Stochastic Partial Differential Equation Models for Spatially Dependent Predator-Prey Equations

TL;DR

This paper develops stochastic partial differential equation models for predator-prey systems to better capture spatial inhomogeneity, incorporating functional responses and analyzing solution properties like existence, uniqueness, permanence, and extinction.

Contribution

It introduces a versatile SPDE framework for predator-prey models, including Beddington-DeAngelis response, and provides theoretical analysis of solutions.

Findings

Existence and uniqueness of solutions established.

Conditions for permanence and extinction derived.

Enhanced modeling of spatial predator-prey dynamics.

Abstract

Stemming from the stochastic Lotka-Volterra or predator-prey equations, this work aims to model the spatial inhomogeneity by using stochastic partial differential equations (SPDEs). Compared to the classical models, the SPDE model is more versatile. To incorporate more qualitative features of the ratio-dependent models, the Beddington-DeAngelis functional response is also used. To analyze the systems under consideration, first existence and uniqueness of solutions of the SPDEs are obtained using the notion of mild solution. Then sufficient conditions for permanence and extinction are derived.