Diffusion approximations for models of population growth with logarithmic interactions

Abstract

In this paper we have studied the diffusion approximations for the stochastic Gompertz and logarithmic models of population growth. These approximations are of particular importance whenever the corresponding Markov population processes are not analytically tractable. For each model we have shown that, as the resource-size tends to infinity, the process on suitable scaling and normalization converges to a non-stationary Ornstein–Uhlenbeck process. Consequently the sizes of species have in the steady state normal distributions whose means and covariance functions are determinable.