The Dynamics of a Predator-Prey System with State-Dependent Feedback Control

and Applied Analysis 3 capacity of the prey population x t is b/a, so it is meaningful that the economical threshold h is less than b/a. Thus, throughout this paper, we set up the following two assumptions: A1 D e < a b , A2 h ≤ b a . 2.1 From the biological point of view, it is reasonable that system 1.1 is considered to control the prey population in the biological meaning space { x, y : x ≥ 0, y ≥ 0}. The smoothness properties of f , which denotes the right hand of 1.1 , guarantee the global existence and uniqueness of a solution of system 1.1 see 25, 26 for the details . Let R −∞,∞ and R2 { x, y | x ≥ 0, y ≥ 0}. Firstly, we denote the distance between the point p and the set S by d p, S infp0∈S|p − p0| and define, for any solution z t x t , y t of system 1.1 , the positive orbit of z t through the point z0 ∈ R2 as O z0, t0 { z ∈ R2 | z z t , t ≥ t0, z t0 z0 } . 2.2 Now, we introduce some definitions cf. 27 . Definition 2.1 orbital stability . z∗ t is said to be orbitally stable if, given > 0, there exists δ δ > 0 such that, for any other solution z t of system 1.1 satisfying |z∗ t0 −z t0 | < δ, then d z t , O z0, t0 < for t > t0. Definition 2.2 asymptotic orbital stability . z∗ t is said to be asymptotically orbitally stable if it is orbitally stable and for any other solution z t of system 1.1 , there exists a constant η > 0 such that, if |z∗ t0 − z t0 | < η, then limt→∞d z t , O z0, t0 0. In order to discuss the orbital asymptotical stability of a positive periodic solution of system 1.1 , a useful lemma, which follows from Corollary 2 of Theorem 1 given in Simeonov and Bainov 28 , is considered as follows. Lemma 2.3 analogue of the Poincaré criterion . The T-periodic solution x φ t , y ζ t of system x′ t P ( x, y ) , y′ t Q ( x, y ) , if φ ( x, y ) / 0, Δx α ( x, y ) , Δy β ( x, y ) , if φ ( x, y ) 0, 2.3 is orbitally asymptotically stable if the multiplier μ2 satisfies the condition |μ2| < 1, where