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March  2008, 9(2): 267-279. doi: 10.3934/dcdsb.2008.9.267

Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales

Mostafa Fazly 1, and  Mahmoud Hesaaraki 1,

1. 

Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran, Iran

Received  April 2007 Revised  November 2007 Published  December 2007

In this paper we explore the existence of periodic solutions of a nonautonomous semi-ratio-dependent predator-prey dynamical system with functional responses on time scales. To illustrate the utility of this work, we should mention that, in our results this system with a large class of monotone functional responses, always has at least one periodic solution. For instance, this system with some celebrated functional responses such as Holling type-II (or Michaelis-Menten), Holling type-III, Ivlev, $mx$ (Holling type I), sigmoidal [e.g., Real and ${mx^2}/{((A+x)(B+x))}$] and some other monotone functions, has always at least one $\omega$-periodic solution. Besides, for some well-known functional responses which are not monotone such as Monod-Haldane or Holling type-IV, the existence of periodic solutions is proved. Our results extend and improve previous results presented in [4], [10], [22], and [38].
Keywords: periodic solution, functional response., coincidence degree, dynamical system on time scales, Nonautonomous predator-prey system.
Mathematics Subject Classification: Primary: 34C25, 39A11; Secondary: 39A12, 92D2.
Citation: Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267
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