On a stochastic Ricker competition model

TL;DR

This paper models the evolution of two competing populations using a stochastic Ricker competition model, analyzing their long-term behavior and coexistence through quasi-stationary distributions as competition effects diminish.

Contribution

It introduces a stochastic two-population model based on the Ricker framework and studies its asymptotic behavior near deterministic coexistence points.

Findings

The stochastic model closely follows the deterministic Ricker model for small competition parameters.

Under mutual invasibility, the deterministic model predicts coexistence in the open first quadrant.

As competition parameters approach zero, the quasi-stationary distributions are analyzed for long-term behavior.

Abstract

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.