Electronic Journal of Qualitative Theory of Differential Equations

This paper is concerned with the existence, uniqueness and global attractivity of positive periodic solution of a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality rate. By some comparison techniques via differential inequalities, we first establish sufficient conditions for the global uniform permanence and dissipativity of the model. We then utilize an extended version of the Lyapunov functional method to show the existence and global attractivity of a unique positive periodic solution of the underlying model. An application to the model with constant coefficients is also presented. Two numerical examples with simulations are given to illustrate the efficacy of the obtained results.


Introduction
Mathematical models are important for describing dynamics of phenomena in the real world [7,12,25]. For example, in [24], Nicholson used the following delay differential equation where α, β, γ are positive constants, to model the laboratory population of the Australian sheep-blowfly. In the biology interpretation of equation (1.1), N(t) is the population size at time t, α is the per capita daily adult mortality rate, β is the maximum per capita daily egg production rate, 1 γ is the size at which the population reproduces at its maximum rate and τ ≥ 0 is the generation time (the time taken from birth to maturity). Model (1.1) is typically referred to the Nicholson's blowflies equation. It is interesting that when the maximum reproducing rate is not limited (i.e. 1 γ → +∞) and the time τ is small which can be ignored, model (1.1) is reduced to a well-known model in population dynamics namely logistic growth model described as (1.2) where K = α β is a constant involving the environment capacity. In the past few years, the qualitative theory for Nicholson model and its variants has been extensively studied and developed [1,2,9]. In particular, the problems associated with asymptotic behavior of positive periodic and almost periodic solutions of Nicholson-type models with delays were studied in [15,19,20,29]. Nicholson-type models with stochastic perturbations and harvesting terms were also investigated in [30] and [6,17,23,27], respectively. Very recently, in [3], the problems of stability and attractivity were studied for a class of ndimensional Nicholson systems with constant coefficients and bounded time-varying delays. Based on some comparison techniques in the theory of monotone dynamical systems, delaydependent sufficient conditions were derived for the existence and global exponential stability of a unique positive equilibrium.
Most of the existing works so far are devoted to Nicholson-type models with linear mortality terms. As discussed in [1], a model of linear density-dependent mortality rate will be most accurate for populations at low densities. According to marine ecologists, many models in fishery such as marine protected areas or models of B-cell chronic lymphocytic leukemia dynamics are suitably described by Nicholson-type delay differential equations with nonlinear density-dependent mortality rate of the form [1]  3) to the case of variable coefficients and delays, which is more realistic in the theory of population dynamics [13,18], is given by N (t) = −D(t, N(t)) + β(t)N(t − τ(t))e −γ(t)N(t−τ(t)) , (1.4) where D(t, N) = a(t) − b(t)e −N or D(t, N) = a(t)N b(t)+N . In model (1.4), D(t, N) is the death rate of the population which depends on time t and the current population level N(t), B(t, N(t − τ(t))) = β(t)N(t − τ(t))e −γ(t)N(t−τ(t)) is the time-dependent birth function which involves a maturation delay τ(t) and gets its maximum Recently, Nicholsontype models with nonlinear density-dependent mortality terms have attracted considerable research attention. In particular, some results on the permanence property of certain types of Nicholson models with delays were established in [16,21]. The existence and exponential stability of positive periodic solutions, almost periodic solutions of various Nicholson-type models with both type-I and type-II nonlinear mortality terms were considered in [4,5,18] and [22,26,32,33], respectively. The author of [14] investigated the problem of global asymptotic stability of zero-equilibrium of the following special model (1.5) It has been shown that under the restrictions 6) the equilibrium N * = 0 of (1.5) is globally asymptotically stable with respect to an admissible phase space called C + . In recent work [31], the effect of delay on the stability and attractivity was studied for the following model where a, b, β j and γ j are constants. Under the restrictions Clearly, condition (1.9) can only be applied for models with small delays. This will restrict the applicability of the obtained results to practical models. Motivated by the above literature review, in this paper we study the problem of existence and global attractivity of positive periodic solution of the following Nicholson model Main contributions and innovation points of this paper are three folds. First, improved conditions on global uniform permanence and dissipativity of time-varying Nicholson models with nonlinear density-dependent mortality term in the form of (1.10) are derived based on new comparison techniques via differential inequalities. We do not impose restrictions on the maximum reproducing rates 1 γ k like (1.6), (1.8) and (1.9). Second, a novel approach to the problem of existence and global attractivity of a unique positive periodic solution of model (1.10) is presented. Third, as an application to Nicholson models with constant coefficients as (1.7), improved results on the existence, uniqueness and global attractivity of a positive equilibrium are obtained.

Preliminaries
For a given scalar ω > 0, a function f : R + → R is said to be ω-periodic if f (t + ω) = f (t) for all t ≥ 0. Let P ω (R + ) denote the set of ω-periodic functions on R + . Clearly, if f : R + → R is a continuous function and f ∈ P ω (R + ) for some ω > 0 then f is bounded on R + . Hereafter, for a bounded function f on [0, +∞), we will denote Consider a Nicholson model with delays and nonlinear density-dependent mortality term of the form where the density-dependent mortality term D(t, N) is of the form and τ M = max 1≤k≤p τ + k represents the upper bound of delays. For more detail on biological explanations of the coefficients of system (2.1)-(2.3), we refer the reader to [1,3,31].
Let C C([−τ M , 0], R) be the Banach space of continuous functions on [−τ M , 0] endowed with the norm ϕ = sup t∈[−τ M ,0] |ϕ(t)| and C + be the cone of nonnegative functions in C, that is, We write ϕ ≥ 0 for ϕ ∈ C + . In addition, a function ϕ ∈ C is said to be positive, write as ϕ > 0, if ϕ(t) is positive for all t ∈ [−τ M , 0]. Due to the biological interpretation, the set of admissible initial conditions in (2.2) is taken as Let us first introduce the following assumptions and conditions. (A2) There exists an ω > 0 such that the functions a, b, β k , γ k and τ k belong to P ω (R + ).

Condition (C):
A preview of our main results is presented in the following table. Conditions There exists a unique positive ω-periodic solution N * (t) and (C4) which is globally attractive in C + 0 .
For a biological interpretation of the proposed conditions, it is reasonable that when the population is absence the death rate is nonpositive (i.e. D(t, 0) ≤ 0) and D(t, N) is always positive when N > 0. This gives rise to condition (C1). On the other hand, in most of biological models, there typically exists a threshold related to the so-called carrying capacity. When the population size is very large, over the carrying capacity, the death rate can be bigger than the maximum birth rate. Similarly to model (1.4), the quantity ∑ p k=1 β k (t) γ k (t)e can be regarded as the maximum birth rate of model (2.1). In addition, when N is large D(t, N) is approximate to a(t). By this observation, we make an assumption to ensure that ∑ p k=1 This reveals the imposing of condition (C2) when considering long-time behavior of the model. (C3) is a testable condition derived from (C2) and (C1a) by taking into account the upper bound of the associated rates. While condition (C3) only guarantees non-extinction and nonblowup behavior, condition (C4) reveals that, by certain scaling coefficients, when maximum per capita daily egg production rates are smaller than the gap between the maximum death rate and birth rate (i.e. = a − − 1 e ∑ p k=1 , the population will be stable around a periodic trajectory (in the case of periodic coefficients) or a positive equilibrium (for time-invariant model).
In the remaining of this section, we present a local existence result of solutions of system 2) can be rewritten in the following abstract form where the function F : Proof. Clearly, the function F(·, ·) defined in (2.5) is continuous and locally Lipschitz with respect to ϕ. Thus, the existence and uniqueness of a local solution of (2.1)-(2.2) is straightforward [8] and the proof is omitted here.

Global existence of positive solutions
Theorem 3.1. Let assumption (A1) hold. Assume that b(t) ≥ a(t) for all t ∈ [0, +∞). Then, for any initial condition ϕ ∈ C + 0 , the solution N(t, t 0 , ϕ) of system The following lemma will be used in the proof of Theorem 3.1.
, be given continuous functions, the unique solution of the initial value problem (IVP) is given by Proof. Definex = e x then (3.1) is written aŝ Observe that (3.3) is an IVP of linear differential equations. Thus, from (3.3) we havê which leads to (3.2). The proof is completed.

Remark 3.4.
To ensure the positiveness of solutions of (2.1)-(2.3) with initial conditions in C + 0 , condition b(t) ≥ a(t) cannot be relaxed. For a counterexample, let n = 1 and assume that Then, it follows from (3.2) that

Uniform permanence
Note also that t T a(s)ds → +∞ as t → +∞.
Thus, let t → +∞ and ↓ 0, we obtain The proof is completed. Remark 3.6. As a special case of (3.8), for bounded functions a(t) and b(t), if b − > a + then the scalar m in (3.8) can be chosen as Thus, Theorem 3.5 in this paper encompasses the result of Lemma 1 in [31].
The following result is obtained as a consequence of Theorems 3.5 and 3.7.
Then, for any ϕ ∈ C + 0 , it holds that Remark 3.9. For bounded coefficients a, b, β k , γ k , it follows from (2.5) that the function F(t, ϕ) maps any bounded set B ⊂ C into a bounded set F(t, B) in R. Thus, by the assumptions of Corollary 3.8, for any ϕ ∈ C + 0 , F(t, N t ) is bounded. Consequently, the corresponding solution N(t, t 0 , ϕ) is uniformly continuous on [0, +∞).

Global attractivity of positive periodic solution
The following lemmas will be used in the proof of our results in this section.
Proof. A detailed proof was presented in [10]. Let us omit it here. This lemma was stated in [10]. To make it easier to follow, in this paper, we will also reconduct the following proof.
Proof. Since u is bounded, there exists a constant u ∞ > 0 satisfying u(t) ≤ u ∞ for all t ≥ 0. Besides that for a given > 0 there exists a δ = δ( ) > 0 such that u(t 1 ) − u(t 2 ) < whenever |t 1 − t 2 | < δ due to the uniform continuity of u. We now defined a sequence of functions u k : R + → R n , k ∈ N 0 , by u k (t) = u(t + kω). Then, we have u k (t) = u(t + kω) ≤ u ∞ for all t ≥ 0, k ∈ N 0 , which shows the uniform boundedness of the sequence {u k }. On the other hand, Thus, the sequence {u k } is uniformly equicontinuous. By the Arzelà-Ascoli Theorem, there exists a subsequence {u k p } that converges uniformly on the interval [0, ω] to a continuous function denoted as u * (t). Hence, for a given > 0, On the other hand, it follows from the assumption that

This ensures ∑ ∞
n=k ω 0 u n (t + ω) − u n (t) dt → 0 as k → ∞. Thus, we can assume without lost of generality that Now, for any k > k p , we have This shows that ω 0 u(t + kω) − u * (t) dt → 0 as k → ∞. Then, it can be verified by similar arguments used in [10] that u k ⇒ u * on [0, ω] as k → ∞. It is clear that for any t ≥ 0 there exist a unique k ∈ N 0 and s ∈ [0, ω) such that t = kω + s. We now span the function u * by defining u * (t) = u * (s) then u * is a continuous and ω-periodic function on [0, ∞) satisfying The proof is completed.
Let f be a differentiable function. Similarly to [28], we define a generalized sign-function σ f as follows Then, it is clear that | f (t)| = f (t)σ f (t). Moreover, by similar lines used in the proof of Lemma 3.1 in [28], we have the following lemma.

Lemma 4.4. For a differentiable function f , it holds that
where D + is the upper-right Dini derivative.
In the following, we assume that assumptions (A1), (A2) and conditions (3.12a)-(3.12b) are satisfied. For convenience, we denote Note that, by (3.12b), b − a + > 1 and hence r * > 0. In addition, since the condition r * < 1 max γ − k is not imposed, 1 − γ − k r * can be positive, negative or zero. For 1 ≤ k ≤ p that 1 − γ − k r * ≤ 0, ν k = 1 e 2 . We are now in a position to present the existence, uniqueness and global attractivity of a positive periodic solution of system (2.1)-(2.3) as in the following theorem.
where is the constant defined in (3.12a). Then, system (2.1)-(2.3) has a unique positive ω-periodic solution N * (t) which is globally attractive in C + 0 .
Proof. We divide the proof of Theorem 4.5 into the following three steps.
Step 1. Since condition (4.3) can be written as ∑ Let N(t) = N(t, t 0 , ϕ) be a solution of system (2.1)-(2.3) with initial condition ϕ ∈ C + 0 . Then, by Corollary 3.8, there exists a T > t 0 such that We define the function N ω (t) = N(t + ω) − N(t) and consider the following Lyapunov-like functional |N ω (s)|ds. (4.4) By virtue of the periodicity of τ k , γ k and other coefficients, and by utilizing Lemma 4.4, the upper-right Dini derivative of V 1 (t) is computed and estimated as follows By mean-value theorem, where ζ(t) is some value between N(t) and N(t + ω). Therefore, In regard to (4.1), we also have The above estimation gives Combining (4.5)-(4.7) we then obtain Similarly to (4.5), for |N ω (s)|ds, It follows from (4.8) and (4.9) that Since ρ > 0, it follows from (4.10) that Note also that |N (t)| is bounded since the right-hand side of (2.1) is bounded. Consequently, N(t) is a uniformly continuous function. By Lemma 4.2, there exists an ω-periodic function N * (t) such that |N(t) − N * (t)| → 0 as t → +∞.
Step 2. It follows from (2.4) that Thus, for any n ∈ N, (4.11) due to the periodicity of the functions a, b, β k and γ k . Since N * (t) is an ω-periodic function, when n tends to infinity, we have Let n → +∞ in (4.11) we obtain This means that N * (t) is an ω-periodic solution of (2.1), which is also a positive solution of (2.1) according to Corollary 3.8.
Finally, for any solution N(t, t 0 , ϕ) of (2.1)-(2.3), it can be deduced from the arguments used in Steps 1 and 3 that |N(t, t 0 , ϕ) − N * (t)| → 0 as t → +∞. This shows the global attractivity of N * (t). The proof is completed. Remark 4.6. Conditions (4.2) and (4.3) are involved a scalar µ > 0 related to the rate of change of delay functions τ k (t). However, this scalar can be relaxed and conditions (4.2), (4.3) are reduced to the following one (4.14) More precisely, we state that in the following theorem. Proof. The proof is similar to that of Theorem 4.5. Specifically, let V(t) = |N(t + ω) − N(t)|, then, similarly to (4.8), we have , by Halanay inequality [11,Corollary 3.1], there exists a λ > 0 such that It follows from (4.16) that The remainder of the proof can be proceeded similarly to that of Theorem 4.5.

Attractivity of positive equilibrium
In this section, we apply our results presented in the preceding sections to the following Nicholson model where β k ≥ 0, γ k > 0 are known coefficients, ∑ Proof. The proof is deduced from (3.13).
. Then, model (5.1) has a unique positive equilibrium N * which is globally attractive in C + 0 .
Proof. An equilibrium of (5.1) is a solution of the following nonlinear equation According to (5.3), any equilibrium point of (5.1) should be within the range [θ 1 , θ 2 ], where there exists an N * ∈ (θ 1 , θ 2 ) such that Φ(N * ) = 0. For any N 1 , N 2 ∈ (θ 1 , θ 2 ), we have where ξ 1 , ξ 2 are mean values between N 1 and N 2 . Note that This shows that the equilibrium N * is unique. In addition, by similar arguments used in the proof of Theorem 4.7, it can be concluded that for any ϕ ∈ C + 0 , which shows the global attractivity of N * in C + 0 . The proof is completed. Remark 5.4. By the methods proposed in [5,18], model (2.1) has a unique positive periodic solution N * (t), which is globally attractive, if there exists a positive constant M such that where κ ∈ (0, 1) andκ > 0 are constants satisfying 1−κ e κ = 1 e 2 and κe −κ =κe −κ . It is clear that condition (5.6b) implies both (C1) and (C2), where σ e ≤ 1 − e −M b a − . In addition, by taking into account the upper bound of coefficients, testable conditions derived from (5.6b) and (5.6c) are given as Thus, conditions (C3) and (C4) are fulfilled. The above verification shows that, for the existence of positive solutions, uniform permanence, uniform dissipativity and global attractivity of positive periodic solution of (2.1), the method presented in this paper can give less conditions with simpler proofs than existing ones in the literatures [5,18].
The admissible region of (a, b) is illustrated in Figure 6.2. For illustrative purpose, let a = 4 e , b = 3. By Theorem 5.2, the unique equilibrium N * = 1 of (5.1) is globally attractive for any bounded delays τ k (t) ∈ [0, τ M ]. The simulation result given in Figure 6.3 is taken with various common delay τ k (t) = τ(t). It can be seen that all the corresponding state trajectories of model (5.1) converge to N * , which validates the theoretical results. a ≤ e. This shows that our conditions are competitive with those of [31].

Conclusion
A delayed Nicholson's blowflies model with nonlinear density-dependent mortality has been studied in this paper. Based on a comparison technique via differential inequalities, sufficient conditions that ensure global uniform permanence and dissipativity of the model have been first derived. The obtained results on uniform permanence and dissipativity have been then utilized to deal with the existence and global attractivity of a unique positive periodic solution of the underlying model. An application to the case of model with constant coefficients has also been presented. Two numerical examples with simulations have been given to illustrate the efficacy of the obtained results.