairwise competition case. A Wentzel-Kramers-Brillouin (WKB) technique yields a general formula for the chance of fixation of a deleterious mutant in the strong selection regime. The possibility of long-term persistence of large [{mathscr{O}}(N)] suboptimal (and extinction-prone) populations is discussed, as well as its relevance to stochastic tunneling between fitness peaks.

Highlights

A fundamental problem in the fields of population genetics, evolution, and community ecology, is to predict the fate of a single mutant introduced in a finite population of wild types
Description number of individuals in the community. number of individuals belonging to the mutant population. fraction of mutants, x = n/N (1 − x is the fraction of wild type). the time-independent component of the fitness. the amplitude of fitness fluctuations. correlation time of the environment, measured in generations. the strength of environmental fluctuations. environmental stochasticity g in units of demographic noise 1/N. scaled selection. useful derived parameter
Using a dominant balance argument we showed that the dynamic is governed by a single second-order equation

Summary

Model B

(n), abundance variations[5], one would expect a substantial impact of these fluctuations on the chance of fixation. The storage effect stabilizes a coexistence state when the fitness affects recruitment but death occurs at random. This is the situation in our model B, which is an individual based version of the lottery game considered by Chesson and Warner[25], see a detailed discussion in[17]. Models A and B are the two extreme scenarios; in general, as long as the effect of fitness on recruitment is larger than its effect on death, one should expect the stabilizing mechanism to affect the system

Model Definitions and the Backward Kolmogorov Equation

Discussion

Findings

Additional Information