Analysis of a stage-structured predator-prey model with Crowley-Martin function
Tóm tắt
Predator-prey interactions may have a large impact on the overall properties of a community. In this paper, we introduce an improved stage-structured predator-prey model with Crowley-Martin function. The possibility of existence of nonnegative equilibria has been considered. The locally asymptotic stability of nonnegative equilibria are considered and the permanence of the system are presented. By stability analysis we obtained sufficient conditions in the parameters for the global stability of the positive equilibrium. Numerical simulations are presented to illustrate the results.
Tài liệu tham khảo
Song, X.Y., Chen, L.S.: Optimal harvesting and stability for a two-species competitive system with stage structure. Math. Biosci. 170, 173–186 (2001)
Hsu, S.B., Huang, T.W.: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 55(3), 763–783 (1995)
Kooij, R.E., Zegeling, A.: A predator-prey model with Ivlev’s functional response. J. Math. Anal. Appl. 198, 473–489 (1996)
Li, Z., Wang, W., Wang, H.: The dynamics of a Beddington-type system with impulsive control strategy. Chaos Solitons Fractals 29, 1229–1239 (2006)
Ruan, S., Xiao, D.: Global analysis in a predator-prey system with non-monotonic functional response. SIAM J. Appl. Math. 61(4), 1445–1472 (2001)
Zhang, L., Ding, X.H.: Solution of the prey-predator problem by He’s parameterized perturbation methods. J. Nonlinear Sci. Numer. Simul. 10, 1223–1226 (2009)
Das, S., Gupta, P.K., Rajeev: A fractional predator-prey model and its solution. J. Nonlinear Sci. Numer. Simul. 10, 873–876 (2009)
Gazi, N.H., Bandyopadhyay, M.: Effect of time delay on a harvested predator-prey model. J. Appl. Math. Comput. 26, 263–280 (2008)
Dai, B.X., Zou, J.Z.: Periodic solutions of a discrete-time nonautonomous predator-prey system with the Beddington-Deangelis functional response. J. Appl. Math. Comput. 24(1), 127–139 (2007)
Zhou, X.Y., Shi, X.Y., Song, X.Y.: Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay. Appl. Math. Comput. 196(1), 129–136 (2008)
Barat, S., Das, S., Gupta, P.K.: A note on fractional Schrodinger equation. Nonlinear Sci. Lett. A 1(1), 91–94 (2010)
He, J.H., Wu, G.C., Austin, F.: The variational iteration method which should be followed. Nonlinear Sci. Lett. A 1, 1–30 (2010)
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)
Upadhyay, R.K., Naji, R.K.: Dynamics of a three species food chain model with Crowley-Martin type functional response. Chaos Solitons Fractals 42(3), 1337–1346 (2009)
Ko, Y.: The asymptotic stability behavior in a Lotka-Volterra type predator-prey system. Bull. Korean Math. Soc. 43(3), 575–587 (2006)
Seifert, G.: Asymptotic behavior in a three-component food chain model. Nonlinear Anal. 32(6), 749–753 (1998)
May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1974)
Freeman, H.I.: Deterministic Mathematical Models in Population Ecology. Dekker, New York (1980)
Hirsch, M.W.: Systems of differential equations which are competitive or cooperative, IV. SIAM J. Math. Anal. 21, 1225–1234 (1990)
Smith, H.L., Zhu, H.R.: Stable periodic orbits for a class of three-dimensional competitive systems. J. Differ. Equ. 110, 143–156 (1994)
Smith, H.L.: Monotone dynamical systems: An Introduction to the theory of competitive and cooperative systems. Trans. Am. Math. Soc. 41 (1995)
Smith, H.L.: Systems of ordinary differential equations which generate an order preserving flow. SIAM Rev. 30, 87–98 (1998)
Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20, 857–872 (1990)
Skalski, G.T., Gilliam, J.F.: Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82(11), 3083–3092 (2001)
Xiao, Y.N., Chen, L.S.: Global stability of a predator-prey system with stage structure for the predator. Acta Math. Sin. (Eng. Ser.) 19(2), 1–11 (2003)