Dynamics of impulsive control in a predator–prey model with the fear effect and group defense

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper investigates a predator–prey model incorporating both the fear effect and a Holling type IV functional response with group defense. In this model, the fear effect characterizes the suppression of prey reproduction caused by predator presence, whereas the Holling type IV functional response describes the enhanced group defense of prey at high densities. First, for the non-impulsive system, we analyze the positivity and boundedness of solutions, as well as the existence and local stability of equilibria. Second, when the equilibrium of the uncontrolled system fails to meet ecological management requirements, we introduce unilateral and bilateral state-feedback impulsive control strategies. The existence of order-one periodic solutions is established by the successor function method, and criteria for orbital asymptotic stability are derived using impulsive Floquet theory. Finally, numerical simulations validate the theoretical results, which provide a theoretical basis for designing ecologically balanced impulsive management strategies that suppress pest outbreaks while maintaining predator and prey populations within acceptable ranges.

Keywords

fear effect / group defense / Holling type IV functional response / state-feedback impulsive control / order-one periodic solutions / orbital asymptotic stability

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] L. Y. Zanette, A. F. White, M. C. Allen, M. Clinchy, Perceived predation risk reduces the number of offspring songbirds produce per year, Science, 334 (2011), 1398-1401. https://doi.org/10.1126/science.1210908

[2] E. L. Preisser, D. I. Bolnick, M. F. Benard, Scared to death? The effects of intimidation and consumption in predator-prey interactions, Ecology, 86 (2005), 501-509. https://doi.org/10.1890/04-0719

[3] M. Clinchy, M. J. Sheriff, L. Y. Zanette, Predator-induced stress and the ecology of fear, Funct. Ecol., 27 (2013), 56-65. https://doi.org/10.1111/1365-2435.12007

[4] E. H. Nelson, C. E. Matthews, J. A. Rosenheim, Predators reduce prey population growth by inducing changes in prey behavior, Ecology, 85 (2004), 1853-1858. https://doi.org/10.1890/03-3109

[5] S. K. Sasmal, Population dynamics with multiple Allee effects induced by fear factors: A mathematical study on prey-predator interactions, Appl. Math. Modell., 64 (2018), 1-14. https://doi.org/10.1016/j.apm.2018.07.021

[6] X. Wang, L. Zanette, X. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179-1204. https://doi.org/10.1007/s00285-016-0989-1

[7] J. P. Suraci, M. Clinchy, L. M. Dill, D. Roberts, L. Y. Zanette, Fear of large carnivores causes a trophic cascade, Nat. Commun., 7 (2016), 10698. https://doi.org/10.1038/ncomms10698

[8] M. Hartbauer, Collective Defense of Aphis nerii and Uroleucon hypochoeridis (Homoptera, Aphididae) against Natural Enemies , PLoS One, 5 (2010), e10417. https://doi.org/10.1371/journal.pone.0010417

[9] G. M. Wu, G. Boivin, J. Brodeur, L. A. Giraldeau, Y. Outreman, Altruistic defence behaviours in aphids, BMC Evol. Biol., 10 (2010), 19. https://doi.org/10.1186/1471-2148-10-19

[10] C. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320. https://doi.org/10.4039/Ent91293-5

[11] M. L. Rosenzweig, R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223. https://doi.org/10.1086/282272

[12] H. I. Freedman, G. S. K. Wolkowicz, Predator-prey systems with group defence: the paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508. https://doi.org/10.1007/BF02462320

[13] J. B. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168. https://doi.org/10.1007/s002850050095

[14] M. C. Kohnke, I. Siekmann, H. Seno, H. Malchow, A type IV functional response with different shapes in a predator-prey model, J. Theor. Biol., 505 (2020), 110419. https://doi.org/10.1016/j.jtbi.2020.110419

[15] X. C. Duan, S. L. Yuan, M. Martcheva, Habitat adaption promotes the evolution of predator species, Z. Angew. Math. Phys., 74 (2023), 10. https://doi.org/10.1007/s00033-022-01904-8

[16] S. L. Yuan, C. Q. Xu, T. H. Zhang, Spatial dynamics in a predator-prey model with herd behavior, Chaos, 23 (2013), 033102. https://doi.org/10.1063/1.4812724

[17] X. B. Zhang, J. Liu, G. L. Wang, Impacts of nonlocal fear effect and delay on a reaction-diffusion predator-prey model, Int. J. Biomath., 18 (2025), 2350089. https://doi.org/10.1142/S1793524523500894

[18] X. Chen, W. Yang, Impact of fear-induced group defense in a Monod-Haldane type prey-predator model, J. Appl. Math. Comput., 70 (2024), 3331-3368. https://doi.org/10.1007/s12190-024-02101-8

[19] R. L. Koch, The multicolored Asian lady beetle, Harmonia axyridis: A review of its biology, uses in biological control, and non-target impacts , J. Insect Sci., 3 (2003), 32. https://doi.org/10.1093/jis/3.1.32

[20] M. Camacho-Cervantes, A. Ortega-Iturriaga, E. del-Val, From effective biocontrol agent to successful invader: The harlequin ladybird (Harmonia axyridis) as an example of good ideas that could go wrong , PeerJ, 5 (2017), e3296. https://doi.org/10.7717/peerj.3296

[21] S. Tang, L. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn. Syst. - Ser. B, 4 (2004), 759-768. https://doi.org/10.3934/dcdsb.2004.4.759

[22] S. Tang, R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257-292. https://doi.org/10.1007/s00285-004-0290-6

[23] G. Jiang, Q. Lu, The dynamics of a prey-predator model with impulsive state feedback control, Discrete Contin. Dyn. Syst. - Ser. B, 6 (2006), 1301-1320. https://doi.org/10.3934/dcdsb.2006.6.1301

[24] T. Zhang, W. Ma, X. Meng, T. Zhang, Periodic solution of a prey-predator model with nonlinear state feedback control, Appl. Math. Comput., 266 (2015), 95-107. https://doi.org/10.1016/j.amc.2015.05.016

[25] L. Zhao, L. Chen, Q. Zhang, The geometrical analysis of a predator-prey model with two state impulses, Math. Biosci., 238 (2012), 55-64. https://doi.org/10.1016/j.mbs.2012.03.011

[26] J. Wang, H. Cheng, X. Meng, B. G. S. A. Pradeep, Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse, Adv. Differ. Equations, 2017 (2017), 252. https://doi.org/10.1186/s13662-017-1300-5

[27] S. M. Barribeau, D. Sok, N. M. Gerardo, Aphid reproductive investment in response to mortality risks, BMC Evol. Biol., 10 (2010), 251. https://doi.org/10.1186/1471-2148-10-251

[28] B. W. Lee, R. E. Clark, S. Basu, D. W. Crowder, Predators affect a plant virus through density and trait-mediated indirect effects on vectors, Food Webs, 33 (2022), e00251. https://doi.org/10.1016/j.fooweb.2022.e00251

[29] J. C. Simon, S. Stoeckel, D. Tagu, Evolutionary and functional insights into reproductive strategies of aphids, C.R. Biol., 333 (2010), 488-496. https://doi.org/10.1016/j.crvi.2010.03.003