Transport equations with resting phases

Abstract

We study a transport model for populations whose individuals move according to a velocity jump process and stop moving in areas which provide shelter or food. This model has direct applications in ecology (e.g. homeranges, territoriality, stream ecosystems, travelling waves) or cellular biology (e.g. movement of bacteria or movement of proteins in the cell nucleus). In this paper we consider a general model from a mathematical point of view. This provides general insight into the features of these models, which in turn is useful in the modelling process. We consider a singular perturbation expansion and show that the leading order term of the outer solution satisfies a reaction-advection-diffusion equation. The advective term describes taxis toward homeranges or toward regions of shelter. The reaction terms are given by “effective” birth and death rates. Within this framework, the parameters of the reactionadvection-diffusion model (like mobility, drift, birth or death rates) are directly related to the individual movement behaviour of the species at hand (like velocity, frequency of directional changes, response to spatial in-homogeneities, death, or reproduction). We prove that in a homogeneous environment the diffusion limit approximates the solution of the resting-phase transport model to second order in the perturbation parameter.