| Citation: |
Lina Zhang, Xue Xiao. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF COEXISTENCE STATES TO A DIFFUSIVE HOLLING TYPE II PREDATOR-PREY MODEL WITH HUNTING COOPERATION[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1413-1429. doi: 10.11948/20240251
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EXISTENCE AND ASYMPTOTIC BEHAVIOR OF COEXISTENCE STATES TO A DIFFUSIVE HOLLING TYPE II PREDATOR-PREY MODEL WITH HUNTING COOPERATION
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Abstract
In this paper, we delve into a diffusive Holling type II predator-prey model, incorporating the element of hunting cooperation, and examine it under Dirichlet boundary conditions. Our primary focus is on addressing two pivotal questions: Firstly, we endeavor to establish the existence of coexistence states across a range of hunting cooperation effects. This exploration aims to reveal how the predator and prey species can maintain their coexistence within the ecological system, regardless of the magnitude of cooperation among predators during hunting. Secondly, we are interested in elucidating the asymptotic behavior of these coexistence states as the cooperation parameter approaches infinity. This analysis will provide insights into how the ecological balance shifts as the predators' cooperation increases indefinitely, offering a deeper understanding of the long-term ecological implications of such cooperation.
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Keywords:
- Predator-prey /
- hunting cooperation /
- coexistence state /
- asymptotic behavior
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References
[1] M. T. Alves and F. M. Hilker, Hunting cooperation and Allee effects in predators, Journal of Theoretical Biology, 2017, 419, 13–22. DOI: 10.1016/j.jtbi.2017.02.002. [2] L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bulletin of Mathematical Biology, 2010, 72(1), 94. DOI: 10.1007/s11538-009-9439-1. [3] C. Boesch, Cooperative hunting in wild chimpanzees, International Journal of Primatology, 1994, 48(3), 653–667. DOI: 10.1006/anbe.1994.1285. [4] R. Cai, Y. Cai and J. Shen, Coexistence of one predator and two prey through rapid evolution in predator's feeding choice, Communications in Nonlinear Science and Numerical Simulation, 2023. DOI: 10.1016/j.cnsns.2023.107454 .[5] X. Cao, Neurobiology, Shanghai Medical University, Shanghai, (Chinese), 1989. [6] F. Capone, M. F. Carfora, R. De. Luca and I. Torcicollo, Turing patterns in a reaction-diffusion system modeling hunting cooperation, Mathematics and Computers in Simulation, 2019, 165, 172–180. doi: 10.1016/j.matcom.2019.03.010 [7] S. Creel and N. M. Creel, Communal hunting and pack size in african wild dogs, lycaon pictus, Animal Behaviour, 1995, 50(5), 1325–1339. doi: 10.1016/0003-3472(95)80048-4 [8] Y. Dong, G. Gao and S. Li, Coexistence states for a prey-predator model with cross-diffusion, Journal of Mathematical Analysis and Applications, 2024, 535(1), 128106. DOI: 10.1016/j.jmaa.2024.128106. [9] S. Fu and H. Zhang, Effect of hunting cooperation on the dynamic behavior for a diffusive Holling type Ⅱ predator-prey model, Communications in Nonlinear Science and Numerical Simulation, 2021(5), 105807. DOI: 10.1016/j.cnsns.2021.105807. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. [11] R. Han, S. Dey and M. Banerjee, Spatio-temporal pattern selection in a preyšCpredator model with hunting cooperation and Allee effect in prey, Chaos, Solitons and Fractals, 2023, 171, 113441. DOI: 10.1016/j.chaos.2023.113441. [12] Y. Liu, Positive solutions to general elliptic systems, Nonlinear Analysis, 1995, 25, 229–246. doi: 10.1016/0362-546X(94)00134-4 [13] J. L$\acute{o}$pez-G$\acute{o}$mez, Positive periodic solution of Lotka-Volterra reaction-diffusion systems, Differential Integral Equations, 1992, 5(1), 55–72. [14] B. Mondal, A. A. Thirthar, N. Sk, M. A. Alqudah and T. Abdeljawad, Complex dynamics in a two species system with Crowley-Martin response function: Role of cooperation, additional food and seasonal perturbations, Mathematics and Computers in Simulation, 2024, 221, 415–434. DOI: 10.1016/j.matcom.2024.03.015. [15] C. Packer, D. Scheel and A. E. Pusey, Why lions form groups: Food is not enough, American Naturalist, 1990, 136(1), 1–19. doi: 10.1086/285079 [16] D. Pal, D. Kesh and D. Mukherjee, Cross-diffusion mediated spatiotemporal patterns in a predator-prey system with hunting cooperation and fear effect, Mathematics and Computers in Simulation, 2024, 220, 128–147. doi: 10.1016/j.matcom.2024.01.003 [17] C. V. Pao, On Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [18] K. Ryu and W. Ko, Asymptotic behavior of positive solutions to a predator-prey, Physica A, 2019, 531, 121726-1-8. doi: 10.1016/j.physa.2019.121726 [19] K. Ryu, W. Ko and M. Haque, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dynamics, 2018, 94, 1639–1656. DOI: 10.1007/s11071-018-4446-0. [20] D. Scheel and C. Packer, Group hunting behaviour of lions: A search for cooperation, Animal Behaviour, 1991, 41(4), 697–709. doi: 10.1016/S0003-3472(05)80907-8 [21] P. A. Schmidt and L. D. Mech, Wolf pack size and food acquisition, American Naturalist, 1997, 150(4), 513–517. doi: 10.1086/286079 [22] T. Singh and S. Banerjee, Spatial aspect of hunting cooperation in predators with Holling type Ⅱ functional response, Journal of Biological Systems, 2018, 26(4), 511–531. doi: 10.1142/S0218339018500237 [23] D. Song, C. Li and Y. Song, Stability and cross-diffusion-driven instability in a diffusive predator-prey system with hunting cooperation functional response, Nonlinear Analysis: Real World Applications, 2020, 54, 103106-1-24. doi: 10.1016/j.nonrwa.2020.103106 [24] D. Y. Wu and M. Zhao, Qualitative analysis for a diffusive predator-prey model with hunting cooperative, Physica A: Statistical Mechanics and Applications, 2019, 515, 299–309. doi: 10.1016/j.physa.2018.09.176 -
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Lina Zhang, Xue Xiao. EXISTENCE AND ASYMPTOTIC BEHAVIOR OF COEXISTENCE STATES TO A DIFFUSIVE HOLLING TYPE II PREDATOR-PREY MODEL WITH HUNTING COOPERATION[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1413-1429. doi: 10.11948/20240251
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