Mathematical insights into chaos in fractional-order fishery model

This study investigates the dynamics of a fractional-order model applied to fishery management to better illustrate the behavior of fish populations over time in a two-zone aquatic environment. The zones consist of unreserve and reserve zones prohibited from fishing. Initially, an integer-order nonlinear differential equation model was modified to fractional order in the Caputo sense modified intrinsic growth rate. Subsequently, the models are analyzed for positivity, boundedness, existence, and uniqueness, and stability analysis within the framework of the Caputo derivative order. The key parameters fishing mortality (ρ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\rho )$$\end{document} and harvesting (Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Lambda )$$\end{document} allowed a detailed exploration of population growth and stability under various harvesting scenarios. The model is numerically solved using the Adam–Bashforth scheme with the Caputo derivatives. This method accounts for fractional order derivatives and provides an efficient numerical solution for nonlinear systems that are commonly observed in biological processes. Numerical simulations, varying the fractional order of the Caputo derivative, examine the impact of model parameters on system dynamics and control. In the fractional case, we establish sufficient conditions to guarantee the model’s uniqueness and existence. An analysis of the dynamics of the system under various parameter settings and under different conditions which is potentially significant for understanding the complex behaviors of diverse biological systems. With the different input factors of the system, a novel numerical technique is presented for the chaotic and dynamic behaviour of the proposed model. Our analysis also shows that fractional order has an impact on the proposed system fishery model. Through numerical simulations, the most critical input parameters are highlighted and control interventions are suggested for policy makers to consider.