Study of memory effects in a fractional order leslie-gower model with holling type IV functional response

Abstract

The use of fractional derivatives in predator–prey models naturally accounts for memory effects, reflecting the fact that population interactions are influenced not only by present conditions but also by their historical states. By incorporating memory through fractional-order operators, these models provide a more realistic description of ecological dynamics and reveal how past population levels can significantly affect stability, persistence, and complex behaviors of predator–prey systems over time. In this paper, following fractional order Caputo derivative approach, here I first convert the integer order three species food chain model to the fractional order model. Some qualitative behaviors of the system like existence and uniqueness, non-negativity and boundedness which are systematically discussed in a feasible region. Local stability criteria of the different equilibrium points have been discussed for fractional order system. Global stability of the interior equilibrium point have been only discussed by defining suitable Lyapunov function. Numerical simulation is performed for different sets of biologically feasible parameter values by using adams-type predictor corrector method (PECE). Numerically it has been observed that the fractional order system shows more complex dynamics, like chaos, bifurcation for lower memory as the fractional order becomes larger and shows more simpler dynamics for higher memory as the order $m$ decreases. Specially, due to memory effect, it becomes stable for lower value of $m$ and the dynamics of the fractional-order system not only depends on system parameters but also depends on fractional order $m$.