Stability and bifurcations in a delayed predator-prey system with prey-taxis and hunting cooperation functional response

Kimun Ryu; Wonlyul Ko; Kimun Ryu; Wonlyul Ko

doi:10.3934/math.2025576

AIMS Mathematics

2025, Volume 10, Issue 6: 12808-12840. doi: 10.3934/math.2025576

Previous Article Next Article

Research article

Stability and bifurcations in a delayed predator-prey system with prey-taxis and hunting cooperation functional response

Kimun Ryu ,
Wonlyul Ko ^,

Department of Mathematics Education, Cheongju University, Cheongju, Chungbuk 28503, Republic of Korea

Received: 24 February 2025 Revised: 16 April 2025 Accepted: 24 April 2025 Published: 03 June 2025
MSC : 35B32, 35K57, 92D25

In this paper, we studied a diffusive predator-prey system that incorporated three ecological features under homogeneous Neumann boundary conditions. These features included a cooperative hunting functional response, prey-taxis, and a time delay effect in the predator growth rate, all of which influenced predator-prey interactions. These three features, respectively, indicated that predators cooperated while hunting their prey, that predators tended to move in the direction of an increasing prey density gradient, and that a certain amount of time was required for the conversion of captured prey into predator growth. We first analyzed the occurrence of steady-state bifurcation by examining the role of the prey-taxis rate $ \chi $. Furthermore, we examined the impact of the time delay effect on the occurrence of Hopf bifurcation, which involved the emergence of spatially homogeneous or nonhomogeneous periodic solutions, as well as stability switches at a positive constant steady state.
- predator-prey,
- prey-taxis,
- Hopf bifurcation,
- steady-state bifurcation,
- hunting cooperation,
- delay
Citation: Kimun Ryu, Wonlyul Ko. Stability and bifurcations in a delayed predator-prey system with prey-taxis and hunting cooperation functional response[J]. AIMS Mathematics, 2025, 10(6): 12808-12840. doi: 10.3934/math.2025576

Related Papers:

Abstract

In this paper, we studied a diffusive predator-prey system that incorporated three ecological features under homogeneous Neumann boundary conditions. These features included a cooperative hunting functional response, prey-taxis, and a time delay effect in the predator growth rate, all of which influenced predator-prey interactions. These three features, respectively, indicated that predators cooperated while hunting their prey, that predators tended to move in the direction of an increasing prey density gradient, and that a certain amount of time was required for the conversion of captured prey into predator growth. We first analyzed the occurrence of steady-state bifurcation by examining the role of the prey-taxis rate $ \chi $. Furthermore, we examined the impact of the time delay effect on the occurrence of Hopf bifurcation, which involved the emergence of spatially homogeneous or nonhomogeneous periodic solutions, as well as stability switches at a positive constant steady state.

References

[1]	M. T. Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13–22. https://doi.org/10.1016/j.jtbi.2017.02.002 doi: 10.1016/j.jtbi.2017.02.002
[2]	R. Arditi, L. R. Ginzburg, Couplng in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 139 (1989), 311–326. https://doi.org/10.1016/S0022-5193(89)80211-5 doi: 10.1016/S0022-5193(89)80211-5
[3]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331–340. https://doi.org/10.2307/3866 doi: 10.2307/3866
[4]	E. Beretta, Y. Kuang, Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal.-Theor., 32 (1998), 381–408. https://doi.org/10.1016/S0362-546X(97)00491-4 doi: 10.1016/S0362-546X(97)00491-4
[5]	L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72 (2010), 94–121. https://doi.org/10.1007/s11538-009-9439-1 doi: 10.1007/s11538-009-9439-1
[6]	R. S. Cantrell, C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206–222. https://doi.org/10.1006/jmaa.2000.7343 doi: 10.1006/jmaa.2000.7343
[7]	J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188–192. https://doi.org/10.2307/1934845 doi: 10.2307/1934845
[8]	C. Cosner, D. L. DeAngelis, J. S. Ault, D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65–75. https://doi.org/10.1006/tpbi.1999.1414 doi: 10.1006/tpbi.1999.1414
[9]	M. G. Crandall, P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321–340. https://doi.org/10.1016/0022-1236(71)90015-2 doi: 10.1016/0022-1236(71)90015-2
[10]	Y. Dai, Y. Zhao, B. Sang, Four limit cycles in a predator–prey system of Leslie type with generalized Holling type Ⅲ functional response, Nonlinear Anal.-Real, 50 (2019), 218–239. https://doi.org/10.1016/j.nonrwa.2019.04.003 doi: 10.1016/j.nonrwa.2019.04.003
[11]	Y. Dai, Y. Zhao, Hopf cyclicity and global dynamics for a predator–prey system of Leslie type with simplified Holling type Ⅳ functional response, Int. J. Bifurcat. Chaos, 28 (2018), 1850166. https://doi.org/10.1142/S0218127418501663 doi: 10.1142/S0218127418501663
[12]	D. L. DeAngelis, R. A. Goldstein, R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881–892. https://doi.org/10.2307/1936298 doi: 10.2307/1936298
[13]	Y. Enatsu, J. Roy, M. Banerjee, Hunting cooperation in a prey-predator model with maturation delay, J. Biol. Dyn., 18 (2024), 2332279. https://doi.org/10.1080/17513758.2024.2332279 doi: 10.1080/17513758.2024.2332279
[14]	S. Fu, H. Zhang, Effect of hunting cooperation on the dynamic behavior for a diffusive Holling tyep Ⅱ predator-prey model, Commun. Nonlinear Sci. Numer. Simulat., 99 (2021), 105807. https://doi.org/10.1016/j.cnsns.2021.105807 doi: 10.1016/j.cnsns.2021.105807
[15]	C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, The Memoirs of the Entomological Society of Canada, 97 (1995), 5–60. https://doi.org/10.4039/entm9745fv doi: 10.4039/entm9745fv
[16]	P. Kareiva, G. Odell, Swarms of predators exhibit "prey-taxis" if individual predators use are-restricted search, Amer. Nat., 130 (1987), 233–270. https://doi.org/10.1086/284707 doi: 10.1086/284707
[17]	W. Ko, K. Ryu, A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: Ⅰ global existence and stability, J. Math. Anal. Appl., 543 (2025), 129005. https://doi.org/10.1016/j.jmaa.2024.129005 doi: 10.1016/j.jmaa.2024.129005
[18]	W. Ko, K. Ryu, A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: Ⅱ stationary pattern formation, J. Math. Anal. Appl., 543 (2025), 128947. https://doi.org/10.1016/j.jmaa.2024.128947 doi: 10.1016/j.jmaa.2024.128947
[19]	Y. Kuang, Delay differential equations, In: Encyclopedia of theoretical ecology, Boston: Academic Press, 1993,163–166. https://doi.org/10.1525/9780520951785-032
[20]	Z. Liu, R. Yuan, Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response, J. Math. Anal. Appl., 296 (2004), 521–537. https://doi.org/10.1016/j.jmaa.2004.04.051 doi: 10.1016/j.jmaa.2004.04.051
[21]	Z. Liu, R. Yuan, The effect of diffusion for a predator-prey system with nonmonotonic functional response, Int. J. Bifurcat. Chaos, 14 (2004), 4309–4316. https://doi.org/10.1142/S0218127404011867 doi: 10.1142/S0218127404011867
[22]	S. Liu, E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J. Appl. Math., 66 (2006), 1101–1129. https://doi.org/10.1137/050630003 doi: 10.1137/050630003
[23]	C. Nie, D. Jin, R. Yang, Hopf bifurcation analysis in a delayed diffusive predator-prey system with nonlocal competition and generalist predator, AIMS Math., 7 (2022), 13344–13360. https://doi.org/10.3934/math.2022737 doi: 10.3934/math.2022737
[24]	L. Přibylová, L. Berec, Predator interference and stability of predator-prey dynamics, J. Math. Biol., 71 (2015), 301–323. https://doi.org/10.1007/s00285-014-0820-9 doi: 10.1007/s00285-014-0820-9
[25]	S. Ruan, D. Xiao, Gobal analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445–1472. https://doi.org/10.1137/S0036139999361896 doi: 10.1137/S0036139999361896
[26]	K. Ryu, W. Ko, On dynamics and stationary pattern formations of a diffusive predator-prey system with hunting cooperation, Discrete. Cont. Dyn. Sys. B, 27 (2022), 6679–6709. https://doi.org/10.3934/dcdsb.2022015 doi: 10.3934/dcdsb.2022015
[27]	K. Ryu, W. Ko, Asymptotic behavior of positive solutions to a predator–prey elliptic system with strong hunting cooperation in predators, Physica A, 531 (2019), 121726. https://doi.org/10.1016/j.physa.2019.121726 doi: 10.1016/j.physa.2019.121726
[28]	K. Ryu, W. Ko, M. Haque, Bifurcation analysis in a predator–prey system with a functional response increasing in both predator and prey densities, Nonlinear Dyn., 94 (2018), 1639–1656. https://doi.org/10.1007/s11071-018-4446-0 doi: 10.1007/s11071-018-4446-0
[29]	D. Sen, S. Ghorai, M. Banerjee, Allee effect in prey versus hunting cooperation on predator – enhancement of stable coexistence, Int. J. Bifurcat. Chaos, 29 (2019), 1950081. https://doi.org/10.1142/S0218127419500810 doi: 10.1142/S0218127419500810
[30]	Q. Shi, Y. Song, Spatially nonhomogeneous periodic patterns in a delayed predator-prey model with predator-taxis diffusion, Appl. Math. Lett., 131 (2022), 108062. https://doi.org/10.1016/j.aml.2022.108062 doi: 10.1016/j.aml.2022.108062
[31]	H. Shu, X. Hu, L. Wang, J. Watmough, Delay induced stability switch, multitype bistability and chaos in an intraguild predation model, J. Math. Biol., 71 (2015), 1269–1298. https://doi.org/10.1007/s00285-015-0857-4 doi: 10.1007/s00285-015-0857-4
[32]	D. Song, C. Li, Y. Song, Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation functional response, Nonlinear Anal.-Real, 54 (2020), 103106. https://doi.org/10.1016/j.nonrwa.2020.103106 doi: 10.1016/j.nonrwa.2020.103106
[33]	Y. Song, J. Shi, H. Wang, Spatiotemporal dynamics of a diffusive consumer-resource model with explicit spatial memory, Stud. Appl. Math., 148 (2021), 373–395. https://doi.org/10.1111/sapm.12443 doi: 10.1111/sapm.12443
[34]	Z. Sun, W. Jiang, Bifurcation and spatial patterns driven by predator-taxis in a predator-prey system with Beddington-DeAngelis functional response, Discrete. Cont. Dyn. Sys. B, 29 (2024), 4043–4070. https://doi.org/10.3934/dcdsb.2024034 doi: 10.3934/dcdsb.2024034
[35]	S. K. Verma, B. Kumar, Bifurcation and pattern formation in a prey–predator model with cooperative hunting, Eur. Phys. J. Plus, 139 (2024), 734. https://doi.org/10.1140/epjp/s13360-024-05543-y doi: 10.1140/epjp/s13360-024-05543-y
[36]	F. Wang, R. Yang, Y. Xie, J. Zhao, Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator, AIMS Math., 8 (2023), 17719–17743. https://doi.org/10.3934/math.2023905 doi: 10.3934/math.2023905
[37]	C. Wang, S. Yuan, H. Wang, Spatiotemporal patterns of a diffusive prey-predator model with spatial memory and pregnancy period in an intimidatory environment, J. Math. Biol., 84 (2022), 12. https://doi.org/10.1007/s00285-022-01716-4 doi: 10.1007/s00285-022-01716-4
[38]	J. Wang, S. Wu, J. Shi, Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis, Discrete. Cont. Dyn. Sys. B, 26 (2021), 1273–1289. https://doi.org/10.3934/dcdsb.2020162 doi: 10.3934/dcdsb.2020162
[39]	X. Wang, S. Li, Y. Dai, K. Wu, Dynamics in a slow-fast Leslie-Gower predator-prey model with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 545 (2025), 129177. https://doi.org/10.1016/j.jmaa.2024.129177 doi: 10.1016/j.jmaa.2024.129177
[40]	S. Wu, J. Shi, B. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differ. Equations, 260 (2016), 5847–5874. https://doi.org/10.1016/j.jde.2015.12.024 doi: 10.1016/j.jde.2015.12.024
[41]	D. Xiao, S. Ruan, Multiple bifurcations in a delayed predator–prey system with nonmonotonic functional response, J. Differ. Equations, 176 (2001), 494–510. https://doi.org/10.1006/jdeq.2000.3982 doi: 10.1006/jdeq.2000.3982
[42]	J.-F. Zhang, W.-T. Li, X.-P. Yan, Multiple bifurcations in a delayed predator–prey diffusion system with a functional response, Nonlinear Anal.-Real, 11 (2010), 2708–2725. https://doi.org/10.1016/j.nonrwa.2009.09.019 doi: 10.1016/j.nonrwa.2009.09.019
[43]	T. Zhang, H. Zang, Delay-induced Turing instability in reaction-diffusion equations, Phys. Rev. E, 90 (2014), 052908. https://doi.org/10.1103/PhysRevE.90.052908 doi: 10.1103/PhysRevE.90.052908

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)