Non local Lotka-Volterra system with cross-diffusion in an heterogeneous medium

TL;DR

This paper develops a stochastic individual-based model for spatially heterogeneous animal populations with competing species, leading to a novel nonlocal, cross-diffusion PDE system with unique existence and convergence properties.

Contribution

It introduces a new stochastic framework and proves existence, uniqueness, and convergence results for a complex nonlocal cross-diffusion Lotka-Volterra system in heterogeneous environments.

Findings

Proved global existence of solutions for the nonlocal PDE system.

Established uniqueness and conditions for solutions in functional spaces.

Derived large population limits using measure-valued process techniques.

Abstract

We introduce a stochastic individual model for the spatial behavior of an animal population of dispersive and competitive species, considering various kinds of biological effects, such as heterogeneity of environmental conditions, mutual attractive or repulsive interactions between individuals or competition between them for resources. As a consequence of the study of the large population limit, global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model of competing species is proved. The main new feature of the corresponding integro-differential equation is the nonlocal nonlinearity appearing in the diffusion terms, which may depend on the spatial densities of all population types. Moreover, the diffusion matrix is generally not strictly positive definite and the cross-diffusion effect allows for linearly growing influences of the subpopulations' sizes. We prove uniqueness of the finite measure-valued solution and give conditions under which the solution takes values in a functional space. We then make the competition kernels converge to a Dirac measure and obtain the existence of a solution to a locally competitive version of the previous equation. The large population limit is obtained by means of weak convergence tools for measure valued processes. The techniques employed in the study of the liming equation are novel, and are essentially based on the underlying stochastic flow related to the dispersive part of the dynamics, together with the use of suitable dual distances in the space of finite measures. This is an extended version of the paper "Non local Lotka-Volterra system with cross-diffusion in a heterogeneous medium" appeared in J. Math. Biol (2015) 70:829-854.