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

The spatial behavior of an animal population of dispersive and competitive species is studied. It may be realized as a result of various kinds of biological effects, as heterogeneity of environmental conditions, mutual attractive or repulsive interactions between individuals, competition between individuals for resources and localization of the offspring during a reproduction event. A stochastic model describing the behavior of each individual of such a population is introduced and a large population limit is studied. As consequence, the global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model for competing species is proved. The main feature of the model is the nonlocal nonlinearity appearing in the diffusion and drift term. The diffusion matrix is non-symmetric and generally not positive definite and the cross-diffusions terms are allowed to be large if subspecies sizes are large. We prove existence and uniqueness of the finite measure-valued solution and give assumptions under which the solution takes values in a functional space. Then we make the competition kernel converge to a Dirac measure and obtain solution to a locally competitive version of the previous equation. The proofs are essentially based on the underlying stochastic flow related to the dispersive part of the dynamics.

Departamento de Matemáticas

Pontificia Universidad Católica de Chile (PUC-Chile)

Av. Vicuña Mackenna 4860, Macul,

Santiago – Chile

(+56 2) 2354 5779

Centro de Modelamiento Matemático (CMM)

Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

Beauchef 851, Edificio Norte, Piso 7,

Santiago – Chile