Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator

Fatao Wang; Ruizhi Yang; Yining Xie; Jing Zhao; Fatao Wang; Ruizhi Yang; Yining Xie; Jing Zhao

doi:10.3934/math.2023905

AIMS Mathematics

2023, Volume 8, Issue 8: 17719-17743. doi: 10.3934/math.2023905

Previous Article Next Article

Research article

Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator

Northeast Forestry University, Harbin 150040, China

Received: 17 March 2023 Revised: 15 May 2023 Accepted: 17 May 2023 Published: 24 May 2023
MSC : 34K18, 35B32, 34K18, 34D23, 34H05, 35B32, 92D25

In this work, a Leslie-Gower model with a weak Allee effect on the prey and a fear effect on the predator is proposed. By using qualitative analyses, the local stability of the coexisting equilibrium and the existence of Turing instable are discussed. By analyzing the distribution of eigenvalues, the existence of a Hopf bifurcation is studied by using the gestation time delay as a bifurcation parameter. By utilizing the normal form method and the center manifold theorem, we calculate the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. We indicate that both the weak Allee effect on the prey and fear effect on the predator have an important impact on the dynamical behaviour of the new Leslie-Gower model. We also verify the obtained results by some numerical examples.
- predator-prey,
- fear effect,
- weak Allee effect,
- nonlocal competition,
- delay,
- Hopf bifurcation
Citation: Fatao Wang, Ruizhi Yang, Yining Xie, Jing Zhao. Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator[J]. AIMS Mathematics, 2023, 8(8): 17719-17743. doi: 10.3934/math.2023905

Related Papers:

Abstract

In this work, a Leslie-Gower model with a weak Allee effect on the prey and a fear effect on the predator is proposed. By using qualitative analyses, the local stability of the coexisting equilibrium and the existence of Turing instable are discussed. By analyzing the distribution of eigenvalues, the existence of a Hopf bifurcation is studied by using the gestation time delay as a bifurcation parameter. By utilizing the normal form method and the center manifold theorem, we calculate the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. We indicate that both the weak Allee effect on the prey and fear effect on the predator have an important impact on the dynamical behaviour of the new Leslie-Gower model. We also verify the obtained results by some numerical examples.

References

[1]	T. Faria, L. T. Magalhaes, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Differ. Equations, 122 (1995), 181–200. https://doi.org/10.1006/JDEQ.1995.1144 doi: 10.1006/JDEQ.1995.1144
[2]	T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217–2238. https://doi.org/10.1090/S0002-9947-00-02280-7 doi: 10.1090/S0002-9947-00-02280-7
[3]	F. Yi, J. Wei, J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equations, 246 (2009), 1944–1977. https://doi.org/10.1016/J.JDE.2008.10.024 doi: 10.1016/J.JDE.2008.10.024
[4]	B. Messaoud, M. B. Almatrafi, Bifurcation and stability of two-dimensional activator-inhibitor model with fractional-order derivative, Fractal Fract., 7 (2023), 344. https://doi.org/10.3390/fractalfract7050344 doi: 10.3390/fractalfract7050344
[5]	A. Q. Khan, S. A. H. Bukhari, M. B. Almatrafi, Global dynamics, Neimark-Sacker bifurcation and hybrid control in a Leslie's prey-predator model, Alexandria Eng. J., 61 (2022), 11391–11404. https://doi.org/10.1016/j.aej.2022.04.042 doi: 10.1016/j.aej.2022.04.042
[6]	A. Q. Khan, F. Nazir, M. B. Almatrafi, Bifurcation analysis of a discrete Phytoplankton-Zooplankton model with linear predational response function and toxic substance distribution, Int. J. Biomath., 16 (2022), 2250095. https://doi.org/10.1142/s1793524522500954 doi: 10.1142/s1793524522500954
[7]	A. Q. Khan, M. Tasneem, M. B. Almatrafi, Discrete-time COVID-19 epidemic model with bifurcation and control, Math. Biosci. Eng., 19 (2022), 1944–1969. https://doi.org/10.3934/mbe.2022092 doi: 10.3934/mbe.2022092
[8]	J. Li, Y. Song, Spatially inhomogeneous periodic patterns induced by distributed memory in the memory-based single population model, Appl. Math. Lett., 137 (2023), 108490. https://doi.org/10.1016/j.aml.2022.108490 doi: 10.1016/j.aml.2022.108490
[9]	H. Shen, Y. Song, H. Wang, Bifurcations in a diffusive resource-consumer model with distributed memory, J. Differ. Equations, 347 (2023), 170–211. https://doi.org/10.1016/j.jde.2022.11.044 doi: 10.1016/j.jde.2022.11.044
[10]	S. Pal, S. Majhi, S. Mandal, N. Pal, Role of fear in a predator-prey model with Beddington-DeAngelis functional response, Z. Nat. A, 74 (2019), 581–595. https://doi.org/10.1515/ZNA-2018-0449 doi: 10.1515/ZNA-2018-0449
[11]	E. L. Preisser, D. I. Bolnick, The many faces of fear: comparing the pathways and impacts of nonconsumptive predator effects on prey populations, PLoS ONE, 3 (2008), e2465. https://doi.org/10.1371/journal.pone.0002465 doi: 10.1371/journal.pone.0002465
[12]	S. Creel, D. Christianson, Relationships between direct predation and risk effects, Trends Ecol. Evol., 23 (2008), 194–201. https://doi.org/10.1016/j.tree.2007.12.004 doi: 10.1016/j.tree.2007.12.004
[13]	R. Yang, Q. Song, Y. An, Spatiotemporal dynamics in a predator-prey model with functional response increasing in both predator and prey densities, Mathematics, 10 (2022), 17. https://doi.org/10.3390/math10010017 doi: 10.3390/math10010017
[14]	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 doi: 10.1111/1365-2435.12007
[15]	Y. Song, Y. Peng, T. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differ. Equations, 300 (2021), 597–624. https://doi.org/10.1016/J.JDE.2021.08.010 doi: 10.1016/J.JDE.2021.08.010
[16]	X. Wang, L. Y. 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 doi: 10.1007/S00285-016-0989-1
[17]	R. Pringle, T. Kartzinel, T. Palmer, T. J. Thurman, K. Fox-Dobbs, C. C. Y. Xu, et al., Predator-induced collapse of niche structure and species coexistence, Nature, 570 (2019), 58–64. https://doi.org/10.1038/s41586-019-1264-6 doi: 10.1038/s41586-019-1264-6
[18]	P. Pandy, N. Pal, S. Samanta, J. Chattopadhyay, A three species food chain model with fear induced trophic cascade, Int. J. Appl. Comput. Math., 5 (2019), 100. https://doi.org/10.1007/s40819-019-0688-x doi: 10.1007/s40819-019-0688-x
[19]	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 doi: 10.1038/ncomms10698
[20]	W. C. Allee, A. Aggregations, A study in general sociology, University of Chicago Press, 1931. https://doi.org/10.2307/2961735
[21]	T. Liu, L. Chen, F. Chen, Z. Li, Dynamics of a Leslie-Gower model with weak Allee effect on prey and fear effect on predator, Int. J. Bifurcation Chaos, 33 (2023), 2350008. https://doi.org/10.1142/s0218127423500086 doi: 10.1142/s0218127423500086
[22]	J. Jiao, C. Chen, Bogdanov-Takens bifurcation analysis of a delayed predator-prey system with double Allee effect, Nonlinear Dyn., 104 (2021), 1697–1707. https://doi.org/10.1007/s11071-021-06338-x doi: 10.1007/s11071-021-06338-x
[23]	P. Aguirre, A general class of predation models with multiplicative Allee effect, Nonlinear Dyn., 78 (2014), 629–648. https://doi.org/10.1007/S11071-014-1465-3 doi: 10.1007/S11071-014-1465-3
[24]	F. Courchamp, T. Clutton-Brock, B. Grenfell, F. Courchamp T. Clutton-Brock, B. Grenfell, et al., Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14 (1999), 405–410. https://doi.org/10.1016/S0169-5347(99)01683-3 doi: 10.1016/S0169-5347(99)01683-3
[25]	P. Feng, Y. Kang, Dynamics of a modified Leslie-Gower model with double Allee effects, Nonlinear Dyn., 80 (2015), 1051–1062. https://doi.org/10.1007/S11071-015-1927-2 doi: 10.1007/S11071-015-1927-2
[26]	N. Iqbal, R. Wu, Turing patterns induced by cross-diffusion in a 2D domain with strong Allee effect, C. R. Math., 357 (2019), 863–877. https://doi.org/10.1016/j.crma.2019.10.011 doi: 10.1016/j.crma.2019.10.011
[27]	D. S. Boukal, L. Berec, Modelling mate-finding Allee effects and populations dynamics, with applications in pest control, Popul. Ecol., 51 (2009), 445–458. https://doi.org/10.1007/s10144-009-0154-4 doi: 10.1007/s10144-009-0154-4
[28]	M. H. Wang, M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83–97. https://doi.org/10.1016/S0025-5564(01)00048-7 doi: 10.1016/S0025-5564(01)00048-7
[29]	T. Liu, L. Chen, F. Chen, Z. Li, Stability analysis of a Leslie-Gower model with strong Allee effect on prey and fear effect on predator, Int. J. Bifurcation Chaos, 32 (2022), 2250082. https://doi.org/10.1142/S0218127422500821 doi: 10.1142/S0218127422500821
[30]	K. Fang, Z. L. Zhu, F. D. Chen, Z. Li, Qualitative and bifurcation analysis in a Leslie-Gower model with Allee effect, Qual. Theory Dyn. Syst., 21 (2022), 86. https://doi.org/10.1007/s12346-022-00591-0 doi: 10.1007/s12346-022-00591-0
[31]	L. M. Zhang, Y. K. Xu, G. Y. Liao, Codimension-two bifurcations and bifurcation controls in a discrete biological system with weak Allee effect, Int. J. Bifurcation Chaos, 32 (2022), 2250036. https://doi.org/10.1142/s0218127422500365 doi: 10.1142/s0218127422500365
[32]	L. Zhao, J. H. Shen, Relaxation oscillations in a slow-fast predator-prey model with weak Allee effect and Holling-Ⅳ functional response, Commun. Nonlin. Sci. Numer. Simul., 112 (2022), 106517. https://doi.org/10.1016/j.cnsns.2022.106517 doi: 10.1016/j.cnsns.2022.106517
[33]	R. Yang, X. Zhao, Y. An, Dynamical analysis of a delayed diffusive predator-prey model with additional food provided and anti-predator behavior, Mathematics, 10 (2022), 469. https://doi.org/10.3390/math10030469 doi: 10.3390/math10030469
[34]	W. Zuo, J. Wei, Stability and Hopf bifurcation in a diffusive predator-prey system with delay effect, Nonlinear Anal.: Real World Appl., 12 (2011), 1998–2011. https://doi.org/10.1016/J.NONRWA.2010.12.016 doi: 10.1016/J.NONRWA.2010.12.016
[35]	R. Yang, D. Jin, W. Wang, A diffusive predator-prey model with generalist predator and time delay, AIMS Math., 7 (2022), 4574–4591. https://doi.org/10.3934/math.2022255 doi: 10.3934/math.2022255
[36]	J. F. Zhang, X. P. Yan, Effects of delay and diffusion on the dynamics of a Leslie-Gower type predator-prey model, Int. J. Bifurcation Chaos, 24 (2014), 1450043. https://doi.org/10.1142/S0218127414500436 doi: 10.1142/S0218127414500436
[37]	Y. Song, Y. Peng, T. Zhang, Double Hopf bifurcation analysis in the memory-based diffusion system, J. Dyn. Differ. Equ., 2022. https://doi.org/10.1007/s10884-022-10180-z doi: 10.1007/s10884-022-10180-z
[38]	M. U. Akhmet, M. Beklioglu, T. Ergenc, V. I. Tkachenko, An impulsive ratio-dependent predator-prey system with diffusion, Nonlinear Anal.: Real World Appl., 7 (2006), 1255–1267. https://doi.org/10.1016/j.nonrwa.2005.11.007 doi: 10.1016/j.nonrwa.2005.11.007
[39]	Y. Liu, J. Wei, Double Hopf bifurcation of a diffusive predator-prey system with strong Allee effect and two delays, Nonlinear Anal.: Model. Control, 26 (2021), 72–92. https://doi.org/10.15388/namc.2021.26.20561 doi: 10.15388/namc.2021.26.20561
[40]	Y. Liu, D. Duan, B. Niu, Spatiotemporal dynamics in a diffusive predator-prey model with group defense and nonlocal competition, Appl. Math. Lett., 103 (2019), 106175. https://doi.org/10.1016/j.aml.2019.106175 doi: 10.1016/j.aml.2019.106175
[41]	R. Yang, F. Wang, D. Jin, Spatially inhomogeneous bifurcating periodic solutions induced by nonlocal competition in a predator-prey system with additional food, Math. Methods Appl. Sci., 45 (2022), 9967–9978. https://doi.org/10.1002/mma.8349 doi: 10.1002/mma.8349
[42]	S. Chen, J. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete Contin. Dyn. Syst., 38 (2018), 43–62. https://doi.org/10.3934/DCDS.2018002 doi: 10.3934/DCDS.2018002
[43]	R. Yang, C. Nie, D. Jin, Spatiotemporal dynamics induced by nonlocal competition in a diffusive predator-prey system with habitat complexity, Nonlinear Dyn., 110 (2022), 879–900. https://doi.org/10.1007/s11071-022-07625-x doi: 10.1007/s11071-022-07625-x
[44]	D. Geng, W. Jiang, Y. Lou, H. Wang, Spatiotemporal patterns in a diffusive predator-prey system with nonlocal intraspecific prey competition, Stud. Appl. Math., 148 (2021), 396–432. https://doi.org/10.1111/sapm.12444 doi: 10.1111/sapm.12444
[45]	M. G. Clerc, D. Escaff, V. M. Kenkre, Analytical studies of fronts, colonies, and patterns: combination of the Allee effect and nonlocal competition interactions, Phys. Rev. E, 82 (2010), 036210. https://doi.org/10.1103/PHYSREVE.82.036210 doi: 10.1103/PHYSREVE.82.036210
[46]	Y. E. Maruvka, T. Kalisky, N. M. Shnerb, Nonlocal competition and the speciation transition on random networks, Phys. Rev. E, 78 (2008), 031920. https://doi.org/10.1103/PHYSREVE.78.031920 doi: 10.1103/PHYSREVE.78.031920
[47]	N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57–66. https://doi.org/10.1016/S0022-5193(89)80189-4 doi: 10.1016/S0022-5193(89)80189-4
[48]	J. Furter, M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65–80. https://doi.org/10.1007/BF00276081 doi: 10.1007/BF00276081

Reader Comments

Your name:*

Email:*
© 2023 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)