Coexistence in Competition-Diffusion Systems

Abstract

It is a central problem in population ecology to understand the mechanism of spatial patterning of ecological communities. In this paper, we will be concerned with regionally segregation of competing species in a homogeneous environment from a theoretical aspect. Suppose the situation where n species are competing with each other and moving by diffusion. Let u _i (t,x) be the population density of the i-th species at time t and position x for i = 1,2,..., n. Then the dynamics of u _i (t,x) are described by

$$ \frac{\partial }{{\partial t}}{{u}_{i}} = {{d}_{i}}\Delta {{u}_{i}} + \left( {{{r}_{i}} - \sum\limits_{{j = 1}}^{n} {{{a}_{{ij}}}{{u}_{j}}} } \right){{u}_{i}}, \left( {t,x} \right) \in \left( {0,\infty } \right) \times \Omega \left( {i = 1,2 \ldots ,n} \right), $$

((1.1))

where Δ is the Laplace operator in R ^N,rj is the intrinsic growth rate, a _ii and a _ii (i ≠ j) are respectively the coefficients of intra- and inter-specific competition and d _i is the diffusion coefficient (i = 1,2...,n). We assume that a habitat Ω is bounded in R ^N. First define a basically homogeneous environment for competing species by the following assumptions

• (A) r _i , a _ij and d_i} (i = 1,2, ...,n) are positive constants;

• (B) The boundary condition at the boundary ∂Ω is of zero flux, i.e.