Incorporating Memory Effects in the Caputo-Fabrizio Framework into a Predator-Prey Model with Structured Prey Dynamics

This study introduces a predator-prey fishery model possessing a distinctive structural composition, featuring a single-stage predator population alongside a two-stage prey population dynamic. Additionally, the model considers harvesting efforts targeting the mature prey and predator populations. The novelty of this work lies in the incorporation of a biologically realistic two-stage prey structure within a fractional-order framework, along with the use of non-singular Caputo-Fabrizio derivatives, which allows for more accurate modeling of memory-dependent processes in ecological systems. To confirm the biological relevance of the proposed model, we first establish the existence and uniqueness of its solution. All equilibrium points of the system are determined and identified. Furthermore, the stability of the system, both locally and globally, is analyzed by employing the Jacobian matrix for system linearization and utilizing the Lyapunov direct approach for a detailed assessment. Lastly, numerical simulations are provided to support the analytical results, along with a graphical analysis of the system's behavior. The numerical simulations are performed employing the three-step Adam-Bashforth technique for fractional-order systems within the Caputo-Fabrizio framework.