Stationary Coexistence in Stochastic Gause Systems with General Trophic Responses

Abstract

We study stationary coexistence in stochastic Gause predator-prey systems with independent multiplicative environmental noise and trophic responses depending on both population densities. A weighted population balance cancels the trophic interaction and yields moments of order greater than one without imposing predator self-regulation. These estimates give a direct occupation-measure proof of a sufficient coexistence criterion based on prey establishment and predator invasion of the prey-only stationary environment. For responses whose rare-predator limit is strictly increasing and whose value at positive predator density never exceeds that limit, we also establish necessity of positive predator invasion whenever the prey-only stationary law exists. If the prey per-capita growth attains its global maximum at zero, interior stationarity itself forces prey establishment, giving a necessary-and-sufficient criterion beyond logistic growth. Logistic and bounded saturating growth laws provide explicit Gamma and beta-prime resident distributions. Applications include Holling, Ivlev, Beddington-DeAngelis, and Crowley-Martin responses. Responses with the same rare-predator limit share the coexistence threshold, although predator interference changes the interior distribution. Reproducible quadrature and simulations illustrate this distinction. The analysis builds on existing stochastic persistence theory and identifies precisely the additional comparison argument needed for necessity, including exclusion of interior invariant measures at zero predator invasion.