Global Asymptotic Stability of Pseudo Almost Periodic Solutions to a Lasota–wazewska Model with Distributed Delays

In this paper, we study a class of Lasota–Wazewska model with distributed delays, new criteria for the existence and global asymptotic stability of positive pseudo almost periodic solutions are established by using the fixed point method and the properties of pseudo almost periodic functions, together with constructing a suitable Lya-punov function. Finally, we present an example with simulations to support the theoretical results. The obtained results are essentially new and they extend previously known results.


Introduction
To describe the survival of red blood cells in an animal, Wa żewska-Czy żewska and Lasota in [20] proposed the following autonomous nonlinear delay differential equation as their appropriate model: where x(t) denotes the number of red blood cells at time t, a > 0 is the probability of death of a red blood cell, b and c are positive constants related to the production of red blood cells per unit time, and τ is the time required to produce a red blood cell.As a classical model of population dynamics, model (1.1) and its modifications have received great attention from both theoretical and mathematical biologists, and have been well studied.In particular, qualitative analysis such as periodicity, almost periodicity and stability of solutions of nonautonomous Lasota-Wazewska models have been studied extensively by many authors, we refer to [5,7,9,11,13,[15][16][17][18][19] and the references therein.
Since the nature is full of all kinds of tiny perturbations, either the periodicity assumption or the almost periodicity assumption is just approximation of some degree of the natural perturbations [21,25].A well-known extension of almost periodicity is the pseudo almost periodicity, which was introduced by C. Zhang in [24,25] and has been widely applied in the theory of ODEs and PDEs, see [2-4, 10, 14] and the references therein.In addition, it is well-known that time delays often occur in realistic biological systems, which can make the dynamic behaviors of the biological model become more complex, and may destabilize the stable equilibria and admit almost periodic oscillation, pseudo almost periodic motion, bifurcation and chaos, compared with the effects of discrete delays, distributed delays are more general and difficult to handle.Therefore, it is important and interesting to study the almost periodic dynamic behaviors of the Lasota-Wazewska model with distributed delays.
Motivated by the above discussions, in this paper, we will consider the following Lasota-Wazewska model with pseudo almost periodic coefficients and distributed delays: where is the probability kernel of the distributed delays, the other variables and parameters have the same biological meanings as those in (1.1) with the difference that they are now time-dependent.
The main purpose of this paper is employing fixed point method and the properties of pseudo almost periodic functions, together with constructing a suitable Lyapunov functional, to establish some sufficient conditions for the existence and global asymptotic stability of a pseudo almost periodic solution for model (1.2).The results obtained in the present paper are completely new and they extend previously known results in the literature.
The structure of this paper is as follows.In Section 2, we give some preliminaries related to our main results.In Section 3, we present the main results on the dynamic behaviors for model (1.2).Section 4 gives an example with simulations to demonstrate the effectiveness of the theoretical results.
Notations: Let BC(R, R) denote the set of bounded continuous functions from R to R, • denote the supremum norm g := sup t∈R |g(t)|, obviously, (BC(R, R), • ) is a Banach space.We generally denote Finally, given a function g ∈ BC(R, R), let g + and g − be defined as

Preliminaries
According to the biological interpretation of model (1.2),only positive solutions are meaningful and therefore admissible.Consequently, the following initial conditions are given by Let us recall some definitions and notations about almost periodicity and pseudo almost periodicity.For more details, we refer the reader to [6,22].Definition 2.1 (see [6]).Let f (t) ∈ BC(R, R).The function f (t) is said to be almost periodic on R if, for any ε > 0, the set T( f , ε) = ς : f (t + ς) − f (t) < ε, for all t ∈ R is relatively dense, i.e., for any ε > 0, it is possible to find a real number l = l(ε) > 0, for any interval with length l(ε), there exists a number ς = ς(ε) in this interval such that f (t + ς) − f (t) < ε, for all t ∈ R. We denote by AP(R, R) the set of all such functions.
In this paper, we denote Definition 2.2 (see [22]).A function f (t) ∈ BC(R, R) is called pseudo almost periodic if it can be expressed as where The collection of such functions will be denoted by PAP(R, R).
Remark 2.3.The functions f 1 and f 2 in Definition 2.2 are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function f .Moreover, the decomposition given in Definition 2.2 is unique.
Definition 2.6 (see [6, 23]).Let x ∈ R and Q(t) be a continuous function defined on R. The linear equation is said to admit an exponential dichotomy on R if there exist positive constants k i , α i , i = 1, 2, projection P and the fundamental solution X(t) of (2.2) satisfying Lemma 2.7 (see [23]).Assume that Q(t) is an almost periodic function and g(t) ∈ PAP(R, R).If the linear equation (2.2) admits an exponential dichotomy, then pseudo almost periodic equation has a unique pseudo almost periodic solution x(t), and Lemma 2.8 (see [6]).Let δ(t) be an almost periodic function on R and Then the linear equation admits an exponential dichotomy on R.
The following lemma is from [1] and will be employed in establishing the asymptotic stability of model (1.2).Lemma 2.9.Let l be a real number and f be a non-negative function defined on [l, ∞) such that f is integrable on [l, ∞) and is uniformly continuous on [l, ∞).Then lim t→∞ f (t) = 0.

Main results
In this section, the main results of this paper are stated as follows.For convenience, we divide this part into two subsections.

Existence of pseudo almost periodic solution
which properly contains the strip where and For the sake of convenience, we denote x(t) = x(t; t 0 , ϕ).Let [t 0 , T) ⊆ [t 0 , η(ϕ)) be an interval such that x(t) > 0, for all t ∈ [t 0 , T), In fact, if (3.2) does not hold, there exists t 1 ∈ (t 0 , T) such that which is a contradiction and implies that (3.2) holds.We next show that Otherwise, there exists t 2 ∈ (t 0 , η(ϕ)) such that In view of (1.2), (3.2) and (3.5), direct calculation produces which is a contradiction and hence (3.4) holds.According to (3.2) and (3.4), one easily see that (3.1) is true, which implies that x(t) is bounded.Therefore, we know from the continuation theorem in [8, Theorem 3.2 on page 46] that the existence interval of each solution for model (1.2) can be extended to [t 0 , ∞).
Lemma 3.2.Suppose assumptions (A 1 )-(A 3 ) are satisfied.Define the nonlinear operator Γ as follows, for each φ ∈ PAP(R, R), (Γφ)(t) := x φ (t), where Proof.Firstly, we claim that F(s) ∈ PAP(R, R).In fact, let φ ∈ PAP(R, R) and e −v is a uniformly continuous function for v ≥ 0, we know from Lemma 3.1 and [22, Corollary 5.4 on page 58] that g φ (v) e −c j (v+s)φ(v) ∈ PAP(R, R), and therefore, g φ can be expressed as We know from the almost periodicity of g 1 φ (v) that for any ε > 0, there exists a number l(ε) such that in any interval [α, α + l(ε)] one can find a number ς, with the property that Then, we have On the other hand, one sees that lim Combining (3.6) and (3.7), we derive that Then, by a standard argument as Lemma 3.2 in [4], we can prove that Γ maps PAP(R, R) into itself.
Our first main result can be stated as follows.
Theorem 3.3.In addition to (A 1 )-(A 3 ), suppose further that Then the model (1.2) admits a unique pseudo almost periodic solution in the region Proof.For any φ ∈ PAP(R, R), we introduce the following auxiliary equation Notice that M[a] > 0, we know from Lemma 2.8 that the linear equation admits an exponential dichotomy on R. Therefore, by Lemmas 2.5 and 2.7, we know that model (1.2) has exactly one solution expressed by one can observe from Lemma 3.2 that x φ (t) ∈ PAP(R, R).
Obviously, to show that model (1.2) has a unique pseudo almost periodic solution, it suffices to prove that Γ has a fixed point in B.
Let us first prove that the operator Γ is a self-mapping from B to B. In fact, for any φ ∈ B, we have

Z. Long
On the other hand, we have which, together with (3.11), means that the mapping Γ is a self-mapping form B to B.
Next, we show that the mapping Γ is a contraction mapping on B. For any φ, φ * ∈ B, one has It follows from (3.12) and (3.13) that 14) shows that Γ is a contraction mapping.Therefore, by virtue of the Banach fixed point theorem, Γ has a unique fixed point which corresponds to the solution of model (1.2) in B ⊂ PAP(R, R).This completes the proof of Theorem 3.3.

Asymptotic stability of pseudo almost periodic solution
In the following, we give the analysis of global asymptotic stability of model (1.2).Proof.Let x(t) be any solution of model (1.2) and x * (t) be a pseudo almost periodic solution of model (1.2), consider a Lyapunov function defined by A direct calculation of the right derivative D + V(t) of V(t) along the solutions of model (1.2), produces It follows from (3.8) and (3.15) that there exists a positive constant µ 1 > 0 such that Integrating on both sides of (3.16) from t 0 to t yields From Lemma 3.1, we can obtain that x(t), x * (t) and their derivatives remain bounded on [t 0 , ∞) (from the equation satisfied by them).Then it follows that |x(t) − x * (t)| is uniformly continuous on [t 0 , ∞).By Lemma 2.9, we conclude that The proof of Theorem 3.4 is complete.Remark 3.5.Very recently, J. Shao in [16] studied the following Lasota-Wazewska model with an oscillating death rate where a : R → R is an almost periodic function which is controlled by a non-negative function, b j , c j , τ j : R → [0, +∞) are pseudo almost periodic functions, if there exist a bounded and continuous function a * : R → [0, ∞) and constants F i , F S , κ such that the author proved that equation (3.17) has a pseudo almost periodic solution which is globally exponentially stable.
One can find that the function of death rate a(t) in equation (3.17) is more general than equation (1.2).However, if K j (s) = δ(s − τ j ), where δ(s) denotes the Dirac-δ function, then equation (1.2) becomes equation (3.17) with constant discrete delays, on the other hand, as pointed out by L. Duan et al. in [5] and Y. Kuang in [12], it is more reasonable and realistic to establish delay-dependent criteria ensuring the dynamics of a system because the delays have important effect on a system, one can clearly see that the stability criteria established here are delay-dependent and the method used here is different from [16].This indicates that these results are complementary to each other.Therefore, our results are new and complement the existing ones.Moreover, it seems that condition (3.8) is easier to verify than (3.18)-(3.19).[5,7,9,13,15,17,19], in particular, one can easily see that the obtained results extend the corresponding ones in [18].To the best of our knowledge, on the other hand, fewer authors have considered the existence and stability of pseudo almost periodic solutions to model (1.2).Therefore, the main results in the present paper are essentially new and they extend previously known results.

Theorem 3 . 4 .
If all the assumptions in Theorem 3.3 are satisfied, then, all solutions of model (1.2) in the region B converge to its unique pseudo almost periodic solution.