# A fractional-order approach to predator-prey interactions: Modeling fear and disease dynamics with memory effects

## TL;DR

This paper introduces a new predator-prey model using fractional calculus to study the effects of fear and disease with memory in ecological systems.

## Contribution

The model introduces fear effects on recruitment, reinfection dynamics, selective predation on infected prey, and memory effects via fractional derivatives.

## Key findings

- The model identifies three equilibrium points representing disease and predator extinction, predator-free, and co-existence states.
- Local and global dynamics are analyzed using Matignon’s condition, Routh-Hurwitz criterion, and Lyapunov functions.
- Numerical simulations confirm forward and Hopf bifurcations showing dynamic changes with parameter variation.

## Abstract

The dynamical behaviors of a predator-prey model with fear effect and disease on prey are studied by employing the fractional-order derivative with a power-law kernel as the operator. The proposed model introduces four novel aspects: the impact of fear on a constant recruitment rate, previously unexplored in the literature; reinfection dynamics where infected prey can re-acquire the disease; selective predation by predators on infected prey due to pathogen-induced vulnerability; and the used of Caputo fractional derivative to include the memory effect. A deterministic approach is provided to establish the mathematical model including its validity by showing the existence, uniqueness, non-negativity, and boundedness. Three types of equilibrium points are obtained which represent the extinction of disease and predator, the predator-free, and the co-existence points. The local dynamics are investigated using Matignon’s condition and generalized Routh-Hurwitz criterion for the Caputo fractional-order model. The Volterra, quadratic, and linear functions are utilized to construct the Lyapunov function, as well as the LaSalle invariance principle, to show global dynamics. Some phenomena are provided namely forward and Hopf bifurcations to show the change of the dynamics when a parameter is varied. These results are supported by numerical ways using the predictor-corrector scheme.