On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals

TL;DR

This paper develops general models for spatially heterogeneous nonlocal diffusion, exploring their connection to local diffusion models, and introduces deciding factors that influence steady-state profiles and individual positioning.

Contribution

It introduces the concept of deciding factors that determine nonlocal diffusion models and analyzes their impact on steady-state solutions and individual positioning.

Findings

Connection established between nonlocal and local diffusion models.

Deciding factors influence the steady-state profiles.

Dependence of individual positioning on model parameters.

Abstract

We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.