Mathematical Study of a Resource-Based Diffusion Model with Gilpin-Ayala Growth and Harvesting

Abstract

This paper focuses on a Gilpin-Ayala growth model with spatial diffusion and Neumann boundary condition to study single species population distribution. In our heterogeneous model, we assume that the diffusive spread of population is proportional to the gradient of population per unit resource, rather than the population density itself. We investigate global well-posedness of the mathematical model, determine conditions on harvesting rate for which non-trivial equilibrium states exist and examine their global stability. We also determine conditions on harvesting that leads to species extinction through global stability of the trivial solution. Additionally, for time periodic growth, resource, capacity and harvesting functions, we prove existence of time-periodic states with the same period. We also present numerical results on the nature of nonzero equilibrium states and their dependence on resource and capacity functions as well as on Gilpin-Ayala parameter [Formula: see text]. We conclude enhanced effects of diffusion for small [Formula: see text] which in particular disallows existence of nontrivial states even in some cases when intrinsic growth rate exceeds harvesting at some locations in space for which a logistic model allows for a nonzero equilibrium density.

Keywords: Directed diffusion; Gilpin–Ayala; Global attraction; Harvesting.