Abstract

We introduce a continuous time Markov chain to model the ecological competition in the population of 3 species with fitness governed by a modified Moran process based on environmental resources in a limited niche of constant total population N. We run simulations to observe population behavior under different N and initial conditions. We then propose a model approximation which for large N converges to an ODE over most of the population space, with the population following a deterministic trajectory until it reaches an asymptotically stable line. We then prove that the approximation converges to a one dimensional diffusion forced onto the stable line until the first extinction occurs. We use the drift and diffusion coefficients of the diffusion to calculate the expected probability of first extinction for specific species, as well as the expected time until first extinction. Finally, we compare these with data obtained via simulations to show that the approximation is a good fit.