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.