Investigating Stochastic Dynamics of the Gilpin-Ayala Model in Dispersed Polluted Environments

Abstract

This research delves into the analysis of a stochastic Gilpin-Ayala model operating within an anxious environment, encompassing the phenomenon of diffusion between two distinct and specified geographical regions that are the subjects of investigation. Initially, we rigorously formulate the essential criteria for ascertaining the survival or extinction of the species. Furthermore, we furnish empirical substantiation for the presence of a stable distribution. A significant milestone of our study involves the discernment and comprehensive delineation of the pivotal determinants that intricately regulate extinction dynamics and persistence within the framework of pollution parameters. This outcome underscores the pronounced impact of pollution on ecological dynamics and affirms the necessity of incorporating pollution parameters into the purview of environmental investigations. This revelation demonstrates that in the absence of pollution, the conventional criteria governing extinction and persistence closely parallel those witnessed in unpolluted environments, thus validating the robustness of our mathematical analysis. A series of numerical depictions are introduced to validate and provide empirical support for the acquired results.