Stability and bifurcation analysis of a fractional-order prey–predator model with ratio-dependent functional response

Ibrahim Alraddadi; Ramesh Perumal; Rizwan Ahmed; Jawad Khan; Youngmoon Lee; Ibrahim Alraddadi; Ramesh Perumal; Rizwan Ahmed; Jawad Khan; Youngmoon Lee

doi:10.3934/math.2026060

AIMS Mathematics

2026, Volume 11, Issue 1: 1412-1448. doi: 10.3934/math.2026060

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Research article

Stability and bifurcation analysis of a fractional-order prey–predator model with ratio-dependent functional response

1.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, Saudi Arabia
2.
Department of Mathematics, Madanapalle Institute of Technology & Science (MITS), Deemed to be University, Madanapalle, Andhra Pradesh, India
3.
Department of Mathematics, Air University Multan Campus, Multan 60001, Pakistan
4.
School of Computing, Gachon University, Seongnam 13120, Republic of Korea
5.
Department of Robotics, Hanyang University, Ansan 15588, Republic of Korea
6.
Department of Applied AI, Hanyang University, Ansan 15588, Republic of Korea

Received: 27 October 2025 Revised: 22 December 2025 Accepted: 05 January 2026 Published: 16 January 2026
MSC : 39A28, 39A30

This paper explores the dynamics of a fractional prey–predator system with a ratio-dependent functional response with memory and hereditary effects in predator–prey interactions. The model is developed by the Caputo fractional derivative, and the existence, uniqueness, positivity, and boundedness of solutions are proven to satisfy biological reality. Stability conditions for local and global stability of both predator-free and coexistence equilibria are proven through linearization and Lyapunov function techniques. The fractional order is used as a bifurcation parameter, and the appearance of Hopf bifurcations is analytically explained with demonstration of the influence of memory on oscillations. To examine discrete-time dynamics, the piecewise constant argument is used to derive a discrete counterpart of the fractional model. The discrete model indicates a wide range of rich complex oscillatory phenomena, including period-doubling and Neimark–Sacker bifurcations, leading to periodic, quasiperiodic, and chaotic oscillations. Numerical computations, including bifurcation diagrams, phase portraits, and Lyapunov exponents, verify the analytical results and describe the routes of transition to chaos. A comparative analysis to compare integer- and fractional-order cases indicates that memory effects enhance dynamical richness and sensitivity to parameters. The study provides a unified framework relating continuous fractional dynamics and their discrete implementations and provides additional insight into how memory and discretization interact to modify stability and bifurcation in ecological models.
- fractional-order,
- prey–predator model,
- stability,
- bifurcation,
- chaos
Citation: Ibrahim Alraddadi, Ramesh Perumal, Rizwan Ahmed, Jawad Khan, Youngmoon Lee. Stability and bifurcation analysis of a fractional-order prey–predator model with ratio-dependent functional response[J]. AIMS Mathematics, 2026, 11(1): 1412-1448. doi: 10.3934/math.2026060

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Abstract

This paper explores the dynamics of a fractional prey–predator system with a ratio-dependent functional response with memory and hereditary effects in predator–prey interactions. The model is developed by the Caputo fractional derivative, and the existence, uniqueness, positivity, and boundedness of solutions are proven to satisfy biological reality. Stability conditions for local and global stability of both predator-free and coexistence equilibria are proven through linearization and Lyapunov function techniques. The fractional order is used as a bifurcation parameter, and the appearance of Hopf bifurcations is analytically explained with demonstration of the influence of memory on oscillations. To examine discrete-time dynamics, the piecewise constant argument is used to derive a discrete counterpart of the fractional model. The discrete model indicates a wide range of rich complex oscillatory phenomena, including period-doubling and Neimark–Sacker bifurcations, leading to periodic, quasiperiodic, and chaotic oscillations. Numerical computations, including bifurcation diagrams, phase portraits, and Lyapunov exponents, verify the analytical results and describe the routes of transition to chaos. A comparative analysis to compare integer- and fractional-order cases indicates that memory effects enhance dynamical richness and sensitivity to parameters. The study provides a unified framework relating continuous fractional dynamics and their discrete implementations and provides additional insight into how memory and discretization interact to modify stability and bifurcation in ecological models.

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