Dynamic complexity of a slow-fast predator-prey model with herd behavior

Ahmad Suleman; Rizwan Ahmed; Fehaid Salem Alshammari; Nehad Ali Shah; Ahmad Suleman; Rizwan Ahmed; Fehaid Salem Alshammari; Nehad Ali Shah

doi:10.3934/math.20231247

AIMS Mathematics

2023, Volume 8, Issue 10: 24446-24472. doi: 10.3934/math.20231247

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Dynamic complexity of a slow-fast predator-prey model with herd behavior

1.
Department of Mathematics, Air University Multan Campus, Multan, Pakistan
2.
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
3.
Department of Mechanical Engineering, Sejong University, Seoul 05006, South Korea
^† These authors contributed equally to this work and are co-first authors

Received: 12 June 2023 Revised: 23 July 2023 Accepted: 09 August 2023 Published: 17 August 2023
MSC : 39A28, 39A30

The complex dynamics of a slow-fast predator-prey interaction with herd behavior are examined in this work. We investigate the presence and stability of fixed points. By employing the bifurcation theory, it is shown that the model undergoes both a period-doubling and a Neimark-Sacker bifurcation at the interior fixed point. Under the influence of period-doubling and Neimark-Sacker bifurcations, chaos is controlled using the hybrid control approach. Moreover, numerical simulations are carried out to highlight the model's complexity and show how well they agree with analytical findings. Employing the slow-fast factor as the bifurcation parameter shows that the model goes through a Neimark-Sacker bifurcation for greater values of the slow-fast factor at the interior fixed point. This makes sense because if the slow-fast factor is large, the growth rates of the predator and its prey will be about identical, automatically causing the interior fixed point to become unstable owing to the predator's slow growth.
- slow-fast,
- predator-prey,
- square root functional response,
- stability,
- bifurcation,
- hybrid control
Citation: Ahmad Suleman, Rizwan Ahmed, Fehaid Salem Alshammari, Nehad Ali Shah. Dynamic complexity of a slow-fast predator-prey model with herd behavior[J]. AIMS Mathematics, 2023, 8(10): 24446-24472. doi: 10.3934/math.20231247

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Abstract

The complex dynamics of a slow-fast predator-prey interaction with herd behavior are examined in this work. We investigate the presence and stability of fixed points. By employing the bifurcation theory, it is shown that the model undergoes both a period-doubling and a Neimark-Sacker bifurcation at the interior fixed point. Under the influence of period-doubling and Neimark-Sacker bifurcations, chaos is controlled using the hybrid control approach. Moreover, numerical simulations are carried out to highlight the model's complexity and show how well they agree with analytical findings. Employing the slow-fast factor as the bifurcation parameter shows that the model goes through a Neimark-Sacker bifurcation for greater values of the slow-fast factor at the interior fixed point. This makes sense because if the slow-fast factor is large, the growth rates of the predator and its prey will be about identical, automatically causing the interior fixed point to become unstable owing to the predator's slow growth.

References

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