International Journal of Bifurcation and Chaos, Vol. 14, No. 11 (2004) 3909–3919 c World Scientific Publishing Company
LOCAL AND GLOBAL HOPF BIFURCATION IN A DELAYED HEMATOPOIESIS MODEL YONGLI SONG∗ Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, P. R. China
[email protected] JUNJIE WEI Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, P. R. China
[email protected] MAOAN HAN Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, P. R. China
[email protected] Received October 10, 2003; Revised January 13, 2004 In this paper, we consider the following nonlinear differential equation q dx = x(t) −p . dt r + xn (t − τ )
(1)
We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second critical value of the delay τ . Finally, numerical simulation results are given to support the theoretical predictions. Keywords: Time delay; local Hopf bifurcation; global Hopf bifurcation; periodic solutions.
1. Introduction For a long time, it has been acknowledged that delays can have very complicated impact on the dynamics of a system (see e.g. [Hale & Lunel, 1993; Kuang, 1993; Wu, 1996]). For example, delays can cause the loss of stability and induce various oscillations and periodic solutions. It is well known that periodic solutions can arise through the Hopf ∗
bifurcation in delay differential equations. However, these periodic solutions bifurcating from Hopf bifurcations are generally local. More specifically, the existence of these periodic solutions remains valid only in a small neighborhood of the critical value and are usually of small amplitudes. Therefore, we wish to know whether these nonconstant periodic solutions which are obtained through local Hopf bifurcations can continue for a large range of parameter values.
Author for correspondence. 3909
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A great deal of research has been devoted to the global existence of nonconstant periodic solutions in delayed equations (see, e.g. [Chow, 1974; MalletParet & Nussbaum, 1986; T´aboas, 1990; Zhao et al., 1997] and references therein). The method of showing the global existence of nonconstant periodic solutions used by the above mentioned researches is Nussbaum’s theorem [1974], which depends in a crucial way on the concept of ejectivity and the decomposition theory of linear functional differential equations. In this paper, instead of applying Nussbaum’s theorem, we shall use a different approach, the degree theory, to study the global existence of nonconstant periodic solutions arising from the following nonlinear delayed differential equation q dx = x(t) −p , t≥0 (2) dt r + xn (t − τ ) with p, q, r, τ ∈ (0, ∞) and n ∈ N = {1, 2, . . .}. Equation (2) is one of the models proposed by Nazarenko [1976] to study the control of a single population of cells. Equation (2) has been recently investigated by several researchers. Kubiaczyk and Saker [2002] have shown that every positive solution of (2) oscillates about its positive equilibrium 1 point x∗ = [(q/p) − r] n if 1 nqxn∗ τ> , 2 n (r + x∗ ) e and x∗ is locally asymptotically stable provided that π nqxn∗ τ< . n 2 (r + x∗ ) 2 Following up the investigation in [Kubiaczyk & Saker, 2002], Saker and Agarwal [2002] considered the oscillation and global attractivity of Eq. (2) with time periodic coefficients. In the present paper, taking the delay τ as a parameter, we investigate the dynamics of Eq. (2). More specifically, we show that Hopf bifurcations occur when the delay τ passes through a critical value, at which a family of periodic solutions bifurcate from the positive equilibrium. Furthermore, we investigate the global existence of bifurcating periodic solutions by applying a global Hopf bifurcation theory due to Wu [1998], which was established using purely topological arguments. This method avoided the sophisticated theory of linear functional differential equations and the difficulty of verifying the ejectivity of fixed points of the flow operator, and has been successfully used to
deal with the global Hopf bifurcation by many researchers (see e.g. [Giannakopoulos & Zapp, 1999; Ruan & Wei, 1999; Song et al., 2001; Song et al., 2004; Song & Wei, 2004; Wei & Ruan, 2002]). In this paper we show that the local Hopf bifurcation of (2) implies the global Hopf bifurcation after the second critical value of the delay τ . The remainder of this paper is organized as follows. In the next section, we consider the stability and the local Hopf bifurcation of the positive equilibrium. In Sec. 3, we use the method of normal forms and center manifold theory introduced by Hassard et al. [1981] to derive the algorithm determining the direction, stability and the period of the bifurcating periodic solution at the critical value of τ . The global existence of these bifurcating periodic solutions will be considered in Sec. 4. In Sec. 5, we give a numerical example. Finally, some conclusions and unsolved problems are stated in Sec. 6.
2. Stability of the Positive Equilibrium and Local Hopf Bifurcations It is easy to verify that for (2), the positive equilib1 rium x∗ ≡ [(q/p) − r] n exists if and only if q > pr. Therefore, we assume this condition holds throughout the paper. By the translation y(t) = x(t) − x∗ , (2) is written as q dy − p . (3) = (y(t) + x∗ ) dt r + (y(t − τ ) + x∗ )n It is well known that the stability of the positive equilibrium and local Hopf bifurcations involve the distribution of the roots associated with the characteristic equation of its linearization at equilibrium x∗ , and the positive equilibrium of (2) is stable if and only if the zero steady state of (3) is stable. Thus, we consider linearization of (3) at y = 0:
dy = −αy(t − τ ) , (4) dt where α = nqxn∗ /(r + xn∗ )2 > 0. The characteristic equation of (4) is λ + αe−λτ = 0 .
(5)
For τ = 0, the only root of (5) is λ = −α < 0. We first examine when this equation has pure imaginary roots λ = ±iω (ω > 0). Obviously, iω is a root of (5) if and only if iω + α(cos ωτ − i sin ωτ ) = 0 .
Local and Global Hopf Bifurcation in a Delayed Hematopoiesis Model
Separating the real and imaginary parts, we obtain α cos ωτ = 0 ,
α sin ωτ = ω ,
(6)
which lead to
(ii) (2 ) undergoes a Hopf bifurcation at the positive equilibrium x∗ when τ = τk , for k = 0, 1, 2, . . . . In Theorem 2.2, we not only obtain the sufficient condition for asymptotical stability of the positive equilibrium of (2), which is the same as that given in [Kubiaczyk & Saker, 2002], but also get the conditions assuring the existence of local Hopf bifurcations. Remark 2.3.
1 π 2kπ + , k = 0, 1, . . . , (7) ω = α , τk = α 2 i.e. when τ = τk , (5) has a pair of purely imaginary roots ±αi. Let λk = ηk (τ ) + iωk (τ ) denote a root of (5) near τ = τk , such that ηk (τk ) = 0, ωk (τk ) = α. Substituting λk into (5) and differentiating with respect to τ , we then have dλ −λτ dλ −α λ+τ e = 0. dτ dτ It follows that
dλ αλe−λτ = , dτ 1 − ατ e−λτ which, together with (6), implies α2 sin ατk + iα2 cos ατk dλ = dτ 1 − ατ cos ατ + iατ sin ατ k
τ =τk
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k
k
k
α2 . 1 + iατk Therefore, we have that the following transversality condition holds: α2 d Re{λk (τk )} = > 0. (8) dτ 1 + (ατk )2 From the above discussion, we can easily obtain the following results about the location of roots of transcendental equation (5). =
Lemma 2.1. For Eq. (5 ), we have the following
results. (i) When τ = τk (k = 0, 1, 2, . . .), Eq. (5 ) has a pair of simple imaginary roots ±iα. (ii) When τ ∈ [0, τ0 ), all roots of Eq. (5 ) have negative real parts, and when τ = τ0 , all roots of Eq. (5 ), except ±iα, have negative real parts. But when τ ∈ (τk , τk+1 ], Eq. (5) has 2(k + 1) roots with positive real parts. From Lemma 2.1 and the transversality condition (8), we may easily obtain the results about the stability of the positive equilibrium and the local Hopf bifurcation for (2). Theorem 2.2. If q > pr, then we have
(i) the positive equilibrium x∗ is asymptotically stable for τ ∈ [0, τ0 ), and unstable for τ > τ0 ; and
3. Direction and Stability of the Hopf Bifurcation In the previous section, we have shown that (2) undergoes Hopf bifurcation at τ = τk , k = 0, 1, . . . . As pointed out in [Hassard et al., 1981], it is interesting to determine the direction, stability and period of these periodic solutions bifurcating from the equilibrium x∗ . In this section, we derive the explicit formulaes which determine these factors at the critical value τk using normal form and center manifold theories due to Hassard et al. [1981]. For convenience, let x(t) = y(τ t) and τ = τk + µ. Then (3) can be written as an FDE in C = C([−1, 0], R) as x(t) ˙ = Lµ (xt ) + f (µ, xt ) ,
(9)
where xt (θ) = x(t + θ) ∈ C, and Lµ : C → R, f : R × C → R are given respectively by Lµ (φ) = −(τk + µ)αφ(−1) ,
(10)
and f (µ, φ) = (τk + µ) α1 φ(0)φ(−1)
1 1 α2 φ(0)φ2 (−1) + α2 x∗ φ2 (−1) 2! 2! 1 3 4 (11) + α3 x∗ φ (−1) + O(φ ) , 3! +
where α = nqxn∗ /(r + xn∗ )2 , αk = (∂ k g(0)/∂y k ) (k = 1, 2, 3) and g(y) = q/(r + (y + x∗ )n ). By the Riesz representation theorem, there exists a function η(θ, µ) of bounded variation for θ ∈ [−1, 0], such that Z 0 dη(θ, 0)φ(θ) for φ ∈ C . Lµ φ = −1
In fact, we can choose η(θ, µ) = (τk + µ)αδ(θ + 1) ,
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where δ is the dirac delta function. For φ ∈ C 1 ([−1, 0], R), define dφ(θ) θ∈[−1, 0) , dθ , A(µ)φ = Z 0 dη(µ, s)φ(s) , θ = 0 .
On the center manifold C0 we have W (t, θ) = W (z(t), z(t), θ) , where z2 + W11 (θ)zz 2 z3 z2 + W02 (θ) + W30 (θ) + · · · , 2 6
W (z, z, θ) = W20 (θ)
−1
and
R(µ)φ =
(
0,
θ ∈ [−1, 0) ,
f (µ, φ) ,
θ = 0.
z and z are local coordinates for center manifold C 0 in the direction of q ∗ and q ∗ . Note that W is real if xt is real. We consider only real solutions. For the solution xt ∈ C0 of (9), since µ = 0, we have
Then (9) is equivalent to x˙ t = A(µ)xt + R(µ)xt ,
(12)
where xt (θ) = x(t + θ) for θ ∈ [−1, 0]. For ψ ∈ C 1 ([0, 1], (R)∗ ), define dψ(s) s ∈ (0, 1] , − ds , A∗ ψ(s) = Z 0 dη(t, 0)ψ(−t) , s = 0 .
= iτk αz+q ∗ (0)f (0, W (z, z, 0)+2 Re{zq(0)}) def
= iτk αz+q ∗ (0)f0 (z, z) .
We rewrite this equation as z(t) ˙ = iτk αz(t) + g(z, z)
−1
with
and a bilinear inner product
g(z, z) = q ∗ (0)f0 (z, z)
hψ(s), φ(θ)i = ψ(0)φ(0) −
Z
0
−1
Z
ψ(ξ − θ)dη(θ)φ(ξ)dξ , ξ=0
(13)
hq ∗ (s), q(θ)i = 0 ,
where D=
1 . 1 + iτk α
z(t) =
ti ,
W (t, θ) = xt (θ) − 2 Re{z(t)q(θ)} .
(15)
Note that xt (θ) = W (t, θ) + zq(θ) + zq(θ) and q(θ) = eiτk αθ . Thus, we may obtain g(z, z) = q ∗ (0)f0 (z, z) 1 = Dτk α1 x(t)x(t−1)+ α2 x(t)x2 (t−1) 2 1 1 + α2 x∗ x2 (t−1)+ α3 x∗ x3 (t−1)+ · · · 2 6 z2 = Dτk (2α1 e−iτk α + α2 x∗ e−2iτk α ) 2
+ (2α1 Re{eiτk α } + α2 x∗ )zz
In the remainder of this section, we will follow the ideas and use the same notations as given in [Hassard et al., 1981] to first compute the coordinates describing the center manifold C 0 at µ = 0. Let xt be the solution of Eq. (9) when µ = 0. Define hq ∗ , x
z2 z2 + g11 zz + g02 2 2 2 z z + g21 + ···. 2
= g20
θ
where η(θ) = η(θ, 0). Then A(0) and A∗ are adjoint operators. By the discussion in Sec. 2, we know that ±iτk α are eigenvalues of A(0). Thus, they are also eigenvalues of A∗ . By the direct computation, it can be verified that q(θ) = eiτk αθ is an eigenvector of A(0) corresponding to iτk α, and q ∗ (s) = Deiτk αs is an eigenvector of A∗ corresponding to −iτk α. Furthermore, hq ∗ (s), q(θ)i = 1 ,
z(t) ˙ = iτk αz+hq ∗ (θ), f (0, W (z, z, θ)+2 Re{zq(θ)})i
(14)
+ (2α1 eiτk α + α2 x∗ e2iτk α )
z2 2
+ [α1 (2W11 (−1)+W20 (−1)+2e−iτk α W11 (0) + eiτk α W20 (0))+ α2 (2 + e−2iτk α ) + α2 x∗ (2e−iτk α W11 (−1) + eiτk α W20 (−1) 2 −iτk α z z + ··· + α 3 x∗ e ] 2
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Comparing the coefficients with (15), we have g20 = Dτk (2α1 e−iτk α + α2 x∗ e−2iτk α ) ; g11 = Dτk (2α1 Re{eiτk α } + α2 x∗ ) ; g02 = Dτk (2α1 eiτk α + α2 x∗ e2iτk α ) ; g21 = Dτk {α1 [2W11 (−1) + W20 (−1) + 2e−iτk α W11 (0) + eiτk α W20 (0)] + α2 (2 + e−2iτk α ) + α2 x∗ [2e−iτk α W11 (−1) + eiτk α W20 (−1)] + α3 x∗ e−iτk α } .
(16)
Note that g21 contains W20 (θ) and W11 (θ) which need to be computed. From (12) and (14), we have ˙ = x˙t − zq ˙ W ˙ − zq ( AW − 2 Re{q ∗ (0)f0 q(θ)} , θ ∈ [−1, 0) , = AW − 2 Re{q ∗ (0)f0 q(0)} + f0 , θ = 0 . def
= AW + H(z, z, θ) , (17)
Similarly, from (19) and (22), we can obtain ig11 iτk αθ e W11 (θ) = − τk α ig + 11 e−iτk αθ + E2 . (24) τk α In what follows, we shall seek appropriate E 1 and E2 . From the definition of A and (20), we obtain Z 0 dη(θ)W20 (θ) = 2iτk αW20 (0) − H20 (0) (25) −1
and
Z
0
dη(θ)W11 (θ) = −H11 (0) ,
where η(θ) = η(0, θ). From (17), we have H20 (0) = −g20 q(0) − g 02 q(0) + 2τk α1 e−iτk α + τk α2 x∗ e−2iτk α H11 (0) = −g11 q(0) − g 11 q(0) + 2τk α1 Re{eiτk α } + τk α2 x∗ .
AW11 (θ) = −H11 (θ), . . . .
(19)
From (18), we know that for θ ∈ [−1, 0), ∗
H(z, z, θ) = −q (0)f0 q(θ) − q (0)f 0 q(θ) = −gq(θ) − gq(θ) .
H20 (θ) = −g20 q(θ) − g 02 q(θ) ,
(21)
H11 (θ) = −g11 q(θ) − g 11 q(θ) .
(22)
and
It follows from (19) and (21) that ˙ 20 (θ) = 2iτk αW20 (θ) + g20 q(θ) + g 02 q(θ) . W
which, together with the definition of η(θ), leads to 2α1 e−iτk α + α2 x∗ e−2iτk α . α(2i + e−2iτk α ) Similarly, substituting (24) and (28) into (26), we get E1 =
ig ig20 iτk αθ e + 02 e−iτk αθ τk α 3τk α
2τk α1 Re{eiτk α } + τk α2 x∗ Z 0 dη(θ) −1
2α1 Re{eiτk α } + α2 x∗ . α Therefore, all gij in (17) have been expressed in terms of the parameters and the delay given in (9). Furthermore, we can compute the following values: i |g02 |2 g21 2 c1 (0) = , g11 g20 − 2|g11 | − + 2τk α 3 2 =−
µ2 = −
Note that q(θ) = eiτk αθ , hence
+ E1 e2iτk αθ .
2τk α1 e−iτk α + τk α2 x∗ e−2iτk α , Z 0 2iτk αθ 2iτk α − e dη(θ) −1
(20)
Comparing the coefficients with (19) gives that
W20 (θ) =
E1 =
E2 =
∗
Re{c1 (0)} , Re{λ0k (τk )}
β2 = 2 Re{c1 (0)} , 0
(23)
(28)
Substituting (23) and (27) into (25), we obtain
H(z, z, θ) = H20 (θ)
(A − 2iτk α)W20 (θ) = −H20 (θ) ,
(27)
and
where z2 + H11 (θ)zz 2 z2 (18) + H02 (θ) + · · · . 2 Expanding the above series and comparing the corresponding coefficients, we obtain
(26)
−1
Im{c1 (0)} + µ2 Im{λk (τk )} T2 = − , τk α
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which determine the quantities of bifurcating periodic solutions on the center manifold at the critical value τk , i.e. µ2 determines the directions of the Hopf bifurcation: if µ2 > 0 (µ2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ > τ k (τ < τk ); β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β 2 < 0 (β2 > 0); and T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2 > 0 (T2 < 0). Note that when τ > τ0 , the corresponding characteristic equation of (3) has at least one root with positive real part. Therefore, when τ = τk (k ≥ 1), the nontrivial periodic solutions bifurcating from the positive equilibrium must be unstable in the phase space, even though they are stable on the center manifold. But when τ = τ 0 , the stability of bifurcating periodic solutions on the center manifold is equivalent to that of bifurcating periodic solutions in the whole phase space.
Remark 3.1.
4. Global Existence of Periodic Solutions In this section, we shall study the global continuation of periodic solutions bifurcating from the positive equilibrium x∗ . Throughout this section, we follow closely the notations used in [Wu, 1998]. To simplify notations, we may rewrite (2) as the following functional differential equation x(t) ˙ = F (xt , τ, T ) ,
(29)
where xt (θ) = x(t + θ) ∈ C([−τ, 0], R) and F (xt , τ, T ) = xt (0)[(q/(r + xnt (−τ ))) − p]. Here we still suppose that q > pr. It is obvious that when n = 2k, (29) has three equilibria x1 ≡ 0, x2 ≡ x∗ and x3 ≡ −x∗ , while (29) has two equilibria x1 ≡ 0, x2 ≡ x∗ for n = 2k + 1. Following the work of Wu [1998], we need to define X = C([−τ, 0], R) , P = Cl{(x, τ, T ) ∈ X × R × R+ ; is a T -periodic solution of (29)} , N = {(x, τ , T ); F (x, τ , T ) = 0} . Let `(x∗ ,τk , 2π ) denote the connected component of α P (x∗ , τk , 2π/α) in , where τk is defined in (7). From Theorem 2.2, we know that `(x∗ ,τk , 2π ) is nonempty. α For the benefit of the readers, we first state the
global Hopf bifurcation theory due to Wu [1998] for functional differential equations. Lemma 4.1 [Wu, 1998].
Assume that (x∗ , τ, T ) is an isolated center satisfying the hypotheses (A 1 ) − (A4 ) in Wu [1998, p. 4814 P ]. Denote by `(x∗ ,τ,T ) the connection of (x∗ , τ, T ) in . Then either (i) `(x∗ ,τ,T ) is unbounded, or (ii) `(x∗ ,τ,T ) is bounded, `(x∗ ,τ,T ) ∩ N is finite and X γm (x∗ , τ, T ) = 0 (x∗ ,τ,T )∈`(x∗ ,τ,T ) ∩N
for all m = 1, 2, . . . , where γm (x∗ , τ, T ) is the mth crossing number of (x∗ , τ, T ) if m ∈ J(x∗ , τ, T ), or it is zero if otherwise. Clearly, if (ii) in Lemma 4.1 is not true, then `(x∗ ,τ,T ) is unbounded. Thus, if the projections of `(x∗ ,τ,T ) onto x-space and onto T -space are bounded, then the projection of `(x∗ ,τ,T ) onto τ space is unbounded. Further, if we can show that the projection of `(x∗ ,τ,T ) onto τ -space is away from zero, then the projection of `(x∗ ,τ,T ) onto τ -space must include interval [τ, ∞). In the following, we follow this ideal to prove our main results about the global Hopf bifurcation. Lemma 4.2. When q > pr, Eq. (29 ) has no non-
trivial τ -periodic solutions. Proof. For a contradiction, suppose that Eq. (29) has τ -periodic solution. Then the ordinary differential equation q − p (30) x(t) ˙ = x(t) r + xn (t)
has a periodic solution. When n = 2k, (30) has three steady states x(t) = 0, x(t) = x∗ and x(t) = −x∗ . From (30), it follows that when either 0 < x(t) < x ∗ or x(t) < −x∗ , x(t) ˙ > 0, and x(t) ˙ < 0 as either x(t) > x∗ or −x∗ < x(t) < 0. Therefore, the ordinary differential equation (30) with n = 2k has no nontrivial periodic solution. However, when n = 2k + 1, (30) has two steady states x(t) = 0 and x(t) = x∗ . Suppose that x(t) is a nontrivial periodic solution of (30). From Z t q x(t) = x(0) exp − p ds , r + xn (s) 0
we have either x(t) > 0 or x(t) < 0. Notice that x(t) ˙ > 0 for 0 < x(t) < x∗ and x(t) ˙ < 0 for
Local and Global Hopf Bifurcation in a Delayed Hematopoiesis Model
x(t) > x∗ . Thus, when x(t) > 0, (30) with n = 2k+1 has no positive periodic solution. On the other hand, suppose that x(t) < 0 is a periodic solution of (30) with n = 2k + 1, we have 0=
q − p, r + xn (ξ)
where x(ξ) = min{x(t)}. Thus, we get x(ξ) = 1 [(q − pr)/p] n > 0, which contradicts x(t) < 0. This implies that (30) with n = 2k + 1 has no negative periodic solution. Thus, the ordinary differential equation (30) with n = 2k + 1 has no nontrivial periodic solution, too. From the above discussion, we known that (30) has no nontrivial periodic solution. This fact implies that Eq. (29) has no nontrivial τ -periodic solutions. Lemma 4.3. If q > pr, then all the nontrivial pe-
riodic solutions of (29 ) are uniformly bounded for τ ∈ [0, τ ∗ ]. Proof. For a periodic function x(t), we define
x(ξ) = min{x(t)} ,
x(η) = max{x(t)} . (31)
Let x(t) be a periodic solution of (2). Then we obtain Z t q − p ds , x(t) = x(0) exp r + xn (s − τ ) 0 which implies that either x(t) ≡ 0 or x(t) 6= 0. Thus, it is sufficient to consider the following two cases. Case 1.
When x(t) > 0, it follows from (2) that x(t) ˙
τ . x(t) < x(t − τ ) exp r (32)
Therefore, from (32) and (33), we have (q − pr)τ x(η) < x(η − τ ) exp r 1 q − pr n (q − pr)τ = exp . p r
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(34)
When x(t) < 0 and n = 2k, we set y(t) = −x(t) > 0. Then (2) can be rewritten as q y(t) ˙ = y(t) −p . r + y n (t − τ ) Case 2.
Thus, by the same arguments as Case 1, we get 1 (q − pr)τ q − pr n , exp y(η) < p r which means
q − pr x(ξ) > − p
1
n
exp
(q − pr)τ r
.
(35)
On the other hand, if x(t) < 0 and n = 2k + 1, then from (2) and (31), we get q −p. 0= n r + x (ξ − τ ) It follows that
q − pr x(ξ − τ ) = p
1
n
> 0,
which obviously contradicts the assumption x(t) < 0. Thus, Eq. (2) with n = 2k + 1 has no negative periodic solution. From (34) and (35), it follows that if x(t) is a nonconstant periodic solution of (2), then 1 q − pr n (q − pr)τ − exp p r 1 q − pr n (q − pr)τ < x(t) < exp , p r which implies that all the nontrivial periodic solutions of (2) are uniformly bounded. The proof is completed.
In addition, from (2) and (31), we may obtain 0=
Although the boundaries of periodic solutions depend on the delay τ , they are uniformly bounded provided that τ is bounded. Remark 4.4.
q − p, n r + x (η − τ )
which implies x(η − τ ) =
q − pr p
1
Theorem 4.5. If q > pr, then, for each τ > τk
n
.
(33)
(k ≥ 1), Eq. (2 ) has at least k periodic solutions, where τk = (1/α)(2kπ + (π/2)).
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It is sufficient to prove that the projection of `(x∗ ,τk , 2π ) onto τ -space is [τ , ∞) for each k ≥ 1, α where τ ≤ τk . The characteristic equation of (29) at an equilibrium x takes the following form Proof.
4(x, τ, T )(λ) = λI − DF (x, τ, T )(eλ· Id) , =λ− −
∂F (x, τ, T ) ∂x(t)
∂F (x, τ, T ) −λτ e . ∂x(t − τ )
(36)
From [Wu, 1998], we know that (x, τ , T ) is called a center if F (x, τ , T ) = 0 and ∆(x, τ , T )((2π/T )i) = 0. A center (x, τ , T ) is said to be isolated if it is the only center in some neighborhood of (x, τ , T ). It follows from (36) that 4(0, τ, T )(λ) = λ −
q − pr , r
which has no purely imaginary roots. Thus, we conclude that (29) does not have a center in the form of (0, τ, T ). In addition, it is easy to verify that when n = 2k, 4(−x∗ , τ, T )(λ) = 4(x∗ , τ, T )(λ) = λ + αe−λτ . On the other hand, from the discussion about the local Hopf bifurcation in Sec. 2, it is easy to verify that (x∗ , τk , 2π/α) and (−x∗ , τk , 2π/α) are isolated centers. However, when n = 2k + 1, (29) has only single isolated center (x∗ , τk , 2π/α). For a isolated center (x∗ , τk , 2π/α), it follows from Sec. 2 that there exist ε > 0, δ > 0 and a smooth curve λ : (τk − δ, τk + δ) → C such that ∆(λ(τ )) = 0, |λ(τ ) − α| < ε for all τ ∈ [τk − δ, τk + δ] and d Re{λ(τ )} > 0. λ(τk ) = αi, dτ τ =τk
Let
Ωε, 2π = α
2π (η, T ) : 0 < η < ε, T − , for k > 1 . α 2 α Thus, there exists an nk ∈ N such that τk 2π < < τk . nk α Now we prove that the projection of ` (x∗ ,τk , 2π ) α onto τ -space is [τ , ∞), where τ ≤ τk . Clearly, it follows from the proof of Lemma 4.2 that Eq. (2) with τ = 0 has no nontrivial periodic solution. Hence, the projection of `(x∗ ,τk , 2π ) onto τ -space is away from α zero. For a contradiction, we suppose that the projection of `(x∗ ,τk , 2π ) onto τ -space is bounded. This α means that the projection of `(x∗ ,τk , 2π ) onto τ α space is included on a interval (0, τ ∗ ). Noticing 2π/α < τk and applying Lemma 4.2 we have T < τ∗ for all (x(t), τ, T ) belonging to `(x∗ ,τk , 2π ) . This imα plies that the projection of `(x∗ ,τk , 2π ) onto T -space α
Local and Global Hopf Bifurcation in a Delayed Hematopoiesis Model
is bounded. Then, applying Lemma 4.3 we get that the connected component `(x∗ ,τk , 2π ) is bounded. α This leads to a contradiction, which completes the proof.
5. A Numerical Example From the algorithm given in Sec. 3, we know that if the values of p, q and r are known, then we can compute the values of µ2 and β2 , which determine the stability and direction of periodic solutions bifurcating from the positive equilibrium at the critical point τk . In this section, we present some numerical results of the following system: 2 −1 , (37) x(t) ˙ = x(t) 1 + x2 (t − τ ) which has a positive equilibrium x∗ ≡ 1. It fol. . lows from (4) that τ0 = 1.5708, τ1 = 7.85398, . τ2 = 14.1372, . . . . By Theorem 2.2, we know that the positive equilibrium x∗ is stable when τ < τ0 , as illustrated by computer simulations, as shown in . Fig. 1, for τ = 1.4. When τ passes through τ0 , x∗ loses its stability and a Hopf bifurcation occurs at τ > τk . Furthermore, from Sec. 3, we can determine the stability and direction of periodic solutions bifurcating from the positive equilibrium at the crit. ical point τk . For instance, when τ = τ0 = 1.5708,
Fig. 1.
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. we find c1 (0) = −0.689745 + 0.723412i, indicating that µ2 > 0 and β2 < 0. Therefore, bifurcating periodic solutions exist at least for the value τ slightly larger than τ0 and the corresponding periodic orbits are orbitally, asymptotically stable, as depicted in Fig. 2, where τ = 1.7. For a larger delay τ = 9, which is relatively far away from τ1 , the bifurcating periodic solutions still exist (see Fig. 3). Numerical simulation results also verify theoretical investigation for the global existence of periodic solutions discussed in Sec. 4.
6. Discussion In this paper, we investigate local and global Hopf bifurcations in a delayed hematopoiesis model. We show that, as the delay τ increases, the positive equilibrium x∗ loses its stability and a sequence of Hopf bifurcations (local) occur at x ∗ . For a local Hopf bifurcation, using center manifold reduction and normal form theories, we derive explicit formulae determining stability, direction and period of bifurcating periodic solutions. Further, existence of periodic solutions for τ far away from the local Hopf bifurcation values is also established. More specifically, we show that, for each τ ∈ [τ1 , ∞), Eq. (2) has at least a periodic solution. Unfortunately, the following questions have not been investigated completely by the authors.
The positive equilibrium Eq. (37) is stable when τ = 1.4 < τ0 .
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Y. Song et al.
Fig. 2.
A periodic solution Eq. (37) is orbitally, asymptotically stable when τ = 1.7.
Fig. 3.
A large periodic solution Eq. (37) for τ1 < τ = 9 < τ2 .
1. For any τ ∈ [τ0 , τ1 ), it is unknown whether Eq. (2) has a periodic solution. 2. From the global Hopf bifurcation theory due to Wu [1998], we obtain the existence of periodic solutions for a large range τ ∈ [τ1 , ∞), but this approach seems to generate little information for the stability.
Here, these two problems are left open.
Acknowledgments We wish to thank two reviewers for their valuable comments that led to truly significant improvement of the manuscript. This research is supported in
Local and Global Hopf Bifurcation in a Delayed Hematopoiesis Model
part by the National Natural Science Foundation of China (Nos. 10471030 and 10371072).
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