Diffusion approximations for lotka – volterra type models

Abstract

We obtain diffusion approximations for two bivariate models of population growth. These are useful whenever the corresponding Markov population processes are analytically intractable. In the context of birth and death processes, we consider stochastic versions of the deterministic models of Lotka – Volterra type, a model of species competition and a prey-predator model, both with logarithmic interactions. We obtain a diffusion approximation which show that, as the resource-sizes tend to infinity, the suitably scaled and normalized processes converge to non-stationary Ornstein – Uhlenbeck processes. Furthermore, we have determined from what order on the processes are well approximated by the limiting diffusions. Finally, in the homogenous situation, we have determined from what order onward the species sizes become, in the steady state, stationary Gaussian processes. We derive the mean vector and covariance for each model