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Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 3988-4006. doi: 10.3934/mbe.2019197
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Bifurcation structure of nonconstant positive steady states for a diffusive predator-prey model

  • School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, P.R. China
  • Received: 31 January 2019 Accepted: 18 April 2019 Published: 06 May 2019

  • In this paper, we make a detailed descriptions for the local and global bifurcation structure of nonconstant positive steady states of a modified Holling-Tanner predator-prey system under homogeneous Neumann boundary condition. We first give the stability of constant steady state solution to the model, and show that the system exhibits Turing instability. Second, we establish the local structure of the steady states bifurcating from double eigenvalues by the techniques of space decomposition and implicit function theorem. It is shown that under certain conditions, the local bifurcation can be extended to the global bifurcation.

    Citation: Dongfu Tong, Yongli Cai, Bingxian Wang, Weiming Wang. Bifurcation structure of nonconstant positive steady states for a diffusive predator-prey model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 3988-4006. doi: 10.3934/mbe.2019197

    Related Papers:

  • In this paper, we make a detailed descriptions for the local and global bifurcation structure of nonconstant positive steady states of a modified Holling-Tanner predator-prey system under homogeneous Neumann boundary condition. We first give the stability of constant steady state solution to the model, and show that the system exhibits Turing instability. Second, we establish the local structure of the steady states bifurcating from double eigenvalues by the techniques of space decomposition and implicit function theorem. It is shown that under certain conditions, the local bifurcation can be extended to the global bifurcation.


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    [1] J. Jang, W.-M. Ni and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dyn. Diff. Equa., 2 (2004), 297–320.
    [2] A. M. Turing, The chemical basis of morphogenesis, Philos. T. R. SOC. B, 237 (1852), 37–72.
    [3] N. F. Britton, Essential Mathematical Biology, Springer, New York, 2003.
    [4] L. A. Segel and J. L. Jackson, Dissipative structure: an explanation and an ecological example, J. Theor. Biol., 37 (1972), 545–559.
    [5] D. Alonso, F. Bartumeus and J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28–34.
    [6] W.-M. Wang, L. Zhang, H. Wang, et al., Pattern formation of a predator-prey system with Ivlev-type functional response, Ecol. Model., 221 (2008), 131–140.
    [7] C. Neuhauser, Mathematical challenges in spatial ecology, Notices AMS, 48 (2001), 1304–1314.
    [8] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, 2001.
    [9] A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, et al., Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev., 44 (2002), 311–370.
    [10] J. D. Murray, Mathematical biology. II: Spatial models and biomedical applications, Springer, New York, 2003.
    [11] P. Y. H. Pang and M. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. R. Soc. Edinb., 133 (2003), 919–942.
    [12] K. Kuto and Y. Yamada. Multiple coexistence states for a prey-predator system with crossdi ffusion, J. Differ. Equations, 197 (2004), 315–348.
    [13] K. Kuto, Stability of steady-state solutions to a prey-predator system with cross-diffusion, J. Differ. Equations, 197 (2004), 293–314.
    [14] X. Zeng and Z. Liu, Non-constant positive steady states of a prey-predator system with cross-diffusions, J. Math. Anal. Appl., 332 (2007), 989–1009.
    [15] R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differ. Equations, 247 (2009), 866–886.
    [16] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equations, 246 (2009), 1944–1977.
    [17] H. Shi,W.-T. Li and G. Lin. Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response, Nonl. Anal. Real, 11 (2010), 3711–3721.
    [18] Y. Cai, M. Banerjee, Y. Kang, et al., Spatiotemporal complexity in a predator-prey model with weak Allee effects, Math. Biosci. Eng., 11 (2014), 1247–1274.
    [19] S. Li, J. Wu and Y. Dong, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differ. Equations, 259 (2015), 1990–2029.
    [20] H. Shi and S. Ruan. Spatial, temporal and spatiotemporal patterns of diffusive predator-prey models with mutual interference, IMA J. Appl. Math., 80 (2015), 1534–1568.
    [21] Y. Cai and W.-M. Wang, Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion, Nonl. Anal. Real, 30 (2016), 99–125.
    [22] T. Kuniya and J.Wang, Global dynamics of an SIR epidemic model with nonlocal diffusion, Nonl. Anal. Real, 43 (2018), 262–282.
    [23] J.Wang, J.Wang and T. Kuniya, Analysis of an age-structured multi-group heroin epidemic model.Appl. Math. Comp., 347 (2019), 78–100.
    [24] Y. Cai, Z. Ding, B. Yang, et al., Transmission dynamics of Zika virus with spatial structure–A case study in Rio de Janeiro, Brazil. Phys. A, 514 (2019), 729–740.
    [25] Y. Cai, K. Wang and W.M. Wang, Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92 (2019), 190–195.
    [26] Y. Cai, X. Lian, Z. Peng, et al., Spatiotemporal transmission dynamics for influenza disease in a heterogenous environment, Nonl. Anal. Real., 46 (2019), 178–194.
    [27] Y. Cai, Z. Gui, X. Zhang, et al., Bifurcations and pattern formation in a predator-prey model,. Inter. J. Bifur. Chaos, 28 (2018), 1850140.
    [28] H. Zhang, Y. Cai, S. Fu, et al., Impact of the fear effect in a prey-predator model incorporating a prey refuge, Appl. Math. Comp., 356 (2019), 328–337.
    [29] X. Cao, Y. Song and T. Zhang. Hopf bifurcation and delay-induced Turing instability in a diffusive Iac Operon model, Inter. J. Bifur. Chaos, 26 (2016), 1650167.
    [30] J. Jiang, Y. Song and P. Yu, Delay-induced Triple-Zero bifurcation in a delayed Leslie-type predator-prey model with additive Allee effect, Inter. J. Bifur. Chaos, 26 (2016), 1650117.
    [31] Y. Song, H. Jiang, Q. Liu, et al., Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation. SIAM J. Appl. Dyn. Sys., 16 (2017), 2030–2062.
    [32] Y. Song and X. Tang, Stability, steady-state bifurcations, and Turing patterns in a predator-prey model with herd behavior and prey-taxis, Stud. Appl. Math., 139 (2017), 371–404.
    [33] S. Wu and Y. Song, Stability and spatiotemporal dynamics in a diffusive predator Cprey model with nonlocal prey competition, Nonl. Anal. Real, 48 (2019), 12–39.
    [34] S.-B. Hsu and T.-W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763–783.
    [35] Y. Lou and W.-M. Ni. Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79–131.
    [36] W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, T. Am. Math. Soc., 357 (2005), 3953–3969.
    [37] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Func. Anal., 7 (1971), 487–513.
    [38] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555–593.
    [39] I. Takagi, Point-condensation for a reaction-diffusion system, J. Differ. Equations, 61 (1986), 208–249.
    [40] A.Chertock, A. Kurganov, X. Wang, et al., On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51–95.
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  • © 2019 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)
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