New results on global exponential stability for a periodic Nicholson’s blowflies model involving time-varying delays

This paper investigates a periodic Nicholson’s blowflies equation with multiple time-varying delays. By using differential inequality techniques and the fluctuation lemma, we establish a criterion to ensure the global exponential stability on the positive solutions of the addressed equation, which improves and complements some existing ones. The effectiveness of the obtained result is illustrated by some numerical simulations.

Based on the above discussions, in this paper, avoiding assumption (1.2) and without adopting [κ, κ] as the existence interval of periodic solutions, we establish the global exponential stability of periodic solutions for system (1.1). The proposed criterion improves and complements some existing results in the recent publications [1][2][3], and its effectiveness is demonstrated by a numerical example.

Main results
The main results in this paper will now be presented as the subsequent proposition and theorem.
which is a contradiction, validating (3.5). This implies that (3.1) holds, and the proof of the Proposition 3.1 is now finished. Proof According to Proposition 3.1 and Lemma 2.3, one can follow the argument of Theorem 3.1 in [1] to demonstrate that x(t + qT) = x(t + qT; t 0 , ϕ) is not only convergent on every compact interval as q → +∞, but also converges uniformly to a continuous function x * (t), where x * is a T-periodic solution of (1.1), and such that Furthermore, by applying a similar argument as in Lemma 2.3, we can validate the global exponential stability of x * (t). This completes the proof of Theorem 3.1.
By applying Theorem 3.1, we can obtain the following result. Then, the classical autonomous Nicholson's blowflies equation,  [10,11] can be concluded from the above Corollary 3.1. In addition, in [10,11], the exponential stability and existence range of a positive equilibrium point have not been considered for the classical autonomous Nicholson's blowflies equation with the conditions (3.8), which implies that the obtained results of this present paper improve and complement some existing ones.

Example
In this section, we present a numerical example to verify the theoretical results derived in the previous section. Obviously, it is observed that (4.1) satisfies (2.2) and (2.3). Therefore, from Theorem 3.1, one can see that (4.1) has exactly one globally exponentially stable positive 2π -periodic solution. This fact is also supported by the numerical simulations in Fig. 1 (numerical solutions of (4.1) for different initial values).

Conclusions
In this paper, we combine the Lyapunov function method with the differential inequality method to establish some new criteria ensuring the existence and exponential stability of positive periodic solutions for a class of Nicholson's blowflies equation with multiple timevarying delays. Avoiding the assumption that the maximum reproduction rate is less than 1, these criteria are obtained without assuming that [κ, κ] ≈ [0.7215355, 1.342276] is the existence region of periodic solutions, and the analogous results in the recently published literature are summarized and refined. The approach presented in this article can be used as a possible way to study other population models involving multiple time-varying delays, for example, neoclassical growth model, Mackey-Glass model, epidemical system or agestructured population model, and so on.