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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

On Predator-Prey Dynamics: Incorporating Prey Vigilance and Predator Competition

Journal of Applied Nonlinear Dynamics 14(3) (2025) 685--704 | DOI:10.5890/JAND.2025.09.012

Eric M. Takyi$^{1}$, Kasey Cooper$^{1}$, Kya Jones$^{1}$, Vincent T. Teyekpiti$^{2}$

$^{1}$ Department of Mathematics, Computer Science and Statistics, Ursinus College, Collegeville, PA 19426, USA

$^{2}$ Department of Mathematics, School of Physical and Mathematical Sciences, University of Ghana, Legon, Ghana

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Abstract

Vigilance is an anti-predator behavior that creates a trade-off between foraging and safety, impacting the growth rate of prey. In this work, we study the dynamics of a predator-prey system where the prey exhibit active vigilance in the presence of predators. Vigilance behavior reduces predation risk and can lead to a competition for resources (prey) among predators. We provide conditions for the feasibility and local stability of equilibrium points as well as global stability results for the system. We demonstrate the possible occurrence of saddle-node and Hopf bifurcations when varying the parameters for the level of vigilance $v$ and predator intraspecific competition $c$. Additionally, we numerically show the possible occurrence of cusp and Bogdanov-Takens bifurcations. Our results indicate that increased vigilance leads to a reduction in prey growth rate and highlight that prey vigilance and predator intraspecific competition play a key role in maintaining ecosystem stability. We also extend our system to include a spatial component, observing the occurrence of spatial Turing patterns in one and two dimensions. We present numerical experiments to support our findings and discuss the applications of our results in species conservation and pest control.

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