On a stochastic Ricker competition model

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We model the evolution of two competing populations Uₜ, Vₜ by a two-dimensional size-dependent branching process. The population characteristics are assumed to be close to each other, as in a resident-mutant situation. Given that Uₜ = m and Vₜ = n the expected values of Uₜ₊₁ and Vₜ₊₁ are given by mer - K(m + bn) and ne r - K (n + am), respectively, where r, r model the intrinsic population growth, K, K model the force of inhibition on the population growth by the present population (such as scarcity of food), and a, b model the interaction between the two populations. For small K, K the process typically follows the corresponding deterministic Ricker competition model closely, for a very long time. Under some conditions, notably a mutual invasibility condition, the deterministic model has a coexistence fixed point in the open first quadrant. The asymptotic behaviour is studied through the quasi-stationary distribution of the process. We initiate a study of those distributions as the inhibitive force K, K approach 0.