Stability and bifurcation analysis of a discrete-time predator-prey model with combined effects of prey harvesting and anti-predator behavior

Asifa Tassaddiq; Youngsoo Seol; Rabab Alharbi; Ruhaila Md. Kasmani; Rizwan Ahmed; Asifa Tassaddiq; Youngsoo Seol; Rabab Alharbi; Ruhaila Md. Kasmani; Rizwan Ahmed

doi:10.3934/math.20261191

AIMS Mathematics

2026, Volume 11, Issue 9: 30065-30090. doi: 10.3934/math.20261191

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Research article

Stability and bifurcation analysis of a discrete-time predator-prey model with combined effects of prey harvesting and anti-predator behavior

1.
Department of Information Technology, College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi Arabia
2.
Department of Mathematics, Dong-A University, Busan 49315, Republic of Korea
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Institute of Mathematical Sciences, Universiti Malaya, Kuala Lumpur 50603, Malaysia
5.
International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore 54792, Pakistan

Received: 01 July 2026 Revised: 02 September 2026 Accepted: 04 September 2026 Published: 16 September 2026
MSC : 39A28, 39A30

This work focuses on examining the joint influence of prey harvesting and anti-predator responses within a discrete-time predator-prey model. Beginning with a continuous formulation, the model is converted into a discrete system via the forward Euler scheme, allowing the representation of stepwise ecological dynamics and enabling more complex behavioral patterns to emerge. The presence and local stability of equilibrium states are investigated using the Jacobian matrix alongside eigenvalue-based criteria. In addition, the emergence of transcritical, period-doubling (PD), and Neimark–Sacker (NS) bifurcations is systematically derived through normal forms and center manifold theory. To support the analytical results, numerical simulations, such as bifurcation plots and computations of the maximum Lyapunov exponent, are conducted. The findings show that the system is capable of exhibiting a rich dynamic behavior, such as periodic behavior, quasi-periodic phenomena, and chaotic dynamics, through changes in the values of parameters. Specifically, the combined effect of prey harvesting intensity and anti-predation strategies plays an important role in the stability of the system and can either increase or disrupt stability. It is pointed out that an increased environmental carrying capacity or unbalanced parameter selection might cause chaotic population dynamics.
- predator-prey system,
- stability,
- bifurcation,
- chaos,
- Lyapunov exponent,
- simulation
Citation: Asifa Tassaddiq, Youngsoo Seol, Rabab Alharbi, Ruhaila Md. Kasmani, Rizwan Ahmed. Stability and bifurcation analysis of a discrete-time predator-prey model with combined effects of prey harvesting and anti-predator behavior[J]. AIMS Mathematics, 2026, 11(9): 30065-30090. doi: 10.3934/math.20261191

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Abstract

This work focuses on examining the joint influence of prey harvesting and anti-predator responses within a discrete-time predator-prey model. Beginning with a continuous formulation, the model is converted into a discrete system via the forward Euler scheme, allowing the representation of stepwise ecological dynamics and enabling more complex behavioral patterns to emerge. The presence and local stability of equilibrium states are investigated using the Jacobian matrix alongside eigenvalue-based criteria. In addition, the emergence of transcritical, period-doubling (PD), and Neimark–Sacker (NS) bifurcations is systematically derived through normal forms and center manifold theory. To support the analytical results, numerical simulations, such as bifurcation plots and computations of the maximum Lyapunov exponent, are conducted. The findings show that the system is capable of exhibiting a rich dynamic behavior, such as periodic behavior, quasi-periodic phenomena, and chaotic dynamics, through changes in the values of parameters. Specifically, the combined effect of prey harvesting intensity and anti-predation strategies plays an important role in the stability of the system and can either increase or disrupt stability. It is pointed out that an increased environmental carrying capacity or unbalanced parameter selection might cause chaotic population dynamics.

References

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