A Class of Stochastic Predation Ecosystems with Nonlinear Predation Rates and Closed-form Densities

Abstract

The Lotka-Volterra model, which exhibits linear predation rates and is deterministic in nature, has been frequently used to describe population dynamics with predator-prey relations. However, real ecosystems are often probabilistic systems rather than deterministic paradigms. In a stochastic framework, the issue of persistence could be best studied through the stationary probability density of the system. This paper presents a class of stochastic predation ecosystem with general functional structures of predation rates and closed-form solution for its stationary probability density. The general functional structures allow different types of nonlinear predation rates to be incorporated and the introduction of stochastic shocks further facilitates the incorporation more realistic characteristics in predation ecosystem modeling. The model developed can be used to derive a variety of predation ecosystems. In particular, some complicated systems like multiple common preys and predators ecosystems, multiple connected food-chains, and food-chains with multiple species in each trophic level can be generated. This is the first time that explicit stationary densities of predation ecosystems at this level of complexity are obtained.