# Dynamics of a harvested prey-predator model with prey refuge depended on both species

## TL;DR

This paper investigates a prey-predator model with prey refuge for both species and independent harvesting, showing how refuge influences system stability and establishing optimal harvesting policies through analytical and numerical methods.

## Contribution

It introduces a prey-predator model incorporating refuge for both species and analyzes how refuge affects stability and harvesting strategies.

## Key findings

- Refuge coefficient significantly influences system stability.
- Refuge can break limit cycles and achieve equilibrium.
- Numerical results support theoretical analysis.

## Abstract

The present paper deals with a prey-predator model with prey refuge proportion to both species and independent harvesting of each species. Our study shows that using refuge as control, it can break the limit circle of the system and reach the required state of equilibrium level. It is established the optimal harvesting policy. The boundedness, feasibility of interior equilibria, bionomic equilibrium have been determined. The main observation is that coefficient of refuge plays an important role in regulating the dynamics of the present system. Moreover the variation of the coefficient of refuge change the system from stable to unstable and vice-versa. Some numerical illustration are given in order to support of our analytical and theoretical findings.