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Periodic solutions for a discrete time predator-prey system with monotone functional responses. (English) Zbl 1122.39005
Summary: Sharp sufficient conditions for the existence of periodic solutions of a nonautonomous discrete time semi-ratio-dependent predator-prey system with functional responses are derived. In our results this system with any monotone functional response bounded by polynomials in $$\mathbb R^{+}$$, always has at least one $$\omega$$-periodic solution. In particular, this system with the most popular functional responses Michaelis-Menten, Holling type-II and III, sigmoidal, Ivlev and some other monotone response functions, always has at least one $$\omega$$-periodic solution.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 92D25 Population dynamics (general)
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##### References:
 [1] Bohner, M.; Fan, M.; Zhang, J., Existence of periodic solutions in predator – prey and competition dynamic systems, Nonlinear anal.: real world appl., 7, 1193-1204, (2006) · Zbl 1104.92057 [2] Fan, M.; Wang, Q., Periodic solutions of a class of nonautonomous discrete time semi-ratio-dependent predator – prey systems, Discrete contin. dyn. syst. ser. B, 4, 3, 563-574, (2004) · Zbl 1100.92064 [3] Fazly, M.; Hesaaraki, M., Periodic solutions for predator – prey systems with beddington – deangelis functional response on time scales, Nonlinear anal.: real world appl., (2007) · Zbl 1145.92035 [4] Freedman, H.I., Deterministic mathematical models in population ecology, Monograph textbooks pure applied math., vol. 57, (1980), Marcel Dekker New York · Zbl 0448.92023 [5] Gaines, R.E.; Mawhin, J.L., Coincidence degree and non-linear differential equations, (1977), Springer Berlin [6] Wang, Q.; Fan, M.; Wang, K., Dynamics of a class of nonautonomous semi-ratio-dependent predator – prey systems with functional responses, J. math. anal. appl., 278, 443-471, (2003) · Zbl 1029.34042
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