CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 18, Number 4, Winter 2010
ASYMPTOTIC STABILITY OF DELAY DIFFERENTIAL EQUATIONS VIA FIXED POINT THEORY AND APPLICATIONS MENG FAN, ZHINAN XIA AND HUAIPING ZHU
ABSTRACT. Nonlinear delay differential equations have been widely used to study the dynamics in biology, but the stability of such equations are challenging. In this paper, new criteria are established for the asymptotic stability of some nonlinear delay differential equations with finite delay via fixed point theory. The techniques involve both the contractive mapping principle and Schauder’s fixed point theorem. The criteria are then applied to obtain the stability of some well known models in physiology and ecology.
1 Introduction Stability plays an important role in the theory of dynamical systems. Since the pioneer work of Liapunov more than 100 years ago, Liapunov’s direct method has been the primary tool to deal with stability problems in various type of dynamical model equations [11, 14, 22]. However, the construction of appropriate Liapunov functions or functionals are technical, empirical and are not universally applicable. Criteria deduced from the direct method usually requires point-wise conditions, while many of the real-world dynamical models call for averages; calculations involved in the direct method are very technical and sophisticated (see [7, 11, 12, 14] and references therein). New methods and techniques are needed to address those difficulties. Recently, in addition to the application in the existence problem, the fixed point theory has been proven to be a powerful tool for dealing with the stability of functional differential equations. Burton [2, 3, 5] The work of the first and the second authors was supported by NSFC, NCET08-07550 and FRFCU. The work of the third author was supported by NSERC of Canada. Keywords: Asymptotic stability, delay differential equations, contraction mapping principle, Schauder’s fixed point theorem, oscillation, hematopoiesis model, Glass and Mackey model, Nicholson’s Blowflies Model. c Copyright Applied Mathematics Institute, University of Alberta.
361
362
M. FAN, Z. XIA AND H. ZHU
was among the first who study the stability using fixed point theory. When applicable, by the fixed point theory, one can try to avoid certain difficulties incurred in applying the Liapunov method but usually achieve conditions of average type [5]. For the comparison of these two approaches, we refer to [4, 5] and the references cited therein. Though there have been studies of the stability of delay differential equations using the fixed point theory, most of the work were restricted to the equations of linear or semi-linear type. Delay differential equations have been used widely to model the dynamics in biology and many other fields. In studying the dynamics of red blood cells and the linearized oscillation in population dynamics, the authors [1, 6, 9, 13, 15, 20] proposed and studied the model equation of the form (1.1)
x0 (t) = −γ(t)x(t) + α(t)e−β(t)x(t−τ (t)) .
The results obtained for this equation are still limited to certain autonomous versions and some special cases. Oscillation is a common phenomena in (1.1), but the stability of the oscillations in equation (1.1) were untouched (see also [21]). In this paper, we will consider the stability of a more general type of delay differential equations of the form (1.2)
x0 (t) = −a(t, xt )x(t) + f (t, xt ),
and (1.3)
x0 (t) = −g(t, x(t)) + f (t, xt ),
where g(t, x(t)) ∈ C 1 (R×R, R), a(t, xt ), f (t, xt ) ∈ C(R+ ×C, R), xt (θ) = x(t + θ) for any θ ∈ [−r, 0], r > 0. Here we use C = C([−r, 0], R) to denote the space of continuous function from [−r, 0] to R with the supremum norm. The equations (1.2) and (1.3) have been studied extensively, especially the existence of periodic solutions (see [6, 19, 21, 23]). Due to the technical difficulties, the stability of the periodic solutions were known, yet it becomes essential to understand the stability of the oscillations in the applications in biology. In this paper, we will study the asymptotic stability of the zero solution, therefore also the periodic solutions of (1.2) and (1.3) by applying
DELAY DIFFERENTIAL EQUATIONS
363
the fixed point theory. By using two different approaches: the contractive mapping principle (Section 2) and the Schauder’s fixed point theorem (Section 3), the sufficient, necessary, and sufficient and necessary criteria for the asymptotic stability will be established. We will then apply the main results in Section 4 to study the stability of the periodic solutions in the model (1.1) for the dynamics of red blood cells. Numerical simulations will be given to support our conclusions. 2 The Contractive mapping principle approach We start with delay differential equation (1.2). First, we make the following assumptions. (H1 ) There exist continuous functions a1 (t), a2 (t) ∈ C(R, R) such that a1 (t) ≤ a(t, xt ) ≤ a2 (t). (H2 ) Assume that f (t, 0) = 0 and there exists a positive constant L and a continuous function b1 (t) ∈ C(R, R+ ) such that |f (t, φ) − f (t, ψ)| ≤ b1 (t)|φ − ψ| for all φ, ψ ∈ CL := {x ∈ C : |x| ≤ L}. Theorem 2.1. Suppose that (H1 ) and (H2 ) hold and if there exists a constant α ∈ (0, 1) such that (H3 )
Zt
e−
Rt s
a1 (u) du
b1 (s) ds ≤ α, t ≥ 0;
0
(H4 )
lim
t→+∞
Zt
a1 (s) ds → +∞,
0
then the zero solution of (1.2) is asymptotically stable. Proof. For any t0 ≥ 0, one can find a constant δ0 ≥ 0 such that δ0 ≤ L and δ0 K1 + αL ≤ L, where K1 := sup {e
−
Rt
t0
a1 (s) ds
}.
t≥t0
For any ϕ ∈ C, it follows from (H2 ) that there is a unique solution of (1.2) x(t) = x(t, t0 , ϕ), xt0 = ϕ
364
M. FAN, Z. XIA AND H. ZHU
defined on [t0 , +∞). Now, we show that the zero solution of (1.2) is attractive, i.e., lim x(t, t0 , ψ) = 0, ∀ψ ∈ Cδ0 := {x ∈ C : |x| ≤ δ0 }.
t→+∞
For any given ψ ∈ Cδ0 , let u(t) = u(t, t0 , ψ) be the unique solution of (1.2) through (t0 , ψ). Consider the initial value problem: x0 (t) = −a(t, ut )x(t) + f (t, xt ),
(2.4)
xt0 = ψ.
Then the unique solution of (2.4) can be written as: x(t) = ψ(0)e
−
Rt
t0
a(s,us ) ds
+
Z
t
e−
Rt s
a(τ,uτ ) dτ
f (s, xs ) ds.
t0
Let Ω = {x ∈ C([t0 −r, +∞), R) : xt0 = ψ, |xt | ≤ L, t ≥ t0 ,
lim x(t) = 0}
t→+∞
and define the mapping P : Ω → C([t0 − r, +∞), R) by ψ(t − t0 ), t0 − r ≤ t ≤ t 0 Rt a(s,us ) ds − (P x)(t) = ψ(0)eZ t0 t Rt e− s a(τ,uτ ) dτ f (s, xs ) ds, t ≥ t0 . + t0
Then Ω is a complete metric space with the metric ρ(x, y) = sup |x(t) − y(t)|. t≥t0 −r
Next, we show that P maps Ω into Ω. It is clear that for x ∈ Ω, P x is continuous, (P x)(t) = ψ(t − t0 ) for t ∈ [t0 − r, t0 ], and |(P x)(t)| ≤ |ψ(0)|e
−
Rt
t0
a1 (s) ds
+
Z
t
e− t0
≤ δ0 K1 + α|xs | ≤ δ0 K1 + αL ≤ L,
t ≥ t0 .
Rt s
a1 (τ ) dτ
b1 (s)|xs | ds
DELAY DIFFERENTIAL EQUATIONS
365
Thus, (P x)t ∈ CL for t ≥ t0 . We now show that (P x)(t) → 0 as t → +∞. By (H4 ), we have lim e
−
Rt
t0
a1 (s) ds
t→+∞
= 0;
so the first term on the right-hand side of (P x)(t) tends to zero. Next, we will show that the second term on the right side of (P x)(t) tends to zero too. The fact x ∈ Ω implies that |x(t)| ≤ L for all t ≥ t0 and for any given ε > 0, there exists a t1 > t0 such that |x(t)| < ε for all t ≥ t1 . It follows from (H4 ) that there exists a t2 > t1 such that e
−
Rt
a1 (s) ds
t1 +r
t2 .
Whence, for t > t2 , we have Z t R − st a(τ,uτ ) dτ e f (s, x ) ds s t0
≤
Z
t1 +r
≤
Z
s
a(τ,uτ ) dτ
t
e−
|f (s, xs )| ds
Rt
a(τ,uτ ) dτ
s
|f (s, xs )| ds
t1 +r
t1 +r
e−
Rt s
a1 (τ ) dτ
b1 (s)|xs | ds
t0
Z
+ ≤e
Rt
t0
+ Z
e−
−
t
e−
t1 +r
≤ Le
Z
−
s
a1 (τ ) dτ
b1 (s)|xs | ds
t1 +r
Rt
+
Rt
a1 (τ ) dτ
t
e−
Rt s
Z
t1 +r
e−
Rt
1 +r
s
a1 (τ ) dτ
b1 (s) |xs | ds
t0
a1 (τ ) dτ
b1 (s)|xs | ds
t1 +r
Rt
t1 +r
+ε
Z
a1 (τ ) dτ
t
e− t1 +r
Z
Rt s
t1 +r
e−
Rt s
1 +r
a1 (τ ) dτ
t0
a1 (τ ) dτ
b1 (s) ds
b1 (s) ds
366
M. FAN, Z. XIA AND H. ZHU
≤ αL e
−
Rt
t1 +r
a1 (τ ) dτ
+αε
< ε + αε = (1 + α)ε. So limt→+∞ (P x)(t) = 0 and hence (P x) ∈ Ω. Now, we are at the position to show that (P x)(t) is a contraction mapping. For any x, y ∈ Ω, |(P x)(t) − (P y)(t)| ≤ ≤ ≤
Z Z Z
t
e−
Rt
a(τ,uτ ) dτ
e−
Rt s
a1 (τ ) dτ
|f (s, xs ) − f (s, ys )| ds
e−
Rt
a1 (τ ) dτ
b1 (s) |xs − ys | ds ≤ αρ(x, y).
s
|f (s, xs ) − f (s, ys )| ds
t0 t t0 t
s
t0
By the Contraction Mapping Principle, P has a unique fixed point in Ω, which is the unique solution of (2.4) and tends to zero as t tends to infinity. Obviously, the unique solution of (2.4) is u(t). Therefore, lim u(t, t0 , ψ) = 0, ∀ψ ∈ Cδ0 .
t→+∞
To obtain the asymptotic stability, we need to show the zero solution of (1.2) is stable. For any given ε > 0, one can choose δ > 0 such that δ < (1−α)ε K1 , then we have |u(t)| ≤ |ψ(0)|e
−
Rt
t0
a1 (s) ds
+
Z
t
e−
Rt s
a1 (τ ) dτ
b1 (s)|us | ds
t0
≤ K1 |ψ| + α|us | ≤ K1 |ψ| + αkuk,
t ≥ t0 ,
where kuk = sup |u(t)|, then |ψ| < δ implies that t≥t0 −r
kuk ≤
δK1 < ε, 1−α
t ≥ t0 .
Hence, we conclude that the zero solution of (1.2) is asymptotically stable. Theorem 2.2. Suppose that the conditions (H1 ), (H2 ) and (H3 ) hold for (1.2) and also
DELAY DIFFERENTIAL EQUATIONS
(H5 ) lim inf t→+∞
Zt
367
a1 (s) ds > −∞.
0
If the zero solution of (1.2) is asymptotically stable, then we have Rt (H6 ) lim 0 a2 (s) ds = +∞. t→+∞
Proof. We will prove this theorem by contradiction argument. Suppose Rt that (H6 ) fails. Then (H5 ) implies lim inf 0 a2 (s) ds > −∞, and there t→+∞
exists a sequence {tn }, tn → +∞ as n → +∞ such that lim
n→+∞
Z
tn
a2 (s) ds = l, 0
for some l ∈ R. We can choose a positive constant q such that −q ≤
Z
tn
a2 (s) ds ≤ q,
n = 1, 2, · · · .
0
Since (H6 ) fails, so (H4 ) fails too. By (H5 ), for the sequence {tn } defined above, one can choose a positive constant R satisfying −R ≤
Z
tn
a1 (s) ds ≤ R, 0
this yields Z
tn
e
Rs 0
a1 (u) du
b1 (s)ds ≤ αe
R tn 0
a1 (u) du
< eR .
0
R t Rs So the sequence { 0 n e 0 a1 (u) du b1 (s)ds} is bounded, hence there exists a convergent For brevity in notation, we still assume the Rs R t subsequence. sequence { 0 n e 0 a1 (u) du b1 (s)ds} is convergent, then we can choose a positive integer k so large that Z for all n ≥ k.
tn
e tk
Rs 0
a1 (u) du
b1 (s)ds ≤
1−α , 2K12 e2q
368
M. FAN, Z. XIA AND H. ZHU
Now we consider the solution x(t) = x(t, tk , ψ) with ψ(s − tk ) ≡ δ0 for s ∈ [tk − r, tk ], and |x(t)| ≤ L for all t ≥ tk , then |x(t)| ≤ δ0 e
−
Rt
tk
a1 (s) ds
+
Z
t
e−
Rt s
a1 (τ ) dτ
b1 (s)|xs | ds
tk
≤ δ0 K1 + α|xt |. 0 K1 Thus, we have |xt | ≤ δ1−α for all t ≥ tk . On the other hand, for n large enough, we also have
x(tn ) = x(tk )e
−
R tn tk
a(s,us ) ds
+
Z
tn
e−
R tn s
a(τ,uτ ) dτ
f (s, xs ) ds,
tk
so |x(tn )| ≥ δ0 e ≥ δ0 e ≥ δ0 e ≥ δ0 e ≥ δ0 e
−
−
−
−
−
R tn tk
R tn tk
R tn tk
R tn tk
R tn tk
a2 (s) ds
a2 (s) ds
a2 (s) ds
a2 (s) ds
a2 (s) ds
Z − −
− − −
Z
Z Z
tn
e
−
R tn s
a(τ,uτ ) dτ
tk tn
f (s, xs ) ds
e−
R tn s
a(τ,uτ ) dτ
|f (s, xs )| ds
e−
R tn
a(τ,uτ ) dτ
b1 (s)|xs | ds
e−
R tn
a1 (τ ) dτ
tk tn
s
tk tn
s
b1 (s)|xs | ds
tk
δ0 K1 − R tn a1 (τ ) dτ e 0 1−α
Z
tn
e
Rs 0
a1 (τ ) dτ
tk
But e
−
R tn tk
a2 (s) ds
=e
R0
tn
= e− and e−
R tn 0
a1 (s) ds
a2 (s) ds
R tn 0
·e
a2 (s) ds
R tk 0
·e
a2 (s) ds
R tk 0
a2 (s) ds
≥ e−2q ,
≤ K1 , so
|x(tn )| ≥ δ0 e−2q −
δ 0 K1 (1 − α) 1 K1 = δ0 e−2q . 1−α 2K12 e2q 2
This implies limt→+∞ x(t) 6= 0. The proof is complete.
b1 (s) ds.
DELAY DIFFERENTIAL EQUATIONS
369
Corollary 2.1. If a(t, xt ) ≡ a(t), and (H2 ), (H3 ) and (H5 ) hold, then the zero solution of (1.2) is asymptotically stable if and only if lim
t→+∞
Zt
a(s) ds = +∞.
0
Now we consider the delay differential equations of the form (1.3). Similar to the equation (1.2), if we assume that (H7 ) g(t, 0) = 0 and there exist continuous functions a1 , a2 ∈ C(R, R) such that ∂g(t, x) a1 (t) ≤ ≤ a2 (t), ∂x then immediately we have Theorem 2.3. Suppose that (H2 )–(H4 ) and (H7 ) hold, then the zero solution of (1.3) is asymptotically stable. Proof. For any g ∈ C 1 , since g(t, 0) = 0, it is easy to see that g(t, x) =
Z1
∂g(t, sx) ds x. ∂x
0
R1
sx) Then if we let a(t, xt ) = 0 ∂g(t, ds, then we can rewrite (1.3) as ∂x (1.2) with a1 (t) ≤ a(t, xt ) ≤ a2 (t). Then the claim is true following Theorem 2.1.
For equation (1.3), we also have Theorem 2.4. Suppose that (H2 ), (H3 ), (H5 ) and (H7 ) hold. If the zero solution of (1.3) is asymptotically stable, then (H6 ) holds. 3 Schauder fixed point theorem approach The assumption (H2 ) plays a key role when using the the Contraction Mapping Principle. In this section, we will relax the condition (H2 ) to study the stability via a different approach, i.e., using the Schauder fixed point theorem. In applications, the Contraction Mapping Principle is different from the Schauder’s fixed point theorem in that the latter requires compactness. Assume that the solutions of (1.2) and (1.3) exist uniquely and there exists a function b2 (t) ∈ C(R, R) such that
370
M. FAN, Z. XIA AND H. ZHU
(H8 ) |f (t, φ)| ≤ |b2 (t)||φ|. Zt (H9 ) lim (a1 (s) − |b2 (s)|) ds = +∞. t→+∞
0
Corresponding to equation (1.2), we consider the scalar linear equation: q 0 (t) = [−a1 (t) + |b2 (t)|] q(t).
(3.5)
For any t0 ≥ 0, it is not difficult to show that the unique solution of (3.5) satisfying q(t0 ) = β > 0 can be written as (3.6)
q(t) = βe
Rt
t0
(|b2 (s)|−a1 (s)) ds
,
t ≥ t0 .
It follows from (H9 ) that we can see that q(t) → 0 as t → +∞ and 0 < q(t) ≤ βeσ := γ, where σ = sup
Z
t ≥ t0 ,
t
(|b2 (s)| − a1 (s)) ds : t ≥ t0 t0
< +∞.
It is then trivial to show that the zero solution of (3.5) is stable. Definition 3.1. [5] (Equi-convergence). The set S of real-value functions on [−r, +∞) is said to be equi-convergent to 0 if there is a function q : R+ → R+ such that q(t) → 0 as t → 0, and that for ϕ ∈ S and t ∈ R+ , |ϕ(t)| ≤ q(t) holds. Here we state a Lemma regarding the equi-convergence, the proof and details can be found in the book of Burton [5]. Lemma 3.1. If the set {ϕk (t)} of real-value functions on [−r, +∞) is equi-convergent to 0 with respect to a function q : R+ → R+ , and is equi-continuous, then the sequence {ϕk (t)} contains a subsequence that converges uniformly on R+ to a continuous function ϕ(t) with |ϕ(t)| ≤ q(t) on R+ . Theorem 3.1. For equation (1.2), if (H1 ), (H8 ) and (H9 ) hold, then the zero solution is asymptotically stable.
DELAY DIFFERENTIAL EQUATIONS
371
Proof. Since the zero solution of (3.5) is stable, for any ε ∈ (0, γ] and t0 ∈ R+ , there exists a η = η(ε, t0 ) ∈ (0, β], such that for any q0 with |q0 | ≤ η, one has |q(t, t0 , q0 )| < ε for all t ≥ t0 . For any ψ ∈ Cη , define n S = u ∈ C([t0 − r, +∞), R) ut0 = ψ, |u(t)| ≤ q(t), t ≥ t0 , |u(t1 ) − u(t2 )| ≤ L|t1 − t2 |
o for t1 , t2 ∈ R+ with t0 ≤ τ1 ≤ t1 , t2 ≤ τ2 . where L = L(τ1 , τ2 ) = max {|a1 (t)|γ + |a2 (t)|γ + |b2 (t)|γ} t∈[τ1 ,τ2 ]
and q(t) is defined by (3.6) with β = η. Since |q(t)| ≤ γ, (3.7)
|q 0 (t)| ≤ |a1 (t)|γ + |b2 (t)|γ ≤ L,
t ≥ t0 .
Define ξ(t) :=
then |ξ(t)| ≤
ψ(t − t0 ), ψ(0)q(t) , η
t0 − r ≤ t ≤ t 0 t ≥ t0 ,
|ψ(0)| |ψ| |q(t)| ≤ |q(t)| ≤ |q(t)| , η η
t ≥ t0 .
It follows from (3.7) that |ξ(t1 ) − ξ(t2 )| =
ψ(0) |q(t1 ) − q(t2 )| ≤ L|t1 − t2 |, η
for τ1 ≤ t1 , t2 ≤ τ2 . Therefore, ξ ∈ S. From Lemma 3.1, we can see that S is a compact convex nonempty subset of C([t0 − r, +∞), R). Let u(t) be the unique solution of (1.2) with ut0 = ψ and consider the equation (3.8)
x0 (t) = −a(t, ut )x(t) + f (t, ut ).
372
M. FAN, Z. XIA AND H. ZHU
Then u(t) is a solution of (3.8) and u(t) can be written as Z t R R t − t a(s,us ) ds u(t) = ψ(0)e t0 + e− s a(τ,uτ ) dτ f (s, us ) ds,
t ≥ t0 .
t0
Define a mapping P on S by (P u)(t) = ψ(t − t0 ), t0 − r ≤ t ≤ t0 , and (P u)(t) = ψ(0)e
−
Rt
t0
a(s,us ) ds
+
Z
t
e−
Rt s
a(τ,uτ ) dτ
f (s, us ) ds, t ≥ t0 .
t0
It is clear that finding a solution of (1.2) is equivalent to finding a fixed point of P in S. Next, we show P maps S to S. Note that (P u)(t) = ψ(t − t0 ) for t0 − r ≤ t ≤ t0 , and |(P u)(t)| ≤ |ψ(0)| e ≤ ηe ≤ ηe = ηe
−
−
−
Rt
t0
Rt
t0
Rt
= q(t),
t0
−
Rt
t0
a1 (s) ds
a1 (s) ds
a1 (s) ds
a1 (s) ds
+ + +
Z Z Z
+
Z
t
t
e−
Rt s
a1 (τ ) dτ
|f (s, us )| ds
t0
e−
Rt s
a1 (τ ) dτ
|b2 (s)| |u(s)| ds
e−
Rt s
a1 (τ ) dτ
|b2 (s)| q(s) ds
e−
Rt
a1 (τ ) dτ
(q 0 (s) + a1 (s)q(s)) ds
t0 t t0 t
s
t0
t ≥ t0 ,
where we have used the fact that q 0 (t) = −a1 (t)q(t) + |b2 (t)|q(t). Moreover, we have (P u)0 (t) = −a(t, ut )(P u)(t) + f (t, ut ), which implies that |(P u)0 (t)| ≤ |a(t, ut )|q(t) + |b2 (t)|q(t) ≤ |a1 (t)|γ + |a2 (t)|γ + |b2 (t)|γ ≤ L,
t ≥ t0 ;
DELAY DIFFERENTIAL EQUATIONS
373
therefore, P maps S into S. It is clear that P is continuous. By the Schauder’s fixed point theorem, P has a fixed point u in S, satisfying |u(t)| ≤ q(t), t ≥ t0 , which implies that the solution u(t, t0 , ψ) of (1.2) satisfies |u(t, t0 , ψ)| ≤ q(t) = q(t, t0 , η) < ε t ≥ t0 . Hence, the zero solution of (1.2) is stable. It follows from q(t) → 0 as t → +∞ that u(t, t0 , ψ) → 0 as t → +∞. The proof is then complete. Theorem 3.2. If (H7 )–(H9 ) hold, then the zero solution of (1.3) is asymptotically stable. 4 Applications In this section, by using the criteria set up in sections 2 and 3, we will study the stability of several equations, in particular the stability of the oscillations in the model for the red blood cells. We first start with a simple but an interesting example. Example 4.1. Consider (4.9) x0 (t) = −γ(t) β(t) +
1 x(t) β(t) + x2 (t − τ (t)) + γ(t)β(t)x(t − τ (t))
where τ (t), γ(t), β(t) ∈ C(R, R+ ) and τ (t) is bounded above by positive constant. Equation (4.9) is a special case of (1.2) with 1 a(t, xt ) = γ(t) β(t) + . β(t) + x2 (t − τ (t)) 1 ), then a1 (t) ≤ One can take a1 (t) = γ(t)β(t) and a2 (t) = γ(t)(β(t)+ β(t) Rt a(t, xt ) ≤ a2 (t). If limt→+∞ 0 γ(s)β(s)ds = +∞, then we can choose b1 (t) = γ(t)β(t). Note that
Z
t
e 0
−
Rt s
a1 (u) du
b1 (s) ds =
Z
t
e−
Rt s
γ(u)β(u) du
γ(s)β(s) ds
0
= 1 − e−
Rt 0
γ(u)β(u) du
< 1,
374
M. FAN, Z. XIA AND H. ZHU
and Z
t
a1 (s) ds = 0
Z
t
γ(s)β(s) ds → +∞,
t → +∞.
0
Thus, (H2 )–(H4 ) are all satisfied. It follows from Theorem 2.1 that the zero solution of (4.9) is asymptotically stable. In (4.9), if we take γ(t) = 1 + t2 and β(t) = 1, then as shown in Figure 4.1, the zero solution of (4.9) is asymptotically stable.
τ=3
1 0.8 x0=0.5
0.6
x0=1
0.4
x(t)
0.2 0 −0.2 x0=−1
−0.4 −0.6 −0.8 −1
x0=−0.5 0
5
10
15 t
20
25
30
FIGURE 4.1: The zero solution of (4.9) is asymptotically stable. Here τ = 3 and x(θ) = x0 , θ ∈ [−τ, 0].
Example 4.2. Now, let us consider a generalized hematopoiesis model (1.1), where x(t) is the count of red blood cells at time t, and α(t), β, γ(t) ∈ C(R, R+ ) are ω-periodic. If the parameters are positive constants, then (1.1) reduced to the original model introduced by WazewskaCzyewska and Lasota [20] modeling the survival of red blood cells in animal. The models of form (1.1) have been extensively studied, e.g., [1, 6, 9, 13, 15, 20]. And later, a generalized model was proposed and studied [8] x0 (t) = −γ(t)x(t) + α(t)f (x(t − τ )).
DELAY DIFFERENTIAL EQUATIONS
375
In [21], it was proved that (1.1) has at least one positive ω-periodic solution x∗ (t) while the stability of x∗ (t) was untouched and unknown. With the criteria established in this paper, we now can claim that Corollary 4.1. If Zt
e−
Rt s
γ(u) du
|α(s)||β(s)| ds ≤ α < 1, t ≥ 0,
0
then the positive ω-periodic solution x∗ (t) is asymptotically stable (see Figure 4.2). Proof. To study the asymptotic stability of the non-trivial periodic solution x∗ (t), we let y(t) = x(t) − x∗ (t). Then we have (4.10)
y 0 (t) = x(t) ˙ − x˙ ∗ (t) = −γ(t)y(t) + α(t)e−β(t)x
∗
(t−τ (t))
(e−β(t)y(t−τ (t)) − 1).
It is clear that the asymptotic stability of x∗ (t) is equivalent to the asymptotic stability of the zero solution of (4.10). Take f (t, φ) = α(t)e−β(t)x
∗
(t−τ (t))
(e−β(t)φ(−τ (t)) − 1),
then |f (t, φ) − f (t, ψ)| ≤ |α(t)||e−β(t)φ(−τ (t)) − e−β(t)ψ(−τ (t))| ≤ |α(t)β(t)||φ − ψ|. In addition, lim
t→+∞
Zt
γ(s) ds = +∞
0
since γ(t) is nonnegative ω-periodic function. Set b1 (t) = |α(t)β(t)|, then by Theorem 2.1, the positive ω-periodic solution x∗ (t) is asymptotically stable. Example 4.3. To study the regulation of hematopoiesis and the formation of blood cell elements in a body, Glass and Mackey [16] proposed a model of delay differential equation. Here we consider a generalization
376
M. FAN, Z. XIA AND H. ZHU
1 0.9 x =0.1 0
0.8 0.7
x(t)
0.6 0.5 0.4 0.3 x
0.2
=1.0
0
x =0.4 0
0.1 0
0
2
4
6
8
10 t
12
14
16
18
20
FIGURE 4.2: The zero solution of (1.1) is asymptotically stable. Here, γ(t) = 1 + sin(2πt), α(t) = 1 + cos(2πt), β(t) ≡ 1, τ = 2 and x(θ) = x0 , θ ∈ [−τ, 0].
of the original one, where some parameters are generalized from positive constants to nonnegative functions, that is, (4.11)
x0 (t) = −γ(t)x(t) +
β(t)θn x(t − τ (t)) , θn + xn (t − τ (t))
where γ, β, τ ∈ C(R, R+ ), θ is a positive constant, and n is a given positive integer. In (4.11), the time delay τ is the time between the production of immature cells in the bone marrow and their maturation for release in the circulating bloodstream, and x(t) denotes the density of mature cells in the blood circulation. Corollary 4.2. If lim
t→+∞
Z
t
(γ(s) − β(s))ds = +∞ 0
then the zero solution of (4.11) is asymptotically stable (see Figure 4.3). Proof. Note that f (t, φ) =
β(t)θn φ(−τ (t)) . θn + φn (−τ (t))
DELAY DIFFERENTIAL EQUATIONS
So
377
β(t)θn φ(−τ (t)) ≤ |β(t)||φ(−τ (t))| ≤ β(t)|φ|. |f (t, φ)| = n θ + φn (−τ (t))
One can take b2 (t) = β(t), then lim
t→+∞
Z
t
(a1 (s) − |b2 (s)|)ds = lim
t→+∞
0
Z
t
(γ(s) − β(s))ds = +∞. 0
The asymptotic stability now follows from Theorem 3.1.
0.4
0.35
0.3
x(t)
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20
25 t
30
35
40
45
50
FIGURE 4.3: The zero solution of (4.11) is asymptotically stable. Here, γ(t) = 4 + sin(2πt), β(t) = 2 + cos(t), θ = 1, n = 5, τ = 3 and x(θ) = 5, 15, 50, 100 > 0, θ ∈ [−τ, 0].
Example 4.4. We close the paper by considering a generalized so-called Nicholson’s Blowflies Model (4.12)
x0 (t) = −γ(t)x(t) + β(t)x(t − τ (t)) exp{−ax(t − τ (t))},
where γ, β, τ ∈ C(R, R+ ) and a is a positive constant. When all the parameters are positive constants, The equation (4.12) reduces to the original model developed by Gurney et al. [10] to describe the population of the Australian sheep-blowfly (Lucila Curpina) that agrees well with the experimental data of Nicholson [17]. Since dynamics of the
378
M. FAN, Z. XIA AND H. ZHU
equation matches well with Nicholson’s data of blowfly, the equation is now referred as the ‘Nicholson’s Blowflies Model’, which has been extensively studied in the literatures. By carrying out proof exactly the same as that of Corollary 4.2, we reach the following claim. Corollary 4.3. If lim
t→+∞
Z
t
(γ(s) − β(s))ds = +∞, 0
then the zero solution of (4.12) is asymptotically stable. The variation of the environment plays an important role in many biological and ecological dynamical systems. It has been suggested by Nicholson [18] that any periodic change of climate tends to impose its periodicity upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes. In view of this, it is realistic to assume that the parameters in the models are periodic. The numerical simulations were carried out by taking γ(t) = 4+cos(πt), β(t) = 3 + sin(t), a = 0.25 and τ = 5. As shown in Figure 4.4, the zero solution of the model (4.12) is asymptotically stable. 1.6
1.4
1.2
x(t)
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50 t
60
70
80
90
100
FIGURE 4.4: The zero solution of (4.12) is asymptotically stable. Here γ(t) = 4 + cos(πt), β(t) = 3 + sin(t), a = 0.25, τ = 5 and x(θ) = 5, 15, 50, 100, θ ∈ [−τ, 0].
DELAY DIFFERENTIAL EQUATIONS
379
REFERENCES 1. O. Arino and M. Kimmel, Stability analysis of models of cell production system, Math. Modelling 17 (1986), 1269–1300. 2. T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations, Proc. Dynamic Syst. Appl. 10 (2001), 89–116. 3. T. A. Burton and T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Proc. Dynamic Syst. Appl. 11 (2002), 499–519. 4. T. A. Burton, Stability by fixed point theory or Liapunov theory: a comparison, Fixed Point Theory 4 (2003), 15–32. 5. T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications Inc., 2006. 6. S. N. Chow, Existence of periodic solutions of autonomous functional differential equations, J. Diff. Equ. 15 (1974), 350–378. 7. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Springer-Verlag, 1992. 8. K. Gopalsamy and S. Trofimchuk, Almost periodic solutions of Lasota-Wazewaska type delay differential equations, J. Math. Anal. Appl. 237 (1999), 106– 127. 9. J. R. Graef, C. Qian and P. W. Spikes, Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull. 38 (1996), 275–283. 10. W. S. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980), 17–21. 11. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 12. Y. Kuang, Delay Differential Equations with Applications to Population Dynamics, Academic Press, 1994. 13. M. R. S. Kulenovic and G. Ladas, Linearized oscillations in population dynamics, Bull. Math. Biol. 49 (1987), 615–627. 14. V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, 1996. 15. G. Liu, A. Zhao and J. Yan, Existence and global attractivity of unique positive periodic solution for a Lasota-Wazewska model, Nonlinear Anal. 64 (2006), 1737–1746. 16. M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science 197 (1977), 287–289. 17. A. J. Nicholsom, The self adjustment of population to change, Cold Spring Harbor Symposium Quantitative Biol. 22 (1957), 153–173. 18. A. J. Nicholson, The balance of animal population, J. Animal Ecol. 2 (1993), 132–178. 19. H. Y. Wang, Positive periodic solutions of functional differential equations, J. Diff. Equ. 202(2) (2004), 354–366. 20. M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of the red blood cells system, Annals of the Polish Mathematical Society Series III, Appl. Math. 17 (1976), 23–40. 21. D. Ye, M. Fan and H. Y. Wang, Periodic solutions for scalar functional differential equations, Nonlinear Anal. TMA 62(7) (2005), 1157–1181. 22. T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Mathematical Society of Japan, Toyko, 1966.
380
M. FAN, Z. XIA AND H. ZHU
23. B. Zhang, Contraction mapping and stability in a delay-differential equation, Proc. Dynamic Syst. Appl. 4 (2004), 183–190. School of Mathematics and Statistics, KLAS, and KLVE, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, The People’s Republic of China E-mail address:
[email protected] Corresponding author: Huaiping Zhu Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada E-mail address:
[email protected] 
