Nonlocal pattern formation effects in evolutionary population dynamics

Abstract

In this work, we derive a deterministic equation for a generalized nonlocal population dynamics that depends fundamentally on two lengths parameters (α,β). We associate these parameters to the reproduction, competition, and pattern formation in bacterial dynamics. We propose that each set of parameters defines a state, and based on that, we formulate a stochastic cellular automaton for describing the evolutionary dynamics of a bacterial colony. It results in a selection of the bacteria more able to share space. In this way, we show that the evolution of the colony towards a maximum population allowed, in a given niche, defines the final pattern.