Switching behaviour of two-phenotype bacteria in varying environment

Abstract

An increasing interest in multi-phenotype species has stimulated both experimental and mathematical research. One example is bacteria which have two phenotypes and can make transitions from one phenotype to the other in response to variations in environmental conditions. We model a population of such bacteria subjected to a stochastic environmental input, which fluctuates between two conditions preferred by the phenotypes. Our interest in this model is how the average growth rate of the total population is affected by alterations to the environmental thresholds at which the transitions between phenotypes are allowed. Under certain conditions, we find that the bacteria achieve a maximum growth rate by adjusting their behavior to act in a similar manner to a non-ideal relay. In this scenario, memory helps to increase fitness. We then extend the model to include multiple competing species with different thresholds and examine the limit of distribution of population among these species and phenotypes as time increases. For this purpose, we formulate a reaction-diffusion model which involves non-ideal relays describing the evolution of the state of different species; and, a Preisach operator with time-dependent density function to account for the integral effect of the species on the environment. Formation of patterns and multiple stationary limits are shown numerically in the multi-species model.