In population ecology, the interactions between predators and prey encompass more than just direct predation and killing. Various indirect effects also significantly influence the system’s dynamics. In this paper, we consider a refuge proportional to the prey population and assume that only unsuccessful refuge prey engage in group defense mechanisms. We introduce an exponential response function to model this behavior, taking into account the negative impact of defense measures on the prey growth rate. Subsequently, we analyze the non-negativity, boundedness, and persistence of the system. Furthermore, we study the existence and stability of different types of equilibrium points, and provide theoretical proofs for bifurcations in the system, such as transcritical, pitchfork, saddle-node, Hopf, and Bogdanov–Takens bifurcations. Finally, we verify our theoretical results through numerical simulations. Our study reveals that variations in refuge size and the intensity of group defense can alter the number and stability of equilibrium points, leading to phenomena such as bubbling and periodic oscillations. Additionally, group defense can induce bistability. Both prey refuge behavior and excessive group defense can lead to a reduction in predator numbers and eventually to their extinction. Moreover, compared to prey refuge, group defense as an anti-predator behavior has a greater effect on predators.

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References

Bhatta, E. A., DeCarli, N. J., Duffy, J. R. & McAdams, B. [2024] “Critical care nursing and mass casualty response during Operation Allies Refuge,” Crit. Care Nurse 44, 42–47. Crossref, Web of Science, Google Scholar
Chatterjee, A., Abbasi, M. A., Venturino, E., Zhen, J. & Haque, M. [2024] “A predator–prey model with prey refuge: Under a stochastic and deterministic environment,” Nonlin. Dyn. 112, 13667–13693. Crossref, Web of Science, Google Scholar
Chen, L., Song, X. & Lu, Z. [2003] Mathematical Ecology Models and Research Methods (Sichuan Science and Technology Press, China). Google Scholar
Chen, F. [2005] “On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay,” J. Comput. Appl. Math. 180, 33–49. Crossref, Web of Science, Google Scholar
Chen, W., Zhu, H., Zhang, X. & An, Q. [2023] “Stability and bifurcation in a predator–prey model with prey refuge,” J. Biol. Syst. 31, 417–435. Link, Web of Science, Google Scholar
Das, S., Biswas, S. & Das, P. [2024] “Impact of fear and prey refuge parameters in a fuzzy prey–predator model with group defense,” New Math. Nat. Comput. 20, 183–220. Link, Web of Science, Google Scholar
Freedman, H. I. & Wolkowicz, G. S. K. [1986] “Predator–prey systems with group defence: The paradox of enrichment revisited,” Bull. Math. Biol. 48, 493–508. Crossref, Web of Science, Google Scholar
Gard, T. C. [1984] “Persistence in food webs,” Mathematical Ecology: Proc. Autumn Course (Springer, Berlin, Heidelberg), pp. 208–219. Crossref, Google Scholar
Ghosh, J., Sahoo, B. & Poria, S. [2017] “Prey–predator dynamics with prey refuge providing additional food to predator,” Chaos Solit. Fract. 96, 110–119. Crossref, Web of Science, Google Scholar
Hartbauer, M. [2010] “Collective defense of Aphis nerii and Uroleucon hypochoeridis (Homoptera, Aphididae) against natural enemies,” PLoS One 5, e10417. Crossref, Web of Science, Google Scholar
Hochberg, M. E. & Holt, R. D. [1995] “Refuge evolution and the population dynamics of coupled host-parasitoid associations,” Evol. Ecol. 9, 633–661. Crossref, Web of Science, Google Scholar
Hoppensteadt, F. [2006] “Predator–prey model,” Scholarpedia 1, 1563. Crossref, Google Scholar
Jawad, S. R. & Al Nuaimi, M. [2023] “Persistence and bifurcation analysis among four species interactions with the influence of competition, predation and harvesting,” Iraqi J. Sci. 64, 1369–1390. Crossref, Google Scholar
Kuznetsov, Y. A. [1998] Elements of Applied Bifurcation Theory (Springer, NY). Google Scholar
Leslie, P. H. & Gower, J. C. [1960] “The properties of a stochastic model for the predator–prey type of interaction between two species,” Biometrika 47, 219–234. Crossref, Web of Science, Google Scholar
Liang, Z. & Meng, X. [2023] “Stability and Hopf bifurcation of a multiple delayed predator–prey system with fear effect, prey refuge and Crowley–Martin function,” Chaos Solit. Fract. 175, 113955. Crossref, Web of Science, Google Scholar
Liu, T., Zhao, X., Zhang, Y. & Liu, L. [2023] “Dynamics of a Leslie–Gower model with weak Allee effect on prey and fear effect on predator,” Int. J. Bifurcation and Chaos 33, 2350008. Link, Web of Science, Google Scholar
Lotka, A. J. [1925] Elements of Physical Biology (Williams and Wilkins, Baltimore, MD). Google Scholar
Ma, Z., Li, W., Wang, S. & Chen, F. [2009] “Effects of prey refuges on a predator–prey model with a class of functional responses: The role of refuges,” Math. Biosci. 218, 73–79. Crossref, Web of Science, Google Scholar
Mandal, S. & Tiwari, P. K. [2024] “Schooling behavior in a generalist predator–prey system: Exploring fear, refuge and cooperative strategies in a stochastic environment,” Eur. Phys. J. Plus 139, 29. Crossref, Web of Science, Google Scholar
Murray, J. D. [2002] Mathematical Biology (Springer, NY). Crossref, Google Scholar
Pal, S., Tiwari, P. K. & Mandal, S. [2024a] “Dynamical behaviors of a constant prey refuge ratio-dependent prey–predator model with Allee and fear effects,” Int. J. Biomath. 17, 2350010. Link, Web of Science, Google Scholar
Pal, S., Tiwari, P. K. & Mandal, S. [2024b] “Impact of fear and group defense on the dynamics of a predator–prey system,” Int. J. Bifurcation and Chaos 34, 2450019. Link, Web of Science, Google Scholar
Roy, S., Sk, N. & Tiwari, P. K. [2024] “Bifurcation analysis of autonomous and nonautonomous modified Leslie–Gower models,” Chaos 34, 2350010. Crossref, Google Scholar
Sasmal, S. K. & Takeuchi, Y. [2020] “Dynamics of a predator–prey system with fear and group defense,” J. Math. Anal. Appl. 481, 123471. Crossref, Web of Science, Google Scholar
Teschl, G. [2024] Ordinary Differential Equations and Dynamical Systems (American Mathematical Society). Google Scholar
Trienens, M. & Rohlfs, M. [2020] “A potential collective defense of Drosophila larvae against the invasion of a harmful fungus,” Front. Ecol. Evol. 8, 79. Crossref, Web of Science, Google Scholar
Volterra, V. [1928] “Variations and fluctuations of the number of individuals in animal species living together,” ICES J. Mar. Sci. 3, 3–51. Crossref, Google Scholar
Yao, W. & Li, X. [2023] “Complicate bifurcation behaviors of a discrete predator–prey model with group defense and nonlinear harvesting in prey,” Appl. Anal. 102, 2567–2582. Crossref, Web of Science, Google Scholar
Zhang, Z. [1992] Qualitative Theory of Differential Equations (American Mathematical Society). Google Scholar
Zhang, X., Zhu, H. & An, Q. [2023] “Dynamics analysis of a diffusive predator–prey model with spatial memory and nonlocal fear effect,” J. Math. Anal. Appl. 525, 127123. Crossref, Web of Science, Google Scholar