Global asymptotical stability for a diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal delay

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• 作者：Wensheng Yang ; Xuepeng Li
• 关键词：Predator–prey system ; Beddington–DeAngelis functional response ; Iterative method ; Global asymptotical stability ; 92D25 ; 35K51 ; 35B35
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：50
• 期：1-2
• 页码：327-347
• 全文大小：484 KB
• 参考文献：1.Tanner, J.T.: The stability and the intrinsic growth rates of prey and predator populations. Ecology 56, 855–867 (1975)CrossRef
  2.Wollkind, D.J., Collings, J.B., Logan, J.A.: Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit flies. Bull. Math. Biol. 50, 379–409 (1988)MathSciNet CrossRef
  3.Saez, E., Gonzalez-Olivares, E.: Dynamics of a predator-prey model. SIAM J. Appl. Math. 59, 1867–1878 (1999)MathSciNet CrossRef
  4.Hassell, M.P.: The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press, Princeton (1978)
  5.Yang, W.S.: Global asymptotical stability and persistent property for a diffusive predator–prey system with modified Leslie–Gower functional response. Nonlinear Anal. Real World Appl. 14, 1323–1330 (2013)MathSciNet CrossRef
  6.Hsu, S.B., Hwang, T.W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55, 763–783 (1995)MathSciNet CrossRef
  7.Hsu, S.B., Hwang, T.W., Kuang, Y.: Global analysis of the Michaelis–Menten-type ratio-dependent predator-prey system. J. Math. Biol. 42, 489–506 (2001)MathSciNet CrossRef
  8.Shi, H.B., Li, W.T., Lin, G.: Positive steady states of a diffusive predator-prey system with modified Holling–Tanner functional response. Nonlinear Anal. Real World Appl. 11, 3711–3721 (2010)MathSciNet CrossRef
  9.Duque, C., Lizana, M.: Global asymptotic stability of a ratio dependent predator-prey system with diffusion and delay. Period. Math. Hung. 56, 11–23 (2008)MathSciNet CrossRef
  10.Wang, Y., Cao, J.D., et al.: Effect of time delay on pattern dynamics in a spatial epidemic model. Phys. A 412, 137–148 (2014)MathSciNet CrossRef
  11.Boshaba, K., Ruan, S.: Instability in diffusive ecological models with nonlocal delay effects. J. Math. Anal. Appl. 258, 269–286 (2001)MathSciNet CrossRef
  12.Xu, R.: A reaction diffusion predator-prey model with stage structure and non local delay. Appl. Math. Comput. 175, 984–1006 (2006)MathSciNet CrossRef
  13.Fan, M., Wang, K.: Global existence of positive periodic solutions of periodic predator-prey system infinity delay. J. Math. Anal. Appl. 262, 1–11 (2001)MathSciNet CrossRef
  14.Kar, T.K.: Selective harvesting in a prey-predator fishery with time delay. Math. Comput. Model. 38, 449–458 (2003)MathSciNet CrossRef
  15.Cosner, C., DeAngelis, D.L., Ault, J.S., Olson, D.B.: Effects of spatial grouping on the functional response of predators. Theor. Popul. Biol. 56, 65–75 (1999)CrossRef
  16.Kuang, Y., Baretta, E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389–406 (1998)MathSciNet CrossRef
  17.Aziz-Alaoui, M.A., Daher Okiye, M.: Boundeness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Appl. Math. Lett. 16, 1069–1075 (2003)MathSciNet CrossRef
  18.Nindjin, A.F., Aziz-Alaoui, M.A., Cadivel, M.: Analysis of a predator-prey model with modified Leslie–Gower and Holling-type II schemes with delay. Nonlinear Anal. Real World Appl. 7, 1104–1118 (2006)MathSciNet CrossRef
  19.Ko, W., Ryu, K.: Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a prey refuge. J. Differ. Equ. 231, 534–550 (2006)MathSciNet CrossRef
  20.Lindstrom, T.: Global stability of a model for competing predators: an extension of the Aradito and Ricciardi Lyapunov function. Nonlinear Anal. 39, 793–805 (2000)MathSciNet CrossRef
  21.Peng, R., Wang, M.X.: Note on a ratio-dependent predator-prey system with diffusion. Nonlinear Anal. Real World Appl. 7, 1–11 (2006)MathSciNet CrossRef
  22.Wang, M.X.: Nonlinear Parabolic Equation of Parabolic Type (in Chinese). Science Press, Beijing (1993)
  23.Yamada, Y.: Global solution for quasilinear parabolic systems with cross-diffusion effects. Nonlinear Anal. TMA 24, 1395–1412 (1995)CrossRef
  24.Ye, Q., Li, Z.: Introduction to Reaction-Diffusion Equations. Science Press, Beijing (1990)
  25.Pao, C.V.: Dynamics of nonlinear parabolic systems with time delays. J. Math. Anal. Appl. 198, 751–779 (1996)MathSciNet CrossRef
  26.Smoller, J.: Shock Waves and Reaction-Diffusion Equation, 2nd edn. Springer, New York (1994)CrossRef
  27.Ko, W., Ryu, K.: Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment. J. Math. Anal. Appl. 327, 539–549 (2007)MathSciNet CrossRef
  28.Peng, R., Wang, M.X.: Global stability of the equilibrium of a diffusive Holling–Tanner prey-predator model. Appl. Math. Lett. 20, 664–670 (2007)MathSciNet CrossRef
  29.Chen, S.S., Shi, J.P.: Global stability in a diffusive Holling–Tanner predator–prey model. Appl. Math. Lett. 25, 614–618 (2012)MathSciNet CrossRef
  30.Li, J.J., Gao, W.J.: A strongly coupled predator–prey system with modified Holling–Tanner functional response. Comput. Math. Appl. 60, 1908–1916 (2010)MathSciNet CrossRef
  31.Xu, C., Boyce, M.S.: Dynamic complexities in a mutual interference host–parasitoid model. Chaos Soliton Fractals 24, 175–182 (2004)CrossRef
  32.Pal, P.J., Mandal, P.K.: Bifurcation analysis of a modified Leslie–Gower predator–prey model with Beddington–DeAngelis functional response and strong Allee effect. Math. Comput. Simul. 97, 123–146 (2014)MathSciNet CrossRef
  33.Walter, W.: Differential inequalities and maximum principles: theory, new methods and applications. Nonlinear Anal. TMA 30, 4695–4711 (1997)CrossRef
  
• 作者单位：Wensheng Yang (1)
  Xuepeng Li (1)
  
  1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, Fujian, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Theory of Computation
  Mathematics of Computing
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1865-2085

文摘

A diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal delay is considered in this work. Sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by constructing new recurrent sequences which are different from Duque and Lizana’s paper and using an iterative method. It is shown that our result supplements and complements one of the main results of Duque and Lizana’s paper. Keywords Predator–prey system Beddington–DeAngelis functional response Iterative method Global asymptotical stability