Skip to main content
Erschienen in: Journal of Applied Mathematics and Computing 1-2/2016

01.02.2016 | Original Research

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

verfasst von: Wensheng Yang, Xuepeng Li

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2016

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Tanner, J.T.: The stability and the intrinsic growth rates of prey and predator populations. Ecology 56, 855–867 (1975)CrossRef Tanner, J.T.: The stability and the intrinsic growth rates of prey and predator populations. Ecology 56, 855–867 (1975)CrossRef
2.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
4.
Zurück zum Zitat Hassell, M.P.: The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press, Princeton (1978)MATH Hassell, M.P.: The Dynamics of Arthropod Predator-Prey Systems. Princeton University Press, Princeton (1978)MATH
5.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
6.
7.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
8.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
9.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
10.
Zurück zum Zitat 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)MathSciNetCrossRef 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)MathSciNetCrossRef
11.
Zurück zum Zitat Boshaba, K., Ruan, S.: Instability in diffusive ecological models with nonlocal delay effects. J. Math. Anal. Appl. 258, 269–286 (2001)MathSciNetCrossRef Boshaba, K., Ruan, S.: Instability in diffusive ecological models with nonlocal delay effects. J. Math. Anal. Appl. 258, 269–286 (2001)MathSciNetCrossRef
12.
Zurück zum Zitat Xu, R.: A reaction diffusion predator-prey model with stage structure and non local delay. Appl. Math. Comput. 175, 984–1006 (2006)MathSciNetCrossRefMATH Xu, R.: A reaction diffusion predator-prey model with stage structure and non local delay. Appl. Math. Comput. 175, 984–1006 (2006)MathSciNetCrossRefMATH
13.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
14.
15.
Zurück zum Zitat 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)CrossRefMATH 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)CrossRefMATH
16.
Zurück zum Zitat Kuang, Y., Baretta, E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389–406 (1998)MathSciNetCrossRefMATH Kuang, Y., Baretta, E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389–406 (1998)MathSciNetCrossRefMATH
17.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
18.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
19.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
20.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
21.
Zurück zum Zitat Peng, R., Wang, M.X.: Note on a ratio-dependent predator-prey system with diffusion. Nonlinear Anal. Real World Appl. 7, 1–11 (2006)MathSciNetCrossRefMATH Peng, R., Wang, M.X.: Note on a ratio-dependent predator-prey system with diffusion. Nonlinear Anal. Real World Appl. 7, 1–11 (2006)MathSciNetCrossRefMATH
22.
Zurück zum Zitat Wang, M.X.: Nonlinear Parabolic Equation of Parabolic Type (in Chinese). Science Press, Beijing (1993) Wang, M.X.: Nonlinear Parabolic Equation of Parabolic Type (in Chinese). Science Press, Beijing (1993)
23.
Zurück zum Zitat Yamada, Y.: Global solution for quasilinear parabolic systems with cross-diffusion effects. Nonlinear Anal. TMA 24, 1395–1412 (1995)CrossRefMATH Yamada, Y.: Global solution for quasilinear parabolic systems with cross-diffusion effects. Nonlinear Anal. TMA 24, 1395–1412 (1995)CrossRefMATH
24.
Zurück zum Zitat Ye, Q., Li, Z.: Introduction to Reaction-Diffusion Equations. Science Press, Beijing (1990)MATH Ye, Q., Li, Z.: Introduction to Reaction-Diffusion Equations. Science Press, Beijing (1990)MATH
26.
Zurück zum Zitat Smoller, J.: Shock Waves and Reaction-Diffusion Equation, 2nd edn. Springer, New York (1994)CrossRefMATH Smoller, J.: Shock Waves and Reaction-Diffusion Equation, 2nd edn. Springer, New York (1994)CrossRefMATH
27.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
28.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
29.
Zurück zum Zitat Chen, S.S., Shi, J.P.: Global stability in a diffusive Holling–Tanner predator–prey model. Appl. Math. Lett. 25, 614–618 (2012)MathSciNetCrossRefMATH Chen, S.S., Shi, J.P.: Global stability in a diffusive Holling–Tanner predator–prey model. Appl. Math. Lett. 25, 614–618 (2012)MathSciNetCrossRefMATH
30.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
31.
Zurück zum Zitat Xu, C., Boyce, M.S.: Dynamic complexities in a mutual interference host–parasitoid model. Chaos Soliton Fractals 24, 175–182 (2004)CrossRefMATH Xu, C., Boyce, M.S.: Dynamic complexities in a mutual interference host–parasitoid model. Chaos Soliton Fractals 24, 175–182 (2004)CrossRefMATH
32.
Zurück zum Zitat 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)MathSciNetCrossRef 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)MathSciNetCrossRef
33.
Zurück zum Zitat Walter, W.: Differential inequalities and maximum principles: theory, new methods and applications. Nonlinear Anal. TMA 30, 4695–4711 (1997)CrossRefMATHMathSciNet Walter, W.: Differential inequalities and maximum principles: theory, new methods and applications. Nonlinear Anal. TMA 30, 4695–4711 (1997)CrossRefMATHMathSciNet
Metadaten
Titel
Global asymptotical stability for a diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal delay
verfasst von
Wensheng Yang
Xuepeng Li
Publikationsdatum
01.02.2016
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2016
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-015-0873-y

Weitere Artikel der Ausgabe 1-2/2016

Journal of Applied Mathematics and Computing 1-2/2016 Zur Ausgabe

Original Research

On Abelian codes over