Dynamics of an Almost Periodic Food Chain System with Impulsive Effects

and Applied Analysis 3 stable positive almost periodic solution of system (5) by applying the Lyapunov method of the impulsive differential equations [11]. 2. Preliminaries Now, let us state the following definitions and lemmas, which will be useful in proving our main result. By I, I = {{τk} ∈ R : τk < τk+1, k ∈ Z, limk→±∞τk = ±∞}, we denote the set of all sequences that are unbounded and strictly increasing. Introduce the following notations. For J ⊂ R, PC(J,R) is the space of all piecewise continuous functions from J to R with points of discontinuity of the first kind τk, at which it is left continuous. By the basic theories of impulsive differential equations in [10, 11], system (5) has a unique solutionX(t) = X(t, X0) ∈ PC([0, +∞),R ). Since the solution of system (5) is a piecewise continuous function with points of discontinuity of the first kind τk, k ∈ Z, we adopt the following definitions for almost periodicity. Definition 1 (see [11]). The set of sequences {τj k }, τ j k = τk+j − τk, k ∈ Z, j ∈ Z, {τk} ∈ I, is said to be uniformly almost periodic if for arbitrary ε > 0 there exists a relatively dense set of ε-almost periods common for any sequences. Definition 2 (see [11]). The function φ ∈ PC(R,R) is said to be almost periodic, if the following hold. (1) The set of sequences {τj k }, τ j k = τk+j − τk, k ∈ Z, j ∈ Z, {τk} ∈ I, is uniformly almost periodic. (2) For any ε > 0, there exists a real number δ > 0 such that if the points t and t belong to one and the same interval of continuity of φ(t) and satisfy the inequality |t − t| < δ, then |φ(t) − φ(t)| < ε. (3) For any ε > 0, there exists a relatively dense setT such that if η ∈ T, then |φ(t + η) − φ(t)| < ε for all t ∈ R satisfying condition |t − τk| > ε, k ∈ Z. The elements of T are called ε-almost periods. Lemma 3 (see [11]). Let {τk} ∈ I. Then, there exists a positive integer A such that, on each interval of length 1, one has no more than A elements of the sequence {τk}; that is, i (s, t) ≤ A (t − s) + A, (6) where i(s, t) is the number of the points τk in the interval (s, t). Theoretically, one can investigate the existence, uniqueness, and stability of almost periodic solution for functional differential equations by using Lyapunov functional as follows [11, P109]. Consider the system of impulsive differential equations as follows: ?̇? (t) = f (t, x (t)) , t ̸ = τk, Δx (τk) = Ikx (τk) , (7) where t ∈ R, {τk} ∈ I, f : R×D → R , Ik : D → R , k ∈ Z, andD is an open set in R. Introduce the following conditions. (C1) Function f(t, x) is almost periodic in t uniformly with respect to x ∈ D. (C2) Sequence {Ik(x)}, k ∈ Z, is almost periodic uniformly with respect to x ∈ D. Lemma4 (see [11, P109]). Suppose that there exists a Lyapunov functional V(t, x, y) defined on R × D × D satisfying the following conditions. (1) u(‖x − y‖) ≤ V(t, x, y) ≤ V(‖x − y‖), where u, V ∈ P withP = {u : R → R | u is continuous increasing function and u(s) → 0 as s → 0}. (2) |V(t, x, y) − V(t, x, y)| ≤ K(‖x − x‖ + ‖y − y‖), where K > 0 is a constant. (3) For t = τk, V(t, x + Ik(x), y + Ik(y)) ≤ V(t, x, y); for t ̸ = τk, ?̇?(2.2)(t, x, y) ≤ −γV(t, x, y), ∀k ∈ Z, where γ > 0 is a constant. Moreover, one assumes that system (7) has a solution that remains in a compact set S ⊂ D. Then, system (7) has a unique almost periodic solution which is uniformly asymptotically stable. 3. Permanence In this section, we establish a permanence result for system (5). Lemma 5 (see [10]). Assume that x ∈ PC(R) with points of discontinuity at t = τk and is left continuous at t = τk for k ∈ Z and ?̇? (t) ≤ f (t, x (t)) , t ̸ = τk, x (τ k ) ≤ Ik (x (τk)) , k ∈ Z , (8) where f ∈ C(R × R,R), Ik ∈ C(R,R), and Ik(x) is nondecreasing in x for k ∈ Z. Let u(t) be the maximal solution of the scalar impulsive differential equation as ?̇? (t) = f (t, u (t)) , t ̸ = τk, u (τ k ) = Ik (u (τk)) ≥ 0, k ∈ Z , u (t 0 ) = u0 (9) existing on [t0,∞). Then, x(t 0 ) ≤ u0 implies x(t) ≤ u (t) for t ≥ t0. Remark 6. If inequalities (8) in Lemma 5 are reversed and u∗(t) is theminimal solution of system (9) existing on [t0,∞), then x(t 0 ) ≥ u0 implies x(t) ≥ u∗(t) for t ≥ t0. Lemma 7. Assume that a, b > 0; then, the following impulsive logistic equation ?̇? (t) = x (t) [a − bx (t)] , t ̸ = τk, Δx (τ k ) = hkx (τk) , k ∈ Z + (10) 4 Abstract and Applied Analysis has a unique globally asymptotically stable positive almost periodic solution x which can be expressed as follows: α eb ≤ x (t) = [b∫ t


Introduction
Let R and Z denote the sets of real numbers and integers, respectively.Related to a continuous function , we use the following notations: As was pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance.Food chain predator-prey system, as one of the most important predator-prey systems, has been extensively studied by many scholars; many excellent results were concerned with the persistent property and positive periodic solution of the system; see [2][3][4][5][6][7][8] and the references cited therein.Recently, Shen considered the following three species food chain predator-prey system with Holling type IV functional response: where   (),  = 1, 2, 3, denotes the density of species   at time ,  2 is the predator of the first species  1 , and  3 is the predator of the second species  2 .By applying the comparison theorem of the differential equation and constructing the suitable Lyapunov function, sufficient conditions which guarantee the permanence and the global attractivity of the system are obtained.
Considering the exploited predator-prey system (harvesting or stocking) is very valuable, for it involves the human activities.It can be referred to [9], in which the human activities always happen in a short time or instantaneously.The continuous action of human is then removed from the model and replaced with an impulsive perturbation.These models are subject to short-term perturbations which are often assumed to be in the form of impulsive in the modelling process.Consequently, impulsive differential equations provide a natural description of such systems [10][11][12][13].Then, in [14], Zhang and Tan studied the following Holling II functional responses food chain system with periodic constant impulsive perturbation of predator: where ℎ > 0 is the release amount of top predator at  =  and  is the period of the impulsive effect.By using the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we consider the local stability of prey and top predator eradication periodic solution.
In real world phenomenon, the environment varies due to the factors such as seasonal effects of weather, food supplies, mating habits, and harvesting.So, it is usual to assume the periodicity of parameters in system (2).However, if the various constituent components of the temporally nonuniform environment are with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since the assumption of almost periodicity is more realistic, more important, and more general when we consider the effects of the environmental factors.In recent years, there have been many mathematical studies for the existence, uniqueness, and stability of positive almost periodic solution of biological models governed by differential equations in the literature (see [11,[15][16][17][18][19][20][21][22][23][24][25] and the references cited therein).Therefore, Bai and wang in [15] studied the following nonautonomous food chains system with Holling's type II functional response: By applying the comparison theorem and the Lyapunov method of ordinary differential equations, some sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of system (4) are obtained.Stimulated by the above reason, this paper is concerned with the following almost periodic food chain system with almost periodic impulsive perturbations and general functional responses: where   (),   (), and   (),  = 1, 2, 3,  = 1, 2, 3, 4, are all continuous almost periodic functions which are bounded above and below by positive constants;  1 ,  Obviously, system (2)-( 4) is special case of system (5).
The main purpose of this paper is to establish some new sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of system (5) by using the comparison theorem and the Lyapunov method of the impulsive differential equations [10,11] (see Theorems 11 and 14 in Sections 3 and 4).
The organization of this paper is as follows.In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections.In Section 3, by using the comparison theorem of the impulsive differential equations [10], we give the permanence of system (5).In Section 4, we study the existence of a unique uniformly asymptotically stable positive almost periodic solution of system (5) by applying the Lyapunov method of the impulsive differential equations [11].

Preliminaries
Now, let us state the following definitions and lemmas, which will be useful in proving our main result.
For  ⊂ R 3 , (, R 3 ) is the space of all piecewise continuous functions from  to R 3 with points of discontinuity of the first kind   , at which it is left continuous.By the basic theories of impulsive differential equations in [10,11], system (5) has a unique solution () = (,  0 ) ∈ ([0, +∞), R 3 ).
Since the solution of system ( 5) is a piecewise continuous function with points of discontinuity of the first kind   ,  ∈ Z, we adopt the following definitions for almost periodicity.
Definition 1 (see [11]).The set of sequences {   },    =  + −   ,  ∈ Z,  ∈ Z, {  } ∈ I, is said to be uniformly almost periodic if for arbitrary  > 0 there exists a relatively dense set of -almost periods common for any sequences.
Definition 2 (see [11]).The function  ∈ (R, R) is said to be almost periodic, if the following hold.
(1) The set of sequences (2) For any  > 0, there exists a real number  > 0 such that if the points   and   belong to one and the same interval of continuity of () and satisfy the inequality (3) For any  > 0, there exists a relatively dense set  such that if  ∈ , then |( + ) − ()| <  for all  ∈ R satisfying condition | −   | > ,  ∈ Z.The elements of  are called -almost periods.
Lemma 3 (see [11]).Let {  } ∈ I.Then, there exists a positive integer  such that, on each interval of length 1, one has no more than  elements of the sequence {  }; that is, where (, ) is the number of the points   in the interval (, ).
Theoretically, one can investigate the existence, uniqueness, and stability of almost periodic solution for functional differential equations by using Lyapunov functional as follows [11, P 109 ].
Consider the system of impulsive differential equations as follows: where Introduce the following conditions.

Permanence
In this section, we establish a permanence result for system (5).
Lemma 7. Assume that ,  > 0; then, the following impulsive logistic equation has a unique globally asymptotically stable positive almost periodic solution  * which can be expressed as follows: where  is defined as that in Lemma 3,  := ln sup ∈Z (1/(1 + ℎ  )),  := −,  : Proof.Let  = 1/; then, system (10) changes to We can easily obtain that system (13) has a unique almost periodic solution which can be expressed as follows: Then, system (10) has a unique almost periodic solution  * which can be expressed by (11).By Lemma 3, we have On the other hand, Suppose that () is another positive solution of system (10).Define a Lyapunov function as For  ̸ =   ,  ∈ Z + , calculating the upper right derivative of () along the solution of system (10), we have For  =   ,  ∈ Z + , we have
Proof.From the first equation of system (5), we have Consider the following auxiliary system: By Lemma 5,  1 () ≤  1 (), where  1 () is the solution of system (26) with  1 (0 + ) =  1 (0 + ).By Lemma 7, system (26) has a unique globally asymptotically stable positive almost periodic solution  * 1 which can be expressed as follows: where Then, for any constant  > 0, there exists For any  > 0, there exists  2 > 0 such that From the second equation of system (5), we have Similar to the above argument as that in (29), one has lim sup Then, there exists  3 >  2 such that By the third equation of system (5), we have Similar to the above argument as that in (32), we have lim sup This completes the proof.
Proof.According to Proposition 8, there exist  > 0 and  4 > 0 such that From the first equation of system (5), we have Consider the following auxiliary system: By Remark 15,  1 () ≥  1 () for  >  4 , where  1 () is the solution of system (39) with  1 ( + 4 ) =  1 ( + 4 ).By Lemma 7, system (39) has a unique globally asymptotically stable positive almost periodic solution  * 1 which can be expressed as follows: Similar to the above argument as that in (29), we have lim inf  → ∞  1 () ≥  1 ().By the arbitrariness of , it leads to lim inf Then, there exist 0 <  1 ≤  and  5 >  4 such that By the second equation of system (5), we have Similar to the above argument as that in (42), one has lim inf Then, there exist 0 <  2 ≤  1 and  6 >  5 such that In view of the third equation of system (5), we have Similar to the above argument as that in (45), one has lim inf This completes the proof.
Remark 10.In view of ( 1 ) in Proposition 9, the values of impulse coefficients ℎ  ( = 1, 2, 3) and the number of the impulse points   in each interval of length 1 have negative effect on the permanence of system (5).
By Propositions 8 and 9, we have the following theorem.
Remark 12.When ℎ  ( = 1, 2, 3) ≡ 0 in system (5), then Theorem 11 changes to the corresponding permanence result in Bai and Wang [15].So, Theorem 11 extends the corresponding result in Bai and Wang [15].Further, Theorem 11 gives the sufficient conditions for the permanence of system (5) with almost periodic impulsive perturbations.Therefore, Theorem 11 provides a possible method to study the permanence of the models with impulsive perturbations in biological populations.

Almost Periodic Solution
The main result of this paper is concerned with the existence of a unique uniformly asymptotically stable positive almost periodic solution for system (5). Let , ∀ ∈ R.
For  ̸ =   ,  ∈ Z + , calculating the upper right derivative of () along the solution of system (51), we have For  =   ,  ∈ Z + , we have
Remark 15.When ℎ  ( = 1, 2, 3) ≡ 0 in system (5), then Theorem 14 changes to the corresponding permanence result in Bai and Wang [15].So, Theorem 14 extends the corresponding result in Bai and Wang [15].Further, Theorem 14 gives the sufficient conditions for the uniform asymptotical stability of a unique positive almost periodic solution of system (5), in which  1 and  2 are allowed to be any realvalued positive number.Therefore, Theorem 14 provides a possible method to study the existence, uniqueness, and stability of positive almost periodic solution of the models with impulsive perturbations in biological populations.

An Example and Numerical Simulations
Example 1.Consider the following food chain system with impulsive perturbations: ).This completes the proof.

Conclusion
By using the comparison theorem and the Lyapunov method of the impulsive differential equations, sufficient conditions   are obtained which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of a food chain system with almost periodic impulsive perturbations.Proposition 9 and Theorem 14 imply that the values of impulse coefficients ℎ  ( = 1, 2, 3) and the number of the impulse points   in each interval of length 1 are harm for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the model.The main results obtained in this paper are completely new and the method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations.

Figure 4 :
Figure 4: Stability of state variable  1 of Example 1.