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Existence of positive steady states for a predator-prey model with diffusion

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  • In this paper, we are concerned with the existence of positive steady states for a diffusive predator-prey model in a spatially heterogeneous environment. We completely determine the intervals of certain parameter of the model in which a positive steady state exists.
    Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60.


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  • [1]

    R. S. Cantrell and C. Cosner, Practical persistence in ecological models via comparison methods, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 247-272.


    E. N. Dancer and Y. H. Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal, 34 (2002), 292-314.doi: 10.1137/S0036141001387598.


    Y. H. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.doi: 10.1016/j.jde.2004.05.010.


    Y. H. Du and S. J. Li, Positive solutions with prescribed patterns in some simple semilinear equations, Differential Integral Equations, 15 (2002), 805-822.


    Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc, 349 (1997), 2443-2475.doi: 10.1090/S0002-9947-97-01842-4.


    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.


    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.


    J. M. Fraile, P. K. Medina, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations, 127 (1996), 295-319.doi: 10.1006/jdeq.1996.0071.


    R. T. Gong and S. B. Hsu, Stability analysis for a class of diffusive coupled system with application to population biology, Can. Appl. Math. Quart, 8 (2000), 79-96.


    K. Hasík, On a predator-prey system of Gause type, J. Math. Biol., 60 (2010), 59-74.doi: 10.1007/s00285-009-0257-8.


    S. B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology, Taiwannese J. Mathematics, 9 (2005), 151-173.


    W. Ko, K. Ryu, A qualitative study on general Gause type predator-prey models with constant diffusion rates, J. Math. Anal. Appl., 344 (2008), 217-230.doi: 10.1016/j.jmaa.2008.03.006.


    W. Ko, K. Ryu, A qualitative study on general Gause-type predator-prey models with non-monotonic functional response, Nonlinear Anal. RWA, 10 (2009), 2558-2573.doi: 10.1016/j.nonrwa.2008.05.012.


    Y. Kuang, Global stability of Gause-type predator-prey systems, J. Math. Biol., 28 (1990), 463-474.doi: 10.1007/BF00178329.


    C. S. Lin, W. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.


    Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426.doi: 10.1016/j.jde.2005.05.010.


    Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.


    J. D. Murray, "Mathematical Biology," Springer-Verlag, Berlin, 1989.


    T. C. Ouyang, On the positive solutions of semilinear equations $\Delta u + \lambda u-hu^p = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503-527.doi: 10.1090/S0002-9947-1992-1055810-7.


    Peter Y. H. Pang and M. X. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London. Math. Soc., 88 (2004), 135-157.doi: 10.1112/S0024611503014321.


    J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, New York, 1994.


    M. X. Wang, Peter Y. H. Pang and W. Y. Chen, Sharp spatial pattern of the diffusive Holling-Tanner prey-predator model in heterogeneous environment, IMA Journal of Applied Mathematics, 73 (2008), 815-835.doi: 10.1093/imamat/hxn016.

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