  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.
