On a stochastic Ricker competition model

We model the evolution of two competing populations $U_t, V_t$ 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 $U_t = m$ and $V_t = n$ the expected values of $U_{t+1}$ and $V_{t+1}$ are given by $me^{r - K(m + bn)}$ and $ne^{\tilde r - \tilde K (n + am)}$, respectively, where $r, \tilde r$ model the intrinsic population growth, $K, \tilde K$ 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 $K, \tilde K$ 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 $K, \tilde K$ approach 0.