Memory effect on prey-predator dynamics: Exploring the role of fear effect, additional food and anti-predator behaviour of prey

Memory over past events has great importance on the survival of species in an ecological system. It is helpful in searching favourite food, escaping from predator attacks, group defence, use of environmental protection in the future time for the species. To explore the role of different ecological effects like fear effect, additional food and anti-predator behaviour of prey in the presence memory on an ecological system, in this paper we have proposed a two-dimensional prey-predator model and formulated mathematically using a system of two fractional order differential equations. Here the Caputo sense fractional derivative is used to construct the fractional order differential equations. The existence-uniqueness, non-negativity and boundedness of system solutions have been established. The detailed mathematical and graphical analysis for the feasibility of different equilibrium points and their analytical conditions for local asymptotic stability has been studied. The sufficient parametric conditions for the global stability of non-trivial equilibrium points are investigated. The Hopf bifurcation analysis of the system around the coexistence equilibrium point is derived in terms of system parameters. Using numerical simulations we have established that the memory effect can stabilize the system from an unstable periodic situation and makes the above-considered ecological effects more positively effective.

Automated summary

Memory of past events is crucial for the survival of species in an ecosystem as it aids in finding food, escaping predators, group defense, and future environmental adaptation. This paper presents a two-dimensional prey-predator model formulated using fractional order differential equations to explore the impact of ecological effects on the system in the presence of memory. The study examines the existence, uniqueness, non-negativity, and boundedness of system solutions, as well as the feasibility and stability conditions of different equilibrium points. It is found that memory effect stabilizes the system and enhances the positive effects of considered ecological factors.