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Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 6753-6768. doi: 10.3934/mbe.2019337
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Positive steady states of a ratio-dependent predator-prey system with cross-diffusion

  • 1.
    School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
  • 2.
    Department of Mathematics, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA
  • Received: 25 January 2019 Accepted: 04 July 2019 Published: 26 July 2019

  • In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack's inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.

    Citation: Xiaoling Li, Guangping Hu, Xianpei Li, Zhaosheng Feng. Positive steady states of a ratio-dependent predator-prey system with cross-diffusion[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6753-6768. doi: 10.3934/mbe.2019337

    Related Papers:

  • In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack's inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.


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    [1] Y. Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, New York: Academic Press, 191 (1993).
    [2] A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, Singapore: World Scientific, (1998).
    [3] B. Li and Y. Kuang, Heteroclinic bifurcation in the Michaelis-Menten type ratiodependent predator-prey system, SIAM J. Appl. Math., 67 (2007), 1453–1464.
    [4] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389–406.
    [5] L. Zhang, J. Liu and M. Banerjee, Hopf and steady state bifurcation analysis in a ratio-dependent predatorprey model, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 52–73.
    [6] M. X. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio- dependent functional responses and diffusion, Phys. D, 196 (2004), 172–192.
    [7] P. Y. H. Pang and M. X. Wang, Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Roy. Soc. Edinburgh A, 133 (2003), 919–942.
    [8] R. Z. Yang, M. Liu and C. R. Zhang, A delayed-diffusive predator-prey model with a ratio-dependent functional response, Commun. Nonlinear Sci. Numer. Simul., 53 (2017), 94–110.
    [9] X. Jiang, Z. K. She, Z. Feng, et al., Bifurcation analysis of a predator-prey system with ratio-dependent functional response, Internat. J. Bifur. Chaos., 27 (2017), 1750222–21.
    [10] L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differ. Equations, 224 (2006), 39–59.
    [11] Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differ. Equations, 203 (2004), 331–364.
    [12] A. Madzvamuse, H. S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domian: the effects of cross-diffusion, Discrete Cont. Dyn. Syst., 36 (2017), 2133–2170.
    [13] W. Ko and K. Ryu, Non-constant positive steady-state of a diffusive predator-prey system in ho- mogeneous environment, J. Math. Anal. Appl., 327 (2007), 539–549.
    [14] Z. Du, Z. Feng and X. Zhang, Traveling wave phenomena of n-dimensional diffusive predator-prey systems, Nonlinear Anal. Real World Appl., 41 (2018), 288-312.
    [15] Z. L. Shen and J. Wei, Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect, Math. Biosci. Eng., 15 (2018), 693–715.
    [16] X. Y. Chang and J. Wei, Stability and Hopf bifurcation in a diffusive predator-prey system incor-porating a prey refuge, Math. Biosci. Eng., 10 (2013), 979–996.
    [17] E. Avila-Vales, G. Garca-Almeida and E. Rivero-Esquivel, Bifurcation and spatiotemporal pat-terns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response, Discrete Cont. Dyn. Syst. Ser. B, 22 (2017), 717–740.
    [18] R. Peng and M. X. Wang, Positive steady states of the Holling-Tanner prey-predator model with diffusion, Proc. Roy. Soc. Edinburgh A, 135 (2005), 149–164.
    [19] P. Y. H. Pang and M. X. Wang, N-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. Lond. Math. Soc., 88 (2004), 135–157.
    [20] H. B. Shi, W. T. Li and G. Lin, Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response, Nonlinear Anal. Real World Appl., 11 (2010), 3711–3721.
    [21] L. J. Hei and Y. Yu, Non-constant positive steady state of one resource and two consumers model with diffusion, J. Math. Anal. Appl., 339 (2008), 566–581.
    [22] C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differ. Equations, 72 (1998), 1–27.
    [23] L. Yang and Y. M. Zhang, Positive steady states and dynamics for a diffusive predator-prey system with a degeneracy, Acta Math. Sci., 36 (2016), 537–548.
    [24] W. M. Ni, Cross-diffusion and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), 9–18.
    [25] X. Z. Zeng, Non-constant positive steady states of a prey-predator system with cross-diffusions, J. Math. Anal. Appl., 332 (2007), 989–1009.
    [26] G. P. Hu and X. L. Li, Turing patterns of a predator-prey model with a modified Leslie-Gower term and cross-diffusion, Int. J. Biomath., 5 (2012), 1250060–1250082.
    [27] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79–131.
    [28] C. R. Tian, Z. Ling and Z.G. Lin, Turing pattern formation in a predator-prey-mutualist system, Nonlinear Anal. Real World Appl., 12 (2011), 3224–3237.
    [29] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, New York: Springer-Verlag, (2001).
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