Multiple positive periodic solutions of a discrete time delayed predator-prey system with harvesting terms. (English) Zbl 1211.92060
Summary: By using Mawhin’s continuation theorem of coincidence degree theory, we establish the existence of at least two positive periodic solutions for a discrete time delayed predator-prey system with harvesting terms. An example is given to illustrate the effectiveness of our results.
References:
| [1] | Ma, Z.: Mathematical Modelling and Studying on Species Ecology. Education Press, Hefei (1996) (in Chinese) |
| [2] | Tian D., Zeng X.: Existence of at least two periodic solutions of a ratio-dependence predator–prey model with exploited term. Acta Math. Appl. Sin. English Ser. 21(3), 489–494 (2005) · Zbl 1098.92072 · doi:10.1007/s10255-005-0256-5 |
| [3] | Thieme, H.R.: Mathematics in population biology. In: Princeton Syries in Theoretial and Computational Biology. Princeton University Press, Princeton, NJ (2003) · Zbl 1054.92042 |
| [4] | Chen Y.: Multiple periodic solutions of delayed predator–prey systems with type IV functional responses. Nonlinear Anal. Real World Appl. 5, 45–53 (2004) · Zbl 1066.92050 · doi:10.1016/S1468-1218(03)00014-2 |
| [5] | Wang Q., Dai B., Chen Y.: Multiple periodic solutions of an impulsive predator–prey model with Holling-type IV functional response. Math. Comput. Model. 49, 1829–1836 (2009) · Zbl 1171.34341 · doi:10.1016/j.mcm.2008.09.008 |
| [6] | Hu, D., Zhang, Z.: Four positive periodic solutions to a Lotka-Volterra cooperative system with harvesting terms. Nonlinear Anal. Real World Appl. doi: 10.1016/j.nonrwa.2009.02.002 |
| [7] | Li Y., Kuang Y.: Periodic solutions of periodic delay Lotka-Volterra eqnarrays and systems. J. Math. Anal. Appl. 255, 260–280 (2001) · Zbl 1024.34062 · doi:10.1006/jmaa.2000.7248 |
| [8] | Zhao, K., Ye, Y.: Four periodic solutions to a periodic Lotka-Volterra system with harvesting terms. Nonlinear Anal. Real World Appl. doi: 10.1016/j.nonrwa.2009.08.001 |
| [9] | Agarwal, R.P.: Difference Equations and Inequalities: Theory, Method and Applications, Monographs and Textbooks in Pure and Applied Mathematics, No. 228, Marcel Dekker, New York (2000) |
| [10] | Agarwal R.P., Wong P.J.Y.: Advance Topics in Difference Equations. Kluwer Publisher, Dordrecht (1997) · Zbl 0878.39001 |
| [11] | Freedman H.I.: Deterministic Mathematics Models in Population Ecology. Marcel Dekker, New York (1980) · Zbl 0448.92023 |
| [12] | Murry J.D.: Mathematical Biology. Springer-Verlag, New York (1989) |
| [13] | Gopalsamy K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, Boston (1992) · Zbl 0752.34039 |
| [14] | Fan M., Wang K.: Periodic solutions of a discrete time nonautonomous ratio-dependent predator–prey system. Math. Comput. Model. 35(9–10), 951C961 (2002) · Zbl 1050.39022 · doi:10.1016/S0895-7177(02)00062-6 |
| [15] | Gaines R., Mawhin J.: Coincidence Degree and Nonlinear Differetial Equitions. Springer Verlag, Berlin (1977) · Zbl 0339.47031 |
| [16] | Zhang R.Y. et al.: Periodic solutions of a single species discrete population modle with periodic harvest/stock. Comput. Math. Appl. 39, 77–90 (2000) · Zbl 0970.92019 · doi:10.1016/S0898-1221(99)00315-6 |
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