Analysis of the spatial-temporal model for harvested predator-prey system with prey refuge and intraspecific competition

Abstract

This study investigated the spatial-temporal dynamics of a predator-prey system incorporating prey refuge, intraspecific competition, harvesting, and diffusion. Spatial pattern formation in a predator-prey system governed by a reaction-diffusion framework was also considered. By incorporating spatial movement through diffusion, the non-spatial model by Mapunda and Sagamiko was extended into a partial differential equation (PDE) framework, enabling the analysis of self-organized spatial patterns such as spots and stripes that arose due to Turing instability. The coexistence equilibrium was analyzed for stability, and numerical simulations were performed using the Implicit-Explicit (IMEX) Euler method for time integration with the finite difference method employed for discretization of the Laplacian operator under Neumann boundary conditions (zero-flux), assuming no population flux across the boundary. The impact of key ecological parameters, particularly the refuge rate m and diffusion coefficients  and  on population distribution and pattern evolution was systematically explored. Simulation results revealed that increasing diffusion coefficients enhanced well-defined spatial aggregation of predators and prey while decreasing the refuge level intensified predation pressure and accelerated pattern formation. Specifically, when   (predators diffused faster than prey), the homogeneous coexistence equilibrium destabilized, leading to Turing-type patterns including isolated hotspots and periodic bands. At higher refuge levels such as  combined with elevated predator diffusion ( the system exhibited stable and matured spot patterns, indicating a balance between protection and mobility. These findings highlight the critical role of species mobility and protective mechanisms in shaping ecosystem structure and offered important insights for conservation strategies and the sustainable management of harvested populations in spatially heterogeneous environments.