The Dynamics of Biological models with Optimal Harvesting

TL;DR

This paper introduces a new equilibrium concept for biological dynamical systems, analyzes a prey-predator model with growth modifications, and develops an optimal harvesting strategy using Pontryagin's maximum principle, supported by numerical simulations.

Contribution

It presents a novel equilibrium concept, analyzes a prey-predator model with growth modifications, and formulates an optimal harvesting strategy using discrete-time Pontryagin's maximum principle.

Findings

Conditions for local stability and equilibrium existence are established.

Optimal harvesting strategies are characterized mathematically.

Numerical simulations confirm theoretical results.

Abstract

This paper aims to introduce a concept of an equilibrium point of a dynamical system which will call it almost global asymptotically stable. A biological prey-predator model is also analyzed with a modification function growth in prey species. The conditions of the local stable and existence of all its equilibria are given. After that the model is extended to an optimal control problem to obtain an optimal harvesting strategy. The discrete time version of Pontryagin's maximum principle is applied to solve the optimality problem. The characterization of the optimal harvesting variable and the adjoint variables are derived. Finally numerical simulations of various set of values of parameters are provided to confirm the theoretical findings.