Global asymptotic behaviour of positive solutions to a non-autonomous Nicholson's blowflies model with delays

This paper addresses the global existence and global asymptotic behaviour of positive solutions to a non-autonomous Nicholson's blowflies model with delays. By using a novel approach, sufficient conditions are derived for the existence and global exponential convergence of positive solutions of the model without any restriction on uniform positiveness of the per capita dead rate. Numerical examples are provided to illustrate the effectiveness of the obtained results.


Introduction
Nicholson [14] used the following delay differential equation: where a, δ, P and τ are positive constants, to model laboratory population of the Australian sheepblowfly. Biologically, x(t) is the size of the population at time t, P is the maximum per capita daily egg production rate, 1/a is the size at which the population reproduces at its maximum rate, δ is the per capita daily adult mortality rate and τ is the generation time, or the time taken from birth to maturity. The dynamics of Equation (1) was later studied in [5,15], where this model was referred to as the Nicholson's blowflies equation. The theory of the Nicholson's blowflies equation has made a remarkable progress in the past 40 years and attracted extensive attention from researchers (see, for example, [1] and the references therein). Many important results on the qualitative properties of the model such as existence of positive solutions, positive periodic/almost periodic solutions, persistence, permanence, oscillation and stability for the classical Nicholson's model and its generalizations (in particular, to variable coefficients, time-varying delays and impulsive equations) have been established in the literature [2][3][4][7][8][9][10][11][12][13][16][17][18][19][20][21][22].
However, it should be noted that, in most of the aforementioned works, the per capita daily adult mortality terms have been restricted to be uniformly positive in order to use the coincidence degree method [3,18,21], fixed point theorems [7,12,13] or comparison principles [9][10][11]17,19,20,22]. Furthermore, it is difficult to study the global asymptotic behaviour of the Nicholson's blowflies model with variable coefficients and time-varying delays. So far, there has been no result in the literature considering the global existence and global exponential convergence to the zero equilibrium point of positive solutions of nonautonomous Nicholson's blowflies model without the assumption on the uniform positiveness of the per capita daily mortality term.
Motivated by the above discussions, in this paper, we first consider the problem of global existence of positive solutions for a non-autonomous Nicholson's blowflies model of the following form: where m is a given positive integer, α, β j , γ j , τ j , j ∈ m := {1, 2, . . . , m}, are continuous functions We then employ a novel proof to establish conditions for the global exponential convergence to the zero equilibrium point of model (2). It is worth noting that, the restriction on the uniform positiveness of α(t) (that means, there is a positive constant α − such that α(t) ≥ α − for all t ≥ 0) as well as the upper and the lower bounds of β j (t), γ j (t), j ∈ m, will be removed. We assume that τ j (t) ≤ τ + j , t ≥ 0, and let τ = max j∈m τ + j . Throughout this paper, let C + = C + ([−τ , 0], R + ) be the set of nonnegative continuous functions with the usual supremum norm . and C + 0 = {ϕ ∈ C + : ϕ(0) > 0}. In the biological interpretation of model (2), only positive solutions are meaningful and admissible. Thus we consider only the admissible initial conditions for Equation (2) as follows: where x t 0 is defined as x t 0 (θ ) = x(t 0 + θ) for all θ ∈ [−τ , 0]. Note that, the function f : is continuous and locally Lipschitz with respect to ϕ ∈ C + 0 . Thus, for each t 0 ∈ R + , ϕ ∈ C + 0 , there exists a unique locally solution x(t; t 0 , ϕ) of Equations (2) and (3) (for more details, see [6]). Let [t 0 , η(ϕ)) be the right maximal interval of existence of x(t; t 0 , ϕ).

Global existence of positive solutions
In this section we will prove the global existence of positive solutions of Equation (2) for admissible initial conditions (3).
Proof Let x(t; t 0 , ϕ) be a solution of Equations (2) and (3). For convenience, let us denote x(t) = x(t; t 0 , ϕ) if it does not make any confusion. We will show that Suppose in contrary that Equation (4) does not hold. Then, there exists t * ∈ (t 0 , η(ϕ)) such that which yields Let t ↑ t * , it follows from Equation (5) that, Next, we will prove the global existence of x(t; t 0 , ϕ), that means η(ϕ) = +∞. Note that Therefore It follows from Equation (6) x for all t ∈ [t 0 , η(ϕ)).

Global exponential convergence to the zero equilibrium point
In this section, we will establish conditions for the global exponential convergence to the zero equilibrium point of positive solutions of model (2).
Let us consider the following assumptions: Proof By (A3), there exists T > 0 such that Let us define then δ is a positive constant. Suppose x(t; t 0 , ϕ) be a solution of Equation (2). Without loss of generality, we assume that t 0 ≤ T . From Equation (7), we have We will prove that Equation (12) holds for all t ∈ [t 0 , +∞). For given > 0, assume that there existst > T such that Then, for t ∈ [T ,t), from Equation (2) we have Therefore, Let t →t, from Equation (13) we obtain which yields a contradiction. Thus, Let → 0 + we finally obtain This completes the proof. (11) holds for all t ∈ R + then every solution x(t; t 0 , ϕ) of Equation (2) satisfies x(t) ≤ ϕ for all t ≥ t 0 .

Remark 3 It can be seen from the proof of Proposition 2.3 that, if Equation
We now prove the global exponential convergence to the zero equilibrium point of positive solutions of Equation (2) as given in the following theorem.  (2). More precisely, there exist positive constants κ, λ, δ such that every solution x(t; t 0 , ϕ) of Equations (2) and (3), with ϕ ∈ C + 0 , satisfies Remark 4 It is worth noting that, estimation of (14) and (A2) guarantee the global exponential convergence to zero of all positive solutions of (2) which we will refer to generalized exponential convergence.
Proof Let x(t) = x(t; t 0 , ϕ) be a solution of Equations (2) and (3). By Proposition 2.3, there exists a constant δ > 0 such that Also, by (A3), and hence, 1−σ 2 > 0. Therefore, there exists T * ≥ T (defined in Equation (11)) such that Furthermore, we can assume that t − τ j (t) ≥ t 0 for all t ≥ T * , j ∈ m. Then, by (A1), we have Now, we consider the following scalar equation: for all t ≥ T * , λ ∈ (0, λ * ], from which we obtain Let us consider the following function: and thus, by Equation (17), We will show that For given > 0, note that (2) and (18), we have Taking integral on both sides of the above inequality we obtain Let t ↑t, we obtain e − t T * α(s) ds ≤ 0, which yields a contradiction. This shows that x(t) < v(t) + for all t ≥ t 0 . Consequently, Equation (19) holds for all t ≥ t 0 , and thus, Equation (14) holds for any 0 < λ ≤ λ * . The proof is now completed.
As a special case, if α(t) is upper-and lower-bounded by positive constants as considered in many other works in the literature (e.g., [3,11,12,17,20,22]) then we obtain the following corollary.

An example
In this section, we give a numerical example to illustrate the effectiveness of our results.
Example 2.6 Consider the following Nicholson's blowflies model with time-varying delay where It should be noted that, for this model, the obtained results in the literature cannot be applied to conclude the convergence of positive solutions of Equation (21). In this case we have where ϕ = sup −1≤θ ≤0 ϕ(θ). Furthermore, it can be seen that, there is no positive constant η such that 2 √ t + 1 ≥ ηt + ν, ν ∈ R, for all t ≥ 0. And thus, a classical exponential estimation, that is, x(t, ϕ) ≤ M ϕ e −ηt , η > 0, M > 0, does not exist. Therefore, the exponential estimation proposed in this paper is less conservative and is expected to relax conditions for the exponential convergence of the model. In the following simulation, we take initial function ϕ(θ) = e −0.2 sin 2 (8θ) , θ ∈ [−1, 0]. As shown in Figure 2, the corresponding state trajectory of Equation (21) satisfies a generalized exponential estimation x(t, ϕ) ≤ e −0.277( √ t+1−1) , t ≥ 0. Furthermore, a classical exponential estimation does not exist. For illustrative purpose, we take η = 0.2 then it can be seen in Figure 2 that x(t, ϕ) ≥ e −0.2t for all t ≥ 20.

Conclusion
This paper has dealt with the global existence and global asymptotic behaviour of positive solutions to a non-autonomous Nicholson's blowflies model with time-varying delays. By using a new approach, we have derived sufficient conditions for the global generalized exponential convergence of positive solutions of the model without any restriction on the uniform positiveness of the per capita dead rate term. Numerical examples have been provided to illustrate the effectiveness of the obtained results.