Stability and Permanence of a Pest Management Model with Impulsive Releasing and Harvesting

and Applied Analysis 3 (1) V is continuous in ((n − 1)T, (n + τ − 1)T] × R + , ((n + τ − 1)T, nT] × R + and for each x ∈ R + , n ∈ N, lim (t,y)→ ((n+τ−1)T + ,x) V(t, y) = V((n + τ − 1)T + , x) and lim (t,y)→ (nT + ,x) V(t, y) = V(nT, x) exist. (2) V is locally Lipschitzian x. Definition 1. Letting V ∈ V 0 , one defines the upper right derivative of V with respect to the impulsive differential system (1) at (t, x) ∈ ((n − 1)T, (n + τ − 1)T] × R + and ((n + τ − 1)T, nT] × R + by D + V (t, x) = lim sup h→0 + 1 h [V (t + h, x + hf (t, x)) − V (t, x)] . (2) Definition 2. The system (1) is said to be permanent if there are positive constants m, M > 0 and a finite time T 0 such that all solutions of (1) with initial values x(0), S(0), I(0), y J (0), y M (0),m ≤ x(t), S(t), I(t), y J (t), y M (t) ≤ M hold for all t ≥ T 0 , where m and M are independent of initial value, T 0 may depend on initial value. Remark 3. The global existence and uniqueness of system (1) is guaranteed by the smoothness properties of f (for details, see [13, 14]). Lemma 4 (see [15]). Let V : R + × R → R + satisfy V i ∈ V 0 , i = 1, 2, . . . , m, and assume that D + V (t, x (t)) ≤ (≥) g (t, V (t, x (t))) , t ̸ = (k + τ − 1) T, kT, V (t, x (t + )) ≤ (≥) ψ τ k (V (t, x (t))) , t = (k + τ − 1) T, V (t, x (t + )) ≤ (≥) ψ k (V (t, x (t))) , t = kT, k ∈ N, x (0 + ) = x 0 , (3) where g : R + × R + → R + is continuous in ((k − 1)T, (k + τ − 1)T] × R and ((k + τ − 1)T, kT] × R, for each p ∈ R, k = 1, 2, . . ., the limit lim (t,q)→ ((k+τ−1)T + ,p) g(t, q) = g((k+ τ − 1)T, p) and lim (t,q)→ ((k−1)T + ,p) g(t, q) = g((k − 1)T, p) exists. g(t, q) is quasimonotone nondecreasing in q. ψ k , ψ k : R + → R + is nondecreasing for all k ∈ N. Let θ(t) be the maximal (minimal) solution of the following impulsive differential equation on [0,∞): w 󸀠 (t) = g (t, w (t)) , t ̸ = (k + τ − 1) T, kT, w (t + ) = ψ τ k (w (t)) , t = (k + τ − 1) T, w (t + ) = ψ k (w (t)) , t = kT, k ∈ N, w (0 + ) = w 0 . (4) Then for any solution x(t) of the system (3), V(0, x 0 ) ≤ (≥)w 0 implies that V(t, x(t)) ≤(≥)θ(t) for all t ≥ 0. Lemma 5 (see [13, 15]). Consider the following system: V 󸀠 (t) ≤ (≥) p (t) V (t) + q (t) , t ̸ = t k , V (t + k ) ≤ (≥) d k V (t k ) + b k , t = t k , k ∈ N, V (0 + ) ≤ (≥) V 0 , (5) where p, q ∈ PC(R + ,R) and d k ≥ 0, V 0 and b k are constants. Suppose that (A1) the sequence t k satisfies 0 ≤ t 1 ≤ t 2 < ⋅ ⋅ ⋅ , with lim t→∞ t k = ∞; (A2) V ∈ PC(R + ,R) and V(t) is left-continuous at t k , k ∈ N. Then, for t > 0, V (t) ≤ (≥) V 0 e ∫ t