Dynamical Analysis of Fractional-Order Bazykin’s Model with Prey Refuge, Gestation Delay and Density-Dependent Mortality Rate

Abstract

The motivation of the present study is to investigate the impact of memory in the framework of ecology employing a Caputo-type fractional-order derivative by means of a fractional-order ecological model that incorporates delay and prey refuge treatment effects. The model’s solutions are shown to exist, to be unique, and to be bounded. The behaviour of various equilibrium points with the memory effect is then examined, and certain necessary requirements are deduced to guarantee the global asymptotic stability of co-existing equilibrium points. Additionally, we looked into the possibility of Hopf bifurcation in relation to the delay parameter, which serves as the suggested system’s bifurcation parameter. This paper’s main contribution is the explanation of the fractional order model’s derivation in terms of the memory impact on population growth, and the application of the Caputo derivative with equal dimensionality to models that include memory. This fractional-order system with unknown dynamics is subject to control chaos, which is addressed by using Bazykin’s prey-predator model. The suggested model is new in that it highlights the importance of the memory effect, which encompasses prey refuge, latency, and predator death rate based on density. We run numerical simulations with various memory parameter, latency, and prey refuge values. Based on the numerical data, it seems that the system is behaving more like a chaotic system with an increasing memory effect, or stable behaviour from a time of chaos.