Dynamical Analysis of a Prey-Predator Model Including Fear, Its Carry-Over Effect and Allee Effect with Memory Effect

In ecological systems, predation is a fundamental interaction influencing species survival and biodiversity. Understanding the long-term dynamics of prey-predator relationships necessitates examining various models that incorporate different biological phenomenas. In this study, we investigate a predator-prey model that combines fear, its carry-over effect and Allee effect. The indirect effects of predators can act as a suppressive factor on the reproduction of prey populations. We incorporate the classic fear factor and extend it to include the concept of carry-over effect, which refers to how an individual’s past experiences and history influence their current performance in specific situations. Additionally, we account for the Allee effect in the prey population, an important ecological mechanism driven by factors like mate limitation, cooperative defense, cooperative feeding, and environmental conditioning. Given that many biological systems inherently exhibit memory effects, we employ the Caputo fractional derivative, which is beneficial due to its non-local properties. This approach helps us better understand how past experiences influence current hunting behavior, leading to more realistic outcomes. Our research provides deeper insights into predator-prey dynamics and introduces a novel method for modeling ecological systems with memory effects. The stability analysis of the equilibrium points of the system is performed and the existence of a Hopf bifurcation around the positive equilibrium point is investigated. Then the theoretical findings are verified by numerical results and interpreted biologically.