# Local and global dynamics of a fractional-order predator-prey system with habitat complexity and the corresponding discretized fractional-order system

## TL;DR

This study investigates the stability and complex dynamics of a fractional-order predator-prey model with habitat complexity, revealing conditions for stability, bifurcations, and the effects of discretization on system behavior.

## Contribution

It introduces a fractional-order predator-prey model with habitat complexity and analyzes its stability, bifurcations, and discretized dynamics, highlighting fractional-order-dependent phenomena.

## Key findings

- Existence of positivity and boundedness of solutions.
- Conditions for local and global stability of equilibrium points.
- Presence of Hopf bifurcation and complex dynamics depending on fractional order and step size.

## Abstract

This paper is focused on local and global stability of a fractional-order predator-prey model with habitat complexity constructed in the Caputo sense and corresponding discrete fractional-order system. Mathematical results like positivity and boundedness of the solutions in fractional-order model is presented. Conditions for local and global stability of different equilibrium points are proved. It is shown that there may exist fractional-order-dependent instability through Hopf bifurcation for both fractional-order and corresponding discrete systems. Dynamics of the discrete fractional-order model is more complex and depends on both step length and fractional-order. It shows Hopf bifurcation, flip bifurcation and more complex dynamics with respect to the step size. Several examples are presented to substantiate the analytical results.