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Journal of Applied Mathematics and Computing

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01.10.2013 | Original Research

Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses

verfasst von: N. Ali, M. Jazar

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2013

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Abstract

In this paper, we study a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. We show the existence of a bounded positive invariant and attracting set. The possibility of existence and uniqueness of positive equilibrium are considered. The asymptotic behavior of the positive equilibrium and the existence of Hopf-bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. The existence and non-existence of periodic solutions are established under suitable conditions. The permanence conditions are also established. We obtained sufficient conditions to ensure the global stability of the unique positive equilibrium, by using suitable Lyapunov functions, LaSalle invariance principle and Dulac’s criterion. We obtained also sufficient conditions for the global stability of the prey-extinction equilibrium when the unique positive equilibrium is not feasible. Finally, numerical simulations are presented to illustrate the analytical results.

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Abrams, P.A.: The nature of predation: prey dependent, ratio-dependent or neither. Trends Ecol. Evol. 15, 337–341 (2000) CrossRef

Arditi, R., Ginzburg, L.R.: Coupling in predator-prey dynamics: ratio-dependence. J. Theor. Biol. 139, 311–326 (1989) CrossRef

Arditi, R., Saiah, H.: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73, 1544–1551 (1992) CrossRef

Aziz-Alaoui, M.A.: Study of a Leslie-Gower type tritrophic population model. Chaos Solitons Fractals 14(8), 1275–1293 (2002) MathSciNetCrossRefMATH

Aziz-Alaoui, M.A., Daher Okiye, M.: Boundedness 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

Balabane, M., Jazar, M., Souplet, Ph.: Oscillatory blow-up in nonlinear second order ODE’s: the critical case. Discrete Contin. Dyn. Syst. 9(3), 577–584 (2003) MathSciNetCrossRefMATH

Berezovskaya, F., Karev, G., Arditi, R.: Parametric analysis of the ratio-dependent predator-prey model. J. Math. Biol. 43, 221–246 (2001) MathSciNetCrossRefMATH

Cantrell, R.S., Cosner, C.: On the dynamics of predator-prey models with the Beddington-DeAngelis functional response. J. Math. Anal. Appl. 257, 206–222 (2001) MathSciNetCrossRefMATH

Crowley, P.H., Martin, E.K.: Functional responses and interference within and between year classes of a dragonfly population. J. North Am. Benthol. Soc. 8, 211–221 (1989) CrossRef

10.

Fan, M., Wang, Y.: Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response. J. Math. Anal. Appl. 295, 15–39 (2004) MathSciNetCrossRefMATH

11.

Fan, M., Wang, Q., Zou, X.F.: Dynamics of a nonautonomous ratio-dependent predator-prey system. Proc. R. Soc. Edinb., Sect. A 133, 97–118 (2003) MathSciNetCrossRefMATH

12.

Freedman, H.I.: Deterministic Mathematical Models in Population Ecology. Dekker, New York (1980) MATH

13.

Gaie, P., Zhang, H.: Qualitative analysis of a prey-predator system with Holling I functional response. J. Jilin Univ. Sci. 44(3), 373–376 (2006) MathSciNet

14.

Hale, J.K.: Ordinary Differential Equations. Wiley-Interscience, New York (1969) MATH

15.

Hale, J.K., Kocak, H.: Dynamics and Bifurcations. Springer, New York (1991) CrossRefMATH

16.

Hale, J.K., Waltman, P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20(2), 388–395 (1989) MathSciNetCrossRefMATH

17.

Holling, C.S.: Some characteristics of simple types of predation and parasitism. Can. Entomol. 91(7), 385–398 (1966) CrossRef

18.

Holling, C.S.: On the dynamics of predator-prey systems with Beddington-DeAngelis functional response. Asian-Eur. J. Math. 4, 35–48 (2011) MathSciNetCrossRef

19.

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

20.

Hsu, S.B., Hwang, T.W., Kuang, Y.: Global dynamics of a predator prey model with Hassell-Varley type functional response. Discrete Contin. Dyn. Syst., Ser. B 10(4), 857–875 (2005) MathSciNet

21.

Huentutripay, J., Jazar, M., Véron, L.: A dynamical system approach to the construction of singular solutions of some degenerate elliptic equations. J. Differ. Equ. 195(1), 175–193 (2003) CrossRefMATH

22.

Hwang, T.W.: Global analysis of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 281, 395–401 (2003) MathSciNetCrossRefMATH

23.

Hwang, T.W.: Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 290, 113–122 (2004) MathSciNetCrossRefMATH

24.

Hwang, J., Xiao, D.: Analyses of bifurcations and stability in a predator-prey system with Holling type-IV functional response. Acta Math. Appl. Sin. 20, 167–178 (2004) CrossRef

25.

Jost, C.: Comparaison qualitative et quantitative de modèles proie-prédateur à des données chronologiques en écologie. Thèse de doctorat, Institut National Agronomique Paris-Grignon

26.

Jost, C., Arini, O., Arditi, R.: About deterministic extinction in ratio-dependent predator-prey models. Bull. Math. Biol. 61(1), 19–32 (1999) CrossRef

27.

Kooij, R.E., Zegeling, A.: A predator-prey model with Ivlev’s functional response. J. Math. Anal. Appl. 198, 473–489 (1996) MathSciNetCrossRefMATH

28.

Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio-dependent predator-prey system. J. Math. Biol. 36, 389–406 (1998) MathSciNetCrossRefMATH

29.

Kuang, Y., Freedman, H.I.: Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math. Biosci. 88(1), 67–84 (1988) MathSciNetCrossRefMATH

30.

Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, Berlin (1998) MATH

31.

Leslie, P.H., Crowley, J.C.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika 47, 219–234 (1960) MathSciNetMATH

32.

Marsden, J., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976) CrossRefMATH

33.

Peng, R., Wang, M.: Positive steady states of the Holling-Tanner prey-predator model with diffusion. Proc. R. Soc. Edinb. A 135, 149–164 (2005) MathSciNetCrossRefMATH

34.

Quilin, T.: A predator-prey system with Ivlev’s functional response. J. Beihua Univ. Nat. Sci. 3(5), 381–384 (2002)

35.

Shi, X., Zhou, X., Song, X.: Analysis of a stage-structured predator-prey model with Crowley-Martin function. J. Appl. Math. Comput. 36(1–2), 459–472 (2011) MathSciNetCrossRefMATH

36.

Sugie, J., Kohno, R., Miyazaki, R.: On a predator-prey system of Holling type. Proc. Am. Math. Soc. 125(7), 2041–2050 (1997) MathSciNetCrossRefMATH

37.

Upadhyay, R.K., Naji, R.K.: Dynamics of a three species food chain model with Crowley-Martin type functional response. Chaos Solitons Fractals 42, 1337–1346 (2009) MathSciNetCrossRefMATH

38.

Upadhyay, R.K., Raw, S.N., Rai, V.: Dynamical complexities in a tri-trophic hybrid food chain model with Holling type II and Crowley-Martin functional responses. Nonlinear Anal. Model. Control 15, 366–375 (2010) MathSciNet

39.

Wang, X.: Dynamics of a predator-prey system with Watt-type functional response. Master’s thesis, Northeast Normal University, Changchun (2005) (in Chinese)

40.

Wang, X., Wang, W., Lin, X.: Chaotic behavior of a Watt-type predator-prey system with impulsive control strategy. Chaos Solitons Fractals 37, 706–718 (2008) MathSciNetCrossRefMATH

41.

Wu, R., Lin, L.: Permanence and global attractivity of discrete predator-prey system with Hassell-Varley type functional response. Discrete Dyn. Nat. Soc., 1–17 (2009)

42.

Xiao, D., Ruan, S.: Global dynamics of a ratio-dependent predator-prey system. J. Math. Biol. 43, 268–290 (2001) MathSciNetCrossRefMATH

43.

Zhou, X., Cui, J.: Global stability of the viral dynamics with Crowley-Martin functional response. Bull. Korean Math. Soc. 48, 555–574 (2011) MathSciNetCrossRefMATH

44.

Zhuang, K., Wen, Z.: Analysis for a food chain model with Crowley-Martin functional response and time delay. World Acad. Sci., Eng. Technol. 61, 562–565 (2010)

Titel: Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses
verfasst von: N. Ali
M. Jazar
Publikationsdatum: 01.10.2013
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Applied Mathematics and Computing / Ausgabe 1-2/2013
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-013-0663-3

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