Global dynamic behaviors for a delayed Nicholson’s blowflies model with a linear harvesting term ∗

In this paper, we study a generalized Nicholson’s blowflies model with a linear harvesting term, which is defined on the positive function space. Under proper conditions, we employ a novel proof to establish some criteria for the global dynamic behaviors on existence of positive solutions, permanence, and exponential stability of the zero equilibrium point for this model. Moreover, we give two examples and their numerical simulations to illustrate our main results.


Introduction
Recently, assuming that a harvesting term is a function of the delayed estimate for the true population, L. Berezansky et al. [1] proposed the following Nicholson's blowflies model x ′ (t) = −δx(t) + px(t − τ )e −ax(t−τ ) − Hx(t − σ), δ, p, τ, a, H, σ ∈ (0, +∞), (1.1) where Hx(t − σ) is the linear harvesting term, x(t) is the size of the population at time t, p is the maximum per capita daily egg production, 1 a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult death rate, and τ is the generation time.Moreover, L. Berezansky et al. [1] formulated an open problem: How about the dynamic behaviors of (1.1).Consequently, some criteria were established in [2−5] to guarantee the existence of positive periodic solutions for (1.1) and its generalized equations by applying the method of coincidence degree; some sufficient conditions were also obtained in [6−8] to ensure that the solutions of its generalized system converge locally exponentially to a positive almost periodic solution.However, it is difficult to study the global dynamic behaviors of the Nicholson's blowflies model with a linear harvesting term.So far, there is no literature considering the global existence of positive solutions and the global permanence for (1.1).In particular, there is no research on the global stability of the zero equilibrium point of (1.1).Thus, it is also a unsolved open problem to reveal the global dynamic behaviors of Nicholson's blowflies model (1.1).
Motivated by the above discussions, the main purpose of this paper is to establish some criteria for the global dynamic behaviors on existence of positive solutions, permanence, and exponential stability of zero equilibrium point for Nicholson's blowflies model with a linear harvesting term.Since the coefficients and delays in differential equations of population and ecology problems are usually time-varying in the real world, we consider the following Nicholson's blowflies model with a linear harvesting term where a(t), H(t), σ(t) and γ j (t) are continuous functions bounded above and below by positive constants, β j (t) and τ j (t) are nonnegative bounded continuous functions, and j = 1, 2, • • • , m. Obviously, (1.1) is a special case of (1.2) with constant coefficients and delays.
For convenience, we introduce some notations.In the following part of this paper, given a bounded continuous function g defined on R, let g + and g − be defined as It will be assumed that Define a continuous map f : R × C + → R by setting Then, f is a locally Lipschitz map with respect to φ ∈ C + , which ensures the existence and uniqueness of the solution of (1.2) with admissible initial conditions (1.4).

Global existence of the positive solutions
In this section, we establish sufficient conditions on the global existence of the positive solutions for (1.2).
Proof.We first show that Suppose, for the sake of contradiction, that (2.2) does not hold.Then, there exists t 1 ∈ which is a contradiction and implies that (2.2) holds.

Global permanence
In this section, we shall derive new sufficient conditions for checking the global permanence of model (1.2).
Theorem 3.1.Suppose that all conditions in Theorem 2.1 are satisfied.Let Then model (1.2) is permanent, i.e., there exist two positive constants k and K such that where x(t) = x(t; t 0 , φ).

Global exponential stability for the zero equilibrium point
In this section, we establish sufficient conditions on the global exponential stability of the zero equilibrium point for (1.2).
Theorem 4.1.Suppose that all conditions in Theorem 2.1 are satisfied.Let Then 0 is a globally exponentially stable equilibrium point on C + , i.e., there exist two constants M > 0 and T > t 0 such that 0 < x(t; t 0 , φ) < M e −λt for all t > T. (4.2) is bounded, and 0 < x(t) for all t > t 0 .( From (4.1), we obtain that there exist T > t 0 and 0 Then, from (4.4), we have which implies that there exist two constants η > 0 and λ ∈ (0, 1] such that We consider the Lyapunov functional Calculating the derivative of V (t) along the solution x(t) of (1.2), in view of (2.1) and (4.3), we have e λt e −γ j (t)x(t−τ j (t)) This completes the proof.

Examples and remarks
In this section, we present two examples to check the validity of our results we obtained in the previous sections.