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Article

Effect of Fear and Time Delay on Predator–Prey Interaction

by
Ibrahim Elmojtaba
1,
Mariam Al-Moqbali
2 and
Nasser Al-Salti
3,*
1
Department of Mathematics, College of Science, Sultan Qaboos University, Muscat P.O. Box 36, Oman
2
Mathematics and Computing Skills Unit, University of Technology and Applied Sciences, Ibri P.O. Box 466, Oman
3
Department of Applied Mathematics and Science, National University of Science and Technology, Muscat P.O. Box 2322, Oman
*
Author to whom correspondence should be addressed.
AppliedMath 2025, 5(4), 175; https://doi.org/10.3390/appliedmath5040175
Submission received: 21 September 2025 / Revised: 29 October 2025 / Accepted: 24 November 2025 / Published: 7 December 2025
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Abstract

In this paper, we incorporate a new type of fear effect into a predator–prey time-delay model to study their combined impact on the system’s dynamics. Without time delay, our results show that the prey-only and coexistence equilibrium points are globally asymptotically stable under certain conditions. We also find that a transcritical bifurcation occurs near the prey-only equilibrium, while a Hopf bifurcation arises near the coexistence equilibrium. The fear effect plays a crucial role in the system’s behavior, as it can lead to predator extinction or near-extinction of the prey. Moreover, the inclusion of time delay influences the coexistence equilibrium, potentially destabilizing it and giving rise to a stable limit cycle.
Keywords: predator–prey; fear; time delay; bifurcation; simulation predator–prey; fear; time delay; bifurcation; simulation

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MDPI and ACS Style

Elmojtaba, I.; Al-Moqbali, M.; Al-Salti, N. Effect of Fear and Time Delay on Predator–Prey Interaction. AppliedMath 2025, 5, 175. https://doi.org/10.3390/appliedmath5040175

AMA Style

Elmojtaba I, Al-Moqbali M, Al-Salti N. Effect of Fear and Time Delay on Predator–Prey Interaction. AppliedMath. 2025; 5(4):175. https://doi.org/10.3390/appliedmath5040175

Chicago/Turabian Style

Elmojtaba, Ibrahim, Mariam Al-Moqbali, and Nasser Al-Salti. 2025. "Effect of Fear and Time Delay on Predator–Prey Interaction" AppliedMath 5, no. 4: 175. https://doi.org/10.3390/appliedmath5040175

APA Style

Elmojtaba, I., Al-Moqbali, M., & Al-Salti, N. (2025). Effect of Fear and Time Delay on Predator–Prey Interaction. AppliedMath, 5(4), 175. https://doi.org/10.3390/appliedmath5040175

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