Bifurcations of a Leslie-Gower predator-prey model with fear, strong Allee effect and hunting cooperation

Weili Kong; Yuanfu Shao; Weili Kong; Yuanfu Shao

doi:10.3934/math.20241520

AIMS Mathematics

2024, Volume 9, Issue 11: 31607-31635. doi: 10.3934/math.20241520

Previous Article Next Article

Research article Special Issues

Bifurcations of a Leslie-Gower predator-prey model with fear, strong Allee effect and hunting cooperation

Weili Kong ¹,
Yuanfu Shao ^{2
,
,}

1.
College of Teacher Education, Qujing Normal University, Qujing, Yunnan 655011, China
2.
School of Mathematics and Statistics, Guilin University of Technology, Guilin, Guangxi 541004, China

Received: 11 September 2024 Revised: 19 October 2024 Accepted: 01 November 2024 Published: 07 November 2024
MSC : 60H10, 92B05

Considering the impact of fear levels, Allee effects and hunting cooperation factors on system stability, a Leslie-Gower predator-prey model was formulated. The existence, stability and bifurcation analysis of equilibrium points were studied by use of topological equivalence, characteristic equations, Sotomayor's theorem, and bifurcation theory. The sufficient conditions of saddle-node, Hopf, and Bogdanov-Takens bifurcations were established, respectively. Numerically, the theoretical findings were validated and some complicated dynamical behaviors as periodic fluctuation and multi-stability were revealed. The parameter critical values of saddle-node, Hopf bifurcation, and Bogdanov-Takens bifurcations were established. Biologically, how these factors of fear, Allee effect, and hunting cooperation affect the existence of equilibria and jointly affect the system dynamics were analyzed.
- fear,
- Allee effect,
- hunting cooperation,
- equilibrium,
- bifurcation
Citation: Weili Kong, Yuanfu Shao. Bifurcations of a Leslie-Gower predator-prey model with fear, strong Allee effect and hunting cooperation[J]. AIMS Mathematics, 2024, 9(11): 31607-31635. doi: 10.3934/math.20241520

Related Papers:

Abstract

Considering the impact of fear levels, Allee effects and hunting cooperation factors on system stability, a Leslie-Gower predator-prey model was formulated. The existence, stability and bifurcation analysis of equilibrium points were studied by use of topological equivalence, characteristic equations, Sotomayor's theorem, and bifurcation theory. The sufficient conditions of saddle-node, Hopf, and Bogdanov-Takens bifurcations were established, respectively. Numerically, the theoretical findings were validated and some complicated dynamical behaviors as periodic fluctuation and multi-stability were revealed. The parameter critical values of saddle-node, Hopf bifurcation, and Bogdanov-Takens bifurcations were established. Biologically, how these factors of fear, Allee effect, and hunting cooperation affect the existence of equilibria and jointly affect the system dynamics were analyzed.

References

[1]	J. D. Murray, Mathematical biology, New York: Springer, 1993.
[2]	P. H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35 (1948), 213–245. http://doi.org/10.2307/2332342 doi: 10.2307/2332342
[3]	P. H. Leslie, J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219–234. http://doi.org/10.2307/2333294 doi: 10.2307/2333294
[4]	P. H. Ye, D. Y. Wu, Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system? Chinese J. Phys., 68 (2020), 49–64. http://doi.org/10.1016/j.cjph.2020.07.021 doi: 10.1016/j.cjph.2020.07.021
[5]	W. Ni, M. Wang, Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey? Discrete Contin. Dyn. Syst.-B, 22 (2017), 3409–3420. http://doi.org/10.3934/dcdsb.2017172 doi: 10.3934/dcdsb.2017172
[6]	D. Melese, S. Feyissa, Stability and bifurcation analysis of a diffusive modified Leslie-Gower prey-predator model with prey infection and Beddington-DeAngelis functional response, Heliyon, 7 (2021), e06193. http://doi.org/10.1016/j.heliyon.2021.e06193 doi: 10.1016/j.heliyon.2021.e06193
[7]	X. Feng, X. Liu, C. Sun, Y. L. Jiang, Stability and Hopf bifurcation of a modified Leslie-Gower predator-prey model with Smith growth rate and Beddington-DeAngelis functional response, Chaos Solitons Fract., 174 (2023), 113794. http://doi.org/10.1016/j.chaos.2023.113794 doi: 10.1016/j.chaos.2023.113794
[8]	M. M. Chen, Y. Takeuchi, J. F. Zhang, Dynamic complexity of a modified Leslie-Gower predator-prey system with fear effect? Commun. Nonlinear Sci. Numer. Simul., 101 (2023), 107109. http://doi.org/10.1016/j.cnsns.2023.107109 doi: 10.1016/j.cnsns.2023.107109
[9]	S. M. Fu, H. S. Zhang, Effect of hunting cooperation on the dynamic behavior for a diffusive Holling type Ⅱ predator-prey model, Commun. Nonlinear Sci. Numer. Simul., 99 (2021), 105807. http://doi.org/10.1016/j.cnsns.2021.105807 doi: 10.1016/j.cnsns.2021.105807
[10]	Z. Wei, Y. H. Xia, T. H. Zhang, Dynamic analysis of multi-factor influence on a Holling type Ⅱ predator-prey model, Qual. Theory Dyn. Syst., 21 (2022), 124. http://doi.org/10.1007/s12346-022-00653-3 doi: 10.1007/s12346-022-00653-3
[11]	Z. H. Shang, Y. H. Qiao, Bifurcation analysis of a Leslie-type predator-prey system with simplified Holling type Ⅳ functional response and strong Allee effect on prey, Nonlinear Anal. Real World Appl., 64 (2022), 103453. http://doi.org/10.1016/j.nonrwa.2021.103453 doi: 10.1016/j.nonrwa.2021.103453
[12]	M. X. He, Z. Li, Global dynamics of a Leslie-Gower predator-prey model with square root response function, Appl. Math. Lett., 140 (2023), 108561. http://doi.org/10.1016/j.aml.2022.108561 doi: 10.1016/j.aml.2022.108561
[13]	W. Kong, Y. Shao, The effects of fear and delay on a predator-prey model with Crowley-Martin functional response and stage structure for predator, AIMS Math., 8 (2023), 29260–29289. http://doi.org/10.3934/math.20231498 doi: 10.3934/math.20231498
[14]	X. Feng, C. Sun, W. Yang, C. T. Li, Dynamics of a predator-prey model with nonlinear growth rate and Beddington-DeAngelis functional response, Nonlinear Anal. Real World Appl., 65 (2023), 103766. http://doi.org/10.1016/j.nonrwa.2022.103766 doi: 10.1016/j.nonrwa.2022.103766
[15]	C. Packer, D. Scheel, A. E. Pusey, Why lions form groups: food is not enough, Amer. Nat., 136 (1990), 1–19. http://doi.org/10.1086/285079 doi: 10.1086/285079
[16]	S. Creel, N. M. Creel, Communal hunting and pack size in African wild dogs, Lycaon Pictus. Anim. Behav., 50 (1995), 1325–1339. http://doi.org/10.1016/0003-3472(95)80048-4 doi: 10.1016/0003-3472(95)80048-4
[17]	P. S. Rodman, Inclusive fitness and group size with a reconsideration of group sizes in lions and wolves, Amer. Nat., 118 (1981), 275–283.
[18]	M. T. Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13–22. http://doi.org/10.1016/j.jtbi.2017.02.002 doi: 10.1016/j.jtbi.2017.02.002
[19]	D. Pal, D. Kesh, D. Mukherjee, Cross-diffusion mediated Spatiotemporal patterns in a predator-prey system with hunting cooperation and fear effect, Math. Comput. Simul., 220 (2024), 128–147. http://doi.org/10.1016/j.matcom.2024.01.003 doi: 10.1016/j.matcom.2024.01.003
[20]	Y. Z. Liu, Z. Y Zhang, Z. Li, The impact of Allee effect on a Leslie-Gower predator-prey model with hunting cooperation, Qual. Theory Dyn. Syst., 23 (2024), 88. http://doi.org/10.1007/s12346-023-00940-7 doi: 10.1007/s12346-023-00940-7
[21]	J. Zhang, W. N. Zhang, Dynamics of a predator-prey model with hunting cooperation and Allee effects in predators, Int. J. Bifurc. Chaos, 30 (2020), 2050199. http://doi.org/10.1142/S0218127420501990 doi: 10.1142/S0218127420501990
[22]	K. Vishwakarma, M. Sen, Role of Allee effect in prey and hunting cooperation in a generalist predator, Math. Comput. Simul., 190 (2021), 622–640. http://doi.org/10.1016/j.matcom.2021.05.023 doi: 10.1016/j.matcom.2021.05.023
[23]	W. C. Allee, Animal aggregations, Quarterly Rev. Biol., 2 (1927), 367–398.
[24]	W. C. Allee, Animal aggregations: a study in general sociology, Chicago: University of Chicago Press, 1931.
[25]	F. Courchamp, L. Berec, J. Gascoigne, Allee effects in ecology and conservation, Oxford: Oxford University Press, 2008.
[26]	G. Mandal, S. Das, G. L. Narayan, C. Santabrata, Dynamic response of a system of interactive species influenced by fear and Allee consequences, Eur. Phys. J. Plus, 138 (2023), 661. http://doi.org/10.1140/epjp/s13360-023-04246-0 doi: 10.1140/epjp/s13360-023-04246-0
[27]	M. Kuussaari, I. Saccheri, M. Camara, I. Hanski, Allee effect and population dynamics in the glanville fritillary butterfly, Oikos, 82 (1998), 384–392. http://doi.org/10.2307/3546980 doi: 10.2307/3546980
[28]	F. Courchamp, B. T. Grenfell, T. H. Clutton-Brock, Impact of natural enemies on obligately cooperative breeders, Oikos, 91 (2000), 311–322. http://doi.org/10.1034/j.1600-0706.2000.910212.x doi: 10.1034/j.1600-0706.2000.910212.x
[29]	A. W. Stoner, M. Ray-Culp, Evidence for Allee effects in an over-harvested marine gastropod: density-dependent mating and egg production, Marine Ecol. Progr. Ser., 202 (2000), 297–302. http://doi.org/10.3354/meps202297 doi: 10.3354/meps202297
[30]	P. A. Stephens, W. J. Sutherland, Consequences of the Allee effect for behaviour ecology and conservation, Trends Ecol. Evolut., 14 (1999), 401–405. http://doi.org/10.1016/S0169-5347(99)01684-5 doi: 10.1016/S0169-5347(99)01684-5
[31]	F. Courchamp, L. Berec, J. Gascoigne, Allee effects in ecology and conservation, New York: Oxford University Press, 2008.
[32]	T. Ma, X. Z. Meng, T. Hayat, Hopf bifurcation induced by time delay and influence of Allee effect in a diffusive predator-prey system with herd behavior and prey chemotaxis, Nonlinear Dyn., 108 (2022), 4581–4598. http://doi.org/10.1007/s11071-022-07401-x doi: 10.1007/s11071-022-07401-x
[33]	R. Yadav, N. Mukherjee, M. Sen, Spatiotemporal dynamics of a prey-predator model with Allee effect in prey and hunting cooperation in a Holling type Ⅲ functional response, Nonlinear Dyn., 107 (2022), 1397–1410. http://doi.org/10.1007/s11071-021-07066-y doi: 10.1007/s11071-021-07066-y
[34]	Y. Shi, J. H. Wu, Q. Cao, Analysis on a diffusive multiple Allee effects predator-prey model induced by fear factors, Nonlinear Anal. Real World Appl., 59 (2021), 103249. http://doi.org/10.1016/j.nonrwa.2020.103249 doi: 10.1016/j.nonrwa.2020.103249
[35]	W. Z. Lidicker, The Allee effect: its history and future importance, Open Ecol. J., 3 (2010), 71–82. http://doi.org/10.2174/1874213001003010071 doi: 10.2174/1874213001003010071
[36]	K. Garain, P. S. Mandal, Bubbling and hydra effect in a population system with Allee effect, Ecol. Complex., 47 (2021), 100939. http://doi.org/10.1016/j.ecocom.2021.100939 doi: 10.1016/j.ecocom.2021.100939
[37]	J. V. Buskirk, K. L. Yurewicz, Effects of predators on prey growth rate: relative contributions of thinning and reduced activity, Oikos, 82 (1998), 20–28. http://doi.org/10.2307/3546913 doi: 10.2307/3546913
[38]	S. Creel, D. Christianson, Relationships between direct predation and risk effects, Trends Ecol. Evol., 23 (2008), 194–201. http://doi.org/10.1016/j.tree.2007.12.004 doi: 10.1016/j.tree.2007.12.004
[39]	S. Creel, D. Christianson, L. Stewart, A. John, J. Winnie, Predation risk affects reproductive physiology and demography of elk, Science, 315 (2007), 960–960. http://doi.org/10.1126/science.1135918 doi: 10.1126/science.1135918
[40]	A. J. Wirsing, M. R. Heithaus, M. D. Lawrence, Fear factor: Do dugongs (Dugong dugon) trade food for safety from tiger sharks (Galeocerdo cuvier)? Oecologia, 153 (2007), 1031–1040. http://doi.org/10.1007/s00442-007-0802-3 doi: 10.1007/s00442-007-0802-3
[41]	X. Y. Wang, L. Zanette, X. F. Zou, Modelling the fear effect in predator-prey interactions, J. Math. Biol., 73 (2016), 1179–1204. http://doi.org/10.1007/s00285-016-0989-1 doi: 10.1007/s00285-016-0989-1
[42]	A. Kumar, B. Dubey, Modeling the effect of fear in a prey-predator system with prey refuge and gestation delay, Int. J. Bifurc. Chaos, 29 (2019), 1950195. http://doi.org/10.1142/S0218127419501955 doi: 10.1142/S0218127419501955
[43]	B. M. Das, D. Sahoo, G. P. Samanta, Impact of fear in a delay-induced predator-prey system with intraspe-cific competition within predator species, Math. Comput. Simul., 191 (2022), 134–156. http://doi.org/10.1016/j.matcom.2021.08.005 doi: 10.1016/j.matcom.2021.08.005
[44]	Y. Shao, Fear and delay effects on a food chain system with two kinds of different functional Responses, Int. J. Biomath., 17 (2024), 2350025. http://doi.org/10.1142/S1793524523500250 doi: 10.1142/S1793524523500250
[45]	B. Mondal, S. Roy, U. Ghosh, T. P. Kumar, A systematic study of autonomous and nonautonomous predator-prey models for the combined effects of fear, refuge, cooperation and harvesting, Eur. Phys. J. Plus, 137 (2022), 724. http://doi.org/10.1140/epjp/s13360-022-02915-0 doi: 10.1140/epjp/s13360-022-02915-0
[46]	X. B. Zhang, H. L. Zhu, Q. An, Dynamics analysis of a diffusive predator-prey model with spatial memory and nonlocal fear effect, J. Math. Anal. Appl., 525 (2023), 127123. http://doi.org/10.1016/j.jmaa.2023.127123 doi: 10.1016/j.jmaa.2023.127123
[47]	C. Chai, Y. Shao, Y. P. Wang, Analysis of a Holling-type Ⅳ stochastic prey-predator system with anti-predatory behavior and Lévy noise, AIMS Math., 8 (2023), 21033–21054. http://doi.org/10.3934/math.20231071 doi: 10.3934/math.20231071
[48]	R. Xue, Y. Shao, M. J. Cui, Analysis of a stochastic predator-prey system with fear effect and Lévy noise, Adv. Contin. Discr. Model., 2022 (2022), 72. http://doi.org/10.1186/s13662-022-03749-x doi: 10.1186/s13662-022-03749-x
[49]	P. Feng, Y. Kang, Dynamics of a modified Leslie-Gower model with double Allee effects, Nonlinear Dyn., 80 (2015), 1051–1062. http://doi.org/10.1007/s11071-015-1927-2 doi: 10.1007/s11071-015-1927-2
[50]	Z. F. Zhang, T. R. Ding, W. Z. Huang, Z. X. Dong, Qualitative theory of differential equations, Beijing: Science Rress, 1992.
[51]	L. Perko, Differential equations and dynamical systems, New York: Springer, 1996.
[52]	H. J. Alsakaji, S. Kundu, F. A. Rihan, Delay differential model of one-predator two-prey system with Monod-Haldane and Holling type Ⅱ functional responses, Appl. Math. Comput., 15 (2021), 125919. http://doi.org/10.1016/j.amc.2020.125919 doi: 10.1016/j.amc.2020.125919
[53]	D. Bai, B. Zheng, Y. Kang, Global dynamics of a predator-prey model with a Smith growth function and the additive predation in prey, Discrete Cont. Dyn.-B, 29 (2024), 1923–1960.
[54]	D. Bai, J. Yu, B. Zheng, J. Wu, Hydra effect and global dynamics of predation with strong Allee effect in prey and intraspecific competition in predator, J. Differ. Equ., 384 (2024), 120–164. http://doi.org/10.1016/j.jde.2023.11.017 doi: 10.1016/j.jde.2023.11.017

Reader Comments

Your name:*

Email:*
© 2024 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)