Complex population patterns in a discrete-time predator-prey model incorporating fear and harvesting and chaos control

Figen Kangalgil; Nilüfer Topsakal; Mehmet Unlu; Figen Kangalgil; Nilüfer Topsakal; Mehmet Unlu

doi:10.3934/math.20261078

AIMS Mathematics

2026, Volume 11, Issue 8: 26868-26896. doi: 10.3934/math.20261078

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Research article

Complex population patterns in a discrete-time predator-prey model incorporating fear and harvesting and chaos control

1.
Bergama Vocational School, Dokuz Eylul University, 35700 Bergama, Izmir, Türkiye
2.
Department of Mathematics, Sivas Cumhuriyet University, 58140 Sivas, Türkiye
3.
Department of Mathematics, Recep Tayyip Erdogan University, 53100 Rize, Türkiye

Received: 17 June 2026 Revised: 31 July 2026 Accepted: 11 August 2026 Published: 26 August 2026
MSC : 37D45, 37G35, 37N25, 39A30, 92D25, 93D15

This study examines the effects of incorporating harvesting and fear effects into a discrete-time predator-prey amensalism population model on system dynamics. To establish a more realistic ecological model framework, the study considers not only classical predator-prey interactions, but also the behavioral responses of prey individuals to the presence of predators and human-induced external interventions. The model was analyzed using mathematical methods and detailed numerical simulations; additionally, bifurcation analysis was performed. The existence and local stability of all equilibrium points have been determined, and a topological classification of these points has been performed. Obtained results indicate that an increase in fear and hunting levels reduces prey population density in the stable region, whereas it has no significant effect on the predator population. Furthermore, it was demonstrated that the system undergoes a flip (period-doubling) bifurcation, which can lead to complex or even chaotic dynamics. An Ott-Grebogi-Yorke (OGY) feedback control strategy was applied to control erratic behavior, and it was shown that this method is effective in stabilizing unstable periodic orbits. Overall, this study explains how fear-induced behavioral responses and the harvesting effect jointly influence prey and predator populations, providing important insights into the stabilization and management of discrete-time ecological systems.
- prey-predator,
- chaos,
- fear effect,
- harvesting effect,
- amensalism
Citation: Figen Kangalgil, Nilüfer Topsakal, Mehmet Unlu. Complex population patterns in a discrete-time predator-prey model incorporating fear and harvesting and chaos control[J]. AIMS Mathematics, 2026, 11(8): 26868-26896. doi: 10.3934/math.20261078

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Abstract

This study examines the effects of incorporating harvesting and fear effects into a discrete-time predator-prey amensalism population model on system dynamics. To establish a more realistic ecological model framework, the study considers not only classical predator-prey interactions, but also the behavioral responses of prey individuals to the presence of predators and human-induced external interventions. The model was analyzed using mathematical methods and detailed numerical simulations; additionally, bifurcation analysis was performed. The existence and local stability of all equilibrium points have been determined, and a topological classification of these points has been performed. Obtained results indicate that an increase in fear and hunting levels reduces prey population density in the stable region, whereas it has no significant effect on the predator population. Furthermore, it was demonstrated that the system undergoes a flip (period-doubling) bifurcation, which can lead to complex or even chaotic dynamics. An Ott-Grebogi-Yorke (OGY) feedback control strategy was applied to control erratic behavior, and it was shown that this method is effective in stabilizing unstable periodic orbits. Overall, this study explains how fear-induced behavioral responses and the harvesting effect jointly influence prey and predator populations, providing important insights into the stabilization and management of discrete-time ecological systems.

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