Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 798961, 9 pages http://dx.doi.org/10.1155/2013/798961 Research Article Permanence, Extinction, and Almost Periodic Solution of a Nicholson’s Blowflies Model with Feedback Control and Time Delay Haihui Wu and Shengbin Yu Sunshine College, Fuzhou University, Fuzhou, Fujian 350015, China Correspondence should be addressed to Haihui Wu; whh3346@sina.com Received 20 February 2013; Accepted 7 April 2013 Academic Editor: Yonghui Xia Copyright © 2013 H. Wu and S. Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A Nicholson’s blowflies model with feedback control and time delay is studied. By applying the comparison theorem of the differential equation and fluctuation lemma and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence, extinction, and existence of a unique globally attractive positive almost periodic solution of the system are obtained. It is proved that the feedback control variable and time delay have no influence on the permanence and extinction of the system. 1. Introduction Let π(π‘) be any continuous bounded function defined on [0, +∞); we set ππ = inf π (π‘) , π‘≥0 ππ’ = supπ (π‘) . π‘≥0 (1) In order to describe the dynamics of Nicholson’s blowflies, Gurney et al. [1] proposed the following mathematical model in 1980: πΜ (π‘) = −πΏπ (π‘) + ππ (π‘ − π) π−ππ(π‘−π) , (2) where π(π‘) is the size of the population at time π‘, π is the maximum per capita daily egg production rate, (1/π) is the size at which the population reproduces at its maximum rate, πΏ is the per capita daily adult death rate, and π is the generation time. KulenovicΜ and Ladas [2], GyoΜri and Ladas [3], and GyoΜri and Trofimchuk [4] investigated the oscillatory behaviors of the solutions of (2). For the attractivity, KulenovicΜ et al. [5] and So and Yu [6] have shown that, when π > πΏ, every positive solution π(π‘) of (2) tends to a positive equilibrium π∗ = (1/π) ln(π/πΏ) as π‘ → ∞ if π (π − 1) ( − 1) < 1. πΏ πΏπ (3) Reference [5] further showed that, for π ≤ πΏ, every nonnegative solution of (2) tends to zero as π‘ → ∞, and for π > πΏ, (2) is uniformly persistent. Furthermore, Li and Fan [7] considered the following nonautonomous equation: π₯Μ (π‘) = π₯ (π‘) (−πΏ (π‘) + π (π‘) exp {−πΌ (π‘) π₯ (π‘)}) , (4) where πΌ(π‘), πΏ(π‘), and π(π‘) are all positive π-periodic functions. The authors show that (4) has a unique globally attractive π-periodic positive solution if π (π‘) > πΏ (π‘) for π‘ ∈ [0, π] . (5) Their results improved the results of Saker and Agarwal [8] who considered system (4) with πΌ(π‘) = π (π is a constant). Recently, Wang and Fan [9] proposed the following discrete Nicholson’s blowflies model with feedback control: π₯ (π + 1) = π₯ (π) exp {−πΏ (π) + π (π) exp {−πΌ (π) π₯ (π)} −π (π) π (π)} , (6) Δπ (π) = −π (π) π (π‘) + π (π) π₯ (π − π) . Sufficient conditions are established for the permanence and the extinction of the system (6). They show that the bounded feedback terms do not have any influence on the permanence 2 Discrete Dynamics in Nature and Society or extinction of (6). The authors in [9] also proposed the following continuous model: π₯Μ (π‘) = π₯ (π‘) (−πΏ (π‘) + π (π‘) exp {−πΌ (π‘) π₯ (π‘)} − π (π‘) π (π‘)) , πΜ (π‘) = −π (π‘) π (π‘) + π (π‘) π₯ (π‘ − π) ; (7) however, they did not discuss the dynamic behaviors of the system (7). Considering that continuous models can excellently show the dynamic behaviors of those populations who have a long life cycle, overlapping generations, and large quantity, sufficient conditions for the permanence, global attractivity, and the existence of a unique, globally attractive, strictly positive almost periodic solution of the system (7) with π = 0 are obtained by Yu [10]. As pointed out by Nindjin et al. [11], time delay plays an important role in many biological dynamical systems, being particularly relevant in ecology and a model with time delay is a more realistic approach to the understanding of dynamics. Hence, it is necessary to study the model (7) which contains time delay. In the following discussion, we always assume that πΏ(π‘), π(π‘), πΌ(π‘), π(π‘), π(π‘), π(π‘) are all continuous, positive almost periodic functions. Also, from the viewpoint of mathematical biology, we consider (7) together with the following initial conditions: π₯ (π) = π (π) ≥ 0, π ∈ [−π, 0] π (0) > 0, π (π) = π (π) ≥ 0, π ∈ [−π, 0] π (0) > 0, (8) where π(π ) and π(π ) are continuous on [−π, 0]. It is not difficult to see that solutions of (7) and (8) are well defined for all π‘ ≥ 0 and satisfy π₯ (π‘) > 0, π (π‘) > 0, for π‘ ≥ 0. (9) The aim of this paper is, by constructing a suitable Lyapunov functional and applying the analysis technique of Feng et al. [12], to obtain sufficient conditions for the existence of a unique globally attractive positive almost periodic solution of the system (7) with initial condition (8). This paper is organized as follows. In Section 2, by applying the analysis technique of [13, 14] and Fluctuation lemma [15, 16], we present the permanence and the extinction of model (7) and (8). In Section 3, by constructing a suitable Lyapunov functional, a sufficient conditions for the existence of a unique globally attractive positive almost periodic solution of the system (7) and (8). Examples together with their numeric simulations are stated in Section 4. For more works on almost periodic solutions of the ecosystem with feedback control, one could refer to [17–23] and the references cited therein. 2. Permanence and Extinction Now let us state several lemmas which will be useful in proving the main result of this section. Lemma 1 (see [13]). Assume that π > 0, π(π‘) > 0 is a boundedness continuous function and π₯(0) > 0. Further suppose that (i) π₯Μ (π‘) ≤ −ππ₯ (π‘) + π (π‘) (10) then for all π‘ ≥ π , π‘ π₯ (π‘) ≤ π₯ (π‘ − π ) exp {−ππ } + ∫ π (π) exp {π (π − π‘)} ππ. π‘−π (11) Particularly, if π(π‘) is bounded above with respect to π, then lim supπ₯ (π‘) ≤ π‘ → +∞ π . π (12) (ii) Also suppose that π₯Μ (π‘) ≥ −ππ₯ (π‘) + π (π‘) ; (13) then for all π‘ ≥ π , π‘ π₯ (π‘) ≥ π₯ (π‘ − π ) exp {−ππ } + ∫ π (π) exp {π (π − π‘)} ππ. π‘−π (14) Particularly, if π(π‘) is bounded above with respect to π, then lim inf π₯ (π‘) ≥ π‘ → +∞ π . π (15) Lemma 2 (see [20]). If π > 0, π > 0 and π₯Μ ≥ π₯(π − ππ₯), when π‘ ≥ 0 and π₯(0) > 0, one has lim inf π₯ (π‘) ≥ π‘ → +∞ π . π (16) If π > 0, π > 0, and π₯Μ ≤ π₯(π − ππ₯), when π‘ ≥ 0 and π₯(0) > 0, one has lim supπ₯ (π‘) ≤ π‘ → +∞ π . π (17) Lemma 3. Let (π₯(π‘), π(π‘))π be any solution of system (7) with initial condition (8); there exists positive numbers π1 and π2 , which are independent of the solution of the system, such that lim supπ₯ (π‘) ≤ π1 , π‘ → +∞ lim supπ (π‘) ≤ π2 . π‘ → +∞ (18) Proof. Let (π₯(π‘), π(π‘))π be any solution of system (7) satisfying initial condition (8). Since π₯ exp{−πΌπ π₯(π‘)} ≤ (1/πΌπ π) for π₯ > 0, according to the positivity of solution and the first equation of system (7), for π‘ ≥ 0, π₯Μ (π‘) ≤ π₯ (π‘) (−πΏπ + ππ’ exp {−πΌπ π₯ (π‘)}) ≤ −πΏπ π₯ (π‘) + ππ’ , πΌπ π where π is the mathematical constant. (19) Discrete Dynamics in Nature and Society 3 By applying Lemma 1(i) to (19), we have lim supπ₯ (π‘) ≤ π‘ → +∞ ππ’ Δ = π1 . πΏπ πΌπ π Substituting (29) into the second equation of system (7) leads to (20) ∀π‘ ≥ π1 . (21) Equation (21) together with the second equation of (7) leads to πΜ (π‘) ≤ −ππ π (π‘) + 2ππ’ π1 , ∀π‘ ≥ π1 + π. 2ππ’ π1 Δ = π2 . lim supπ (π‘) ≤ ππ π‘ → +∞ (30) π (π‘) ≤ π (π‘ − π ) exp {−ππ π } π‘ π π‘−π π−π + ∫ ππ’ π₯ (π) exp {∫ (22) from (28) ≤ (23) π (π’) ππ’} exp {ππ (π − π‘)} ππ π (π‘ − π ) exp {−ππ π } π‘ π‘ π π‘−π π π−π + ∫ ππ’ π₯ (π‘) exp {∫ π (π’) ππ’} exp {∫ Obviously, ππ (π = 1, 2) are independent of the solution of system (7). Equations (20) and (23) show that the conclusion of Lemma 3 holds. The proof is completed. π (π’) ππ’} × exp {ππ (π − π‘)} ππ Lemma 4. Assume that π‘ π‘ π‘−π π ≤ π (π‘ − π ) exp {−ππ π } + ππ’ π₯ (π‘) ∫ exp {∫ π (π’) ππ’} π (π»1 ) π (π’) ππ’} ππ, × exp {∫ π (π‘) > πΏ (π‘) , π‘ ≥ 0, lim inf π (π‘) ≥ π2 . π‘ → +∞ π−π (24) holds. Then there exists positive constants π1 and π2 , which are independent of the solution of system (7), such that π‘ → +∞ π (π ) ππ } . Applying Lemma 1(i) to the above differential inequality, for 0 ≤ π ≤ π‘, one has Using Lemma 1(i) again, one has lim inf π₯ (π‘) ≥ π1 , π‘ π‘−π Hence, there exists π1 > 0 such that π₯ (π‘) ≤ 2π1 , πΜ (π‘) ≤ −ππ π (π‘) + ππ’ π₯ (π‘) exp {∫ (25) Proof. Let (π₯(π‘), π(π‘))π be any solution of system (7) satisfying initial condition (8). From Lemma 3, there exists a π2 > π1 + π such that for all π‘ ≥ π2 , π₯(π‘) ≤ π, π(π‘) ≤ π, where π = 2 max{π1 , π2 }. According to the first equation of system (7) and the positivity of solution, for π‘ ≥ π2 , (31) where we used the fact maxπ∈[π‘−π ,π‘] exp{ππ (π−π‘)} = exp{0} = 1. Note that there exists a πΎ, such that 2ππ’ π exp{−ππ π } < (π½/2), as π ≥ πΎ, where π½ = inf π‘≥0 (π(π‘) − πΏ(π‘)). In fact, we can choose πΎ > (1/ππ ) ln(4ππ’ π/π½). And so, fixing πΎ, combined with (31), we can obtain π (π‘) ≤ π exp {−ππ πΎ} + ππ’ π₯ (π‘) ∫ π‘ π‘−πΎ π × exp {∫ π₯Μ (π‘) = π₯ (π‘) (−πΏ (π‘) + π (π‘) exp {−πΌ (π‘) π₯ (π‘)} − π (π‘) π (π‘)) π−π π‘ exp {∫ π (π’) ππ’} π π (π’) ππ’} ππ ≤ π exp {−ππ πΎ} + π·π₯ (π‘) , ≥ π₯ (π‘) (−πΏ (π‘) + π (π‘) exp {−πΌ (π‘) π} − π (π‘) π) (32) Δ = −π (π‘) π₯ (π‘) , (26) where π(π‘) = πΏ(π‘) − π(π‘) exp{−πΌ(π‘)π} + π(π‘)π. Integrating both sides of (26) from π (π ≤ π‘) to π‘ leads to π‘ π₯ (π‘) ≥ exp {− ∫ π (π ) ππ } , π₯ (π) π (27) π‘ π π‘ π‘ > π₯ (π‘ − π) ≤ π₯ (π‘) exp {∫ π‘−π π (π ) ππ } . π = supπ‘≥π3 (ππ’ ∫π‘−πΎ exp{∫π π(π’)ππ’} exp{∫π−π π(π’)ππ’}ππ) > 0. Considering that π−π₯ ≥ 1 − π₯, for π₯ > 0, from the first equation of system (7) and the positivity of the solution, for π‘ > π2 + πΎ, we can get (28) ≥ π₯ (π‘) ( − πΏ (π‘) + π (π‘) − π (π‘) πΌ (π‘) π₯ (π‘) −2ππ’ π exp {−ππ πΎ} − ππ’ π·π₯ (π‘)) Particularly, taking π = π‘ − π, one can get π‘ π2 + πΎ, where π· π₯Μ (π‘) = π₯ (π‘) (−πΏ (π‘) + π (π‘) exp {−πΌ (π‘) π₯ (π‘)} − π (π‘) π (π‘)) or π₯ (π) ≤ π₯ (π‘) exp {∫ π (π ) ππ } . for all π‘ ≥ π₯ (π‘) ( (29) π½ − (ππ’ πΌπ’ + ππ’ π·) π₯ (π‘)) . 2 (33) 4 Discrete Dynamics in Nature and Society By Lemma 2, we have lim inf π₯ (π‘) ≥ π‘ → +∞ π½ Δ = π1 . 2 (ππ’ πΌπ’ + ππ’ π·) (34) Thus, there exists π3 > π2 + πΎ such that for all π‘ > π3 , π₯ (π‘) ≥ π1 . 2 (35) Equation (35) together with the second equation of (7) leads to πΜ (π‘) ≥ −ππ’ π (π‘) + ππ π1 , 2 ∀π‘ > π3 . (36) By applying Lemma 1(ii) to the above differential inequality, we have lim inf π (π‘) ≥ π‘ → +∞ ππ π1 Δ = π2 . 2ππ’ (37) Obviously, ππ (π = 1, 2) are independent of the solution of system (7). Equations (34) and (37) show that the conclusion of Lemma 4 holds. The proof is completed. Μ <0 If the latter case of (38) holds, from (40) we have π₯(π‘) or π₯(π‘) is decreasing; therefore, limπ‘ → +∞ π₯(π‘) = π ∈ [0, +∞). Hence lim supπ‘ → +∞ = lim inf π‘ → +∞ π₯(π‘) = π. We only need to show that π = 0. Otherwise, if π > 0, then there exists a π4 > 0, such that π₯(π‘) > (π/2) for π‘ ≥ π4 . According to the Fluctuation lemma, there exists a sequence ππ → ∞ as Μ π ) → 0, π₯(ππ ) → lim supπ‘ → ∞ = π, as π → ∞ such that π₯(π π → ∞. We can choose a large enough number π such that ππ > π4 for π > π; hence, π₯(ππ ) > (π/2) for all π > π. For π > π, π(π‘) ≤ πΏ(π‘) together with the first equation of (7) leads to π₯Μ (ππ ) ≤ π₯ (ππ ) (−πΏ (ππ ) + π (ππ ) exp {−πΌ (ππ ) π₯ (ππ )}) π ≤ π₯ (ππ ) (−πΏ (ππ ) + πΏ (ππ ) exp { −πΌ (ππ ) )}) 2 π ≤ π₯ (ππ ) (−1 + exp { −πΌπ )}) πΏπ . 2 Let π → ∞; we obtain that 0 ≤ π(−1 + exp{−πΌπ (π/2)})πΏπ or exp{−πΌπ (π/2))} > 1 which is impossible. Hence, π = 0 or lim π₯ (π‘) = 0. π‘ → +∞ From Lemmas 3–4 and the definition of permanence, we can obtain the following conclusion. Corollary 6. Suppose that (π»1 ) holds; then system (7) admits at least one positive π-periodic solution if πΏ(π‘), π(π‘), πΌ(π‘), π(π‘), π(π‘), π(π‘) are all continuous positive π-periodic functions. (43) Now, we come to prove that lim π (π‘) = 0. Theorem 5. Assume that (π»1 ) holds; then system (7) with initial condition (8) is permanent. As a direct corollary of Theorem 2 in [24], from Theorem 5, we have the following. (42) π‘ → +∞ (44) For any π > 0, according to (43), there exists a π5 > 0, such that π₯ (π‘) < π ∀π‘ > π5 . (45) Then, for π‘ > π5 + π, πΜ (π‘) ≤ −ππ π (π‘) + ππ’ π. (46) π Theorem 7. Let (π₯(π‘), π(π‘)) be any positive solution of the system (7) with initial condition (8). Assume that ∞ ∫ (π (π‘) − πΏ (π‘)) ππ‘ = −∞ 0 or π (π‘) ≤ πΏ (π‘) , Thus, by applying Lemma 1(i) to the above differential inequality, we have π‘≥0 ππ’ π ππ (47) lim π (π‘) = 0. (48) 0 < π (π‘) ≤ (38) which implies that holds; then lim π₯ (π‘) = 0, π‘ → +∞ lim π (π‘) = 0. π‘ → +∞ (39) The proof is complete. Proof. Firstly, from the the first equation of (7), π₯Μ (π‘) ≤ π₯ (π‘) (π (π‘) − πΏ (π‘)) . (40) If the former case of (38) holds, then π‘ 0 < π₯ (π‘) ≤ π₯ (0) exp [∫ (π (π‘) − πΏ (π‘)) ππ‘] σ³¨→ 0, 0 as π‘ σ³¨→ ∞, which shows that limπ‘ → +∞ π₯(π‘) = 0. π‘ → +∞ 3. Existence of a Unique Almost Periodic Solution Now, we give the definition of the almost periodic function. (41) Definition 8 (see [25, 26]). A function π(π‘, π₯), where π is an π-vector, π‘ is a real scalar, and π₯ is an π-vector, is said to be almost periodic in π‘ uniformly with respect to π₯ ∈ π ⊂ π π , if π(π‘, π₯) is continuous in π‘ ∈ π and π₯ ∈ π, and if for any π > 0, Discrete Dynamics in Nature and Society 5 there is a constant π(π) > 0, such that in any interval of length π(π), there exists π such that the inequality π σ΅© σ΅¨ σ΅¨ σ΅©σ΅© σ΅©σ΅©π (π‘ + π) − π (π‘)σ΅©σ΅©σ΅© = ∑ σ΅¨σ΅¨σ΅¨ππ (π‘ + π, π₯) − ππ (π‘, π₯)σ΅¨σ΅¨σ΅¨ < π (49) π=1 Definition 9 (see [25, 26]). A function π : π → π is said to be an asymptotically almost periodic function if there exists an almost periodic function π(π‘) and a continuous function π(π‘) such that π‘ ∈ π , π (π‘) σ³¨→ 0 as π‘ σ³¨→ ∞. (50) We denote by π(πΈ) the set of all solutions π§(π‘) = (π₯(π‘), π(π‘))π of system (7) satisfying π1 ≤ π₯(π‘) ≤ π1 , π2 ≤ π(π‘) ≤ π2 for all π‘ ∈ π . Lemma 10. One has π(πΈ) =ΜΈ 0. Proof. Since πΏ(π‘), π(π‘), πΌ(π‘), π(π‘), π(π‘), π(π‘) are almost periodic functions, there exists a sequence {π‘π }, π‘π → ∞ as π → ∞ such that πΏ (π‘ + π‘π ) σ³¨→ πΏ (π‘) , π (π‘ + π‘π ) σ³¨→ πΏ (π‘) , πΌ (π‘ + π‘π ) σ³¨→ πΏ (π‘) , π (π‘ + π‘π ) σ³¨→ πΏ (π‘) , π (π‘ + π‘π ) σ³¨→ πΏ (π‘) , π (π‘ + π‘π ) σ³¨→ πΏ (π‘) , (51) π as π → ∞ uniformly on π . Suppose π§(π‘) = (π₯(π‘), π(π‘)) is a solution of (7) satisfying π1 ≤ π₯(π‘) ≤ π1 , π2 ≤ π(π‘) ≤ π2 for π‘ > π. Obviously, the sequence (π§(π‘ + π‘π )) is uniformly bounded and equicontinuous on each bounded subset of π . Therefore, by the Ascoli-Arzela theorem, there exists a subsequence of {π‘π }, which we still denote by {π‘π }, such that π₯(π‘ + π‘π ) → π(π‘), π(π‘ + π‘π ) → π(π‘), as π → ∞ uniformly on each bounded subset of π . For any π1 ∈ π , we may assume that π‘π + π1 ≥ π for all π. For π‘ ≥ 0, we have π₯ (π‘ + π‘π + π1 ) − π₯ (π‘π + π1 ) =∫ π‘+π1 π1 π₯ (π + π‘π ) (−πΏ (π + π‘π ) + π (π + π‘π ) × exp {−πΌ (π + π‘π ) π₯ (π + π‘π )} π (π‘ + π‘π + π1 ) − π (π‘π + π1 ) =∫ π1 π‘+π1 π1 π (π ) (−πΏ (π ) + π (π ) × exp {−πΌ (π ) π (π )} − π (π ) π (π )) ππ , π (π‘ + π1 )−π (π1 ) = ∫ π‘+π1 π1 (−π (π ) π (π ) + π (π ) π (π − π)) ππ , (53) for all π‘ ≥ 0. Since π1 ∈ π is arbitrarily given, (π(π‘), π(π‘))π is a solution of system (7) on π . It is clear that π1 ≤ π(π‘) ≤ π1 , π2 ≤ π(π‘) ≤ π2 for π‘ ∈ π . That is to say, (π(π‘), π(π‘))π ∈ π(πΈ). This completes the proof. Lemma 11 (see [27]). Let π be a nonnegative function defined on [0, +∞) such that π is integrable on [0, +∞) and is uniformly continuous on [0, +∞). Then, limπ‘ → +∞ π(π‘) = 0. Theorem 12. In addition to (π»1 ), further suppose that (π»2 ) there exists a β > 0, such that ππ πΌπ exp (πΌπ’ π1 ) − ππ’ > β, ππ − ππ’ > β, (54) where π1 is defined in (23); then system (7) with initial conditions (8) is globally attractive. That is to say, for any two positive solutions, one has σ΅¨ σ΅¨ lim σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨ = 0, σ΅¨ σ΅¨ lim σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ = 0. π‘ → +∞ σ΅¨ π‘ → +∞ σ΅¨ (55) Proof. Let (π₯∗ (π‘), π∗ (π‘))π and (π₯(π‘), π(π‘))π be any two positive solutions of system (7)-(8). Theorem 5 implies there exist positive constants π, ππ , and ππ (π = 1, 2) such that for π‘ ≥ π π1 ≤ π₯ (π‘) ≤ π1 , π1 ≤ π₯∗ (π‘) ≤ π1 , π2 ≤ π (π‘) ≤ π2 , π2 ≤ π∗ (π‘) ≤ π2 , (56) where ππ and ππ (π = 1, 2) are defined in Lemma 3 and Lemma 4. Set π(π‘) = π1 (π‘) + π2 (π‘), where −π (π + π‘π ) π (π )) ππ , π‘+π1 π (π‘ + π1 ) − π (π1 ) =∫ is satisfied for all π‘ ∈ (−∞, +∞), π₯ ∈ π. The number π is called an π-translation number of π(π‘, π₯). π (π‘) = π (π‘) + π (π‘) , Applying Lebesgue’s dominated convergence theorem and letting π → ∞ in to previous equations, we obtain (−π (π + π‘π ) π (π + π‘π ) + π (π + π‘π ) π₯ (π + π‘π − π)) ππ . (52) σ΅¨ σ΅¨ π1 (π‘) = σ΅¨σ΅¨σ΅¨ln π₯ (π‘) − ln π₯∗ (π‘)σ΅¨σ΅¨σ΅¨ , π‘ σ΅¨ σ΅¨ π2 (π‘) = σ΅¨σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ + ππ’ ∫ π‘−π σ΅¨ σ΅¨σ΅¨ ∗ σ΅¨σ΅¨π₯ (π’) − π₯ (π’)σ΅¨σ΅¨σ΅¨ ππ’. (57) 6 Discrete Dynamics in Nature and Society Calculating the upper right derivatives of π1 (π‘) along the solution of (7) leads to Then, π‘ π (π) σ΅¨ σ΅¨ σ΅¨ σ΅¨ < +∞, ∫ [σ΅¨σ΅¨σ΅¨π (π ) − π∗ (π )σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π₯ (π ) − π₯∗ (π )σ΅¨σ΅¨σ΅¨] ππ < β π π·+ π1 (π‘) = sgn [π₯ (π‘) − π₯∗ (π‘)] π‘ ≥ π. (64) ∗ × (π (π‘) (exp {−πΌ (π‘) π₯ (π‘)} − exp {−πΌ (π‘) π₯ (π‘)}) +π (π‘) (π∗ (π‘) − π (π‘))) = sgn [π₯ (π‘) − π₯∗ (π‘)] (58) × (π (π‘) (−πΌ (π‘) exp {−π (π‘)} (π₯ (π‘) − π₯∗ (π‘))) +π (π‘) (π∗ (π‘) − π (π‘))) Hence, |π(π‘) − π∗ (π‘)| + |π₯(π‘) − π₯∗ (π‘)| ∈ πΏ1 ([π, +∞)). By system (7) and Theorem 5, we get π(π‘), π∗ (π‘), π₯(π‘), π₯∗ (π‘), and their derivatives are bounded on [π, +∞), which implies that |π(π‘) − π∗ (π‘)| + |π₯(π‘) − π₯∗ (π‘)| is uniformly continuous on [π, +∞). By Lemma 11, we obtain σ΅¨ σ΅¨ lim σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨ = 0, σ΅¨ σ΅¨ ≤ −π (π‘) πΌ (π‘) (exp {−π (π‘)} σ΅¨σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨) π‘ → +∞ σ΅¨ σ΅¨ σ΅¨ lim σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ = 0. π‘ → +∞ σ΅¨ (65) σ΅¨ σ΅¨ + π (π‘) σ΅¨σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ , where we used the elementary mean value theorem of differential calculus and π(π‘) lies between πΌ(π‘)π₯(π‘) and πΌ(π‘)π₯∗ (π‘). Then, for π‘ ≥ π, we have πΌπ π1 ≤ π (π‘) ≤ πΌπ’ π1 . (59) Hence, by (58), we can have σ΅¨ σ΅¨ π·+ π1 (π‘) ≤ −ππ πΌπ exp (πΌπ’ π1 ) σ΅¨σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨ + ππ’ σ΅¨σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ . (60) Calculating the upper right derivatives of π2 (π‘) along the solution of (7), one has The proof of Theorem 12 is complete. Theorem 13. Suppose all conditions of Theorem 12 hold; then there exists a unique almost periodic solution of systems (7) and (8). Proof. According to Lemma 10, there exists a bounded positive solution π’(π‘) = (π’1 (π‘), π’2 (π‘))π of (7) with initial condition (8). Then there exists a sequence {π‘πσΈ }, {π‘πσΈ } → ∞ as π → ∞, such that (π’1 (π‘ + π‘πσΈ ), π’2 (π‘ + π‘πσΈ ))π is a solution of the following system: π₯Μ (π‘) = π₯ (π‘) (−πΏ (π‘ + π‘πσΈ ) + π (π‘ + π‘πσΈ ) exp {−πΌ (π‘ + π‘πσΈ ) π₯ (π‘)} −π (π‘ + π‘πσΈ ) π (π‘)) , π·+ π2 (π‘) = sgn [π (π‘) − π∗ (π‘)] πΜ (π‘) = −π (π‘ + π‘πσΈ ) π (π‘) + π (π‘ + π‘πσΈ ) π₯ (π‘ − π) . × (−π (π‘) (π (π‘) − π∗ (π‘)) (66) +π (π‘) (π₯ (π‘ − π) − π₯∗ (π‘ − π))) σ΅¨ σ΅¨ σ΅¨ σ΅¨ + ππ’ (σ΅¨σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨ − σ΅¨σ΅¨σ΅¨π₯ (π‘ − π) − π₯∗ (π‘ − π)σ΅¨σ΅¨σ΅¨) σ΅¨ σ΅¨ σ΅¨ σ΅¨ ≤ −ππ σ΅¨σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ + ππ’ σ΅¨σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨ . (61) According to (58), (66), and condition (π»2 ), we can obtain σ΅¨ σ΅¨ π·+ π (π‘) ≤ (ππ’ − ππ πΌπ exp (πΌπ’ π1 )) σ΅¨σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨ + (ππ’ − ππ ) σ΅¨σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ (62) σ΅¨ σ΅¨ σ΅¨ σ΅¨ < β [σ΅¨σ΅¨σ΅¨π (π‘) − π∗ (π‘)σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π₯ (π‘) − π₯∗ (π‘)σ΅¨σ΅¨σ΅¨] . Integrating both sides of (62) from π to π‘ leads to π‘ σ΅¨ σ΅¨ σ΅¨ σ΅¨ π (π‘) + β ∫ [σ΅¨σ΅¨σ΅¨π (π ) − π∗ (π )σ΅¨σ΅¨σ΅¨ + σ΅¨σ΅¨σ΅¨π₯ (π ) − π₯∗ (π )σ΅¨σ΅¨σ΅¨] ππ π < π (π) < +∞, π‘ ≥ π. (63) According to Theorem 5 and the fact that πΏ(π‘), π(π‘), πΌ(π‘), π(π‘), π(π‘), π(π‘) are all continuous, positive almost periodic functions, we know that both {π’π (π‘ + π‘πσΈ )} (π = 1, 2) and Μ + π‘πσΈ )} (π = 1, 2) are uniformly its derivative function {π’(π‘ σΈ bounded; thus, {π’π (π‘ + π‘π )} (π = 1, 2) are uniformly bounded and equi-continuous. By Ascoli’s theorem, there exists a uniformly convergent subsequence {π’π (π‘ + π‘π )} ⊆ {π’π (π‘ + π‘πσΈ )} such that for any π > 0, there exists a πΎ(π) > 0 with the property that if π, π ≥ πΎ(π), then σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨π’π (π‘ + π‘π ) − π’π (π‘ + π‘π )σ΅¨σ΅¨σ΅¨ < π, π = 1, 2. (67) That is to say, π’π (π‘) (π = 1, 2) are asymptotically almost periodic functions. Hence there exists two almost periodic functions ππ (π‘ + π‘π ) (π = 1, 2) and two continuous functions π π (π‘ + π‘π ) (π = 1, 2) such that π’π (π‘ + π‘π ) = ππ (π‘ + π‘π ) + π π (π‘ + π‘π ) , π = 1, 2, (68) Discrete Dynamics in Nature and Society 7 where lim ππ (π‘ + π‘π ) = ππ (π‘) , π → +∞ lim π π (π‘ + π‘π ) = 0, π → +∞ π = 1, 2, (69) ππ (π‘) (π = 1, 2) are also almost periodic functions. Therefore, lim π’π (π‘ + π‘π ) = ππ (π‘) , π → +∞ π = 1, 2. 4. Examples and Numeric Simulations (70) On the other hand, lim π’Μπ (π‘ + π‘π ) = lim lim π → +∞ π → +∞ β → 0 = lim lim β → 0 π → +∞ π’π (π‘ + π‘π + β) − π’π (π‘ + π‘π ) β π’π (π‘ + π‘π + β) − π’π (π‘ + π‘π ) β ππ (π‘ + β) − ππ (π‘) . β→0 β Remark 15. Li and Fan in [7] show that (73) has a unique globally attractive π-periodic positive solution if π(π‘) > πΏ(π‘) for π‘ ∈ [0, π], which is the same as Corollary 14. Thus, Theorem 13 supplements and generalizes results in [7, 8]. Now we give several examples together with their numeric simulations to show the feasibility of our main results. Example 16. Consider the following example: π₯Μ (π‘) = π₯ (π‘) (−5 − sin (√5π‘) + 10 exp {− (30 + cos (√11π‘)) π₯ (π‘)} = lim (71) So ππΜ (π‘) (π = 1, 2) exist. Moreover, − (1 + 0.5 sin (√7π‘)) π (π‘)) , (74) πΜ (π‘) = − (2.5 + 0.5 cos (√7π‘)) π (π‘) π1Μ (π‘) = lim π’Μ1 (π‘ + π‘π ) π → +∞ + (5.8 + 0.2 sin (√3π‘)) π₯ (π‘ − 2) . = lim {π’1 (π‘ + π‘π ) π → +∞ × (−πΏ (π‘ + π‘π ) − π (π‘ + π‘π ) π’2 (π‘ + π‘π ) + π (π‘ + π‘π ) × exp {−πΌ (π‘ + π‘π ) π’1 (π‘ + π‘π )})} = π1 (π‘) (−πΏ (π‘) + π (π‘) exp {−πΌ (π‘) π1 (π‘)} − π (π‘) π2 (π‘)) , π2Μ (π‘) = lim π’Μ2 (π‘ + π‘π ) In this case, corresponding to system (7), we have πΏ(π‘) = 5 + sin(√5π‘), π(π‘) = 10, πΌ(π‘) = 30 + cos(√11π‘), π(π‘) = 2.5 + 0.5 cos(√7π‘), π(π‘) = 5.8 + 0.2 sin(√3π‘), π(π‘) = 1 + 0.5 sin(√7π‘), π = 2. According to the proof of Lemmas 3 and 4, one has π → +∞ = lim {−π (π‘ + π‘π ) π2 (π‘ + π‘π ) π → +∞ +π (π‘ + π‘π ) π1 (π‘ + π‘π − π)} = −π (π‘) π2 (π‘) + π (π‘) π1 (π‘ − π) . (72) These show that (π1 (π‘), π2 (π‘))π satisfied system (7). Hence, (π1 (π‘), π2 (π‘))π is a positive almost periodic solution of (7). Then, it follows from Theorem 12 that system (7) has a unique positive almost periodic solution. The proof is completed. Without the feedback terms, that is π(π‘) = 0, π(π‘) = 0, π(π‘) = 0, and (7) becomes the following equation: π₯Μ (π‘) = π₯ (π‘) (−πΏ (π‘) + π (π‘) exp {−πΌ (π‘) π₯ (π‘)}) . (73) Equation (73) with periodic coefficients has been studied by Li and Fan [7] and Saker and Agarwal [8] with πΌ(π‘) = π. Since the periodic case is a special case of almost periodic, hence, as a direct corollary of Theorem 13, we have the following. Corollary 14. Suppose πΏ(π‘), π(π‘), πΌ(π‘), π(π‘), π(π‘), π(π‘) are all continuous positive π-periodic functions and π(π‘) > πΏ(π‘) for π‘ ∈ [0, π]; then (73) has a unique globally attractive π-periodic positive solution. π1 = 3ππ’ = 0.04757, 2πΏπ πΌπ π π1 = ππ − πΏπ’ − ππ’ π2 = 0.005934, 2ππ πΌπ’ π2 = ππ π1 = 0.0055384. 2ππ’ π2 = 3ππ’ π1 = 0.214066, 2ππ (75) Hence, π (π‘) > πΏ (π‘) , ππ πΌπ exp (πΌπ’ π1 ) − ππ’ ≅ 1261.193896 > 0, ππ − ππ’ = 0.5 > 0. (76) Thus, all the conditions of Theorem 13 are satisfied, and so, there exists a unique almost periodic solution of systems (74). Figure 1 shows this property. 8 Discrete Dynamics in Nature and Society 0.12 0.09 0.08 0.1 0.07 0.06 Solution Solution 0.08 0.06 0.04 0.03 0.04 0.02 0.02 0 0.05 0.01 0 20 40 60 80 100 0 0 20 Time π‘ 40 60 80 100 Time π‘ π₯ π π₯ π Figure 1: Dynamics of the π₯(π‘)) and π(π‘) of system (74) with the initial values (π₯(0), π(0))π = (0.02, 0.05)π , (0.03, 0.03)π , (0.04, 0.08)π , (0.005, 0.1)π ; here π‘ ∈ [0, 100]. Figure 2: The dynamic behavior of system (77) with initial condition (π₯(0), π(0))π = (0.03, 0.05)π , (0.02, 0.04)π , (0.005, 0.01)π , (0.002, 0.003)π ; here π‘ ∈ [0, 100]. Example 17. Consider the following example: model with feedback control and time delay. At the end of this paper, two examples together with their numerical simulations show the verification of our main results. Our results supplement and generalize the results in [7, 8]. π₯Μ (π‘) = π₯ (π‘) (−2 − 1 (π‘ + 1)2 + 2 exp {− (30 + cos (√11π‘)) π₯ (π‘)} − (1 + 0.5 sin (√7π‘)) π (π‘) ) , References (77) πΜ (π‘) = − (2.5 + 0.5 cos (√7π‘)) π (π‘) + (5.8 + 0.2 sin (√3π‘)) π₯ (π‘ − 2) . In this case, we have π (π‘) < πΏ (π‘) . (78) Hence, By Theorem 7, we know that any positive solution of system (77) satisfies limπ‘ → +∞ π₯(π‘) = 0, limπ‘ → +∞ π(π‘) = 0. Numerical simulation also confirms our result (see Figure 2). 5. Conclusion In this paper, we consider a Nicholson’s blowflies model with feedback control and time delay. 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