Competition systems in a random environment: a convergence result

Abstract

In this article, a competition system in a random environment is considered. There are two species of particles and each will propagate as follows. An individual particle will move according to a Poisson jump process on the lattice Z ^d , split into two at a rate which is random, depending on the environment and die off at a rate which is random, depending on the environment. The main result is that, under the mass/speed rescaling (the particles are of mass {\rm ε } while the reproduction and death rate are rescaled accordingly), as the mass of an individual particle tends to zero, the densities of the species are given precisely by the pair of coupled stochastic partial differential equations

where Δ is the ‘Lattice Laplacian’. Here the {\rm κ } _i are the diffusion rates of each species (which are assumed to be constant) and the {\rm γ } _i are the parameters measuring the competitive effects of one species on the other. The quantities u and v denote the densities of the first and second species respectively. {\rm ξ } ⁽¹⁾ and {\rm ξ } ⁽²⁾ denote ‘noise’ terms and are the rescaled differences between the natural birth rates and death rates respectively (i.e. the differences between the birth rates and the death rates in the absence of any other species).

In the mass/speed rescaling, the variance of the densities of each species has vanished, so that these equations give the precise evolution of the zero-mass limit.

*The work is based substantially upon the dissertation for the Master of Science degree written by Elke Thönnes under the supervision of John Noble while both authors were at the Department of Statistics, University College Cork, Ireland.

Acknowledgments

Notes

*The work is based substantially upon the dissertation for the Master of Science degree written by Elke Thönnes under the supervision of John Noble while both authors were at the Department of Statistics, University College Cork, Ireland.