Dynamics in a discrete predator-prey system with infected prey

Xu, Changjin ; Li, Peiluan

Mathematica Bohemica, Tome 139 (2014) no. 3, pp. 511-534

Abstract (VO)

In this paper, a discrete version of continuous non-autonomous predator-prey model with infected prey is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and global asymptotical stability of positive periodic solution of difference equations in consideration are established. An example shows the feasibility of the main results.

Détail

DOI : 10.21136/MB.2014.143939

Classification : 34C25, 34K20, 39A23, 39A30, 92D25
Keywords: predator-prey model; periodic solution; topological degree; global asymptotic stability

Bibliographie

[1] Agiza, H. N., Elabbasy, E. M., El-Metwally, H., Elsadany, A. A.: Chaotic dynamics of a discrete prey-predator model with Holling type II. Nonlinear Anal., Real World Appl. 10 116-129 (2009). | MR | Zbl

[2] Apreutesei, N., Dimitriu, G.: On a prey-predator reaction-diffusion system with Holling type III functional response. J. Comput. Appl. Math. 235 366-379 (2010). | DOI | MR | Zbl

[3] Dai, B., Zou, J.: Periodic solutions of a discrete-time diffusive system governed by backward difference equations. Adv. Difference Equ. 2005 263-274 (2005). | MR | Zbl

[4] Ding, X., Lu, C.: Existence of positive periodic solution for ratio-dependent $N$-species difference system. Appl. Math. Modelling 33 2748-2756 (2009). | DOI | MR | Zbl

[5] Fan, M., Wang, K.: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Math. Comput. Modelling 35 951-961 (2002). | DOI | MR | Zbl

[6] Fazly, M., Hesaaraki, M.: Periodic solutions for a discrete time predator-prey system with monotone functional responses. C. R., Math., Acad. Sci. Paris 345 199-202 (2007). | DOI | MR | Zbl

[7] Freedman, H. I.: Deterministic Mathematical Models in Population Ecology. Monographs and Textbooks in Pure and Applied Mathematics 57 Marcel Dekker, New York (1980). | MR | Zbl

[8] Gaines, R. E., Mawhin, J. L.: Coincidence Degree, and Nonlinear Differential Equations. Lecture Notes in Mathematics 568 Springer, Berlin (1977). | DOI | MR | Zbl

[9] Hilker, F. M., Malchow, H.: Strange periodic attractors in a prey-predator system with infected prey. Math. Popul. Stud. 13 119-134 (2006). | DOI | MR | Zbl

[10] Jiao, J., Cai, S., Chen, L.: Dynamical behaviors of a biological management model with impulsive stocking juvenile predators and continuous harvesting adult predators. J. Appl. Math. Comput. 35 483-495 (2011). | DOI | MR | Zbl

[11] Jiao, J., Meng, X., Chen, L.: Harvesting policy for a delayed stage-structured Holling II predator-prey model with impulsive stocking prey. Chaos Solitons Fractals 41 103-112 (2009). | DOI | MR | Zbl

[12] Kar, T. K., Misra, S., Mukhopadhyay, B.: A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting. J. Appl. Math. Comput. 22 387-401 (2006). | DOI | MR

[13] Ko, W., Ryu, K.: Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 e1109--e1115 (2009). | DOI | MR | Zbl

[14] Li, Y.: Positive periodic solutions of a discrete mutualism model with time delays. Int. J. Math. Math. Sci. 2005 499-506 (2005). | DOI | MR | Zbl

[15] Li, Y., Zhao, K., Ye, Y.: Multiple positive periodic solutions of $n$ species delay competition systems with harvesting terms. Nonlinear Anal., Real World Appl. 12 1013-1022 (2011). | MR | Zbl

[16] Liu, Q., Xu, R.: Periodic solutions for a delayed one-predator and two-prey system with Holling type-{II} functional response. Ann. Differential Equations 21 14-28 (2005). | MR

[17] Liu, Z., Zhong, S., Liu, X.: Permanence and periodic solutions for an impulsive reaction-diffusion food-chain system with Holling type III functional response. J. Franklin Inst. 348 277-299 (2011). | DOI | MR | Zbl

[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 time delay. Nonlinear Anal., Real World Appl. 7 1104-1118 (2006). | MR | Zbl

[19] Pei, Y., Chen, L., Zhang, Q., Li, C.: Extinction and permanence of one-prey multi-predators of Holling type {II} function response system with impulsive biological control. J. Theoret. Biol. 235 495-503 (2005). | DOI | MR

[20] Scheffer, M.: Fish and nutrients interplay determines algal biomass: a minimal model. Oikos 62 271-282 (1991). | DOI

[21] Song, X., Li, Y.: Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect. Nonlinear Anal., Real World Appl. 9 64-79 (2008). | MR | Zbl

[22] Wang, G.-Q., Cheng, S. S.: Positive periodic solutions for nonlinear difference equations via a continuation theorem. Adv. Difference Equ. 2004 311-320 (2004). | MR | Zbl

[23] Wang, L.-L., Li, W.-T., Zhao, P.-H.: Existence and global stability of positive periodic solutions of a discrete predator-prey system with delays. Adv. Difference Equ. 2004 321-336 (2004). | MR | Zbl

[24] Wiener, J.: Differential equations with piecewise constant delays. Trends in Theory and Practice of Nonlinear Differential Equations V. Lakshmikantham Proc. Int. Conf., Arlington/Tex. 1982. Lecture Notes in Pure and Appl. Math. 90 Dekker, New York 547-552 (1984). | MR | Zbl

[25] Xu, R., Chen, L., Hao, F.: Periodic solutions of a discrete time Lotka-Volterra type food-chain model with delays. Appl. Math. Comput. 171 91-103 (2005). | DOI | MR | Zbl

[26] Zhang, J., Fang, H.: Multiple periodic solutions for a discrete time model of plankton allelopathy. Adv. Difference Equ. (electronic only) 2006 Article ID 90479, 14 pages (2006). | MR | Zbl

[27] Zhang, R. Y., Wang, Z. C., Chen, Y., Wu, J.: Periodic solutions of a single species discrete population model with periodic harvest/stock. Comput. Math. Appl. 39 77-90 (2000). | DOI | MR | Zbl

[28] Zhang, W., Zhu, D., Bi, P.: Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses. Appl. Math. Lett. 20 1031-1038 (2007). | DOI | MR | Zbl

[29] Zhang, Z., Hou, Z.: Existence of four positive periodic solutions for a ratio-dependent predator-prey system with multiple exploited (or harvesting) terms. Nonlinear Anal., Real World Appl. 11 1560-1571 (2010). | MR | Zbl

[30] Zhang, Z., Luo, J.: Multiple periodic solutions of a delayed predator-prey system with stage structure for the predator. Nonlinear Anal., Real World Appl. 11 4109-4120 (2010). | MR | Zbl

[31] Zhuang, K., Wen, Z.: Dynamics of a discrete three species food chain system. Int. J. Comput. Math. Sci. 5 13-15 (2011). | MR