On positive solutions and the Omega limit set for a class of delay differential equations

This paper studies the positive solutions of a class of delay differential equations with two delays. These equations originate from the modeling of hematopoietic cell populations. We give a sufficient condition on the initial function for $t\leq 0$ such that the solution is positive for all time $t>0$. The condition is"optimal". We also discuss the long time behavior of these positive solutions through a dynamical system on the space of continuous functions. We give a characteristic description of the $\omega$ limit set of this dynamical system, which can provide informations about the long time behavior of positive solutions of the delay differential equation.


Introduction
Delay differential equations are extensively used in modeling biological control systems, where the retardation usually originates from a maturation processes or finite signaling velocities [2,8,9,11]. In this paper, we will consider the delay differential equation of form where τ > τ 0 > 0, f ∈ C(R + , R + ) and φ (t) > 0. This type of equation has been used to describe the dynamics of circulating blood cells [10]. For obvious biological reasons, we are only interested in positive solutions of the equation. In this paper, we consider conditions on the initial function φ (t) such that the equation (1) has positive solutions for all t > 0, and the long time behavior of these positive solutions.
The present work was motivated by investigating the dynamics of a mathematical model of hematopoiesis [4,5,6]. This model consists of a set of four nonlinear delay differential equations, describing the dynamics of proliferation and differentiation of hematopoietic stem cells, production of the three major types of circulating cells, leukocytes (white blood cells), erythrocytes (red blood cells) and platelets, and the feedback regulation of proliferation and differentiation [4,5]. These delay differential equations are obtained from age-structured models of each cell line using the method of characteristic line. For this set of equations, only positive solutions are biologically possible. In particular, the long time behavior of this system can be described by the ω-limit set of these solutions, which is important for understanding the possible states of a system under given initial conditions.
Bifurcations and bistability of the model of hematopoietic regulation have been studied numerically in [3,6,10]. However, in numerical simulations we found that, in some cases, solutions of the delay differential equation system with a positive initial function can become negative at some time t > 0. This indicates that the delay differential equation model is not equivalent to the original age-structured model, which always yield positive solutions. The negative populations usually occur for erythrocytes and platelets, but not for stem cells and leukocytes. This is because the circulating erythrocytes and platelets are actively destroyed at a fixed time from their entering the circulating component [1,12]. In out recent study [10], we proposed a set of initial functions such that the solutions are always positive. The current study provides a theoretical foundation for the findings in [10].
Equation (1) can be obtained from a model of the population dynamics of erythrocytes or platelets [10]. In this model u(t) measures the population of circulating cells. The cell is produced from the differentiation of stem cells and amplification of precursor cells at a rate f (u(t)). After differentiation, the cells undergo a stage of maturation of duration τ 0 , and then enter into the circulation. The circulating cells are lost at a rate γ, and are actively destroyed at a fixed time (τ − τ 0 ) from their time of entry the circulating compartment.
This paper is organized as follows. In section 2, we prove a sufficient condition for the existence of positive solutions of (1). At the end of this section, we will show that the condition is "optimal" by an example. In section 3, we will introduce an iterated map on the space of continuous functions and study the ω-limit set of positive solutions of (1) through this map. The paper concludes with an example in section 4.

Sufficient condition for positive solutions
In this section, we develop a sufficient condition for the initial function φ (t) such that the solution of equation (1) is positive for all time t > 0.
Throughout this paper, we define an operator H : and let H 0 : C([−τ, +∞), R + ) → R be defined as Theorem 1 Consider the equation (1). If the initial condition φ (t) satisfies then (1) has a unique solution u(t) in t ∈ [−τ, ∞), and the solution is positive for all t > 0.
Proof First, the existence and uniqueness of the solution is straightforward using the method of steps Next, we will show that the solution u(t) of equation (1) can be given iteratively through From (1), it is easy to obtain the following when t ≥ 0, Thus, we have Here, the initial condition u(t) = φ (t) has been applied in the last equality. Now, we have derived (5), from which the solution is always positive for all t > 0 provide H 0 (φ ) > 0. Thus the theorem is proved.
⊓ ⊔ Remark 1 Theorem 1 can be extended to the case of time-dependent function f = f (u,t).
In this case, equation (1) becomes Accordingly, the operator H is redefined as and H 0 (u) = H(u, 0). Similar to the proof of Theorem 1, we can show that the solution of (7) is given by Thus, if H 0 (φ ) ≥ 0, the solution of (7) is positive for all t > 0.

Remark 2
The condition (4) is not necessary for a given function f . However, we argue that this condition is 'optimal' for general f , i.e. for any ε > 0, there exist a pair of functions, f and φ , such that H 0 (φ ) < −ε, and the solution of (1) with initial condition φ (t) becomes 0 at some t * > 0. A construction of f and φ is given in Proposition 1 below.

Proposition 1 For any
where δ > 0 and Let u δ (t) be the solution of Then when δ is small enough, the solution satisfies u δ (τ) < 0.
Proof When δ → 0, the function f δ (u) converges to the following function Let u 0 (t) be the solution of (11) with f δ replaced by f 0 . We will divide the proof into two steps. First we will prove that for all t > 0. Next, we will show that u 0 (τ) < 0. Therefore, we have u δ (τ) < 0 when δ is sufficiently small. To prove (13), we first note that the proof of Theorem 1 is also valid when f (u) is replaced by f 0 (u). Thus, u 0 (t) also satisfies the iterative equation (5). Therefore, for any t > 0, we have In the last step, we note that | f δ (u δ (t − a)) − f 0 (u 0 (t − a))| ≤ 2A, and therefore the equality holds according to the Lebesgue dominated convergence theorem. Now, we will prove (13) inductively. When Now, we assume that (13) is valid for 0 ≤ t ≤ nτ 0 . When nτ 0 < t ≤ (n + 1)τ 0 and τ 0 ≤ a ≤ τ, we have (t − a) ≤ nτ 0 , and therefore and (13) holds for 0 ≤ t ≤ (n + 1)τ 0 . Therefore, we have (13) for all t > 0.
To prove that u 0 (τ) < 0, we will show that In fact, we will prove a stronger result. Let where [τ/τ 0 ] denotes the integer part of τ/τ 0 . We claim that in each interval I n , there is at most one point t n ∈ I n such that u 0 (t n ) = φ 0 . First, we note that (10) is equivalent to From (5), when 0 ≤ t ≤ τ 0 , Thus, u(t) = φ 0 for any t ∈ I 0 , and hence, Now let n ≤ [τ/τ 0 ] + 1, and assume that for any integer k < n, there is at most one point t k ∈ I k such that u 0 (t k ) = φ 0 . For t ∈ I n and t ≤ τ, we have We note that when t ∈ I n and τ 0 ≤ a ≤ τ, 0 ≤ t − a ≤ nτ 0 . From our assumption, there are at most n points in t ∈ [0, nτ 0 ] such that u 0 (t) = φ 0 . Therefore, the first integral Hence, and u 0 (t) is strictly monotonically decreasing for t ∈ I n . Therefore, there is at most one point t n ∈ I n , t ≤ τ such that u(t n ) = φ 0 , and the claim is proved.
and the proposition is proved.
⊓ ⊔ Figure 1 shows the solution of (11) when δ is sufficiently small.

ω-limit set of positive solutions
In this section, we always assume that f is bounded, and discuss the ω-limit set of positive solutions. It is easy to see that any solution u(t) of (1) is associated with a sequence of functions {u n (t)} n∈N in C([−τ, 0], R + ) such that u n (t) = u(t + nτ). From (5), we define a map T : , 0], R + ) such that u n = Tu n−1 . Explicitly, T is given by where as defined in (3). Therefore, T defines a dynamical system in the function space C([−τ, 0], R + ).
For any u 0 ∈ C([−τ, 0], R + ), the ω-limit set of u 0 under the map T is defined as The structure of the ω-limit set ω T (u 0 ) provides characteristic descriptions of the long time behavior of the solution of (1) with a given initial function u(t) = u 0 (t) when −τ ≤ t ≤ 0. According to the previous discussions, we define two subsets of C([−τ, 0], R + ) as follows.
Further, we equip D and D 0 with the norm: It is obvious that D 0 ⊆ D. To prove Theorem 2, we need several lemmas. In particularly, H 0 (u) is continuous.
Proof From the definition of T in (16), we have for any u ∈ D, Hence,

Lemma 3
For any u 0 (t) ∈ D and the sequence {T n u 0 } n∈N , let the function u(t) on [0, +∞) be defined by Then u(t) is the solution of (1) with initial function u(t) = u 0 (t) when −τ ≤ t ≤ 0.

Lemma 4 For any u ∈ D, (Tu)(t) is continuously differentiable.
Proof For any u ∈ D, it is easy to see that (Tu)(t) is continuous in t ∈ [−τ, 0]. From the proof of Lemma 3, we have Therefore, Tu is continuously differentiable. ⊓ ⊔ Now we are ready to prove Theorem 2.
(2) To prove that T is completely continuous, we only need to prove that for any bounded sequence {u n } n≥1 ⊆ D, {Tu n } n≥1 has a convergent subsequence. To this end, from the Arzelà-Ascoli theorem, we only need to show that the set of continuous functions {Tu n } n≥1 is uniformly bounded and equicontinuous.
Assume that u n ≤ M < ∞ for any n ≥ 1. From the fact that f is bounded, i.e. the inequality (15), and the definition of T in (16), we have

Thus,
which implies that {Tu n } n≥1 is uniformly bounded.
The sequence {Tu n } is equicontinuous if the sequence d dt (Tu n ) is uniformly bounded in t ∈ [−τ, 0], which is proved below. From (21) in Lemma 4, it is easy to have that is bounded. Thus, {Tu n } is uniformly bounded and equicontinuous, and therefore T is completely continuous.
(3) From (16), for any u ∈ D, we have which implies that {T n u} n∈N is uniformly bounded.
(4) Since the sequence {T n u} n∈N is bounded in C([−τ, 0], R + ) and T is completely continuous, there exists a subsequence n k → +∞ such that T n k u converges to a function in C([−τ, 0], R + ). Hence for any u ∈ D, ω T (u) is nonempty.
(5) For any v ∈ ω T (u), there exists a sequence n k → +∞ such that v = lim k→+∞ T n k u. From Lemma 1, H 0 (u) is continuous with respect to u. Thus, we have and hence v ∈ D 0 . Therefore ω T (u) ⊆ D 0 . ⊓ ⊔ From Theorem 2, informations about the long time behavior of positive solutions of (1) are contained in D 0 . In particular, all solutions of (1) with initial function in D 0 will be of great interest. In rest of this section, we will focus our discussion on D 0 .
From Lemma 2, we immediately have the following corollary.

Proposition 2 Assume that u(t) is a solution of (1), then
In particular, if there is t * > 0 such that then (25) is satisfied for any t ∈ R + .
Proof From (5), we have

Thus we have
The second part is obvious when we take t 2 = t * in (24). ⊓ ⊔

Remark 4 In
We note that H 0 (u) ≥ 0 for any u ∈ D. Thus, H(u,t) is a 'Lyapunov functional' of the equation (1) on D. Furthermore, for any solution of (1) with initial function φ ∈ D, the solution approaches D 0 on a time scale 1/γ.
In the proof of Theorem 2, we have shown that D 0 is nonempty. The simplest case is for a constant function u(t) ≡ u 0 . Then u 0 should satisfy the equation i.e., u ≡ u 0 is a steady state of the equation It is not trivial to find a non-constant function in D 0 , i.e. a function u(t) such that Proposition 3 gives an explicit way to construct a function in D 0 , which is important when we solve equation (1) numerically and perform a bifurcation analysis. Let g(t) be a function in C([0, τ], R + ). We define an operator F g : C([0, τ], R + ) → C([−τ, 0], R) as follows.

Proposition 3
For any g ∈ C([0, τ], R + ), let u(t) (−τ ≤ t ≤ 0) be the solution of the differential equation Then u(t) ∈ D 0 . If further, g satisfies then, u(t) is continuously differentiable, and Proof We only prove this for the case τ 1 ≥ τ 0 , the τ 1 < τ 0 case is similar. We just need to solve the equation (31) directly and verify (28). Using variation of parameters, we have Letting t = 0, we have Thus, u ∈ D 0 and the first part is proved.
To prove the continuity of du dt , we only need to show that du dt is continuous at t = −τ 1 . From (32), Thus, du dt is continuous at t = −τ 1 from (31) and (29). To verify (33), we note that The theorem has been proved.
⊓ ⊔ Biologically, the function g(a) in Proposition 3 corresponds to the initial age distribution of cells differentiation from stem cells. From Propostion 3, if we choose a function g(a) that satisfies (32), then the solution of (31) can serve as an initial function of the equation (1), and the solution of (1) therewith is positive for all t > 0, differentiable in t ∈ (−τ, +∞) and satisfies H(u,t) = 0 for all t > 0.

Examples
In this section, we will show numerical results for two examples, with and f (u) = 2 π arctan(α sin(2πu)) + 1, respectively. The ω-limit set is illustrated at figure 4 in which N(ω T ) is a map from R to subsets of R defined as In the simulation, we choose u ∈ D 0 randomly according to Theorem 3 and generate a sequence {T n u} with n large enough (we calculate to n = 600) till the sequence converges. We take the last 100 functions from {T n ω} as the ω-limit set ω T (u). For each v ∈ ω T (u), we calculate the L 1 norm v L 1 = 0 −τ v(t)dt and then give one point in Figure 4. Figure 4 can be thought of as a 'bifurcation diagram' of the delay differential equation (1). When N(ω T ) has only one point, it means that the system has a globally stable steady state. When N(ω T ) contains a finite number of points, then either the system has multiple steady states, or there is a periodic solution, with period qτ for some q a rational number. When N(ω T ) contains an infinite number of points, then the system either has a periodic solution with period qτ and q is an irrational number, or chaotic solutions. Chaotic solutions have been found in many simple delay differential equations [7,11,13], including the well known Mackey-Glass equations, which correspond to the case with τ = +∞ in the present study. The approach in this study provide a way to study complex behaviors of such systems from the point of view of dynamical systems in a function space.