Abstract
In this paper, we study a diffusive predator–prey system with the Allee effect and threshold hunting. First, the number of interior equilibrium points is determined by discussing the relation of parameters. Then, preliminary analysis on the local asymptotic stability and bifurcations of non-spatial system based on ordinary differential equations is presented. It is noted that four stable equilibrium points coexist due to the Allee effect and threshold hunting. The stability of interior equilibrium points and the existence of Turing instability induced by the diffusion, spatially homogeneous and inhomogeneous Hopf bifurcation, Turing–Hopf bifurcation are studied by analyzing the corresponding characteristic equation for spatial system. By constructing generalized Jacobian matrix, we analyze the stability of interior equilibrium point where u-component is equal to the threshold of functional response. These results show that the Allee effect, threshold hunting and diffusion have significant impacts on the dynamics. Last, we present some numerical simulations that supplement the analytic results.
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References
Allee, W.C.: Animal aggregations. Q. Rev. Biol. 2(3), 367–398 (1927)
Aronson, D.G.: The Role of Diffusion in Mathematical Population Biology: Skellam Revisited, Mathematics in Biology and Medicine. Springer-Verlag, Berlin, Heidelberg (1985)
Berec, L., Angulo, E., Courchamp, F.: Multiple Allee effects and population management. Trends Ecol. Evol. 22(4), 185–191 (2007)
Boukal, D.S., Berec, L.: Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. J. Theor. Biol. 218(3), 375–394 (2002)
Boukal, D.S., Sabelis, M.W., Berec, L.: How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor. Popul. Biol. 72(1), 136–147 (2007)
Burgman, M.A., Ferson, S., Akcakaya, H.R.: Risk Assessment in Conservation Biology. Springer Science & Business Media, New York (1993)
Cantrell, R.S., Cosner, C.: On the dynamics of predator–prey models with the Beddington–DeAngelis functional response. J. Math. Anal. Appl. 257(1), 206–222 (2001)
Cantrell, R.S., Cosner, C.: Spatial ecology via reaction–diffusion equations. Bull. Am. Math. Soc. 41, 551–557 (2004)
Cao, X., Jiang, W.: Turing-Hopf bifurcation and spatiotemporal patterns in a diffusive predator–prey system with Crowley–Martin functional response. Nonlinear Anal-Theor. 43, 428–450 (2018)
Carr, J.: Applications of Centre Manifold Theory. Springer Science & Business Media, New York (2012)
Chen, S., Wei, J., Zhang, J.: Dynamics of a diffusive predator–prey model: the effect of conversion rate. J. Dyn. Differ. Equ. 30(4), 1683–1701 (2018)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., et al.: Nonsmooth Analysis and Control Theory. Springer Science & Business Media, New York (2008)
Courchamp, F., Clutton-Brock, T., Grenfell, B.: Inverse density dependence and the Allee effect. Trends Ecol. Evol. 14(10), 405–410 (1999)
Courchamp, F., Berec, L., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press, Oxford (2008)
Gurtin, M.E., MacCamy, R.C.: On the diffusion of biological populations. Math. Biosci. 33(1–2), 35–49 (1977)
Hassard, B.D., Kazarinoff, N.D. Wan, Y.H.: Theory and Applications of Hopf Bifurcation, in: London Mathematical Society Lecture Note Series, vol. 41, Cambridge University Press, Cambridge (1981)
Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91(5), 293–320 (1959)
Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91(7), 385–398 (1959)
Jiang, W., An, Q., Shi, J.: Formulation of the normal forms of Turing–Hopf bifurcation in reaction–diffusion systems with time delay. (2018) arXiv preprint arXiv:1802.10286
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer Science & Business Media, New York (2013)
Mishra, P., Raw, S.N., Tiwari, B.: Study of a Leslie–Gower predator-prey model with prey defense and mutual interference of predators. Chaos Soliton. Fract. 120, 1–16 (2019)
Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Springer Science & Business Media, New York (2012)
Peng, R., Yi, F., Zhao, X.: Spatiotemporal patterns in a reaction–diffusion model with the Degn–Harrison reaction scheme. J. Differ. Equ. 254(6), 2465–2498 (2013)
Perko, L.: Differential Equations and Dynamical Systems. Springer Science & Business Media, New York (2013)
Petrovskii, S.V., Morozov, A.Y., Venturino, E.: Allee effect makes possible patchy invasion in a predator–prey system. Ecol. Lett. 5(3), 345–352 (2002)
Seo, G., DeAngelis, D.L.: A predator-prey model with a Holling type I functional response including a predator mutual interference. J. Nonlinear Sci. 21(6), 811–833 (2011)
Seo, G., Kot, M.: A comparison of two predator-prey models with Holling’s type I functional response. Math. Biosci. 212(2), 161–179 (2008)
Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38(1–2), 196–218 (1951)
Song, Y., Zhang, T., Peng, Y.: Turing–Hopf bifurcation in the reaction–diffusion equations and its applications. Commun. Nonlinear Sci. 33, 229–258 (2016)
Song, Y., Jiang, H., Liu, Q., et al.: Spatiotemporal dynamics of the diffusive Mussel–Algae model near Turing–Hopf bifurcation. SIAM. J. Appl. Dyn. Syst. 16(4), 2030–2062 (2017)
Taylor, C.M., Hastings, A.: Allee effects in biological invasions. Ecol. Lett. 8(8), 895–908 (2005)
Verma, M., Misra, A.K.: Modeling the effect of prey refuge on a ratio-dependent predator-prey system with the Allee effect. B. Math. Biol. 80(3), 626–656 (2018)
Wang, M., Kot, M.: Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 171(1), 83–97 (2001)
Wang, X., Wei, J.: Dynamics in a diffusive predator–prey system with strong Allee effect and Ivlev-type functional response. J. Math. Anal. Appl. 422(2), 1447–1462 (2015)
Wang, J., Shi, J., Wei, J.: Predator-prey system with strong Allee effect in prey. J. Math. Biol. 62(3), 291–331 (2011)
Wang, J., Shi, J., Wei, J.: Dynamics and pattern formation in a diffusive predator–prey system with strong Allee effect in prey. J. Differ. Equ. 251(4–5), 1276–1304 (2011)
Wu, D., Zhao, H., Yuan, Y.: Complex dynamics of a diffusive predator-prey model with strong Allee effect and threshold harvesting. J. Math. Anal. Appl. 469(2), 982–1014 (2019)
Xiao, D., Ruan, S.: Global analysis in a predator-prey system with nonmonotonic functional response. SIAM. J. Appl. Math. 61(4), 1445–1472 (2001)
Ye, Q., Li, Z., Wang, M., et al.: Introduction to Reaction–Diffusion Equations. Chinese Science Press, Beijing (1990)
Acknowledgements
The authors would like to thank the editor and anonymous referees for their valuable comments and suggestions which lead to an improvement of the manuscript. The work is partially supported by the National Natural Science Foundation of China (Nos. 11571170, 31570417); the Natural Science Foundation of Anhui Province of China (Nos. 1608085MA14, 1908085MA01); the Key Project of Natural Science Research of Anhui Higher Education Institutions of China (No. KJ2018A0365).
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Communicated by Dr. Anthony Bloch.
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Wu, D., Zhao, H. Spatiotemporal Dynamics of a Diffusive Predator–Prey System with Allee Effect and Threshold Hunting. J Nonlinear Sci 30, 1015–1054 (2020). https://doi.org/10.1007/s00332-019-09600-0
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DOI: https://doi.org/10.1007/s00332-019-09600-0