Advanced Search
Article Contents
Article Contents

Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator

  • * Corresponding author: Jianshe Yu

    * Corresponding author: Jianshe Yu 
Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value $ \rho $ of the ratio $ \frac{d_2}{d_1} $ of diffusions of predator to prey is obtained, such that if $ \frac{d_2}{d_1}>\rho $, then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if $ \frac{d_2}{d_1}<\rho $, both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio $ \frac{d_2}{d_1} $. As $ \frac{d_2}{d_1} $ increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.

    Mathematics Subject Classification: Primary: 35B36, 45M10; Secondary: 92C15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Stationary hot spots pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.226) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $.

    Figure 2.  Stationary stripes pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.27) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $.

    Figure 3.  Stationary spots-stripes pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.45) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $

  • [1] J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equation, SIAM J. Math. Anal., 17 (1986), 1339-1353.  doi: 10.1137/0517094.
    [2] J. L. BrownV. Morales and K. Summers, Tactical reproductive parasitism via larval cannibalism in Peruvian poison frogs, Biol. Lett., 5 (2009), 148-151.  doi: 10.1098/rsbl.2008.0591.
    [3] Y. L. CaiM. BanerjeeY. Kang and W. M. Wang, Spationtemporal complexity in a predator-prey model with weak Allee effects, Math. Biosci. Eng., 11 (2014), 1247-1274.  doi: 10.3934/mbe.2014.11.1247.
    [4] Y. L. Cai, Z. J. Gui, X. B. Zhang, H. B. Shi and W. M. Wang, Bifurcations and pattern formation in a predator-prey model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850140, 17 pp. doi: 10.1142/S0218127418501407.
    [5] S. Chakraborty, The infuence of generalist predators in spatially extended predator-prey systems, Ecological Complexity, 23 (2015), 50-60.  doi: 10.1016/j.ecocom.2015.06.003.
    [6] D. W. Crowder and W. E. Snyder, Eating their way to the top? Mechanisms underlying the success of invasive generalist predators, Biol. Invas., 12 (2010), 2857-2876.  doi: 10.1007/s10530-010-9733-8.
    [7] Y. H. Du and J. P. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Tran. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.
    [8] Y. H. Du and J. P. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.
    [9] A. ErbachF. Lutscher and G. Seo, Bistability and limit cycles in generalist predator-prey dynamics, Ecological Complexity, 14 (2014), 48-55.  doi: 10.1016/j.ecocom.2013.02.005.
    [10] Y.-H. Fan and W.-T. Li, Global asymptotic stability of a ratio-dependent predator-prey system with diffusion, J. Comput. Appl. Math., 188 (2006), 205-227.  doi: 10.1016/j.cam.2005.04.007.
    [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Grundlehren der Mathematischen Wissenschaften, 224. Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-96379-7.
    [12] R. J. Han and B. X. Dai, Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect, Nonlinear Anal. Real World Appl., 45 (2019), 822-853.  doi: 10.1016/j.nonrwa.2018.05.018.
    [13] I. HanskiL. Hansson and H. Henttonen, Specialist predators, generalist predators, and the microtine rodent cycle, J. Anim. Ecol., 60 (1991), 353-367.  doi: 10.2307/5465.
    [14] M. P. HassellThe Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology, 13. Princeton University Press, Princeton, N.J., 1978.  doi: 10.2307/3280305.
    [15] X. He and S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017), 239-257.  doi: 10.1007/s00285-016-1082-5.
    [16] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/BFb0089649.
    [17] T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401.  doi: 10.1016/j.jmaa.2005.11.065.
    [18] Y. Kang and J. H. Fewell, Co-evolutionary dynamics of a social parasite-host interaction model: obligate versus facultative social parasitism, Nat. Resour. Model., 28 (2015), 398-455.  doi: 10.1111/nrm.12078.
    [19] Y. Kang and L. Wedekin, Dynamics of a intraguild predation model with generalist or specialist predator, J. Math. Biol., 67 (2013), 1227-1259.  doi: 10.1007/s00285-012-0584-z.
    [20] X. L. Li, G. P. Hu and Z. S. Feng, Pattern dynamics in a spatial predator-prey model with nonmonotonic response function, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850077, 17 pp. doi: 10.1142/S0218127418500773.
    [21] C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.
    [22] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.
    [23] S. MadecJ. CasasG. Barles and C. Suppo, Bistability induced by generalist natural enemies can reverse pest invasions, J. Math. Biol., 75 (2017), 543-575.  doi: 10.1007/s00285-017-1093-x.
    [24] C. MagalC. CosnerS. G. Ruan and J. Casas, Control of invasive hosts by generalist parasitoids, Math. Medi. Biol., 25 (2008), 1-20.  doi: 10.1093/imammb/dqm011.
    [25] Z. Mei, Numerical Bifurcation Analysis for Reaction-Diffusion Equations, Springer Series in Computational Mathematics, 28. Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-662-04177-2.
    [26] P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 919-942.  doi: 10.1017/S0308210500002742.
    [27] R. Peng and J. P. Shi, Non-existence of non-constant positive steady states of two Holling type-Ⅱ predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886.  doi: 10.1016/j.jde.2009.03.008.
    [28] R. PengM. X. Wang and M. Yang, Positive solutions of a diffusive prey-predator model in a heterogeneous environment, Math. Comput. Modelling, 46 (2007), 1410-1418.  doi: 10.1016/j.mcm.2007.02.001.
    [29] J. A. RosenheimH. K. KayaL. E. EhlerJ. J. Marois and B. A. Jaffee, Intraguild predation among biological control agents: Theory and evidence, Biol. Control, 5 (1995), 303-335.  doi: 10.1006/bcon.1995.1038.
    [30] S. I. Rothstein, Evolutionary rates and host defenses against avian brood parasitism, The American Naturalist, 109 (1975), 161-176.  doi: 10.1086/282984.
    [31] W. E. Snyder and A. R. Ives, Interactions between specialist and generalist natural enemies: parasitoids, predators, and pea aphid biocontrol, Ecology, 84 (2003), 91-107. 
    [32] Y. L. SongH. P. JiangQ.-X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 16 (2017), 2030-2062.  doi: 10.1137/16M1097560.
    [33] M. D. Sorenson, Effects of intra- and interspecific brood parasitism on a precocial host, the canvasback, Aythya valisineria, Behavioral Ecology, 8 (1997), 153-161.  doi: 10.1093/beheco/8.2.153.
    [34] C. N. SpottiswoodeR. M. Kilner and  N. B. DaviesBrood Parasitism, The Evolution of Parental Care, Oxford University Press, Oxford, 2012.  doi: 10.1093/acprof:oso/9780199692576.003.0001.
    [35] W. O. C. SymondsonK. D. Sunderland and M. H. Greenstone, Can generalist predators be effective biocontrol agents?, Annu. Rev. Entomol., 47 (2002), 561-594.  doi: 10.1146/annurev.ento.47.091201.145240.
    [36] Q. WangC. Y. Gai and J. D. Yan, Qualitative analysis of a Lotka-Volterra competition system with advection, Discrete Contin. Dyn. Syst., 35 (2015), 1239-1284.  doi: 10.3934/dcds.2015.35.1239.
    [37] W. M. WangX. Y. GaoY. L. CaiH. B. Shi and S. M. Fu, Turing patterns in a diffusive epidemic model with saturated infection force, J. Franklin Inst., 355 (2018), 7226-7245.  doi: 10.1016/j.jfranklin.2018.07.014.
    [38] S. H. Wu and Y. L. Song, Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition, Nonlinear Anal. Real World Appl., 48 (2019), 12-39.  doi: 10.1016/j.nonrwa.2019.01.004.
  • 加载中



Article Metrics

HTML views(659) PDF downloads(434) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint