Mathematical Analysis of a Resource-Based Dispersal Model With Gompertz Growth and Optimal Harvesting

Abstract

Gompertz dynamics offer significant applications for the growth of invasive species, cancer modeling, optimal harvesting policies, sustainable yield, and maintaining population levels due to its pattern formation in low-density cases. This paper examines a widely applicable nonhomogeneous diffusive Gompertz law with zero Neumann boundary conditions, where all coefficients are smooth periodic functions. The analytical approach explains the ubiquitous stability of a time-periodic solution and seeks the optimal strategy for harvesting under the Gompertz growth law, potentially generalizing the results for many small organisms, including plants and wild populations. The proposed model successfully investigates the dynamics with and without diffusion. Moreover, the spatio-temporal equation more precisely describes the population’s evolutionary processes using a generalized classical reaction-diffusion equation. Finally, we observe several potential applications, outlining the optimal strategies for real-world scenarios and related fields where optimal harvesting is utilized.