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Stability and Hopf bifurcation for a delay competition diffusion system
Chaos, Solitons and Fractals 14 (2002) 1201–1225 www.elsevier.com/locate/chaos
Stability and Hopf bifurcation for a delay competition diffusion system q Li Zhou
a,*
, Yanbin Tang a, Shawgy Hussein
b
a b
Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Mathematics, Sudan University of Science and Technology, Khartoum, P.O. Box 407, Sudan Accepted 12 March 2002
Abstract This paper investigates the stability and Hopf bifurcation of a delay competition diffusion system. Firstly we discuss the existence and stability of the corresponding steady state solutions. Secondly our purpose is to give more detail information about the Hopf bifurcation of this system. We derive the basis of the eigenfunction subspace and then convert the existence of periodic solutions to the study of the existence of the implicit function. Finally, we analyze the stability of the periodic solutions by reducing the original system on the center manifold. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Stability; Hopf bifurcation; Competition diffusion
1. Introduction Situation involving a negative delayed feedback control occur in many mathematical models of biological systems with spatial diffusion. For a review of these models, see [6,14]. For a set of different species interacting with each other in an ecological community, perhaps the simplest and probably the most important question from a practical point of view is whether all the species in the system survive in the long term. Therefore, the periodic phenomena of biological system are often discussed, for example, the time delay is considered as a parameter and questions of stability of equilibria and periodicity are discussed in [2,5]. For this reason, in the present paper we consider the spatially nonhomogeneous periodic solutions, that is, the so-called Hopf bifurcation which arises from the spatially non-homogeneous steady state solution, of the following autonomous competition diffusion equations with delay: Ut ¼ Uxx þ kU ðt; xÞ½1 b1 U ðt r; xÞ c1 V ðt r; xÞ; Vt ¼ Vxx þ kV ðt; xÞ½1 b2 U ðt r; xÞ c2 V ðt r; xÞ; t > 0; 0 < x < p; U ðt; 0Þ ¼ U ðt; pÞ ¼ V ðt; 0Þ ¼ V ðt; pÞ ¼ 0; t P 0; r 6 t 6 0; 0 6 x 6 p; ðU ; V Þ ¼ ð/1 ; /2 Þ;
ð1:1Þ
with b1 =b2 > 1 > c1 =c2 :
ð1:2Þ
Systems of type (1.1) have been extensively studied in recent years, see [1–3,9,11–13]. The closely related problems with Neumann boundary conditions have also been studied by Zhou and Hussein [11], as we know in these cases there
q *
The project supported by National Natural Science Foundation of China. Corresponding author. E-mail addresses: [email protected] (L. Zhou), [email protected] (Y. Tang).
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 6 8 - 1
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
always exist spatially homogeneous stationary solutions therefore the stability and bifurcation clearly are treated. However, under Dirichlet boundary conditions such homogeneous stationary solutions do not exist in the problem (1.1). We will present a detailed analysis of the occurrence of the Hopf bifurcation arising from the spatially nonhomogeneous steady state of system (1.1). Our first consideration here is to study the existence and stability of nontrivial spatially non-homogeneous steady state solutions, and then a remaining interesting question is whether an increase in delay r will destabilize the steady state ðUk ðxÞ; Vk ðxÞÞ and lead to the occurrence of periodic solutions. Finally, the stability of the bifurcation periodic solution will be considered. In this paper we get the conclusions as follows. The asymptotic behavior of the delay competition diffusion system is determined by the coefficients bi ; ci , and the time delay r. This means that there is a threshold rk0 > 0, the steady state ðUk ðxÞ; Vk ðxÞÞ will attract the solutions of the initial boundary value problem if 0 6 r < rk0 , and the periodic oscillator ðUk;r ðxÞ; Vk;r ðxÞÞ ¼ Uk;r ðxÞ þ w1 ðx; t=ð1 þ bÞÞ; Vk;r ðxÞ þ w2 ðx; t=ð1 þ bÞÞ is an attractor as r > rk0 . The conclusion shows the persistent dynamics of this kind of systems and the method in this paper will be useful for other ecological systems or physical systems. Specially the study of periodicity can be developed by this approach as J.K. Hale mentioned in [3]. This paper is organized as follows. In Section 2, we present the existence of steady state solutions. Section 3 contains the eigenvalue problem. In Section 4 we describe the stability of the positive equilibrium. In Section 5 we study the Hopf bifurcation which arises from the positive equilibrium as the delay r goes through a critical point r0 , and determine a formula that establishes the stability of the bifurcation periodic solutions. Finally some numerical simulations are given to demonstrate the asymptotic behavior of the species.
2. The existence of steady state solutions Consider the following steady state problem: U 00 ðxÞ þ kU ð1 b1 U c1 V Þ ¼ 0; 0 < x < p; V 00 ðxÞ þ kV ð1 b2 U c2 V Þ ¼ 0; 0 < x < p; U ð0Þ ¼ U ðpÞ ¼ V ð0Þ ¼ V ðpÞ ¼ 0; U > 0; V > 0; 0 < x < p:
ð2:1Þ
Here k is restricted in some neighborhood of 1. Suppose that the solution ðUk ðxÞ; Vk ðxÞÞ of (2.1) has the following expressions: Uk ðxÞ ¼ ðk 1Þaðsin x þ ðk 1ÞnðxÞÞ; hn; sin xi ¼ 0; ð2:2Þ Vk ðxÞ ¼ ðk 1Þbðsin x þ ðk 1ÞgðxÞÞ; hg; sin xi ¼ 0; Rp where hgðxÞ; sin xi ¼ 0 gðxÞ sin x dx: Substitute (2.2) into (2.1) we get ðD2 þ 1Þn þ sin x þ ðk 1Þn k½b1 aðsin x þ ðk 1ÞnÞ2 þ c1 bðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞ ¼ 0; ðD2 þ 1Þg þ sin x þ ðk 1Þg k½b2 aðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞ þ c2 bðsin x þ ðk 1ÞgÞ2 ¼ 0: It is easy to check that c2 c1 b1 b2 a ; b1 ¼ a a1 ¼ b1 c2 b2 c1 0 b1 c2 b2 c1 0 solve the equations b1 a þ c1 b ¼ a 0 ; where a 0 ¼
Z
b2 a þ c2 b ¼ a 0 ;
p
sin2 x dx= 0
Z
p 0
sin3 x dx ¼ 38p:
Let n1 ¼ g1 be the solution of the following boundary value problem: ðD2 þ 1Þy þ sin x a 0 sin2 x ¼ 0; yð0Þ ¼ yðpÞ ¼ 0;
0 < x < p;
hy; sin xi ¼ 0:
From the expression of a 0 it is obvious that n1 ; g1 are well defined.
ð2:3Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
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Theorem 2.1. There are a constant k > 1 and a continuously differentiable mapping k ! ðnk ; gk ; ak ; bk Þ from ½1; k to ðH02 Þ2 R2 such that (2.3) holds and hnk ; sin xi ¼ 0;
hgk ; sin xi ¼ 0:
Proof. Let F ¼ ðF1 ; F2 ; F3 ; F4 Þ : ðH02 Þ2 R3 ! ðL2 Þ2 R2 be defined as F1 ðn; g; a; b; kÞ ¼ ðD2 þ 1Þn þ sin x þ ðk 1Þn kðb1 aðsin x þ ðk 1ÞnÞ2 þ c1 bðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞÞ; F2 ðn; g; a; b; kÞ ¼ ðD2 þ 1Þg þ sin x þ ðk 1Þg kðb2 aðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞ þ c2 bðsin x þ ðk 1ÞgÞ2 Þ; F3 ðn; g; a; b; kÞ ¼ hn; sin xi; F4 ðn; g; a; b; kÞ ¼ hg; sin xi: Substitute n1 ; g1 ; a1 ; b1 ; 1 into Fi ; i ¼ 1; 2; 3; 4; we obtain Fi ðn1 ; g1 ; a1 ; b1 ; 1Þ ¼ 0;
i ¼ 1; . . . ; 4:
The Frechet derivative of F at ðn1 ; g1 ; a1 ; b1 ; 1Þ is 0 2 D þ1 0 b1 sin2 x B 2 oF 0 D þ 1 b2 sin2 x B ¼B oðn; g; a; bÞ ðn1 ;g1 ;a1 ;b1 ;1Þ @ h; sin xi 0 0 0 h; sin xi 0
1 c1 sin2 x C c2 sin2 x C C; A 0 0
hence the system oF oðn; g; a; bÞ
0 1 0 1 n f ðxÞ B g C B gðxÞ C B C¼B C @aA @ c A ðn1 ;g1 ;a1 ;b1 ;1Þ b d
ð2:4Þ
has a unique bounded solution for f ðxÞ; gðxÞ 2 L2 ð0; pÞ; c; d 2 R. Therefore the implicit function theorem implies that there exist a constant k > 1 and a continuously differentiable mapping k ! ðnk ; gk ; ak ; bk Þ 2 ðH02 ð0; pÞ \ RðD2 þ 1ÞÞ2 R2 for k 2 ½1; k such that Fi ðnk ; gk ; ak ; bk ; kÞ ¼ 0; i ¼ 1; 2; 3; 4: Corollary 2.1. For every k 2 ½1; k , (2.1) has a positive solution ðUk ; Vk Þ which has the asymptotic expressions (2.2). Remark. The stability of steady state ðUk ; Vk Þ to the system (1.1) without delay is studied in [7,8], and the analysis with delay will be given later for comparison purpose.
3. Eigenvalue problem Let 1 < k 6 k and ðUk ðxÞ; Vk ðxÞÞ be the positive equilibrium of the system (1.1) described in Section 2. The linearized system of (1.1) around ðUk ðxÞ; Vk ðxÞÞ has the form ut ¼ uxx þ ku kðb1 Uk þ c1 Vk Þu kUk ðb1 uðt r; xÞ þ c1 vðt r; xÞÞ; vt ¼ vxx þ kv kðb2 Uk þ c2 Vk Þv kVk ðb2 uðt r; xÞ þ c2 vðt r; xÞÞ; uðt; 0Þ ¼ uðt; pÞ ¼ vðt; 0Þ ¼ vðt; pÞ ¼ 0; ðu; vÞ ¼ ð/1 ; /2 Þ;
t P 0;
ðt; xÞ 2 ½r; 0 ½0; p;
where ð/1 ; /2 Þ 2 Cð½r; 0; X X Þ;
X ¼ L2 ð0; pÞ:
t > 0; 0 < x < p;
ð3:1Þ
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Introduce the operator AðkÞ : DðAðkÞÞ ! X X defined by
b1 Uk þ c1 Vk 0 2 ; AðkÞ ¼ ðD þ kÞ k 0 b2 Uk þ c2 Vk where D2 ¼ o2 =ox2 , with domain DðAðkÞÞ ¼ ðH02 Þ2 : Set ðuðtÞ; vðtÞÞ ¼ ðuðt; Þ; vðt; ÞÞ, ð/1 ðtÞ; /2 ðtÞÞ ¼ ð/1 ðt; Þ; /2 ðt; ÞÞ, then (3.1) can be written as ! ! ! ! u b1 Uk c1 Uk uðt rÞ d u ¼ AðkÞ k ; t > 0; dt v v b2 Vk c2 Vk vðt rÞ ! ! u /1 ; t 2 ½r; 0: ¼ /2 v
ð3:2Þ
AðkÞ is an infinitesimal generator of a compact C0 semigroup [4]. The analysis of the stability of ðUk ; Vk Þ therefore yields the eigenvalue problem
y Dðk; k; rÞ ¼ 0; ð0; 0Þ 6¼ ðy; zÞ 2 DðAðkÞÞ; ð3:3Þ z where
bU Dðk; k; rÞ ¼ AðkÞ k 1 k b2 Vk
c1 Uk kr e k: c2 Vk
Let Ar ðkÞ be the infinitesimal generator of the semigroup induced by the solutions of (3.2) with
d /1 /1 ¼ ; r 6 h 6 0; Ar ðkÞ /2 dh /2 and ( DðAr ðkÞÞ ¼
0
0
ð/1 ; /2 Þ 2 C; ð/1 ; /2 Þ 2 C; ð/1 ð0Þ; /2 ð0ÞÞ 2 ðH02 Þ2 ; /01 ð0Þ /02 ð0Þ
! ¼ AðkÞ
/1 ð0Þ /2 ð0Þ
! k
b1 Uk
c1 U k
b2 Vk
c2 Vk
!
/1 ðrÞ /2 ðrÞ
!) :
Since the eigenvalues of Ar ðkÞ depend continuously on r [6], the point spectrum Pr ðAr ðkÞÞ contains a pair of imaginary eigenvalues will play a key role in the analysis of the stability and bifurcation of periodic solutions. Furthermore, it is obvious that Ar ðkÞ has an imaginary eigenvalue k ¼ im ðm 6¼ 0Þ for some r if and only if that
y b U c1 U k AðkÞ im keih 1 k ¼ 0; ðy; zÞ 6¼ ð0; 0Þ ð3:4Þ b2 Vk c2 Vk z is solvable for some value m > 0; h 2 ð0; 2pÞ. If we find ðm; hÞ such that (3.4) has a solution ðy; zÞ, then
h þ 2np y Dðk; im; rn Þ ; n ¼ 0; 1; 2; . . . ; ¼ 0; rn ¼ z m and hence rn will possibly be the candidate, we will just consider r ¼ r0 , at which the stability of steady state solution changes and stable periodic solutions occur. So the interesting question is how many pairs of ðm; hÞ 2 Rþ ð0; 2pÞ are there such that (3.4) is solvable. It will be shown that if 0 < k 1 1, there is a pair of ðm; hÞ which solves (3.4). First we will prove several lemmas which will be used to conclude our assertion. Denote by N ðBÞ and RðBÞ the null and range spaces of the operator B respectively, therefore L2 ð0; pÞ ¼ N ðD2 þ 1Þ RðD2 þ 1Þ; where N ðD2 þ 1Þ ¼ Spanfsin xg;
RðD2 þ 1Þ ¼ fy 2 L2 ð0; pÞ; hsin x; yi ¼ 0g:
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Lemma 3.1. If ðm; h; y; zÞ solves Eq. (3.4) with ð0; 0Þ 6¼ ðy; zÞ 2 bounded for k 2 ð1; k , and
ðH02 Þ2 ,
1205
then m ¼ Oðk 1Þ, and m=ðk 1Þ is uniformly
Z p mðhy; yi þ hz; ziÞ ¼ Im keih ½ðb1 y þ c1 zÞUk y þ ðb2 y þ c2 zÞVk z dx : 0
Proof. From Eq. (3.4), we have hðD2 þ k kðb1 Uk þ c1 Vk ÞÞy keih Uk ðb1 y þ c1 zÞ imy; yi þ hðD2 þ k kðb2 Uk þ c2 Vk ÞÞz keih Vk ðb2 y þ c2 zÞ imz; zi ¼ 0; since both hðD2 þ k kðb1 Uk þ c1 Vk ÞÞy; yi and hðD2 þ k kðb2 Uk þ c2 Vk ÞÞz; zi are real, the imaginary part of the above identity is as follows:
Z p ih ½ðb1 y þ c1 zÞUk y þ ðb2 y þ c2 zÞVk z dx ¼ 0; mðhy; yi þ hz; ziÞ Im ke 0
Z p m 1 ¼ ½ak ðsin x þ ðk 1Þnk Þðb1 y þ c1 zÞy þ bk ðsin x þ ðk 1Þgk Þðb2 y þ c2 zÞz dx : Im keih k1 ðhy; yi þ hz; ziÞ 0 The boundedness of m=ðk 1Þ follows from the continuity of k ! ðnk ; gk ; ak ; bk Þ and the lemma is now proved.
Lemma 3.2. If Z 2 H02 and hsin x; Zi ¼ 0, then jhðD2 þ 1ÞZ; Zij P 3kZk2L2 : Proof. By the assumption of Z, it can be represented by Z¼
1 X
cn sin nx;
kZk2L2 ¼
n¼2
1 X n¼2
p jcn j2 ; 2
and now, immediately, it follows that jhðD2 þ 1ÞZ; Zij ¼
1 X
jcn j2 ðn2 1Þ
n¼2
This completes the proof of the lemma.
p P 3kZk2L2 : 2
Now, for k 2 ð1; k , suppose that ðm; h; y; zÞ is a solution of (3.4) with ðy; zÞ 6¼ ð0; 0Þ. If we ignore a scalar factor, ðy; zÞ can be represented as y ¼ sin x þ ðk 1ÞdðxÞ;
hsin x; di ¼ 0;
z ¼ ðL þ iMÞ sin x þ ðk 1ÞvðxÞ;
hsin x; vi ¼ 0; L > 0:
ð3:5Þ
Substitute Uk ; Vk , (3.5) and m ¼ ðk 1Þh into (3.4), noting that ðD2 þ 1Þ sin x ¼ 0, we obtain the equivalent system to (3.4) g1 ðd; v; h; h; L; M; kÞ ¼ ðD2 þ 1Þd þ ð1 ihÞðsin x þ ðk 1ÞdÞ kðb1 ak ðsin x þ ðk 1Þnk Þ þ c1 bk ðsin x þ ðk 1Þgk ÞÞðsin x þ ðk 1ÞdÞ keih ak ðsin x þ ðk 1Þnk Þðb1 ðsin x þ ðk 1ÞdÞ þ c1 ððL þ iMÞ sin x þ ðk 1ÞvÞÞ ¼ 0;
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g2 ðd; v; h; h; L; M; kÞ ¼ ðD2 þ 1Þv þ ð1 ihÞððL þ iMÞ sin x þ ðk 1ÞvÞ kðb2 ak ðsin x þ ðk 1Þnk Þ þ c2 bk ðsin x þ ðk 1Þgk ÞÞððL þ iMÞ sin x þ ðk 1ÞvÞ keih bk ðsin x þ ðk 1Þgk Þðb2 ðsin x þ ðk 1ÞdÞ þ c2 ððL þ iMÞ sin x þ ðk 1ÞvÞÞ ¼ 0; g3 ðd; v; h; h; L; M; kÞ ¼ Rehsin x; di ¼ 0; g4 ðd; v; h; h; L; M; kÞ ¼ Imhsin x; di ¼ 0; g5 ðd; v; h; h; L; M; kÞ ¼ Rehsin x; vi ¼ 0; g6 ðd; v; h; h; L; M; kÞ ¼ Imhsin x; vi ¼ 0:
ð3:6Þ
Theorem 3.1. If 0 < k 1 1, then there is a continuously differentiable mapping k ! ðdk ; vk ; hk ; hk ; Lk ; Mk Þ from ½1; k to ðH02 Þ2 R4 such that d1 ¼ ð1 ih1 Þn1 , v1 ¼ ð1 ih1 ÞL1 n1 , h1 ¼ p=2, L1 is the positive root of the following equation: c1 ðc2 c1 ÞL2 þ ðb2 c2 b1 c1 ÞL ðb1 b2 Þb2 ¼ 0; and M1 ¼ 0;
h1 ¼
c2 c1 ðb1 þ c1 L1 Þ; b1 c2 b2 c1
and ðdk ; vk ; hk ; hk ; Lk ; Mk Þ solves system (3.6) for k 2 ð1; k with n1 is defined as in Section 2. Moreover, if k 2 ð1; k , and ðdk ; vk ; hk ; hk ; Lk ; M k Þ solves, (3.6) with hk > 0 and hk 2 ð0; 2pÞ, then ðdk ; vk ; hk ; hk ; k L ; M k Þ ¼ ðdk ; vk ; hk ; hk ; Lk ; Mk Þ. Proof. Define G : ðH02 Þ2 R5 ! ðL2 Þ2 R4 by G ¼ ðg1 ; . . . ; g6 Þ, where gi ; i ¼ 1; . . . ; 6; are given in (3.6). Then it follows from the definitions of d1 ; v1 ; h1 ; h1 ; L1 ; M1 that g1 ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ ¼ ð1 ih1 Þ½ðD2 þ 1Þn1 þ sin x a 0 sin2 x ¼ 0; and gi ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ ¼ 0;
i ¼ 2; 3; 4; 5; 6:
It means that Gðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ ¼ 0: Let J ¼ ðJ1 ; . . . ; J6 Þ : ðH02 Þ2 R4 ! ðL2 Þ2 R4 be defined as J ¼ Dðd;v;h;h;L;MÞ Gðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ; it easy to verify that J1 ðd; v; h; h; L; MÞ ¼ ðD2 þ 1Þd ih sin x þ ha1 ðb1 þ c1 L1 Þ sin2 x þ ia1 c1 ðL þ iMÞ sin2 x; J2 ðd; v; h; h; L; MÞ ¼ ðD2 þ 1Þv iL1 h sin x þ hb1 ðb2 þ c2 L1 Þ sin2 x þ ð1 ih1 ÞðL þ iMÞ sin x a 0 ðL þ iMÞ sin2 x þ ib1 c2 ðL þ iMÞ sin2 x; J3 ðd; v; h; h; L; MÞ ¼ Rehsin x; di; J4 ðd; v; h; h; L; MÞ ¼ Imhsin x; di; J5 ðd; v; h; h; L; MÞ ¼ Rehsin x; vi; J6 ðd; v; h; h; L; MÞ ¼ Imhsin x; vi: Noting that both sin x and sin2 x do not belong to RðD2 þ 1Þ, this enables us to show that J is one to one and onto from ðH02 Þ2 R4 ! ðL2 Þ2 R4 . Hence our first conclusion follows from the implicit function theorem. To obtain the second conclusion, it suffices to show that ðdk ; vk ; hk ; hk ; Lk ; M k Þ ! ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 Þ as k ! 1 in the norm of ðH02 Þ2 R4 , where Lk > 0.
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
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First, Lemma 3.1 implies that fhk g is bounded. Using Lemma 3.2 we get from the first equation of (3.6) that Z p
Z p sin xdk dx þ ðk 1Þkdk k2L2 þ k ðb1 ak ðsin x þ ðk 1Þnk Þ 3kdk k2L2 6 j1 ihk j 0
0
Z p þ c1 bk ðsin x þ ðk 1Þgk ÞÞ sin xdk dx þ k ak ðsin x þ ðk 1Þnk Þ½b1 þ c1 ðLk þ iM k Þ sin xdk dx 0 Z p ðb1 ak ðsin x þ ðk 1Þnk Þ þ c1 bk ðsin x þ ðk 1Þgk ÞÞjdk j2 dx þ kðk 1Þ Z þ
0
0
p
Z ak ðsin x þ ðk 1Þnk Þb1 jdk j2 dx þ
p 0
ak ðsin x þ ðk 1Þnk Þc1 vk dk dx ;
it can be written as 3kdk k2L2 6 k1 kdk kL2 þ ðk2 ðjLk j þ jM k jÞ þ k3 ðk 1Þðkdk kL2 þ kvk kL2 ÞÞkdk kL2 ; that is kdk kL2 6 k1 þ k2 ðjLk j þ jM k jÞ þ k3 ðk 1Þðkdk kL2 þ kvk kL2 Þ:
ð3:7Þ
k
Here k1 ; k2 ; k3 are independent of d ; vk ; hk ; Lk ; M k . Similarly, we have kvk kL2 6 k1 þ k2 ðjLk j þ jM k jÞ þ k3 ðk 1Þðkdk kL2 þ kvk kL2 Þ:
ð3:8Þ
For k 1 1, it follows from (3.7) and (3.8) that kdk kL2 þ kvk kL2 6 k1 þ k2 ðjLk j þ jM k jÞ:
ð3:9Þ
On the other hand we get from the solvable condition of the first two equations of (3.6) that hð1 ihk Þðsin x þ ðk 1Þdk Þ kðb1 ak ðsin x þ ðk 1Þnk Þ þ c1 bk ðsin x þ ðk 1Þgk ÞÞðsin x þ ðk 1Þdk Þ k
keih ak ðsin x þ ðk 1Þnk Þðb1 ðsin x þ ðk 1Þdk Þ þ c1 ððLk þ iM k Þ sin x þ ðk 1Þvk ÞÞ; sin xi ¼ 0; hð1 ihk ÞððLk þ iM k Þ sin x þ ðk 1Þvk Þ kðb2 ak ðsin x þ ðk 1Þnk Þ þ c2 bk ðsin x þ ðk 1Þgk ÞÞ
ð3:10Þ
k
ððLk þ iM k Þ sin x þ ðk 1Þvk Þ keih bk ðsin x þ ðk 1Þgk Þðb2 ðsin x þ ðk 1Þdk Þ þ c2 ððLk þ iM k Þ sin x þ ðk 1Þvk ÞÞ; sin xi ¼ 0: Taking real part and imaginary part respectively in the first equality of (3.10) we get jLk j 6 k4 þ k5 ðk 1Þðkdk kL2 þ kvk kL2 Þ;
ð3:11Þ
jM k j 6 k4 þ k5 ðk 1Þðkdk kL2 þ kvk kL2 Þ:
If 0 < k 1 1 then it follows that kdk kL2 , kvk kL2 , jLk j and jM k j are bounded from (3.9) and (3.11). Return to the first two equations of (3.6), then we get that fdk g and fvk g are compact in H02 . Therefore it implies that fðdk ; vk ; hk ; hk ; Lk ; M k Þ; k 2 ð1; k g is precompact in ðH02 Þ2 R4 . Let fdkn ; vkn ; hkn ; hkn ; Lkn ; M kn g be any convergent subsequence such that ðdkn ; vkn ; hkn ; hkn ; Lkn ; M kn Þ ! ðd1 ; v1 ; h1 ; h1 ; L1 ; M 1 Þ; kn ! 1
as n ! 1:
We claim that ðd1 ; v1 ; h1 ; h1 ; L1 ; M 1 Þ ¼ ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 Þ: In fact, taking the limit to the first two equations of (3.6), we obtain 1
ðD2 þ 1Þd1 þ ð1 ih1 Þ sin x ðb1 a1 þ c1 b1 Þ sin2 x eih a1 ðb1 þ c1 ðL1 þ iM 1 ÞÞ sin2 x ¼ 0; 1
ðD2 þ 1Þv1 þ ðL1 þ iM 1 Þð1 ih1 Þ sin x ðb2 a1 þ c2 b1 Þ sin2 x eih b1 ðb2 þ c2 ðL1 þ iM 1 ÞÞ sin2 x ¼ 0:
ð3:12Þ
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Eq. (3.12) implies that 1
ih1 a 0 ¼ eih a1 ðb1 þ c1 ðL1 þ iM 1 ÞÞ and 1
ih1 ðL1 þ iM 1 Þa 0 ¼ eih b1 ðb2 þ c2 ðL1 þ iM 1 ÞÞ; thus L1 þ iM 1 is a root of c1 ðc2 c1 ÞL2 þ ðc2 b2 c1 b1 ÞL ðb1 b2 Þb2 ¼ 0: Therefore M 1 ¼ 0, h1 ¼ p=2, L1 ¼ L1 and h1 ¼ h1 . Then we conclude that ðdk ; vk ; hk ; hk ; Lk ; M k Þ ! ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 Þ
as k ! 1
in ðH02 Þ2 R4 . This completes the proof of Theorem 3.1.
The following corollary is an immediate consequence of Theorem 3.1. Corollary 3.1. If 0 < k 1 1, then for each k 2 ð1; k the eigenvalue problem
y Dðk; im; rÞ ¼ 0; m > 0; r > 0; ðy; zÞ 6¼ ð0; 0Þ; z has a solution ðm; r; y; zÞ or equivalently im 2 Pr ðAr ðkÞÞ if and only if mk ¼ ðk 1Þhk ;
r ¼ r kn ¼
hk þ 2np ; n ¼ 0; 1; 2; . . . ; mk
and
sin x þ ðk 1Þdk y y ; ¼c k ¼c zk ðLk þ iMk Þ sin x þ ðk 1Þvk z here c is an arbitrary non-zero constant, and dk ; vk ; hk ; hk ; Lk ; Mk are described in Theorem 3.1.
4. Stability of the positive equilibrium In this section we study the stability of the positive equilibrium ðUk ; Vk Þ with k 2 ð1; k fixed, and the delay r considered as a parameter. To describe stability of ðUk ; Vk Þ it suffices to investigate how the eigenvalue k ¼ im varies as the delay r passing through rkn ; n ¼ 0; 1; . . . First we have to solve the adjoint problem of (3.4) as
yk b U b2 Vk ¼ 0: ð4:1Þ AðkÞ imk eihk k 1 k z k c1 Uk c2 Vk Similarly we let yk ¼ sin x þ ðk 1Þd k ;
z k ¼ ðL k þ iMk Þ sin x þ ðk 1Þv k :
ð4:2Þ
After the same argument, we obtain, there is a continuously differentiable mapping k ! ðd k ; v k ; L k ; Mk Þ, from ½1; k to ðH02 Þ2 R2 such that (4.2) satisfies (4.1), and a1 c1 L1 ; d 1 ¼ ð1 ih1 Þn1 ; v 1 ¼ ð1 ih1 ÞL 1 n1 ; M1 ¼ 0: L 1 ¼ b1 b2 Denote C ¼ Cð½0; rkn ; X X Þ, the basis of eigenspace in C of A rkn ðkÞ, with k ¼ imk is expressed as !
~ð1Þ ðyk ; z k Þeimk h w ~ w ¼ ~ð2Þ ¼ ; 0 6 h 6 r kn : imk h ðyk ; zk Þe w
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1209
Similarly we choose a basis of eigenspace in C of Arkn ðkÞ with k ¼ imk as
yk imk h yk imk h /~ ¼ ð/~ð1Þ ; /~ð2Þ Þ ¼ e ; e ; rkn 6 h 6 0: zk zk Then we calculate the respective inner product Z p Z w~ð1Þ ð0Þ/~ð1Þ ð0Þ dx k Skn ¼ ðw~ð1Þ ; /~ð1Þ Þ ¼
bU w~ð1Þ ðs þ rkn Þ 1 k b2 Vk 0 rkn 0
Z p Z p yk b U c1 Uk dx: ðyk yk þ z k zk Þ dx keihk rkn ðyk ; z k Þ 1 k ¼ b2 Vk c2 Vk zk 0 0 Z
0
p
c1 Uk ~ð1Þ / ðsÞ dx ds c2 Vk ð4:3Þ
Lemma 4.1. If 0 < k 1 1, then for each k 2 ð1; k , Skn 6¼ 0. Proof. Since rkn ¼ ðhk þ 2npÞ=½hk ðk 1Þ; and Uk ¼ ðk 1Þak sin x þ Oðk 1Þ2 ;
Vk ¼ ðk 1Þbk sin x þ Oðk 1Þ2 :
Therefore p
Im Skn ! 2
þ 2np ba ð1; L 1 Þ 1 1 b2 b1 h1
c1 a1 c2 b1
This completes the proof of Lemma 4.1.
1 L1
Z
p
sin3 x dx 6¼ 0
as k ! 1:
0
With the aid of Lemma 4.1, we can prove the following lemma, see Appendix A. Lemma 4.2. For each k 2 ð1; k ð0 < k 1 1Þ and n ¼ 0; 1; . . . ; k ¼ imk is a simple eigenvalue of Arkn ðkÞ. Now by the implicit function theorem, it is not difficult to show that there are a neighborhood of ðrkn ; imk ; yk ; zk Þ in 2 2 2 2 Dkn Ckn ðH0k Þ R C ðH02 Þ2 and a continuously differentiable mapping: Dkn ! Ckn ðH0k Þ such that for n n r 2 Dkn the only eigenvalue of Ar ðkÞ in Ckn is kðrÞ and kðrkn Þ ¼ imk ; yðrkn Þ ¼ yk ; zðrkn Þ ¼ zk ;
yðrÞ Dðk; kðrÞ; rÞ ¼ 0; r 2 Dkn : zðrÞ Differentiating the above equality with respect to r at rkn , we have
0
yk y ðr Þ b1 Uk c1 Uk b1 Uk k0 ðrkn Þ 1 þ krkn eihk þ imk keihk þ Dðk; imk ; rkn Þ 0 kn b2 Vk c2 Vk zk b2 Vk z ðrkn Þ Multiplying by ðyk ; z k Þ, and integrating on ð0; pÞ and noting that
0 Z p y ðr Þ dx ¼ 0; ðyk ; z k ÞDðk; imk ; rkn Þ 0 kn z ðrkn Þ 0 then we obtain k0 ðrkn ÞSkn ¼
Z
p
imk keihk ðyk ; z k Þ 0
b1 Uk b2 Vk
c1 U k c2 Vk
yk zk
dx;
noting that Skn ¼
Z
p
ðyk yk þ z k zk Þ krkn eihk ðyk ; z k Þ
0
therefore, k0 ðrkn Þ ¼ ðI1 þ I2 Þ=jSkn j2 ;
b1 Uk b2 Vk
c1 U k c2 Vk
yk zk
dx;
c1 U k c2 Vk
yk zk
¼ 0:
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
where I1 ¼
Z
p
ðyk yk þ z k zk Þ dx
0
and
Z
p
imk keihk ðyk ; z k Þ
0
Z p
bU ðyk ; z k Þ 1 k I2 ¼ ik 2 mk rkn b2 Vk 0
c1 U k c2 Vk
yk zk
b1 Uk b2 Vk
c1 U k c2 Vk
yk zk
dx;
2 dx :
Here I2 is a pure imaginary, so we have: Lemma 4.3. For each k 2 ð1; k ð0 < k 1 1Þ, Re k0 ðrkn Þ > 0;
n ¼ 0; 1; . . . :
Proof. Since Re k0 ðrkn Þ ¼ ReðI1 =jSkn j2 Þ and ieihk ! 1 as k ! 1, Z p p as k ! 1; ðyk yk þ z k zk Þ dx ! ð1 þ L1 L 1 Þ 2 0 and Z
p
ðyk ; z k Þ 0
b1 Uk b2 Vk
c1 Uk c2 Vk
yk zk
dx ¼
ba ð1; L 1 Þ 1 k b2 bk
c1 ak c2 b k
0
1 L1
Z
p
sin3 x dx ðk 1Þ þ Oðk 1Þ2 :
0
Therefore, if 0 < k 1 1 we have Re k ðrkn Þ > 0: This completes the proof.
From Corollary 3.1 and Lemma 4.3 we have Theorem 4.1. The positive equilibrium ðUk ; Vk Þ is locally asymptotically stable if 0 6 r < rk0 , and unstable if r > rk0 .
5. Hopf bifurcation In this section we study the Hopf bifurcation which arises from the positive equilibrium ðUk ðxÞ; Vk ðxÞÞ as the delay r crosses rk0 ¼ r0 . The generic Hopf bifurcation theorem is well established [11]. Our purpose here is to give more detailed information about the Hopf bifurcation for the delay competition diffusion system (1.1). In fact we are going to prove the following theorem. Theorem 5.1. For each fixed k 2 ð1; k , Hopf bifurcation occurs as the delay increasingly passes through r0 . Specifically for r0 , there is a d0 > 0 such that for each r 2 ðr0 ; r0 þ d0 Þ, system (1.1) has a periodic solution ðUk;r ; Vk;r Þ near ðUk ðxÞ; Vk ðxÞÞ with period 2p=mk . Furthermore, ðUk;r ; Vk;r Þ is locally asymptotically stable. 5.1. Asymptotic expression For fixed k 2 ð1; k and r ¼ r0 þ a, let uðtÞ ¼ uðt; Þ ¼ U ðt; Þ Uk ðxÞ;
vðtÞ ¼ vðt; Þ ¼ V ðt; Þ Vk ðxÞ:
Thus we have a system equivalent to the original one (1.1)
d u u b U c1 Uk u u b uðtÞ c1 uðtÞ ¼ AðkÞ k 1 k ðt r0 aÞ: ðt r0 aÞ k 1 v b2 Vk c2 Vk v v b2 vðtÞ c2 vðtÞ dt v
ð5:1Þ
ð5:2Þ
Further, letting xk ¼ 2p=mk and w1 ðtÞ ¼ uðð1 þ bÞtÞ; w2 ðtÞ ¼ vðð1 þ bÞtÞ, and ðuðtÞ; vðtÞÞ is an xk ð1 þ bÞ periodic solution of (5.2) if and only if ðw1 ðtÞ; w2 ðtÞÞ is an xk periodic solution of
d w1 w1 b U c1 Uk w1 ðt r0 Þ þ Gða; b; wt Þ; ð5:3Þ ¼ AðkÞ k 1 k w2 ðt r0 Þ w2 b2 Vk c2 Vk dt w2
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
where
w1 Gða; b; wt Þ ¼ bAðkÞ w2
ð1 þ bÞk
1211
1 0 r0 þa c1 Uk @ w1 ðt r0 Þ ð1 þ bÞw1 t 1þb A c2 Vk 0 þa w2 ðt r0 Þ ð1 þ bÞw2 t r1þb 1 0 r0 þa c1 w1 ðtÞ @ w1 t 1þb A : c2 w2 ðtÞ w2 t r0 þa
bU þk 1 k b2 Vk b1 w1 ðtÞ b2 w2 ðtÞ
1þb
We use the following notations:
Rp y ¼ 0 ðy1 ðxÞy2 ðxÞ þ z1 ðxÞz2 ðxÞÞ dx: 1. ðy1 ; z1 Þ; 2 z2
R0 b1 Uk c1 Uk /ðsÞ ds: 2. ðw; /Þ ¼ hwð0Þ; /ð0Þi k r0 wðs þ r0 Þ; b2 Vk c2 Vk
y y k k ~ ðhÞ ¼ eimk h ; eimk h ; r0 6 h 6 0; 3. U zk zk ! 1 ðy ; z Þeimk h ~ ðhÞ ¼ Sk10 k k im h ; 0 6 h 6 r0 ; W ðyk ; zk Þe k S k0
1 i 1 1 ~ ~ : U ¼ UH ; W ¼ H W; H ¼ 2 1 i 4. Let K be the eigenspace of Ar0 ðkÞ corresponding to the eigenvalues k ¼ imk : 5. Let Pxk be a Banach space defined as f1 P xk ¼ 2 CðR; X X Þ; fi ðt þ xk Þ ¼ fi ðtÞ; i ¼ 1; 2; t 2 R : f2 6. qi : Pxk ! R; i ¼ 1; 2; are defined by Z xk Z p ðiÞ ðiÞ ðW1 f1 þ W2 f2 Þ dx dt; i ¼ 1; 2; qi f ¼
q¼
0
0
q1 : q2
With the above notations, we can verify that U is a real basis of K, and W is a real basis of the eigenfunction subspace of the formal adjoint operator. By directly computing it is clear that ðW; UÞ ¼ I2 , an identity matrix. It is known by [2,6] that for each f 2 Pxk , the equation
dw b U c1 U k ¼ AðkÞw k 1 k wðt r0 Þ þ f ðtÞ ð5:4Þ b2 Vk c2 Vk dt has an xk periodic solution if and only if f 2 N ðqÞ, that is qi f ¼ 0; i ¼ 1; 2. Let K : N ðqÞ ! Pxk be the linear operator such that Kf is the xk periodic solution of (5.4) satisfying ðW; ðKf Þ0 Þ ¼ 0; where ðKf Þ0 is defined by ðKf Þ0 ðhÞ ¼ Kf ðhÞ;
h 2 ½r0 ; 0:
Then up to time translation, Eq. (5.3) has an xk periodic solution wðtÞ if and only if there is a constant c such that qGða; b; wÞ ¼ 0;
ð5:5Þ
wðtÞ ¼ cUð1Þ ðtÞ þ ½KGða; b; wÞðtÞ;
t 2 R;
where Uð1Þ ¼
1 2
yk imk t y k imk t e þ e ; zk zk
t 2 R:
Introducing the change of variables a ¼ cc; b ¼ cd and wðtÞ ¼ c½Uð1Þ ðtÞ þ cW ðtÞ;
t 2 R; W ðtÞ 2 Pxk ; ðW; ðW Þ0 Þ ¼ 0;
ð5:6Þ
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
then (5.5) and (5.6) are equivalent to Z xk hWðsÞ; N ðc; c; d; W Þi ds ¼ 0 J ðc; c; d; W Þ ¼
ð5:7Þ
0
and
W ¼ KN ðc; c; d; W Þ ¼ K
N1 N2
ð5:8Þ
with ð1Þ
N1 ¼ dðD2 þ k kðb1 Uk þ c1 Vk ÞÞðU1 þ cW1 Þ þ ka
Z
1
ð1Þ
ð1Þ
½b1 Uk U_ 1 ðt r0 hcaÞ þ c1 Uk U_ 2 ðt r0 hcaÞ dh
0 ð1Þ
ð1Þ
dkðb1 Uk ðU1 þ cW1 Þ þ c1 Uk ðU2 þ cW2 ÞÞjtq þ k½b1 Uk ðW1 ðt r0 Þ W1 ðt qÞÞ þ c1 Uk ðW2 ðt r0 Þ ð1Þ
ð1Þ
ð1Þ
W2 ðt qÞÞ kð1 þ cdÞðU1 þ cW1 Þðb1 ðU1 ðt qÞ þ cW1 ðt qÞÞ þ c1 ðU2 ðt qÞ þ cW2 ðt qÞÞÞ and ð1Þ
N2 ¼ dðD2 þ k kðb2 Uk þ c2 Vk ÞÞðU2 þ cW2 Þ þ ka
Z
1
ð1Þ ð1Þ ½b2 Vk U_ 1 ðt r0 hcaÞ þ c2 Vk U_ 2 ðt r0 hcaÞ dh
0 ð1Þ
ð1Þ
dkðb2 Vk ðU1 þ cW1 Þ þ c2 Vk ðU2 þ cW2 ÞÞjtq þ k½b2 Vk ðW1 ðt r0 Þ W1 ðt qÞÞ þ c2 Vk ðW2 ðt r0 Þ ð1Þ
ð1Þ
ð1Þ
W2 ðt qÞÞ kð1 þ cdÞðU2 þ cW2 Þðb2 ðU1 ðt qÞ þ cW1 ðt qÞÞ þ c2 ðU2 ðt qÞ þ cW2 ðt qÞÞÞ; where a ¼ ðc dr0 Þ=ð1 þ cdÞ; q ¼ ðcc þ r0 Þ=ð1 þ cdÞ: Since a periodic solution is a C 1 function without loss of generality, we can restrict our discussion on (5.7) and (5.8) for W 2 Px1k ¼ ff j f 2 Pxk ; f_ 2 Pxk g: Thus it is easy to see that the mapping J : Ip Ip Ip Px1k ! R2 is continuous and continuously differentiable, where Ip ¼ ðp; þpÞ with 0 < p < 1. Lemma 5.1. J ð0; 0; 0; W Þ ¼ 0: Lemma 5.2.
oJ ð0; 0; 0; W Þ Re k0 ðr0 Þ 0 ¼ xk ; Im k0 ðr0 Þ mk oðc; dÞ where kðrÞ and k0 ðrÞ are defined in the previous section. Proof. By directly calculating oJ ð0; 0; 0; W Þ ¼ oc
Z
xk
Z xk
oN ð0; c; 0; W Þ 1 ~ ; b1 Uk W ds ¼ kH b2 Vk oc 0 c¼0
1 A11 A12 dt H ; A21 A22 0
W; 0
¼ H 1
Z
xk 0
where yk dx b1 Uk c1 Uk ¼ k0 ðr0 Þ; A11 ¼ ke imk b V c V zk Sk0 2 k 2 k 0
Z p b U c1 U k y k dx A12 ¼ keihk e2imk t ðyk ; z k Þ 1 k ; b2 Vk c2 Vk zk Sk0 0 ihk
A21 ¼ A12 ;
Z
p
ðyk ; z k Þ
A22 ¼ A11 :
1 c1 Uk ~_ ð1Þ U ðt r0 Þ dt H c2 Vk 0
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1213
Therefore
oJ ð0; 0; 0; W Þ Re k0 ðr0 Þ : ¼ xk 0 Im k ðr0 Þ oc
ð5:9Þ
Similar calculation yields
oJ ð0; 0; 0; W Þ 0 : ¼ xk mk od Hence Lemma 5.2 follows from (5.9) and (5.10).
ð5:10Þ
Lemma 5.3. Let ! ð1Þ ð1Þ ð1Þ U1 ðb1 U1 ðt r0 Þ þ c1 U2 ðt r0 ÞÞ : ð1Þ ð1Þ ð1Þ U2 ðb2 U1 ðt r0 Þ þ c2 U2 ðt r0 ÞÞ
Wk ¼ kK
ð5:11Þ
Then Wk ¼ n1k e2imk t þ n2k þ n1k e2imk t þ UðtÞd;
ð5:12Þ
where n1k
¼
n2k ¼
2hk i
AðkÞ ke
AðkÞ k
1
k yk ðb1 yk þ c1 zk Þeihk c1 U k 2imk ; c2 Vk 4 zk ðb2 yk þ c2 zk Þeihk 1
k yk ðb1 y k þ c1 zk Þeihk þ y k ðb1 yk þ c1 zk Þeihk c1 U k ; c2 Vk 4 zk ðb2 y k þ c2 zk Þeihk þ zk ðb2 yk þ c2 zk Þeihk
b1 Uk b2 Vk
b1 Uk b2 Vk
and
d¼
d1 d2
¼ ðW; n1k e2imk t þ n2k þ n1k e2imk t Þ:
This lemma can be proved by the separate variable method. Furthermore, we can verify that ðW; Wk Þ ¼ 0:
ð5:13Þ
The detail is omitted. Lemma 5.4. Let qk ¼
1 y ½yk ðb1 n2k1 þ c1 n2k2 Þ þ n2k1 ðb1 yk þ c1 zk Þeihk þ y k ðb1 n1k1 þ c1 n1k2 Þe2ihk þ n1k1 ðb1 y k þ c1 zk Þeihk Sk0 k þ z k ½zk ðb2 n2k1 þ c2 n2k2 Þ þ n2k2 ðb2 yk þ c2 zk Þeihk þ zk ðb2 n1k1 þ c2 n1k2 Þe2ihk þ n1k2 ðb2 y k þ c2 zk Þeihk ;
ð5:14Þ
and oJ ð0; 0; 0; Wk Þ ¼ oc
T1 : T2
ð5:15Þ
Reqk dx:
ð5:16Þ
Then 1 T1 ¼ kxk 2
Z
p 0
Lemma 5.5. Let n1k and n2k be defined as in Lemma 5.3. Then lim n1k ðk 1Þ ¼ m11 sin x; k!1
lim n2k ðk 1Þ ¼ 0; k!1
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
in X X , where
i b1 a1 2ih1 a 0 m11 ¼ b2 b1 4
c1 a1 c2 b1 2ih1 a 0
1
b1 þ c1 L1 : L1 ðb2 þ c2 L1 Þ
ð5:17Þ
The proofs of both Lemmas 5.4 and 5.5 will be given in Appendix A. Rp In order to prove the following main theorem, we need some information about sign of Re 0 qk ðxÞ dx, but the expression of qk ðxÞ is complicate, for the sake of simplicity, we verify it by some numerical program but for a kind of special case, which requires the parameters b1 , b2 , c1 and c2 satisfying the following relations: b1 ¼ c2 ;
b2 ¼ c1 ;
and
b1 > b2 :
ðSÞ
Now we give the complete analysis. Condition (S) indicates a symmetric property. Lemma 5.6. Suppose that condition (S) holds, then a1 ¼ b1 ¼ a 0 =ðb1 þ b2 Þ
ð5:18Þ
L1 ¼ h1 ¼ L 1 ¼ 1:
ð5:19Þ
and
The proof of this lemma is simple. Lemma 5.7. If 0 < k 1 1 and ðSÞ holds, then Z p 1 qk ðxÞ dx < 0: q k ¼ Re 2 0
ð5:20Þ
Proof. From lemmas established in the previous it follows that qk ¼
1 ðy ðy ðb1 m111 þ cm112 Þe2ihk þ m111 ðb1 y k þ c1 zk Þeihk Þ þ z k ðzk ðb2 m111 þ c2 m112 Þe2ihk þ m112 ðb2 y k Sk0 ðk 1Þ k k þ c2 zk Þeihk ÞÞ sin x þ Oðk 1Þ :
By the condition (S) we have
ð2 iÞðb1 þ b2 Þ 1 ; m11 ¼ 1 20a 0
8 pi Sk0 ¼ a 0 1 þ þ Oðk 1Þ: 3 2
It follows that q k ¼
1 2jSk0 j2 ðk 1Þ
Re
p 16 3 ðb1 þ b2 Þ2 p1þ þ 3 i þ Oðk 1Þ: 45 2 2
The proof of the lemma is completed.
5.2. Proof of Theorem 5.1 We are now in the position to prove our main result Theorem 5.1. First it follows from Lemmas 5.1 and 5.2 that there are neighborhoods B R of the origin and V0 Pxk of Wk ; c0 > 0, and continuous differentiable functions c : ½c0 ; c0 V0 ! B, d : ½c0 ; c0 V0 ! B such that cð0; Wk Þ ¼ dð0; Wk Þ ¼ 0, and for each ðc; W Þ 2 ½c0 ; c0 V0 ; ðc; dÞ 2 B B, J ðc; c; d; W Þ ¼ 0 if and only if c ¼ cðc; W Þ;
d ¼ dðc; W Þ:
We now define a mapping F : ½c0 ; c0 V0 ! Px1k by F ðc; W Þ ¼ W KN ðc; cðc; W Þ; dðc; W Þ; W Þ:
ð5:21Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1215
Then F ð0; Wk Þ ¼ Wk KN ð0; 0; 0; Wk Þ ¼ 0:
ð5:22Þ
Moreover, Lemma 5.1 implies that oJ ð0; 0; 0; Wk Þ ¼ 0: oW
ð5:23Þ
So ocð0;Wk Þ oW odð0;Wk Þ oW
!
¼
oJ ð0; 0; 0; Wk Þ oðc; dÞ
1
oJ ð0; 0; 0; Wk Þ ¼ 0; oW
this yields that oF ð0; Wk Þ oN ð0; 0; 0; Wk Þ ¼I K ¼ I: oW oW Therefore, oF ð0; Wk Þ=oW : Px1k ! Px1k is one to one and onto. It follows from the implicit function theorem that there are a constant c1 2 ð0; c0 , a neighborhood V1 V0 of Wk and a function W : ½c1 ; c1 ! V1 , such that W ð0Þ ¼ Wk ; and for each ðc; W Þ 2 ½c1 ; c1 V1 ; F ðc; W Þ ¼ 0 if and only if W ¼ W ðcÞ: Consequently, Eq. (5.3) has an xk periodic solution wðtÞ near zero for small a and b if and only if wðtÞ ¼ cðUð1Þ ðtÞ þ cW ðcÞðtÞÞ;
ð5:24Þ
a ¼ ccðc; W ðcÞÞ;
ð5:25Þ
b ¼ cdðc; W ðcÞÞ
ð5:26Þ
and
for some value of c 2 ½c1 ; c1 . Now let c ðcÞ ¼ cðc; W ðcÞÞ, d ðcÞ ¼ dðc; W ðcÞÞ, since J ðc; c ðcÞ; d ðcÞ; W ðcÞÞ 0, c 2 ½c1 ; c1 . Differentiating both sides of this equality at c ¼ 0 gives oJ ð0; 0; 0; Wk Þ oJ ð0; 0; 0; Wk Þ þ oc oðc; dÞ
dc ð0Þ dc dd ð0Þ dc
! ¼ 0:
Lemma 5.2 implies that !
1 dc ð0Þ 1 oJ ð0; 0; 0; Wk Þ Re k0 ðr0 Þ 0 dc ¼ : 0 dd ð0Þ ðr Þ m Im k x oc 0 k k dc Applying Lemma 5.4 we obtain dc ð0Þ T1 q k ¼ ¼ > 0: 0 dc xk Re k ðr0 Þ 2Re k0 ðr0 Þ Therefore, for sufficiently small c, a ¼ cc ðcÞ ¼
q k c2 þ Oðc3 Þ: 2Re k0 ðr0 Þ
Hence we know the bifurcation direction that the Hopf bifurcation occurs as the delay r increasingly passes through r0 . About the stability of periodic solutions, we have to reduce the nonlinear system to two-dimensional center manifold corresponding to the eigenspace with eigenvalue k ¼ imk of the operator Ar0 ðkÞ. By the basic theory in [6], we can use the center manifold theorem to approach the detail about stability, but we give the discussion in Appendix A.
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6. Numerical results In this section we give some numerical result for the asymptotic behavior in the time delay logistic equation on a onedimensional spatial domain X ¼ ð0; 1Þ: ut ¼ uxx þ kuðt; xÞ½1 uðt r; xÞ; t > 0; 0 < x < 1; uðt; 0Þ ¼ uðt; 1Þ ¼ 0; t P 0; uðt; xÞ ¼ 0:1ð1 þ t=rÞ sinðpxÞ; r 6 t 6 0; 0 6 x 6 1:
ð6:1Þ
By discreting system (6.1) into a finite difference system, numerical solution is then solved through the forward elimination and backward substitution. When X ¼ ð0; 1Þ, the principal eigenvalue of the Laplacian operator under Dirichlet boundary conditions is k0 ¼ p2 . Our numerical simulations for (6.1) are given in Figs. 1–3. We choose k ¼ 9:8 < p2 ; r ¼ 30 in (6.1), Fig. 1 demonstrates that the solution converges asymptotically to the trivial steady state. When k ¼ 10:1 > p2 ; r ¼ 310, Fig. 2 shows that the solution converges asymptotically to the non-constant steady state. Now we choose k ¼ 10:2 > p2 ; r ¼ 313, Fig. 3 shows that there exists a Hopf bifurcation periodic solution.
Appendix A A.1. Proof of Lemma 4.2 Firstly, it follows from Corollary 3.1 that dim N ½Arkn ðkÞ imk ¼ 1;
n ¼ 0; 1; . . . ;
with
N ½Arkn ðkÞ imk ¼ Span
yk imk h e ; h 2 ½ rkn ; 0 : zk
Fig. 1. The solution of (6.1) converges to the trivial steady state.
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Fig. 2. The solution of (6.1) converges to the non-constant steady state.
Fig. 3. The solution of (6.1) converges to a Hopf bifurcation periodic solution.
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Now, we suppose that / 2 DðArkn ðkÞÞ \ Dð½Arkn ðkÞ2 Þ; and ½Arkn ðkÞ imk 2 / ¼ 0:
ðA:1Þ
It follows that ðArkn ðkÞ imk Þ/ ¼ ceimk h
yk ; zk
ðA:2Þ
where c is a constant, hence we have
y /_ ðhÞ ¼ imk / þ ceimk h k ; h 2 ½rkn ; 0; zk
ðA:3Þ
and
bU /_ ð0Þ ¼ AðkÞ/ð0Þ k 1 k b2 Vk
c1 U k /ðrkn Þ: c2 Vk
ðA:4Þ
From (A.3) we get
y /_ ð0Þ ¼ imk /ð0Þ þ c k zk
ðA:5Þ
y /ðhÞ ¼ eimk h /ð0Þ þ ch k eimk h ; zk
ðA:6Þ
and
set h ¼ rkn in (A.6) we get /ðrkn Þ ¼ eihk /ð0Þ crkn eihk
yk : zk
ðA:7Þ
Substituting (A.5) and (A.7) into (A.4), we have
b1 Uk c keihk rkn b2 Vk
c1 Uk c2 Vk
yk zk
¼
AðkÞ keihk
b1 Uk b2 Vk
c1 U k c2 Vk
imk /ð0Þ:
Multiplying (A.8) by ðyk ; z k Þ and integrating on ½0; p, then we obtain cSkn ¼
Z
p
AðkÞ keihk
0
b1 Uk c1 U k
b2 Vk c2 Vk
imk
yk z k
T
/1 ð0Þ /2 ð0Þ
dx ¼ 0:
Hence by the Lemma 4.1 we get c ¼ 0. So ½Arkn ðkÞ imk / ¼ 0;
/ 2 N ðArkn ðkÞ imk Þ:
By induction we have N ð½Arkn ðkÞ imk j Þ ¼ N ðArkn ðkÞ imk Þ;
j ¼ 1; 2; . . . ; n ¼ 0; 1; . . .
Therefore, k ¼ imk is a simple eigenvalue of Arkn ðkÞ for n ¼ 0; 1; . . . This completes the proof.
A.2. Proof of Lemma 5.4 Let W~k ¼ n1k e2imk t þ n2k þ n1k e2imk t . Then Wk ¼ W~k þ Ud:
ðA:8Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
It is obvious that ! Z xk * ð1Þ ð1Þ ðUdÞ1 ðb1 U1 ðt r0 Þ þ c1 U2 ðt r0 ÞÞ þ W; ð1Þ ð1Þ ðUdÞ2 ðb2U1 ðt r0 Þ þ c2 U2 ðt r0 ÞÞ 0
ð1Þ
U1 ðb1 ðUdÞ1 ðt r0 Þ þ c1 ðUdÞ2 ðt r0 ÞÞ ð1Þ U2 ðb2 ðUdÞ1 ðt r0 Þ þ c2 ðUdÞ2 ðt r0 ÞÞ
Also from (5.7) it is clear that Z xk * oJ ð0; 0; 0; Wk Þ ¼ k W; oc 0
ð1Þ U1 ðsÞðb1 W~k1 ðs r0 Þ þ c1 W~k2 ðs r0 ÞÞ ð1Þ U2 ðsÞðb2 W~k1 ðs r0 Þ þ c2 W~k2 ðs r0 ÞÞ !+ ð1Þ ð1Þ W~k1 ðb1 U1 ðs r0 Þ þ c1 U2 ðs r0 ÞÞ ds þ ð1Þ ð1Þ W~k2 ðb2 U1 ðs r0 Þ þ c2 U2 ðs r0 ÞÞ Z xk D1 W; ds; ¼ k D2 0
1219
!+ dt ¼ 0:
!
where D1 ¼
1h yk ðb1 n2k1 þ c1 n2k2 Þeimk t þ y k ðb1 n2k1 þ c1 n2k2 Þeimk t þ n2k1 ððb1 yk þ c1 zk Þeimk tihk þ ðb1 y k þ c1 zk Þeimk tþihk Þ 2 þ yk ðb1 n1k1 þ c1 n1k2 Þe2ihk imk t þ y k ðb1 n1k1 þ c1 n1k2 Þe2ihk þimk t þ n1k1 ðb1 y k þ c1 zk Þeihk þimk t i þ n1k1 ðb1 yk þ c1 zk Þeihk imk t þ R 1 ;
D2 ¼
1h zk ðb2 n2k1 þ c2 n2k2 Þeimk t þ zk ðb2 n2k1 þ c2 n2k2 Þeimk t þ n2k2 ððb2 yk þ c2 zk Þeimk tihk þ ðb2 y k þ c2 zk Þeimk tþihk Þ 2 þ zk ðb2 n1k1 þ c2 n1k2 Þe2ihk imk t þ zk ðb2 n1k1 þ c2 n1k2 Þe2ihk þimk t þ n1k2 ðb2 y k þ c2 zk Þeihk þimk t i þ n1k2 ðb2 yk þ c2 zk Þeihk imk t þ R 2 ;
where each term of R 1 and R 2 has a factor as e2imk t ; 1 or e2imk t . Recall (5.14), (5.15) and
imk t yk e y eimk t z k eimk t z k eimk t þ k ; ; þ Wð1Þ ¼ Sk0 Sk0 Sk0 Sk0 we have 1 T1 ¼ kxk 2
Z
p
Reqk dx:
0
A.3. Proof of Lemma 5.5 For k 2 ð1; k we decompose nik ; i ¼ 1; 2; as
i 1 n mik sin x þ ðk 1Þ k1 nik ¼ ; i ¼ 1; 2; ni k1 k2
ðA:9Þ
where mikj ; i ¼ 1; 2; j ¼ 1; 2; are complex numbers, and hni kj ; sin xi ¼ 0;
i; j ¼ 1; 2:
Since
b U þ c1 Vk ðD2 þ 1Þ þ ðk 1Þ k 1 k 0
k yk ðb1 yk þ c1 zk Þ ihk e ; ¼ 4 zk ðb2 yk þ c2 zk Þ
0 b2 Uk þ c2 Vk
ke2ihk
b1 Uk b2 Vk
c1 Uk c2 Vk
2imk n1k
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
b U þ c1 Vk 0 ðD2 þ 1Þ þ ðk 1Þ k 1 k 0 b2 Uk þ c2 Vk
k yk ðb1 y k þ c1 zk Þeihk þ y k ðb1 yk þ c1 zk Þeihk ; ¼ 4 zk ðb2 y k þ c2 zk Þeihk þ zk ðb2 yk þ c2 zk Þeihk
k
b1 Uk b2 Vk
c1 U k c2 Vk
n2k
by substituting (A.7) into above two equations then using the solvable condition hh; sin xi ¼ 0 of ðD2 þ 1Þn þ h ¼ 0; n 2 H02 and hn; sin xi ¼ 0, we obtain k a 0 ðm1k1 ð1 2ihk ÞÞ ¼ eihk ðb1 þ c1 L1 Þ þ kðb1 ak þ c1 bk Þm1k1 þ ke2ihk b1 ak m1k1 þ ke2ihk c1 ak m1k2 þ Oðk 1Þ; 4 k a 0 ðm1k2 ð1 2ihk ÞÞ ¼ eihk L1 ðb2 þ c2 L1 Þ þ kðb2 ak þ c2 bk Þm1k2 þ ke2ihk ðb2 bk m1k1 þ c2 bk m1k2 Þ þ Oðk 1Þ; 4 therefore we have i m111 ð1 2ih1 Þa 0 ¼ ðb1 þ c1 L1 Þ þ a 0 m111 a1 ðb1 m111 þ c1 m112 Þ; 4 i m112 ð1 2ih1 Þa 0 ¼ L1 ðb2 þ c2 L1 Þ þ a 0 m112 b1 ðb2 m111 þ c2 m112 Þ; 4 that is m11
¼
m111 m112
i ¼ 4
b1 a1 2ih1 a 0 b2 b1
c1 a1 c2 b1 2ih1 a 0
1
b1 þ c1 L1 : L1 ðb2 þ c2 L1 Þ
We now compute m21 1 a 0 m211 ¼ ððb1 þ c1 L1 Þi þ ðb1 þ c1 L1 ÞðiÞÞ þ a 0 m211 þ a1 ðb1 m211 þ c1 m212 Þ; 4 1 a 0 m212 ¼ ðL1 ðb2 þ c2 L1 Þi þ L1 ðb2 þ c2 L1 ÞðiÞÞ þ a 0 m212 þ b1 ðb2 m211 þ c2 m212 Þ; 4 therefore m21 ¼ 0. The proof is completed.
A.4. Stability of periodic solutions We rewrite the system as follows:
d ut u ¼ Ar0 ðkÞ t þ X0 g; vt dt vt
ðA:10Þ
where X0 : ½r0 ; 0 ! BðX ; X Þ is given by X0 ðhÞ ¼ 0; if r0 6 h < 0; X0 ð0Þ ¼ I; ut ; vt 2 C and g : C ! X is a nonlinear operator defined by
u ð0Þðb1 ut ðr0 Þ þ c1 vt ðr0 ÞÞ g : ðA:11Þ g ¼ k 1 ¼ k t g2 vt ð0Þðb2 ut ðr0 Þ þ c2 vt ðr0 ÞÞ Noting that the solution of (5.2) on the center manifold is represented as
ut ~ ðx1 ; x2 ; kÞ; ¼ x1 Uð1Þ þ x2 Uð2Þ þ x vt
ðA:12Þ
where
Z p
Z 0 Z p ut ut ð0Þ bU ðiÞ ðiÞ ðiÞ xi ¼ W ; dx k ¼ W ð0Þ W ðs þ r0 Þ 1 k vt ð0Þ b2 Vk v t 0 r0 0
c1 Uk c2 Vk
ut ðsÞ vt ðsÞ
dx ds;
i ¼ 1; 2;
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1221
~ ðx1 ; x2 ; kÞ is represented as and when we denote z ¼ x1 ix2 ; x ~ ðx1 ; x2 ; kÞ ¼ x20 ðkÞ x
z2 z2 þ x11 zz þ x02 þ 2 2
We can obtain the complex form of two-dimensional equations on center manifold as Z p z0 ¼ mk iz þ ðWð1Þ ð0Þ iWð2Þ ð0ÞÞg dx:
ðA:13Þ
ðA:14Þ
0
Noting that
1 x sin x þ Oðk 1Þ ReðqzÞ; U 1 ¼ x2 L1 where qz ¼ eimk h ðx1 ix2 Þ; and ðRe qzÞh¼0 ¼
zþz ; 2
ðRe qzÞh¼r0 ¼
i þ Oðk 1Þ ðz zÞ: 2
Therefore
zþz z2 i z2 20 þ x20 ðz þ b þ ðkÞð0Þ ðsin x þ Oðk 1ÞÞ zÞ þ x ðkÞð r Þ g1 ¼ ðsin x þ Oðk 1ÞÞ 1 0 1 1 2 2 2 2
2 i z þ ; ðA:15Þ þ c1 ðL1 sin x þ Oðk 1ÞÞ ðz zÞ þ x20 2 ðkÞð r0 Þ 2 2
zþz z2 i z2 20 þ x20 ðz þ b þ ðkÞð0Þ ðsin x þ Oðk 1ÞÞ zÞ þ x ðkÞð r Þ g2 ¼ ðL1 sin x þ Oðk 1ÞÞ 2 0 2 1 2 2 2 2
2 i z þ : ðA:16Þ þ c2 ðL1 sin x þ Oðk 1ÞÞ ðz zÞ þ x20 2 ðkÞð r0 Þ 2 2
In the next, we need to determine xij . By the definition of the operator X0 , we have 8
< ki ðb1 þ c1 L1 Þ sin2 x þ Oðk 1Þ ðz2 z2 Þ þ ; h ¼ 0; 4 X0 g ¼ L1 ðb2 þ c2 L1 Þ sin2 x þ Oðk 1Þ : 0; r0 6 h < 0; and ki ðW; X0 gÞ ¼ 4
Z
p 3
sin x dx 0
ðS1k þ S1 Þ 0
k0
ðSki S i Þ 0
! ðb þ Oðk 1ÞÞðz2 z2 Þ þ ;
k0
where Sk0 ¼ S1 þ Oðk 1Þ;
b ¼ b1 þ c1 L1 þ L 1 L1 ðb2 þ c2 L1 Þ;
and S1 ¼
Z 0
p
ð1 þ
L1 L 1 Þ sin2
pi 3 xþ ðb1 a1 þ c1 a1 L1 þ b2 b1 L1 þ c2 b1 L1 L1 Þ sin x dx: 2h1
We now denote that
C~ ðW; X0 gÞ ¼ ~1 iðz2 z2 Þ þ h:o:t:; C2 where h.o.t. stands briefly for high order terms, and
Z p
Z p b 1 1 b i i sin3 x dx; C~2 ¼ sin3 x dx: þ C~1 ¼ 4 S1 S 1 4 S1 S 1 0 0
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Let C3 ¼ 12ðC~1 i þ C~2 Þ; C4 ¼ 12ðC~1 i C~2 Þ, we can show that
8
b1 þ c1 L1 y y > > sin2 x C3 k þ C4 k þ h:o:t: ðz2 z2 Þ þ ; < 4i ðb2 þ c2 L1 Þ zk zk L1 X0s g ¼ y y > k > : C3 eimk h þ C4 k eimk h þ h:o:t: ðz2 z2 Þ þ ; zk zk
h ¼ 0; r0 6 h < 0:
Denote X0s g ¼ H 20 ðhÞ
z2 z2 þ H 11 ðhÞzz þ H 02 ðhÞ þ ; 2 2
therefore 8
> > < 2i
b1 þ c1 L1 y y sin2 x þ 2C3 k þ 2C4 k þ h:o:t:; L 1 ðb2 þ c2 L1 Þ z zk k 20
H ðhÞ ¼ y y > > : 2C3 k eimk h þ 2C4 k eimk h þ h:o:t:; zk zk
h ¼ 0; ðA:17Þ r0 6 h < 0;
and H 11 ðhÞ ¼ 0;
H 02 ðhÞ ¼ H 20 ðhÞ:
Now we derive the equations by which we can determine x20 ðkÞðhÞ; x11 ðkÞðhÞ; . . . these are ½2imk Ar0 ðkÞx20 ðkÞðhÞ ¼ H 20 ðhÞ; Ar0 ðkÞx11 ¼ H 11 ;
ðA:18Þ
02
02
½2imk Ar0 ðkÞx ðkÞðhÞ ¼ H ðhÞ; . . . ; it is obvious that x11 ¼ 0: Substituting (A.17) into (A.18) we get
y y ½2imk Ar0 ðkÞx20 ðhÞ ¼ 2C3 k eimk h þ 2C4 k eimk h þ h:o:t:; zk zk
r0 6 h < 0:
Let x20 ðhÞ ¼ A1 eimk h þ A2 eimk h þ Ee2imk h :
ðA:19Þ
Then we have A1 ¼
2C3 imk
yk zk
þ h:o:t:;
A2 ¼
2C4 3imk
yk zk
þ h:o:t:;
ðA:20Þ
and we can use the following relation, at h ¼ 0, to determine E " 20
½2imk Ar0 ðkÞx ð0Þ ¼ 2imk ðA1 þ A2 þ EÞ imk A1 imk A2 þ k
b1 Uk
c1 Uk
b2 Vk
c2 Vk
!
2
ðD þ kÞ
k
! #
b1 Uk þ c1 Vk
0
0
b2 Uk þ c2 Vk
!
e2ihk E ¼ H 20 ð0Þ:
Now substituting (A.17), at h ¼ 0, into the above equality we get
0 b1 Uk b U þ c1 Vk D2 þ k k 1 k ke2ihk 0 b2 Uk þ c2 Vk b2 Vk
i b1 þ c1 L1 ¼ sin2 x þ h:o:t: 2 L1 ðb2 þ c2 L1 Þ
c1 U k c2 Vk
2imk E ðA:21Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1223
We set E¼
1 k1
E11 E21
sin x þ E ;
ðA:22Þ
where E ¼ Oð1Þ as 0 < k 1 1, and Z p Ei sin x dx ¼ 0; i ¼ 1; 2: 0
Therefore by the solvable condition of (A.21) we get Z p Z p i f½ð1 2h1 iÞ sin x c1 b1 sin2 xE11 þ c1 a1 sin2 x E21 g sin x dx ¼ ðb1 þ c1 L1 Þ sin3 x dx þ h:o:t:; 2 0 0 Z p Z p i f½ð1 2h1 iÞ sin x b2 a1 sin2 xE21 þ b2 b1 sin2 x E11 g sin x dx ¼ L1 ðb2 þ c2 L1 Þ sin3 x dx þ h:o:t:; 2 0 0 that is
ð1 2h1 iÞa 0 þ c1 b1 b2 b1
i c1 a1 b1 þ c1 L1 1 þ h:o:t:; E ¼ ð1 2h1 iÞa 0 þ b2 a1 2 L1 ðb2 þ c2 L1 Þ
this implies that
½ð1 2h1 iÞa 0 b2 a1 ðb1 þ c1 L1 Þ þ c1 a1 L1 ðb2 þ c2 L1 Þ i 2 ½ð1 2h1 iÞa0 c1 b1 L1 ðb2 þ c2 L1 Þ þ b2 b1 ðb1 þ c1 L1 Þ þ h:o:t: E1 ¼ ðð1 2h1 iÞa 0 c1 b1 Þðð1 2h1 iÞa 0 b2 a1 Þ c1 b2 a1 b1 Now substituting (A.20), (A.22) and (A.23) into (A.19) we obtain
1 2C3 1 2C4 1 þ þ E1 sin x þ Oð1Þ x20 ð0Þ ¼ 3h1 i L1 k 1 ih1 L1
ðA:23Þ
ðA:24Þ
and x20 ðr0 Þ ¼
1 2C3 1 2C4 1 þ E1 sin x þ Oð1Þ: L1 k1 h1 3h1 L1
ðA:25Þ
Here we rewrite (A.14) as follows
z2 z2 z0 ¼ mk iz þ ðWð1Þ ð0Þ iWð2Þ ð0Þ; gÞ ¼ mk iz þ k g20 þ g11 zz þ g02 þ : 2 2 Hence we find that g20 ¼
4i ðb1 þ c1 L1 þ L1 L 1 ðb2 þ c2 L1 ÞÞ þ h:o:t:; 3S1
g11 ¼ 0; g02 ¼ g20 ; Z p 4 1 i 20 1 20 11 11 g21 ¼ ðb1 x20 1 ð r0 Þ þ c1 x2 ð r0 ÞÞ þ x1 ð0Þðb1 þ c1 L1 Þ þ ðb1 x1 ð r0 Þ þ c1 x2 ð r0 ÞÞ S1 0 4 4 2 i i 20 L1 L1 20 ðb2 x20 ðb2 x11 x11 1 ð0Þðb1 þ c1 L1 Þ þ L1 1 ð r0 Þ þ c2 x2 ð r0 ÞÞ þ x2 ð0Þðb2 þ c2 L1 Þ þ 1 ð r0 Þ 4 2 2 4 i 11 sin2 x dx þ h:o:t: þ c2 x11 2 ð r0 ÞÞ x2 ð0Þðb2 þ c2 L1 Þ 2 For determining the Floquet exponent we transform Eq. (A.14) into the Poincare normal form [6,10] n_ ¼ imk n þ c1 ð0Þnjnj2 þ Oðjnj5 Þ;
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where " # i jg02 j2 g21 2 2jg11 j þ : c1 ð0Þ ¼ g20 g11 3 2 2mk Moreover r r0 ¼ r2 e2 þ Oðe3 Þ; bðeÞ ¼ b2 e2 þ Oðe3 Þ;
r2 ¼
Re c1 ð0Þ ; a0 ðr0 Þ
b2 ¼ Re c1 ð0Þ:
Note that here b2 is the coefficient of the main part of the Floquet exponent and r2 determines the bifurcation direction, since g11 ¼ 0, that is 1 Re c1 ð0Þ ¼ Re g21 : 2 In the general case, we can use a numerical program to compute the value of g21 , and for a special kind of parameters which satisfy the property ðSÞ, we can determine the sign of Re c1 ð0Þ directly. Recall the Lemma 5.6 in Section 5.1, we have a1 ¼ b1 ¼ 1=ðb1 þ b2 Þ;
L1 ¼ L 1 ¼ h1 ¼ 1:
Direct calculations yield that p 16ðb1 þ b2 Þ ; S1 ¼ p 1 þ i ; C~1 ¼ 2 3pð4 þ p2 Þ 4ðb1 þ b2 Þð2i pÞ ; C4 ¼ 3pð4 þ p2 Þ
8ðb1 þ b2 Þ C~2 ¼ ; 3ð4 þ p2 Þ
C3 ¼
4ðb1 þ b2 Þð2i þ pÞ ; 3pð4 þ p2 Þ
and E11 ðkÞ ¼ E21 ðkÞ ¼
ðb1 þ b2 Þð2 þ iÞ þ h:o:t: 10
Substituting C3 ; C4 and E1 into (5.41), (5.42) we get x20 ð0Þ ¼
b1 þ b2 8ið2i þ pÞ 8ið2i pÞ 2 i 1 sin x þ Oð1Þ þ þ 1 10 k 1 3pð4 þ p2 Þ 9pð4 þ p2 Þ
and x20 ðr0 Þ ¼
b1 þ b2 8ð2i þ pÞ 8ð2i pÞ 2i 1 sin x þ Oð1Þ: 1 k 1 3pð4 þ p2 Þ 9pð4 þ p2 Þ 10
20 Noting that x11 ¼ 0; x20 1 ¼ x2 we have Z p 4 1 20 ð r Þ þ ix ð0ÞÞ sin2 x dx þ h:o:t: g21 ¼ ðb1 þ b2 Þðx20 0 1 1 S1 0 2 Z p 20 2 4ð2 piÞ ðb þ b Þ x1 ð r0 Þ þ ix20 ¼ 1 2 1 ð0Þ sin x dx þ h:o:t: pð4 þ p2 Þ 0
Denote D¼
Z
p 2 20 ½x20 1 ðr0 Þ þ ix1 ð0Þ sin x dx ¼ 0
b1 þ b2 16ð2i pÞ ð2 iÞði 1Þ 4 þ þ Oð1Þ: k1 9pð4 þ p2 Þ 10 3
Since Reð2 piÞð2i pÞ ¼ 0, we obtain Reðg21 Þ ¼
8ðb1 þ b2 Þ2 ð3p þ 2Þ þ Oð1Þ: 15pð4 þ p2 Þðk 1Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1225
Therefore Reðg21 Þ < 0;
0 < k 1 1:
It means that the bifurcation periodic solutions are stable.
References [1] Busenberg S, Mahaffy J. Interaction of spatial diffusion and delay in models of genetic control by repression. J Math Biol 1985;22:313–33. [2] Busenberg S, Huang WZ. Stability and Hopf bifurcation for a population delay model with diffusion effects. J Differential Equations 1996;104:80–107. [3] Hale J. Theory of functional differential equations. Berlin: Springer; 1977. [4] Pazy A. Semigroup of linear operators and applications to partial differential equations. Berlin: Springer; 1983. [5] Travis C, Webb G. Existence and stability for partial functional differential equations. Trans Am Math Soc 1974;204:395–418. [6] Wu J. Theory and applications of partial functional differential equations. Berlin: Springer; 1996. [7] Zhou L, Pao CV. Asymptotic behavior of a competition-diffusion system in population dynamics. Nonlinear Anal TMA 1982;6(11):1163–84. [8] Zhou L. Bifurcation method for analysis of coexistence of competition diffusion system. Acta Appl Math (Chinese) 1991;14:102– 10. [9] Kanel JI, Zhou L. Existence of wave front solutions and estimates of wave speed for a competition-diffusin system. Nonlinear Anal TMA 1996;27:579–87. [10] Hassard BD, Kazarinoff ND, Wan YH. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981. [11] Zhou L, Hussein S. Stability and Hopf bifurcation for a competition delay model with diffusion. J Math (Chinese) 1999;19:441–6. [12] Pao CV. Couple nonlinear parabolic system with time delay. J Math Anal Appl 1995;196:237–65. [13] Zhou L, Hussein S. The existence and stability of nontrivial steady state for a delay competition-diffusion model. Wuhan Univ J Natural Sci 1999;4(4):377–81. [14] Hutson V, Schmitt K. Permanence and the dynamics of biological systems. Math Biosci 1992;111:1–71.
Stability and Hopf bifurcation for a delay competition diffusion system q Li Zhou
a,*
, Yanbin Tang a, Shawgy Hussein
b
a b
Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, PR China Department of Mathematics, Sudan University of Science and Technology, Khartoum, P.O. Box 407, Sudan Accepted 12 March 2002
Abstract This paper investigates the stability and Hopf bifurcation of a delay competition diffusion system. Firstly we discuss the existence and stability of the corresponding steady state solutions. Secondly our purpose is to give more detail information about the Hopf bifurcation of this system. We derive the basis of the eigenfunction subspace and then convert the existence of periodic solutions to the study of the existence of the implicit function. Finally, we analyze the stability of the periodic solutions by reducing the original system on the center manifold. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Stability; Hopf bifurcation; Competition diffusion
1. Introduction Situation involving a negative delayed feedback control occur in many mathematical models of biological systems with spatial diffusion. For a review of these models, see [6,14]. For a set of different species interacting with each other in an ecological community, perhaps the simplest and probably the most important question from a practical point of view is whether all the species in the system survive in the long term. Therefore, the periodic phenomena of biological system are often discussed, for example, the time delay is considered as a parameter and questions of stability of equilibria and periodicity are discussed in [2,5]. For this reason, in the present paper we consider the spatially nonhomogeneous periodic solutions, that is, the so-called Hopf bifurcation which arises from the spatially non-homogeneous steady state solution, of the following autonomous competition diffusion equations with delay: Ut ¼ Uxx þ kU ðt; xÞ½1 b1 U ðt r; xÞ c1 V ðt r; xÞ; Vt ¼ Vxx þ kV ðt; xÞ½1 b2 U ðt r; xÞ c2 V ðt r; xÞ; t > 0; 0 < x < p; U ðt; 0Þ ¼ U ðt; pÞ ¼ V ðt; 0Þ ¼ V ðt; pÞ ¼ 0; t P 0; r 6 t 6 0; 0 6 x 6 p; ðU ; V Þ ¼ ð/1 ; /2 Þ;
ð1:1Þ
with b1 =b2 > 1 > c1 =c2 :
ð1:2Þ
Systems of type (1.1) have been extensively studied in recent years, see [1–3,9,11–13]. The closely related problems with Neumann boundary conditions have also been studied by Zhou and Hussein [11], as we know in these cases there
q *
The project supported by National Natural Science Foundation of China. Corresponding author. E-mail addresses: [email protected] (L. Zhou), [email protected] (Y. Tang).
0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 6 8 - 1
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
always exist spatially homogeneous stationary solutions therefore the stability and bifurcation clearly are treated. However, under Dirichlet boundary conditions such homogeneous stationary solutions do not exist in the problem (1.1). We will present a detailed analysis of the occurrence of the Hopf bifurcation arising from the spatially nonhomogeneous steady state of system (1.1). Our first consideration here is to study the existence and stability of nontrivial spatially non-homogeneous steady state solutions, and then a remaining interesting question is whether an increase in delay r will destabilize the steady state ðUk ðxÞ; Vk ðxÞÞ and lead to the occurrence of periodic solutions. Finally, the stability of the bifurcation periodic solution will be considered. In this paper we get the conclusions as follows. The asymptotic behavior of the delay competition diffusion system is determined by the coefficients bi ; ci , and the time delay r. This means that there is a threshold rk0 > 0, the steady state ðUk ðxÞ; Vk ðxÞÞ will attract the solutions of the initial boundary value problem if 0 6 r < rk0 , and the periodic oscillator ðUk;r ðxÞ; Vk;r ðxÞÞ ¼ Uk;r ðxÞ þ w1 ðx; t=ð1 þ bÞÞ; Vk;r ðxÞ þ w2 ðx; t=ð1 þ bÞÞ is an attractor as r > rk0 . The conclusion shows the persistent dynamics of this kind of systems and the method in this paper will be useful for other ecological systems or physical systems. Specially the study of periodicity can be developed by this approach as J.K. Hale mentioned in [3]. This paper is organized as follows. In Section 2, we present the existence of steady state solutions. Section 3 contains the eigenvalue problem. In Section 4 we describe the stability of the positive equilibrium. In Section 5 we study the Hopf bifurcation which arises from the positive equilibrium as the delay r goes through a critical point r0 , and determine a formula that establishes the stability of the bifurcation periodic solutions. Finally some numerical simulations are given to demonstrate the asymptotic behavior of the species.
2. The existence of steady state solutions Consider the following steady state problem: U 00 ðxÞ þ kU ð1 b1 U c1 V Þ ¼ 0; 0 < x < p; V 00 ðxÞ þ kV ð1 b2 U c2 V Þ ¼ 0; 0 < x < p; U ð0Þ ¼ U ðpÞ ¼ V ð0Þ ¼ V ðpÞ ¼ 0; U > 0; V > 0; 0 < x < p:
ð2:1Þ
Here k is restricted in some neighborhood of 1. Suppose that the solution ðUk ðxÞ; Vk ðxÞÞ of (2.1) has the following expressions: Uk ðxÞ ¼ ðk 1Þaðsin x þ ðk 1ÞnðxÞÞ; hn; sin xi ¼ 0; ð2:2Þ Vk ðxÞ ¼ ðk 1Þbðsin x þ ðk 1ÞgðxÞÞ; hg; sin xi ¼ 0; Rp where hgðxÞ; sin xi ¼ 0 gðxÞ sin x dx: Substitute (2.2) into (2.1) we get ðD2 þ 1Þn þ sin x þ ðk 1Þn k½b1 aðsin x þ ðk 1ÞnÞ2 þ c1 bðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞ ¼ 0; ðD2 þ 1Þg þ sin x þ ðk 1Þg k½b2 aðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞ þ c2 bðsin x þ ðk 1ÞgÞ2 ¼ 0: It is easy to check that c2 c1 b1 b2 a ; b1 ¼ a a1 ¼ b1 c2 b2 c1 0 b1 c2 b2 c1 0 solve the equations b1 a þ c1 b ¼ a 0 ; where a 0 ¼
Z
b2 a þ c2 b ¼ a 0 ;
p
sin2 x dx= 0
Z
p 0
sin3 x dx ¼ 38p:
Let n1 ¼ g1 be the solution of the following boundary value problem: ðD2 þ 1Þy þ sin x a 0 sin2 x ¼ 0; yð0Þ ¼ yðpÞ ¼ 0;
0 < x < p;
hy; sin xi ¼ 0:
From the expression of a 0 it is obvious that n1 ; g1 are well defined.
ð2:3Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1203
Theorem 2.1. There are a constant k > 1 and a continuously differentiable mapping k ! ðnk ; gk ; ak ; bk Þ from ½1; k to ðH02 Þ2 R2 such that (2.3) holds and hnk ; sin xi ¼ 0;
hgk ; sin xi ¼ 0:
Proof. Let F ¼ ðF1 ; F2 ; F3 ; F4 Þ : ðH02 Þ2 R3 ! ðL2 Þ2 R2 be defined as F1 ðn; g; a; b; kÞ ¼ ðD2 þ 1Þn þ sin x þ ðk 1Þn kðb1 aðsin x þ ðk 1ÞnÞ2 þ c1 bðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞÞ; F2 ðn; g; a; b; kÞ ¼ ðD2 þ 1Þg þ sin x þ ðk 1Þg kðb2 aðsin x þ ðk 1ÞnÞðsin x þ ðk 1ÞgÞ þ c2 bðsin x þ ðk 1ÞgÞ2 Þ; F3 ðn; g; a; b; kÞ ¼ hn; sin xi; F4 ðn; g; a; b; kÞ ¼ hg; sin xi: Substitute n1 ; g1 ; a1 ; b1 ; 1 into Fi ; i ¼ 1; 2; 3; 4; we obtain Fi ðn1 ; g1 ; a1 ; b1 ; 1Þ ¼ 0;
i ¼ 1; . . . ; 4:
The Frechet derivative of F at ðn1 ; g1 ; a1 ; b1 ; 1Þ is 0 2 D þ1 0 b1 sin2 x B 2 oF 0 D þ 1 b2 sin2 x B ¼B oðn; g; a; bÞ ðn1 ;g1 ;a1 ;b1 ;1Þ @ h; sin xi 0 0 0 h; sin xi 0
1 c1 sin2 x C c2 sin2 x C C; A 0 0
hence the system oF oðn; g; a; bÞ
0 1 0 1 n f ðxÞ B g C B gðxÞ C B C¼B C @aA @ c A ðn1 ;g1 ;a1 ;b1 ;1Þ b d
ð2:4Þ
has a unique bounded solution for f ðxÞ; gðxÞ 2 L2 ð0; pÞ; c; d 2 R. Therefore the implicit function theorem implies that there exist a constant k > 1 and a continuously differentiable mapping k ! ðnk ; gk ; ak ; bk Þ 2 ðH02 ð0; pÞ \ RðD2 þ 1ÞÞ2 R2 for k 2 ½1; k such that Fi ðnk ; gk ; ak ; bk ; kÞ ¼ 0; i ¼ 1; 2; 3; 4: Corollary 2.1. For every k 2 ½1; k , (2.1) has a positive solution ðUk ; Vk Þ which has the asymptotic expressions (2.2). Remark. The stability of steady state ðUk ; Vk Þ to the system (1.1) without delay is studied in [7,8], and the analysis with delay will be given later for comparison purpose.
3. Eigenvalue problem Let 1 < k 6 k and ðUk ðxÞ; Vk ðxÞÞ be the positive equilibrium of the system (1.1) described in Section 2. The linearized system of (1.1) around ðUk ðxÞ; Vk ðxÞÞ has the form ut ¼ uxx þ ku kðb1 Uk þ c1 Vk Þu kUk ðb1 uðt r; xÞ þ c1 vðt r; xÞÞ; vt ¼ vxx þ kv kðb2 Uk þ c2 Vk Þv kVk ðb2 uðt r; xÞ þ c2 vðt r; xÞÞ; uðt; 0Þ ¼ uðt; pÞ ¼ vðt; 0Þ ¼ vðt; pÞ ¼ 0; ðu; vÞ ¼ ð/1 ; /2 Þ;
t P 0;
ðt; xÞ 2 ½r; 0 ½0; p;
where ð/1 ; /2 Þ 2 Cð½r; 0; X X Þ;
X ¼ L2 ð0; pÞ:
t > 0; 0 < x < p;
ð3:1Þ
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Introduce the operator AðkÞ : DðAðkÞÞ ! X X defined by
b1 Uk þ c1 Vk 0 2 ; AðkÞ ¼ ðD þ kÞ k 0 b2 Uk þ c2 Vk where D2 ¼ o2 =ox2 , with domain DðAðkÞÞ ¼ ðH02 Þ2 : Set ðuðtÞ; vðtÞÞ ¼ ðuðt; Þ; vðt; ÞÞ, ð/1 ðtÞ; /2 ðtÞÞ ¼ ð/1 ðt; Þ; /2 ðt; ÞÞ, then (3.1) can be written as ! ! ! ! u b1 Uk c1 Uk uðt rÞ d u ¼ AðkÞ k ; t > 0; dt v v b2 Vk c2 Vk vðt rÞ ! ! u /1 ; t 2 ½r; 0: ¼ /2 v
ð3:2Þ
AðkÞ is an infinitesimal generator of a compact C0 semigroup [4]. The analysis of the stability of ðUk ; Vk Þ therefore yields the eigenvalue problem
y Dðk; k; rÞ ¼ 0; ð0; 0Þ 6¼ ðy; zÞ 2 DðAðkÞÞ; ð3:3Þ z where
bU Dðk; k; rÞ ¼ AðkÞ k 1 k b2 Vk
c1 Uk kr e k: c2 Vk
Let Ar ðkÞ be the infinitesimal generator of the semigroup induced by the solutions of (3.2) with
d /1 /1 ¼ ; r 6 h 6 0; Ar ðkÞ /2 dh /2 and ( DðAr ðkÞÞ ¼
0
0
ð/1 ; /2 Þ 2 C; ð/1 ; /2 Þ 2 C; ð/1 ð0Þ; /2 ð0ÞÞ 2 ðH02 Þ2 ; /01 ð0Þ /02 ð0Þ
! ¼ AðkÞ
/1 ð0Þ /2 ð0Þ
! k
b1 Uk
c1 U k
b2 Vk
c2 Vk
!
/1 ðrÞ /2 ðrÞ
!) :
Since the eigenvalues of Ar ðkÞ depend continuously on r [6], the point spectrum Pr ðAr ðkÞÞ contains a pair of imaginary eigenvalues will play a key role in the analysis of the stability and bifurcation of periodic solutions. Furthermore, it is obvious that Ar ðkÞ has an imaginary eigenvalue k ¼ im ðm 6¼ 0Þ for some r if and only if that
y b U c1 U k AðkÞ im keih 1 k ¼ 0; ðy; zÞ 6¼ ð0; 0Þ ð3:4Þ b2 Vk c2 Vk z is solvable for some value m > 0; h 2 ð0; 2pÞ. If we find ðm; hÞ such that (3.4) has a solution ðy; zÞ, then
h þ 2np y Dðk; im; rn Þ ; n ¼ 0; 1; 2; . . . ; ¼ 0; rn ¼ z m and hence rn will possibly be the candidate, we will just consider r ¼ r0 , at which the stability of steady state solution changes and stable periodic solutions occur. So the interesting question is how many pairs of ðm; hÞ 2 Rþ ð0; 2pÞ are there such that (3.4) is solvable. It will be shown that if 0 < k 1 1, there is a pair of ðm; hÞ which solves (3.4). First we will prove several lemmas which will be used to conclude our assertion. Denote by N ðBÞ and RðBÞ the null and range spaces of the operator B respectively, therefore L2 ð0; pÞ ¼ N ðD2 þ 1Þ RðD2 þ 1Þ; where N ðD2 þ 1Þ ¼ Spanfsin xg;
RðD2 þ 1Þ ¼ fy 2 L2 ð0; pÞ; hsin x; yi ¼ 0g:
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Lemma 3.1. If ðm; h; y; zÞ solves Eq. (3.4) with ð0; 0Þ 6¼ ðy; zÞ 2 bounded for k 2 ð1; k , and
ðH02 Þ2 ,
1205
then m ¼ Oðk 1Þ, and m=ðk 1Þ is uniformly
Z p mðhy; yi þ hz; ziÞ ¼ Im keih ½ðb1 y þ c1 zÞUk y þ ðb2 y þ c2 zÞVk z dx : 0
Proof. From Eq. (3.4), we have hðD2 þ k kðb1 Uk þ c1 Vk ÞÞy keih Uk ðb1 y þ c1 zÞ imy; yi þ hðD2 þ k kðb2 Uk þ c2 Vk ÞÞz keih Vk ðb2 y þ c2 zÞ imz; zi ¼ 0; since both hðD2 þ k kðb1 Uk þ c1 Vk ÞÞy; yi and hðD2 þ k kðb2 Uk þ c2 Vk ÞÞz; zi are real, the imaginary part of the above identity is as follows:
Z p ih ½ðb1 y þ c1 zÞUk y þ ðb2 y þ c2 zÞVk z dx ¼ 0; mðhy; yi þ hz; ziÞ Im ke 0
Z p m 1 ¼ ½ak ðsin x þ ðk 1Þnk Þðb1 y þ c1 zÞy þ bk ðsin x þ ðk 1Þgk Þðb2 y þ c2 zÞz dx : Im keih k1 ðhy; yi þ hz; ziÞ 0 The boundedness of m=ðk 1Þ follows from the continuity of k ! ðnk ; gk ; ak ; bk Þ and the lemma is now proved.
Lemma 3.2. If Z 2 H02 and hsin x; Zi ¼ 0, then jhðD2 þ 1ÞZ; Zij P 3kZk2L2 : Proof. By the assumption of Z, it can be represented by Z¼
1 X
cn sin nx;
kZk2L2 ¼
n¼2
1 X n¼2
p jcn j2 ; 2
and now, immediately, it follows that jhðD2 þ 1ÞZ; Zij ¼
1 X
jcn j2 ðn2 1Þ
n¼2
This completes the proof of the lemma.
p P 3kZk2L2 : 2
Now, for k 2 ð1; k , suppose that ðm; h; y; zÞ is a solution of (3.4) with ðy; zÞ 6¼ ð0; 0Þ. If we ignore a scalar factor, ðy; zÞ can be represented as y ¼ sin x þ ðk 1ÞdðxÞ;
hsin x; di ¼ 0;
z ¼ ðL þ iMÞ sin x þ ðk 1ÞvðxÞ;
hsin x; vi ¼ 0; L > 0:
ð3:5Þ
Substitute Uk ; Vk , (3.5) and m ¼ ðk 1Þh into (3.4), noting that ðD2 þ 1Þ sin x ¼ 0, we obtain the equivalent system to (3.4) g1 ðd; v; h; h; L; M; kÞ ¼ ðD2 þ 1Þd þ ð1 ihÞðsin x þ ðk 1ÞdÞ kðb1 ak ðsin x þ ðk 1Þnk Þ þ c1 bk ðsin x þ ðk 1Þgk ÞÞðsin x þ ðk 1ÞdÞ keih ak ðsin x þ ðk 1Þnk Þðb1 ðsin x þ ðk 1ÞdÞ þ c1 ððL þ iMÞ sin x þ ðk 1ÞvÞÞ ¼ 0;
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
g2 ðd; v; h; h; L; M; kÞ ¼ ðD2 þ 1Þv þ ð1 ihÞððL þ iMÞ sin x þ ðk 1ÞvÞ kðb2 ak ðsin x þ ðk 1Þnk Þ þ c2 bk ðsin x þ ðk 1Þgk ÞÞððL þ iMÞ sin x þ ðk 1ÞvÞ keih bk ðsin x þ ðk 1Þgk Þðb2 ðsin x þ ðk 1ÞdÞ þ c2 ððL þ iMÞ sin x þ ðk 1ÞvÞÞ ¼ 0; g3 ðd; v; h; h; L; M; kÞ ¼ Rehsin x; di ¼ 0; g4 ðd; v; h; h; L; M; kÞ ¼ Imhsin x; di ¼ 0; g5 ðd; v; h; h; L; M; kÞ ¼ Rehsin x; vi ¼ 0; g6 ðd; v; h; h; L; M; kÞ ¼ Imhsin x; vi ¼ 0:
ð3:6Þ
Theorem 3.1. If 0 < k 1 1, then there is a continuously differentiable mapping k ! ðdk ; vk ; hk ; hk ; Lk ; Mk Þ from ½1; k to ðH02 Þ2 R4 such that d1 ¼ ð1 ih1 Þn1 , v1 ¼ ð1 ih1 ÞL1 n1 , h1 ¼ p=2, L1 is the positive root of the following equation: c1 ðc2 c1 ÞL2 þ ðb2 c2 b1 c1 ÞL ðb1 b2 Þb2 ¼ 0; and M1 ¼ 0;
h1 ¼
c2 c1 ðb1 þ c1 L1 Þ; b1 c2 b2 c1
and ðdk ; vk ; hk ; hk ; Lk ; Mk Þ solves system (3.6) for k 2 ð1; k with n1 is defined as in Section 2. Moreover, if k 2 ð1; k , and ðdk ; vk ; hk ; hk ; Lk ; M k Þ solves, (3.6) with hk > 0 and hk 2 ð0; 2pÞ, then ðdk ; vk ; hk ; hk ; k L ; M k Þ ¼ ðdk ; vk ; hk ; hk ; Lk ; Mk Þ. Proof. Define G : ðH02 Þ2 R5 ! ðL2 Þ2 R4 by G ¼ ðg1 ; . . . ; g6 Þ, where gi ; i ¼ 1; . . . ; 6; are given in (3.6). Then it follows from the definitions of d1 ; v1 ; h1 ; h1 ; L1 ; M1 that g1 ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ ¼ ð1 ih1 Þ½ðD2 þ 1Þn1 þ sin x a 0 sin2 x ¼ 0; and gi ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ ¼ 0;
i ¼ 2; 3; 4; 5; 6:
It means that Gðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ ¼ 0: Let J ¼ ðJ1 ; . . . ; J6 Þ : ðH02 Þ2 R4 ! ðL2 Þ2 R4 be defined as J ¼ Dðd;v;h;h;L;MÞ Gðd1 ; v1 ; h1 ; h1 ; L1 ; M1 ; 1Þ; it easy to verify that J1 ðd; v; h; h; L; MÞ ¼ ðD2 þ 1Þd ih sin x þ ha1 ðb1 þ c1 L1 Þ sin2 x þ ia1 c1 ðL þ iMÞ sin2 x; J2 ðd; v; h; h; L; MÞ ¼ ðD2 þ 1Þv iL1 h sin x þ hb1 ðb2 þ c2 L1 Þ sin2 x þ ð1 ih1 ÞðL þ iMÞ sin x a 0 ðL þ iMÞ sin2 x þ ib1 c2 ðL þ iMÞ sin2 x; J3 ðd; v; h; h; L; MÞ ¼ Rehsin x; di; J4 ðd; v; h; h; L; MÞ ¼ Imhsin x; di; J5 ðd; v; h; h; L; MÞ ¼ Rehsin x; vi; J6 ðd; v; h; h; L; MÞ ¼ Imhsin x; vi: Noting that both sin x and sin2 x do not belong to RðD2 þ 1Þ, this enables us to show that J is one to one and onto from ðH02 Þ2 R4 ! ðL2 Þ2 R4 . Hence our first conclusion follows from the implicit function theorem. To obtain the second conclusion, it suffices to show that ðdk ; vk ; hk ; hk ; Lk ; M k Þ ! ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 Þ as k ! 1 in the norm of ðH02 Þ2 R4 , where Lk > 0.
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1207
First, Lemma 3.1 implies that fhk g is bounded. Using Lemma 3.2 we get from the first equation of (3.6) that Z p
Z p sin xdk dx þ ðk 1Þkdk k2L2 þ k ðb1 ak ðsin x þ ðk 1Þnk Þ 3kdk k2L2 6 j1 ihk j 0
0
Z p þ c1 bk ðsin x þ ðk 1Þgk ÞÞ sin xdk dx þ k ak ðsin x þ ðk 1Þnk Þ½b1 þ c1 ðLk þ iM k Þ sin xdk dx 0 Z p ðb1 ak ðsin x þ ðk 1Þnk Þ þ c1 bk ðsin x þ ðk 1Þgk ÞÞjdk j2 dx þ kðk 1Þ Z þ
0
0
p
Z ak ðsin x þ ðk 1Þnk Þb1 jdk j2 dx þ
p 0
ak ðsin x þ ðk 1Þnk Þc1 vk dk dx ;
it can be written as 3kdk k2L2 6 k1 kdk kL2 þ ðk2 ðjLk j þ jM k jÞ þ k3 ðk 1Þðkdk kL2 þ kvk kL2 ÞÞkdk kL2 ; that is kdk kL2 6 k1 þ k2 ðjLk j þ jM k jÞ þ k3 ðk 1Þðkdk kL2 þ kvk kL2 Þ:
ð3:7Þ
k
Here k1 ; k2 ; k3 are independent of d ; vk ; hk ; Lk ; M k . Similarly, we have kvk kL2 6 k1 þ k2 ðjLk j þ jM k jÞ þ k3 ðk 1Þðkdk kL2 þ kvk kL2 Þ:
ð3:8Þ
For k 1 1, it follows from (3.7) and (3.8) that kdk kL2 þ kvk kL2 6 k1 þ k2 ðjLk j þ jM k jÞ:
ð3:9Þ
On the other hand we get from the solvable condition of the first two equations of (3.6) that hð1 ihk Þðsin x þ ðk 1Þdk Þ kðb1 ak ðsin x þ ðk 1Þnk Þ þ c1 bk ðsin x þ ðk 1Þgk ÞÞðsin x þ ðk 1Þdk Þ k
keih ak ðsin x þ ðk 1Þnk Þðb1 ðsin x þ ðk 1Þdk Þ þ c1 ððLk þ iM k Þ sin x þ ðk 1Þvk ÞÞ; sin xi ¼ 0; hð1 ihk ÞððLk þ iM k Þ sin x þ ðk 1Þvk Þ kðb2 ak ðsin x þ ðk 1Þnk Þ þ c2 bk ðsin x þ ðk 1Þgk ÞÞ
ð3:10Þ
k
ððLk þ iM k Þ sin x þ ðk 1Þvk Þ keih bk ðsin x þ ðk 1Þgk Þðb2 ðsin x þ ðk 1Þdk Þ þ c2 ððLk þ iM k Þ sin x þ ðk 1Þvk ÞÞ; sin xi ¼ 0: Taking real part and imaginary part respectively in the first equality of (3.10) we get jLk j 6 k4 þ k5 ðk 1Þðkdk kL2 þ kvk kL2 Þ;
ð3:11Þ
jM k j 6 k4 þ k5 ðk 1Þðkdk kL2 þ kvk kL2 Þ:
If 0 < k 1 1 then it follows that kdk kL2 , kvk kL2 , jLk j and jM k j are bounded from (3.9) and (3.11). Return to the first two equations of (3.6), then we get that fdk g and fvk g are compact in H02 . Therefore it implies that fðdk ; vk ; hk ; hk ; Lk ; M k Þ; k 2 ð1; k g is precompact in ðH02 Þ2 R4 . Let fdkn ; vkn ; hkn ; hkn ; Lkn ; M kn g be any convergent subsequence such that ðdkn ; vkn ; hkn ; hkn ; Lkn ; M kn Þ ! ðd1 ; v1 ; h1 ; h1 ; L1 ; M 1 Þ; kn ! 1
as n ! 1:
We claim that ðd1 ; v1 ; h1 ; h1 ; L1 ; M 1 Þ ¼ ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 Þ: In fact, taking the limit to the first two equations of (3.6), we obtain 1
ðD2 þ 1Þd1 þ ð1 ih1 Þ sin x ðb1 a1 þ c1 b1 Þ sin2 x eih a1 ðb1 þ c1 ðL1 þ iM 1 ÞÞ sin2 x ¼ 0; 1
ðD2 þ 1Þv1 þ ðL1 þ iM 1 Þð1 ih1 Þ sin x ðb2 a1 þ c2 b1 Þ sin2 x eih b1 ðb2 þ c2 ðL1 þ iM 1 ÞÞ sin2 x ¼ 0:
ð3:12Þ
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Eq. (3.12) implies that 1
ih1 a 0 ¼ eih a1 ðb1 þ c1 ðL1 þ iM 1 ÞÞ and 1
ih1 ðL1 þ iM 1 Þa 0 ¼ eih b1 ðb2 þ c2 ðL1 þ iM 1 ÞÞ; thus L1 þ iM 1 is a root of c1 ðc2 c1 ÞL2 þ ðc2 b2 c1 b1 ÞL ðb1 b2 Þb2 ¼ 0: Therefore M 1 ¼ 0, h1 ¼ p=2, L1 ¼ L1 and h1 ¼ h1 . Then we conclude that ðdk ; vk ; hk ; hk ; Lk ; M k Þ ! ðd1 ; v1 ; h1 ; h1 ; L1 ; M1 Þ
as k ! 1
in ðH02 Þ2 R4 . This completes the proof of Theorem 3.1.
The following corollary is an immediate consequence of Theorem 3.1. Corollary 3.1. If 0 < k 1 1, then for each k 2 ð1; k the eigenvalue problem
y Dðk; im; rÞ ¼ 0; m > 0; r > 0; ðy; zÞ 6¼ ð0; 0Þ; z has a solution ðm; r; y; zÞ or equivalently im 2 Pr ðAr ðkÞÞ if and only if mk ¼ ðk 1Þhk ;
r ¼ r kn ¼
hk þ 2np ; n ¼ 0; 1; 2; . . . ; mk
and
sin x þ ðk 1Þdk y y ; ¼c k ¼c zk ðLk þ iMk Þ sin x þ ðk 1Þvk z here c is an arbitrary non-zero constant, and dk ; vk ; hk ; hk ; Lk ; Mk are described in Theorem 3.1.
4. Stability of the positive equilibrium In this section we study the stability of the positive equilibrium ðUk ; Vk Þ with k 2 ð1; k fixed, and the delay r considered as a parameter. To describe stability of ðUk ; Vk Þ it suffices to investigate how the eigenvalue k ¼ im varies as the delay r passing through rkn ; n ¼ 0; 1; . . . First we have to solve the adjoint problem of (3.4) as
yk b U b2 Vk ¼ 0: ð4:1Þ AðkÞ imk eihk k 1 k z k c1 Uk c2 Vk Similarly we let yk ¼ sin x þ ðk 1Þd k ;
z k ¼ ðL k þ iMk Þ sin x þ ðk 1Þv k :
ð4:2Þ
After the same argument, we obtain, there is a continuously differentiable mapping k ! ðd k ; v k ; L k ; Mk Þ, from ½1; k to ðH02 Þ2 R2 such that (4.2) satisfies (4.1), and a1 c1 L1 ; d 1 ¼ ð1 ih1 Þn1 ; v 1 ¼ ð1 ih1 ÞL 1 n1 ; M1 ¼ 0: L 1 ¼ b1 b2 Denote C ¼ Cð½0; rkn ; X X Þ, the basis of eigenspace in C of A rkn ðkÞ, with k ¼ imk is expressed as !
~ð1Þ ðyk ; z k Þeimk h w ~ w ¼ ~ð2Þ ¼ ; 0 6 h 6 r kn : imk h ðyk ; zk Þe w
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1209
Similarly we choose a basis of eigenspace in C of Arkn ðkÞ with k ¼ imk as
yk imk h yk imk h /~ ¼ ð/~ð1Þ ; /~ð2Þ Þ ¼ e ; e ; rkn 6 h 6 0: zk zk Then we calculate the respective inner product Z p Z w~ð1Þ ð0Þ/~ð1Þ ð0Þ dx k Skn ¼ ðw~ð1Þ ; /~ð1Þ Þ ¼
bU w~ð1Þ ðs þ rkn Þ 1 k b2 Vk 0 rkn 0
Z p Z p yk b U c1 Uk dx: ðyk yk þ z k zk Þ dx keihk rkn ðyk ; z k Þ 1 k ¼ b2 Vk c2 Vk zk 0 0 Z
0
p
c1 Uk ~ð1Þ / ðsÞ dx ds c2 Vk ð4:3Þ
Lemma 4.1. If 0 < k 1 1, then for each k 2 ð1; k , Skn 6¼ 0. Proof. Since rkn ¼ ðhk þ 2npÞ=½hk ðk 1Þ; and Uk ¼ ðk 1Þak sin x þ Oðk 1Þ2 ;
Vk ¼ ðk 1Þbk sin x þ Oðk 1Þ2 :
Therefore p
Im Skn ! 2
þ 2np ba ð1; L 1 Þ 1 1 b2 b1 h1
c1 a1 c2 b1
This completes the proof of Lemma 4.1.
1 L1
Z
p
sin3 x dx 6¼ 0
as k ! 1:
0
With the aid of Lemma 4.1, we can prove the following lemma, see Appendix A. Lemma 4.2. For each k 2 ð1; k ð0 < k 1 1Þ and n ¼ 0; 1; . . . ; k ¼ imk is a simple eigenvalue of Arkn ðkÞ. Now by the implicit function theorem, it is not difficult to show that there are a neighborhood of ðrkn ; imk ; yk ; zk Þ in 2 2 2 2 Dkn Ckn ðH0k Þ R C ðH02 Þ2 and a continuously differentiable mapping: Dkn ! Ckn ðH0k Þ such that for n n r 2 Dkn the only eigenvalue of Ar ðkÞ in Ckn is kðrÞ and kðrkn Þ ¼ imk ; yðrkn Þ ¼ yk ; zðrkn Þ ¼ zk ;
yðrÞ Dðk; kðrÞ; rÞ ¼ 0; r 2 Dkn : zðrÞ Differentiating the above equality with respect to r at rkn , we have
0
yk y ðr Þ b1 Uk c1 Uk b1 Uk k0 ðrkn Þ 1 þ krkn eihk þ imk keihk þ Dðk; imk ; rkn Þ 0 kn b2 Vk c2 Vk zk b2 Vk z ðrkn Þ Multiplying by ðyk ; z k Þ, and integrating on ð0; pÞ and noting that
0 Z p y ðr Þ dx ¼ 0; ðyk ; z k ÞDðk; imk ; rkn Þ 0 kn z ðrkn Þ 0 then we obtain k0 ðrkn ÞSkn ¼
Z
p
imk keihk ðyk ; z k Þ 0
b1 Uk b2 Vk
c1 U k c2 Vk
yk zk
dx;
noting that Skn ¼
Z
p
ðyk yk þ z k zk Þ krkn eihk ðyk ; z k Þ
0
therefore, k0 ðrkn Þ ¼ ðI1 þ I2 Þ=jSkn j2 ;
b1 Uk b2 Vk
c1 U k c2 Vk
yk zk
dx;
c1 U k c2 Vk
yk zk
¼ 0:
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
where I1 ¼
Z
p
ðyk yk þ z k zk Þ dx
0
and
Z
p
imk keihk ðyk ; z k Þ
0
Z p
bU ðyk ; z k Þ 1 k I2 ¼ ik 2 mk rkn b2 Vk 0
c1 U k c2 Vk
yk zk
b1 Uk b2 Vk
c1 U k c2 Vk
yk zk
dx;
2 dx :
Here I2 is a pure imaginary, so we have: Lemma 4.3. For each k 2 ð1; k ð0 < k 1 1Þ, Re k0 ðrkn Þ > 0;
n ¼ 0; 1; . . . :
Proof. Since Re k0 ðrkn Þ ¼ ReðI1 =jSkn j2 Þ and ieihk ! 1 as k ! 1, Z p p as k ! 1; ðyk yk þ z k zk Þ dx ! ð1 þ L1 L 1 Þ 2 0 and Z
p
ðyk ; z k Þ 0
b1 Uk b2 Vk
c1 Uk c2 Vk
yk zk
dx ¼
ba ð1; L 1 Þ 1 k b2 bk
c1 ak c2 b k
0
1 L1
Z
p
sin3 x dx ðk 1Þ þ Oðk 1Þ2 :
0
Therefore, if 0 < k 1 1 we have Re k ðrkn Þ > 0: This completes the proof.
From Corollary 3.1 and Lemma 4.3 we have Theorem 4.1. The positive equilibrium ðUk ; Vk Þ is locally asymptotically stable if 0 6 r < rk0 , and unstable if r > rk0 .
5. Hopf bifurcation In this section we study the Hopf bifurcation which arises from the positive equilibrium ðUk ðxÞ; Vk ðxÞÞ as the delay r crosses rk0 ¼ r0 . The generic Hopf bifurcation theorem is well established [11]. Our purpose here is to give more detailed information about the Hopf bifurcation for the delay competition diffusion system (1.1). In fact we are going to prove the following theorem. Theorem 5.1. For each fixed k 2 ð1; k , Hopf bifurcation occurs as the delay increasingly passes through r0 . Specifically for r0 , there is a d0 > 0 such that for each r 2 ðr0 ; r0 þ d0 Þ, system (1.1) has a periodic solution ðUk;r ; Vk;r Þ near ðUk ðxÞ; Vk ðxÞÞ with period 2p=mk . Furthermore, ðUk;r ; Vk;r Þ is locally asymptotically stable. 5.1. Asymptotic expression For fixed k 2 ð1; k and r ¼ r0 þ a, let uðtÞ ¼ uðt; Þ ¼ U ðt; Þ Uk ðxÞ;
vðtÞ ¼ vðt; Þ ¼ V ðt; Þ Vk ðxÞ:
Thus we have a system equivalent to the original one (1.1)
d u u b U c1 Uk u u b uðtÞ c1 uðtÞ ¼ AðkÞ k 1 k ðt r0 aÞ: ðt r0 aÞ k 1 v b2 Vk c2 Vk v v b2 vðtÞ c2 vðtÞ dt v
ð5:1Þ
ð5:2Þ
Further, letting xk ¼ 2p=mk and w1 ðtÞ ¼ uðð1 þ bÞtÞ; w2 ðtÞ ¼ vðð1 þ bÞtÞ, and ðuðtÞ; vðtÞÞ is an xk ð1 þ bÞ periodic solution of (5.2) if and only if ðw1 ðtÞ; w2 ðtÞÞ is an xk periodic solution of
d w1 w1 b U c1 Uk w1 ðt r0 Þ þ Gða; b; wt Þ; ð5:3Þ ¼ AðkÞ k 1 k w2 ðt r0 Þ w2 b2 Vk c2 Vk dt w2
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
where
w1 Gða; b; wt Þ ¼ bAðkÞ w2
ð1 þ bÞk
1211
1 0 r0 þa c1 Uk @ w1 ðt r0 Þ ð1 þ bÞw1 t 1þb A c2 Vk 0 þa w2 ðt r0 Þ ð1 þ bÞw2 t r1þb 1 0 r0 þa c1 w1 ðtÞ @ w1 t 1þb A : c2 w2 ðtÞ w2 t r0 þa
bU þk 1 k b2 Vk b1 w1 ðtÞ b2 w2 ðtÞ
1þb
We use the following notations:
Rp y ¼ 0 ðy1 ðxÞy2 ðxÞ þ z1 ðxÞz2 ðxÞÞ dx: 1. ðy1 ; z1 Þ; 2 z2
R0 b1 Uk c1 Uk /ðsÞ ds: 2. ðw; /Þ ¼ hwð0Þ; /ð0Þi k r0 wðs þ r0 Þ; b2 Vk c2 Vk
y y k k ~ ðhÞ ¼ eimk h ; eimk h ; r0 6 h 6 0; 3. U zk zk ! 1 ðy ; z Þeimk h ~ ðhÞ ¼ Sk10 k k im h ; 0 6 h 6 r0 ; W ðyk ; zk Þe k S k0
1 i 1 1 ~ ~ : U ¼ UH ; W ¼ H W; H ¼ 2 1 i 4. Let K be the eigenspace of Ar0 ðkÞ corresponding to the eigenvalues k ¼ imk : 5. Let Pxk be a Banach space defined as f1 P xk ¼ 2 CðR; X X Þ; fi ðt þ xk Þ ¼ fi ðtÞ; i ¼ 1; 2; t 2 R : f2 6. qi : Pxk ! R; i ¼ 1; 2; are defined by Z xk Z p ðiÞ ðiÞ ðW1 f1 þ W2 f2 Þ dx dt; i ¼ 1; 2; qi f ¼
q¼
0
0
q1 : q2
With the above notations, we can verify that U is a real basis of K, and W is a real basis of the eigenfunction subspace of the formal adjoint operator. By directly computing it is clear that ðW; UÞ ¼ I2 , an identity matrix. It is known by [2,6] that for each f 2 Pxk , the equation
dw b U c1 U k ¼ AðkÞw k 1 k wðt r0 Þ þ f ðtÞ ð5:4Þ b2 Vk c2 Vk dt has an xk periodic solution if and only if f 2 N ðqÞ, that is qi f ¼ 0; i ¼ 1; 2. Let K : N ðqÞ ! Pxk be the linear operator such that Kf is the xk periodic solution of (5.4) satisfying ðW; ðKf Þ0 Þ ¼ 0; where ðKf Þ0 is defined by ðKf Þ0 ðhÞ ¼ Kf ðhÞ;
h 2 ½r0 ; 0:
Then up to time translation, Eq. (5.3) has an xk periodic solution wðtÞ if and only if there is a constant c such that qGða; b; wÞ ¼ 0;
ð5:5Þ
wðtÞ ¼ cUð1Þ ðtÞ þ ½KGða; b; wÞðtÞ;
t 2 R;
where Uð1Þ ¼
1 2
yk imk t y k imk t e þ e ; zk zk
t 2 R:
Introducing the change of variables a ¼ cc; b ¼ cd and wðtÞ ¼ c½Uð1Þ ðtÞ þ cW ðtÞ;
t 2 R; W ðtÞ 2 Pxk ; ðW; ðW Þ0 Þ ¼ 0;
ð5:6Þ
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
then (5.5) and (5.6) are equivalent to Z xk hWðsÞ; N ðc; c; d; W Þi ds ¼ 0 J ðc; c; d; W Þ ¼
ð5:7Þ
0
and
W ¼ KN ðc; c; d; W Þ ¼ K
N1 N2
ð5:8Þ
with ð1Þ
N1 ¼ dðD2 þ k kðb1 Uk þ c1 Vk ÞÞðU1 þ cW1 Þ þ ka
Z
1
ð1Þ
ð1Þ
½b1 Uk U_ 1 ðt r0 hcaÞ þ c1 Uk U_ 2 ðt r0 hcaÞ dh
0 ð1Þ
ð1Þ
dkðb1 Uk ðU1 þ cW1 Þ þ c1 Uk ðU2 þ cW2 ÞÞjtq þ k½b1 Uk ðW1 ðt r0 Þ W1 ðt qÞÞ þ c1 Uk ðW2 ðt r0 Þ ð1Þ
ð1Þ
ð1Þ
W2 ðt qÞÞ kð1 þ cdÞðU1 þ cW1 Þðb1 ðU1 ðt qÞ þ cW1 ðt qÞÞ þ c1 ðU2 ðt qÞ þ cW2 ðt qÞÞÞ and ð1Þ
N2 ¼ dðD2 þ k kðb2 Uk þ c2 Vk ÞÞðU2 þ cW2 Þ þ ka
Z
1
ð1Þ ð1Þ ½b2 Vk U_ 1 ðt r0 hcaÞ þ c2 Vk U_ 2 ðt r0 hcaÞ dh
0 ð1Þ
ð1Þ
dkðb2 Vk ðU1 þ cW1 Þ þ c2 Vk ðU2 þ cW2 ÞÞjtq þ k½b2 Vk ðW1 ðt r0 Þ W1 ðt qÞÞ þ c2 Vk ðW2 ðt r0 Þ ð1Þ
ð1Þ
ð1Þ
W2 ðt qÞÞ kð1 þ cdÞðU2 þ cW2 Þðb2 ðU1 ðt qÞ þ cW1 ðt qÞÞ þ c2 ðU2 ðt qÞ þ cW2 ðt qÞÞÞ; where a ¼ ðc dr0 Þ=ð1 þ cdÞ; q ¼ ðcc þ r0 Þ=ð1 þ cdÞ: Since a periodic solution is a C 1 function without loss of generality, we can restrict our discussion on (5.7) and (5.8) for W 2 Px1k ¼ ff j f 2 Pxk ; f_ 2 Pxk g: Thus it is easy to see that the mapping J : Ip Ip Ip Px1k ! R2 is continuous and continuously differentiable, where Ip ¼ ðp; þpÞ with 0 < p < 1. Lemma 5.1. J ð0; 0; 0; W Þ ¼ 0: Lemma 5.2.
oJ ð0; 0; 0; W Þ Re k0 ðr0 Þ 0 ¼ xk ; Im k0 ðr0 Þ mk oðc; dÞ where kðrÞ and k0 ðrÞ are defined in the previous section. Proof. By directly calculating oJ ð0; 0; 0; W Þ ¼ oc
Z
xk
Z xk
oN ð0; c; 0; W Þ 1 ~ ; b1 Uk W ds ¼ kH b2 Vk oc 0 c¼0
1 A11 A12 dt H ; A21 A22 0
W; 0
¼ H 1
Z
xk 0
where yk dx b1 Uk c1 Uk ¼ k0 ðr0 Þ; A11 ¼ ke imk b V c V zk Sk0 2 k 2 k 0
Z p b U c1 U k y k dx A12 ¼ keihk e2imk t ðyk ; z k Þ 1 k ; b2 Vk c2 Vk zk Sk0 0 ihk
A21 ¼ A12 ;
Z
p
ðyk ; z k Þ
A22 ¼ A11 :
1 c1 Uk ~_ ð1Þ U ðt r0 Þ dt H c2 Vk 0
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1213
Therefore
oJ ð0; 0; 0; W Þ Re k0 ðr0 Þ : ¼ xk 0 Im k ðr0 Þ oc
ð5:9Þ
Similar calculation yields
oJ ð0; 0; 0; W Þ 0 : ¼ xk mk od Hence Lemma 5.2 follows from (5.9) and (5.10).
ð5:10Þ
Lemma 5.3. Let ! ð1Þ ð1Þ ð1Þ U1 ðb1 U1 ðt r0 Þ þ c1 U2 ðt r0 ÞÞ : ð1Þ ð1Þ ð1Þ U2 ðb2 U1 ðt r0 Þ þ c2 U2 ðt r0 ÞÞ
Wk ¼ kK
ð5:11Þ
Then Wk ¼ n1k e2imk t þ n2k þ n1k e2imk t þ UðtÞd;
ð5:12Þ
where n1k
¼
n2k ¼
2hk i
AðkÞ ke
AðkÞ k
1
k yk ðb1 yk þ c1 zk Þeihk c1 U k 2imk ; c2 Vk 4 zk ðb2 yk þ c2 zk Þeihk 1
k yk ðb1 y k þ c1 zk Þeihk þ y k ðb1 yk þ c1 zk Þeihk c1 U k ; c2 Vk 4 zk ðb2 y k þ c2 zk Þeihk þ zk ðb2 yk þ c2 zk Þeihk
b1 Uk b2 Vk
b1 Uk b2 Vk
and
d¼
d1 d2
¼ ðW; n1k e2imk t þ n2k þ n1k e2imk t Þ:
This lemma can be proved by the separate variable method. Furthermore, we can verify that ðW; Wk Þ ¼ 0:
ð5:13Þ
The detail is omitted. Lemma 5.4. Let qk ¼
1 y ½yk ðb1 n2k1 þ c1 n2k2 Þ þ n2k1 ðb1 yk þ c1 zk Þeihk þ y k ðb1 n1k1 þ c1 n1k2 Þe2ihk þ n1k1 ðb1 y k þ c1 zk Þeihk Sk0 k þ z k ½zk ðb2 n2k1 þ c2 n2k2 Þ þ n2k2 ðb2 yk þ c2 zk Þeihk þ zk ðb2 n1k1 þ c2 n1k2 Þe2ihk þ n1k2 ðb2 y k þ c2 zk Þeihk ;
ð5:14Þ
and oJ ð0; 0; 0; Wk Þ ¼ oc
T1 : T2
ð5:15Þ
Reqk dx:
ð5:16Þ
Then 1 T1 ¼ kxk 2
Z
p 0
Lemma 5.5. Let n1k and n2k be defined as in Lemma 5.3. Then lim n1k ðk 1Þ ¼ m11 sin x; k!1
lim n2k ðk 1Þ ¼ 0; k!1
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
in X X , where
i b1 a1 2ih1 a 0 m11 ¼ b2 b1 4
c1 a1 c2 b1 2ih1 a 0
1
b1 þ c1 L1 : L1 ðb2 þ c2 L1 Þ
ð5:17Þ
The proofs of both Lemmas 5.4 and 5.5 will be given in Appendix A. Rp In order to prove the following main theorem, we need some information about sign of Re 0 qk ðxÞ dx, but the expression of qk ðxÞ is complicate, for the sake of simplicity, we verify it by some numerical program but for a kind of special case, which requires the parameters b1 , b2 , c1 and c2 satisfying the following relations: b1 ¼ c2 ;
b2 ¼ c1 ;
and
b1 > b2 :
ðSÞ
Now we give the complete analysis. Condition (S) indicates a symmetric property. Lemma 5.6. Suppose that condition (S) holds, then a1 ¼ b1 ¼ a 0 =ðb1 þ b2 Þ
ð5:18Þ
L1 ¼ h1 ¼ L 1 ¼ 1:
ð5:19Þ
and
The proof of this lemma is simple. Lemma 5.7. If 0 < k 1 1 and ðSÞ holds, then Z p 1 qk ðxÞ dx < 0: q k ¼ Re 2 0
ð5:20Þ
Proof. From lemmas established in the previous it follows that qk ¼
1 ðy ðy ðb1 m111 þ cm112 Þe2ihk þ m111 ðb1 y k þ c1 zk Þeihk Þ þ z k ðzk ðb2 m111 þ c2 m112 Þe2ihk þ m112 ðb2 y k Sk0 ðk 1Þ k k þ c2 zk Þeihk ÞÞ sin x þ Oðk 1Þ :
By the condition (S) we have
ð2 iÞðb1 þ b2 Þ 1 ; m11 ¼ 1 20a 0
8 pi Sk0 ¼ a 0 1 þ þ Oðk 1Þ: 3 2
It follows that q k ¼
1 2jSk0 j2 ðk 1Þ
Re
p 16 3 ðb1 þ b2 Þ2 p1þ þ 3 i þ Oðk 1Þ: 45 2 2
The proof of the lemma is completed.
5.2. Proof of Theorem 5.1 We are now in the position to prove our main result Theorem 5.1. First it follows from Lemmas 5.1 and 5.2 that there are neighborhoods B R of the origin and V0 Pxk of Wk ; c0 > 0, and continuous differentiable functions c : ½c0 ; c0 V0 ! B, d : ½c0 ; c0 V0 ! B such that cð0; Wk Þ ¼ dð0; Wk Þ ¼ 0, and for each ðc; W Þ 2 ½c0 ; c0 V0 ; ðc; dÞ 2 B B, J ðc; c; d; W Þ ¼ 0 if and only if c ¼ cðc; W Þ;
d ¼ dðc; W Þ:
We now define a mapping F : ½c0 ; c0 V0 ! Px1k by F ðc; W Þ ¼ W KN ðc; cðc; W Þ; dðc; W Þ; W Þ:
ð5:21Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1215
Then F ð0; Wk Þ ¼ Wk KN ð0; 0; 0; Wk Þ ¼ 0:
ð5:22Þ
Moreover, Lemma 5.1 implies that oJ ð0; 0; 0; Wk Þ ¼ 0: oW
ð5:23Þ
So ocð0;Wk Þ oW odð0;Wk Þ oW
!
¼
oJ ð0; 0; 0; Wk Þ oðc; dÞ
1
oJ ð0; 0; 0; Wk Þ ¼ 0; oW
this yields that oF ð0; Wk Þ oN ð0; 0; 0; Wk Þ ¼I K ¼ I: oW oW Therefore, oF ð0; Wk Þ=oW : Px1k ! Px1k is one to one and onto. It follows from the implicit function theorem that there are a constant c1 2 ð0; c0 , a neighborhood V1 V0 of Wk and a function W : ½c1 ; c1 ! V1 , such that W ð0Þ ¼ Wk ; and for each ðc; W Þ 2 ½c1 ; c1 V1 ; F ðc; W Þ ¼ 0 if and only if W ¼ W ðcÞ: Consequently, Eq. (5.3) has an xk periodic solution wðtÞ near zero for small a and b if and only if wðtÞ ¼ cðUð1Þ ðtÞ þ cW ðcÞðtÞÞ;
ð5:24Þ
a ¼ ccðc; W ðcÞÞ;
ð5:25Þ
b ¼ cdðc; W ðcÞÞ
ð5:26Þ
and
for some value of c 2 ½c1 ; c1 . Now let c ðcÞ ¼ cðc; W ðcÞÞ, d ðcÞ ¼ dðc; W ðcÞÞ, since J ðc; c ðcÞ; d ðcÞ; W ðcÞÞ 0, c 2 ½c1 ; c1 . Differentiating both sides of this equality at c ¼ 0 gives oJ ð0; 0; 0; Wk Þ oJ ð0; 0; 0; Wk Þ þ oc oðc; dÞ
dc ð0Þ dc dd ð0Þ dc
! ¼ 0:
Lemma 5.2 implies that !
1 dc ð0Þ 1 oJ ð0; 0; 0; Wk Þ Re k0 ðr0 Þ 0 dc ¼ : 0 dd ð0Þ ðr Þ m Im k x oc 0 k k dc Applying Lemma 5.4 we obtain dc ð0Þ T1 q k ¼ ¼ > 0: 0 dc xk Re k ðr0 Þ 2Re k0 ðr0 Þ Therefore, for sufficiently small c, a ¼ cc ðcÞ ¼
q k c2 þ Oðc3 Þ: 2Re k0 ðr0 Þ
Hence we know the bifurcation direction that the Hopf bifurcation occurs as the delay r increasingly passes through r0 . About the stability of periodic solutions, we have to reduce the nonlinear system to two-dimensional center manifold corresponding to the eigenspace with eigenvalue k ¼ imk of the operator Ar0 ðkÞ. By the basic theory in [6], we can use the center manifold theorem to approach the detail about stability, but we give the discussion in Appendix A.
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6. Numerical results In this section we give some numerical result for the asymptotic behavior in the time delay logistic equation on a onedimensional spatial domain X ¼ ð0; 1Þ: ut ¼ uxx þ kuðt; xÞ½1 uðt r; xÞ; t > 0; 0 < x < 1; uðt; 0Þ ¼ uðt; 1Þ ¼ 0; t P 0; uðt; xÞ ¼ 0:1ð1 þ t=rÞ sinðpxÞ; r 6 t 6 0; 0 6 x 6 1:
ð6:1Þ
By discreting system (6.1) into a finite difference system, numerical solution is then solved through the forward elimination and backward substitution. When X ¼ ð0; 1Þ, the principal eigenvalue of the Laplacian operator under Dirichlet boundary conditions is k0 ¼ p2 . Our numerical simulations for (6.1) are given in Figs. 1–3. We choose k ¼ 9:8 < p2 ; r ¼ 30 in (6.1), Fig. 1 demonstrates that the solution converges asymptotically to the trivial steady state. When k ¼ 10:1 > p2 ; r ¼ 310, Fig. 2 shows that the solution converges asymptotically to the non-constant steady state. Now we choose k ¼ 10:2 > p2 ; r ¼ 313, Fig. 3 shows that there exists a Hopf bifurcation periodic solution.
Appendix A A.1. Proof of Lemma 4.2 Firstly, it follows from Corollary 3.1 that dim N ½Arkn ðkÞ imk ¼ 1;
n ¼ 0; 1; . . . ;
with
N ½Arkn ðkÞ imk ¼ Span
yk imk h e ; h 2 ½ rkn ; 0 : zk
Fig. 1. The solution of (6.1) converges to the trivial steady state.
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Fig. 2. The solution of (6.1) converges to the non-constant steady state.
Fig. 3. The solution of (6.1) converges to a Hopf bifurcation periodic solution.
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Now, we suppose that / 2 DðArkn ðkÞÞ \ Dð½Arkn ðkÞ2 Þ; and ½Arkn ðkÞ imk 2 / ¼ 0:
ðA:1Þ
It follows that ðArkn ðkÞ imk Þ/ ¼ ceimk h
yk ; zk
ðA:2Þ
where c is a constant, hence we have
y /_ ðhÞ ¼ imk / þ ceimk h k ; h 2 ½rkn ; 0; zk
ðA:3Þ
and
bU /_ ð0Þ ¼ AðkÞ/ð0Þ k 1 k b2 Vk
c1 U k /ðrkn Þ: c2 Vk
ðA:4Þ
From (A.3) we get
y /_ ð0Þ ¼ imk /ð0Þ þ c k zk
ðA:5Þ
y /ðhÞ ¼ eimk h /ð0Þ þ ch k eimk h ; zk
ðA:6Þ
and
set h ¼ rkn in (A.6) we get /ðrkn Þ ¼ eihk /ð0Þ crkn eihk
yk : zk
ðA:7Þ
Substituting (A.5) and (A.7) into (A.4), we have
b1 Uk c keihk rkn b2 Vk
c1 Uk c2 Vk
yk zk
¼
AðkÞ keihk
b1 Uk b2 Vk
c1 U k c2 Vk
imk /ð0Þ:
Multiplying (A.8) by ðyk ; z k Þ and integrating on ½0; p, then we obtain cSkn ¼
Z
p
AðkÞ keihk
0
b1 Uk c1 U k
b2 Vk c2 Vk
imk
yk z k
T
/1 ð0Þ /2 ð0Þ
dx ¼ 0:
Hence by the Lemma 4.1 we get c ¼ 0. So ½Arkn ðkÞ imk / ¼ 0;
/ 2 N ðArkn ðkÞ imk Þ:
By induction we have N ð½Arkn ðkÞ imk j Þ ¼ N ðArkn ðkÞ imk Þ;
j ¼ 1; 2; . . . ; n ¼ 0; 1; . . .
Therefore, k ¼ imk is a simple eigenvalue of Arkn ðkÞ for n ¼ 0; 1; . . . This completes the proof.
A.2. Proof of Lemma 5.4 Let W~k ¼ n1k e2imk t þ n2k þ n1k e2imk t . Then Wk ¼ W~k þ Ud:
ðA:8Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
It is obvious that ! Z xk * ð1Þ ð1Þ ðUdÞ1 ðb1 U1 ðt r0 Þ þ c1 U2 ðt r0 ÞÞ þ W; ð1Þ ð1Þ ðUdÞ2 ðb2U1 ðt r0 Þ þ c2 U2 ðt r0 ÞÞ 0
ð1Þ
U1 ðb1 ðUdÞ1 ðt r0 Þ þ c1 ðUdÞ2 ðt r0 ÞÞ ð1Þ U2 ðb2 ðUdÞ1 ðt r0 Þ þ c2 ðUdÞ2 ðt r0 ÞÞ
Also from (5.7) it is clear that Z xk * oJ ð0; 0; 0; Wk Þ ¼ k W; oc 0
ð1Þ U1 ðsÞðb1 W~k1 ðs r0 Þ þ c1 W~k2 ðs r0 ÞÞ ð1Þ U2 ðsÞðb2 W~k1 ðs r0 Þ þ c2 W~k2 ðs r0 ÞÞ !+ ð1Þ ð1Þ W~k1 ðb1 U1 ðs r0 Þ þ c1 U2 ðs r0 ÞÞ ds þ ð1Þ ð1Þ W~k2 ðb2 U1 ðs r0 Þ þ c2 U2 ðs r0 ÞÞ Z xk D1 W; ds; ¼ k D2 0
1219
!+ dt ¼ 0:
!
where D1 ¼
1h yk ðb1 n2k1 þ c1 n2k2 Þeimk t þ y k ðb1 n2k1 þ c1 n2k2 Þeimk t þ n2k1 ððb1 yk þ c1 zk Þeimk tihk þ ðb1 y k þ c1 zk Þeimk tþihk Þ 2 þ yk ðb1 n1k1 þ c1 n1k2 Þe2ihk imk t þ y k ðb1 n1k1 þ c1 n1k2 Þe2ihk þimk t þ n1k1 ðb1 y k þ c1 zk Þeihk þimk t i þ n1k1 ðb1 yk þ c1 zk Þeihk imk t þ R 1 ;
D2 ¼
1h zk ðb2 n2k1 þ c2 n2k2 Þeimk t þ zk ðb2 n2k1 þ c2 n2k2 Þeimk t þ n2k2 ððb2 yk þ c2 zk Þeimk tihk þ ðb2 y k þ c2 zk Þeimk tþihk Þ 2 þ zk ðb2 n1k1 þ c2 n1k2 Þe2ihk imk t þ zk ðb2 n1k1 þ c2 n1k2 Þe2ihk þimk t þ n1k2 ðb2 y k þ c2 zk Þeihk þimk t i þ n1k2 ðb2 yk þ c2 zk Þeihk imk t þ R 2 ;
where each term of R 1 and R 2 has a factor as e2imk t ; 1 or e2imk t . Recall (5.14), (5.15) and
imk t yk e y eimk t z k eimk t z k eimk t þ k ; ; þ Wð1Þ ¼ Sk0 Sk0 Sk0 Sk0 we have 1 T1 ¼ kxk 2
Z
p
Reqk dx:
0
A.3. Proof of Lemma 5.5 For k 2 ð1; k we decompose nik ; i ¼ 1; 2; as
i 1 n mik sin x þ ðk 1Þ k1 nik ¼ ; i ¼ 1; 2; ni k1 k2
ðA:9Þ
where mikj ; i ¼ 1; 2; j ¼ 1; 2; are complex numbers, and hni kj ; sin xi ¼ 0;
i; j ¼ 1; 2:
Since
b U þ c1 Vk ðD2 þ 1Þ þ ðk 1Þ k 1 k 0
k yk ðb1 yk þ c1 zk Þ ihk e ; ¼ 4 zk ðb2 yk þ c2 zk Þ
0 b2 Uk þ c2 Vk
ke2ihk
b1 Uk b2 Vk
c1 Uk c2 Vk
2imk n1k
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
b U þ c1 Vk 0 ðD2 þ 1Þ þ ðk 1Þ k 1 k 0 b2 Uk þ c2 Vk
k yk ðb1 y k þ c1 zk Þeihk þ y k ðb1 yk þ c1 zk Þeihk ; ¼ 4 zk ðb2 y k þ c2 zk Þeihk þ zk ðb2 yk þ c2 zk Þeihk
k
b1 Uk b2 Vk
c1 U k c2 Vk
n2k
by substituting (A.7) into above two equations then using the solvable condition hh; sin xi ¼ 0 of ðD2 þ 1Þn þ h ¼ 0; n 2 H02 and hn; sin xi ¼ 0, we obtain k a 0 ðm1k1 ð1 2ihk ÞÞ ¼ eihk ðb1 þ c1 L1 Þ þ kðb1 ak þ c1 bk Þm1k1 þ ke2ihk b1 ak m1k1 þ ke2ihk c1 ak m1k2 þ Oðk 1Þ; 4 k a 0 ðm1k2 ð1 2ihk ÞÞ ¼ eihk L1 ðb2 þ c2 L1 Þ þ kðb2 ak þ c2 bk Þm1k2 þ ke2ihk ðb2 bk m1k1 þ c2 bk m1k2 Þ þ Oðk 1Þ; 4 therefore we have i m111 ð1 2ih1 Þa 0 ¼ ðb1 þ c1 L1 Þ þ a 0 m111 a1 ðb1 m111 þ c1 m112 Þ; 4 i m112 ð1 2ih1 Þa 0 ¼ L1 ðb2 þ c2 L1 Þ þ a 0 m112 b1 ðb2 m111 þ c2 m112 Þ; 4 that is m11
¼
m111 m112
i ¼ 4
b1 a1 2ih1 a 0 b2 b1
c1 a1 c2 b1 2ih1 a 0
1
b1 þ c1 L1 : L1 ðb2 þ c2 L1 Þ
We now compute m21 1 a 0 m211 ¼ ððb1 þ c1 L1 Þi þ ðb1 þ c1 L1 ÞðiÞÞ þ a 0 m211 þ a1 ðb1 m211 þ c1 m212 Þ; 4 1 a 0 m212 ¼ ðL1 ðb2 þ c2 L1 Þi þ L1 ðb2 þ c2 L1 ÞðiÞÞ þ a 0 m212 þ b1 ðb2 m211 þ c2 m212 Þ; 4 therefore m21 ¼ 0. The proof is completed.
A.4. Stability of periodic solutions We rewrite the system as follows:
d ut u ¼ Ar0 ðkÞ t þ X0 g; vt dt vt
ðA:10Þ
where X0 : ½r0 ; 0 ! BðX ; X Þ is given by X0 ðhÞ ¼ 0; if r0 6 h < 0; X0 ð0Þ ¼ I; ut ; vt 2 C and g : C ! X is a nonlinear operator defined by
u ð0Þðb1 ut ðr0 Þ þ c1 vt ðr0 ÞÞ g : ðA:11Þ g ¼ k 1 ¼ k t g2 vt ð0Þðb2 ut ðr0 Þ þ c2 vt ðr0 ÞÞ Noting that the solution of (5.2) on the center manifold is represented as
ut ~ ðx1 ; x2 ; kÞ; ¼ x1 Uð1Þ þ x2 Uð2Þ þ x vt
ðA:12Þ
where
Z p
Z 0 Z p ut ut ð0Þ bU ðiÞ ðiÞ ðiÞ xi ¼ W ; dx k ¼ W ð0Þ W ðs þ r0 Þ 1 k vt ð0Þ b2 Vk v t 0 r0 0
c1 Uk c2 Vk
ut ðsÞ vt ðsÞ
dx ds;
i ¼ 1; 2;
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1221
~ ðx1 ; x2 ; kÞ is represented as and when we denote z ¼ x1 ix2 ; x ~ ðx1 ; x2 ; kÞ ¼ x20 ðkÞ x
z2 z2 þ x11 zz þ x02 þ 2 2
We can obtain the complex form of two-dimensional equations on center manifold as Z p z0 ¼ mk iz þ ðWð1Þ ð0Þ iWð2Þ ð0ÞÞg dx:
ðA:13Þ
ðA:14Þ
0
Noting that
1 x sin x þ Oðk 1Þ ReðqzÞ; U 1 ¼ x2 L1 where qz ¼ eimk h ðx1 ix2 Þ; and ðRe qzÞh¼0 ¼
zþz ; 2
ðRe qzÞh¼r0 ¼
i þ Oðk 1Þ ðz zÞ: 2
Therefore
zþz z2 i z2 20 þ x20 ðz þ b þ ðkÞð0Þ ðsin x þ Oðk 1ÞÞ zÞ þ x ðkÞð r Þ g1 ¼ ðsin x þ Oðk 1ÞÞ 1 0 1 1 2 2 2 2
2 i z þ ; ðA:15Þ þ c1 ðL1 sin x þ Oðk 1ÞÞ ðz zÞ þ x20 2 ðkÞð r0 Þ 2 2
zþz z2 i z2 20 þ x20 ðz þ b þ ðkÞð0Þ ðsin x þ Oðk 1ÞÞ zÞ þ x ðkÞð r Þ g2 ¼ ðL1 sin x þ Oðk 1ÞÞ 2 0 2 1 2 2 2 2
2 i z þ : ðA:16Þ þ c2 ðL1 sin x þ Oðk 1ÞÞ ðz zÞ þ x20 2 ðkÞð r0 Þ 2 2
In the next, we need to determine xij . By the definition of the operator X0 , we have 8
< ki ðb1 þ c1 L1 Þ sin2 x þ Oðk 1Þ ðz2 z2 Þ þ ; h ¼ 0; 4 X0 g ¼ L1 ðb2 þ c2 L1 Þ sin2 x þ Oðk 1Þ : 0; r0 6 h < 0; and ki ðW; X0 gÞ ¼ 4
Z
p 3
sin x dx 0
ðS1k þ S1 Þ 0
k0
ðSki S i Þ 0
! ðb þ Oðk 1ÞÞðz2 z2 Þ þ ;
k0
where Sk0 ¼ S1 þ Oðk 1Þ;
b ¼ b1 þ c1 L1 þ L 1 L1 ðb2 þ c2 L1 Þ;
and S1 ¼
Z 0
p
ð1 þ
L1 L 1 Þ sin2
pi 3 xþ ðb1 a1 þ c1 a1 L1 þ b2 b1 L1 þ c2 b1 L1 L1 Þ sin x dx: 2h1
We now denote that
C~ ðW; X0 gÞ ¼ ~1 iðz2 z2 Þ þ h:o:t:; C2 where h.o.t. stands briefly for high order terms, and
Z p
Z p b 1 1 b i i sin3 x dx; C~2 ¼ sin3 x dx: þ C~1 ¼ 4 S1 S 1 4 S1 S 1 0 0
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
Let C3 ¼ 12ðC~1 i þ C~2 Þ; C4 ¼ 12ðC~1 i C~2 Þ, we can show that
8
b1 þ c1 L1 y y > > sin2 x C3 k þ C4 k þ h:o:t: ðz2 z2 Þ þ ; < 4i ðb2 þ c2 L1 Þ zk zk L1 X0s g ¼ y y > k > : C3 eimk h þ C4 k eimk h þ h:o:t: ðz2 z2 Þ þ ; zk zk
h ¼ 0; r0 6 h < 0:
Denote X0s g ¼ H 20 ðhÞ
z2 z2 þ H 11 ðhÞzz þ H 02 ðhÞ þ ; 2 2
therefore 8
> > < 2i
b1 þ c1 L1 y y sin2 x þ 2C3 k þ 2C4 k þ h:o:t:; L 1 ðb2 þ c2 L1 Þ z zk k 20
H ðhÞ ¼ y y > > : 2C3 k eimk h þ 2C4 k eimk h þ h:o:t:; zk zk
h ¼ 0; ðA:17Þ r0 6 h < 0;
and H 11 ðhÞ ¼ 0;
H 02 ðhÞ ¼ H 20 ðhÞ:
Now we derive the equations by which we can determine x20 ðkÞðhÞ; x11 ðkÞðhÞ; . . . these are ½2imk Ar0 ðkÞx20 ðkÞðhÞ ¼ H 20 ðhÞ; Ar0 ðkÞx11 ¼ H 11 ;
ðA:18Þ
02
02
½2imk Ar0 ðkÞx ðkÞðhÞ ¼ H ðhÞ; . . . ; it is obvious that x11 ¼ 0: Substituting (A.17) into (A.18) we get
y y ½2imk Ar0 ðkÞx20 ðhÞ ¼ 2C3 k eimk h þ 2C4 k eimk h þ h:o:t:; zk zk
r0 6 h < 0:
Let x20 ðhÞ ¼ A1 eimk h þ A2 eimk h þ Ee2imk h :
ðA:19Þ
Then we have A1 ¼
2C3 imk
yk zk
þ h:o:t:;
A2 ¼
2C4 3imk
yk zk
þ h:o:t:;
ðA:20Þ
and we can use the following relation, at h ¼ 0, to determine E " 20
½2imk Ar0 ðkÞx ð0Þ ¼ 2imk ðA1 þ A2 þ EÞ imk A1 imk A2 þ k
b1 Uk
c1 Uk
b2 Vk
c2 Vk
!
2
ðD þ kÞ
k
! #
b1 Uk þ c1 Vk
0
0
b2 Uk þ c2 Vk
!
e2ihk E ¼ H 20 ð0Þ:
Now substituting (A.17), at h ¼ 0, into the above equality we get
0 b1 Uk b U þ c1 Vk D2 þ k k 1 k ke2ihk 0 b2 Uk þ c2 Vk b2 Vk
i b1 þ c1 L1 ¼ sin2 x þ h:o:t: 2 L1 ðb2 þ c2 L1 Þ
c1 U k c2 Vk
2imk E ðA:21Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1223
We set E¼
1 k1
E11 E21
sin x þ E ;
ðA:22Þ
where E ¼ Oð1Þ as 0 < k 1 1, and Z p Ei sin x dx ¼ 0; i ¼ 1; 2: 0
Therefore by the solvable condition of (A.21) we get Z p Z p i f½ð1 2h1 iÞ sin x c1 b1 sin2 xE11 þ c1 a1 sin2 x E21 g sin x dx ¼ ðb1 þ c1 L1 Þ sin3 x dx þ h:o:t:; 2 0 0 Z p Z p i f½ð1 2h1 iÞ sin x b2 a1 sin2 xE21 þ b2 b1 sin2 x E11 g sin x dx ¼ L1 ðb2 þ c2 L1 Þ sin3 x dx þ h:o:t:; 2 0 0 that is
ð1 2h1 iÞa 0 þ c1 b1 b2 b1
i c1 a1 b1 þ c1 L1 1 þ h:o:t:; E ¼ ð1 2h1 iÞa 0 þ b2 a1 2 L1 ðb2 þ c2 L1 Þ
this implies that
½ð1 2h1 iÞa 0 b2 a1 ðb1 þ c1 L1 Þ þ c1 a1 L1 ðb2 þ c2 L1 Þ i 2 ½ð1 2h1 iÞa0 c1 b1 L1 ðb2 þ c2 L1 Þ þ b2 b1 ðb1 þ c1 L1 Þ þ h:o:t: E1 ¼ ðð1 2h1 iÞa 0 c1 b1 Þðð1 2h1 iÞa 0 b2 a1 Þ c1 b2 a1 b1 Now substituting (A.20), (A.22) and (A.23) into (A.19) we obtain
1 2C3 1 2C4 1 þ þ E1 sin x þ Oð1Þ x20 ð0Þ ¼ 3h1 i L1 k 1 ih1 L1
ðA:23Þ
ðA:24Þ
and x20 ðr0 Þ ¼
1 2C3 1 2C4 1 þ E1 sin x þ Oð1Þ: L1 k1 h1 3h1 L1
ðA:25Þ
Here we rewrite (A.14) as follows
z2 z2 z0 ¼ mk iz þ ðWð1Þ ð0Þ iWð2Þ ð0Þ; gÞ ¼ mk iz þ k g20 þ g11 zz þ g02 þ : 2 2 Hence we find that g20 ¼
4i ðb1 þ c1 L1 þ L1 L 1 ðb2 þ c2 L1 ÞÞ þ h:o:t:; 3S1
g11 ¼ 0; g02 ¼ g20 ; Z p 4 1 i 20 1 20 11 11 g21 ¼ ðb1 x20 1 ð r0 Þ þ c1 x2 ð r0 ÞÞ þ x1 ð0Þðb1 þ c1 L1 Þ þ ðb1 x1 ð r0 Þ þ c1 x2 ð r0 ÞÞ S1 0 4 4 2 i i 20 L1 L1 20 ðb2 x20 ðb2 x11 x11 1 ð0Þðb1 þ c1 L1 Þ þ L1 1 ð r0 Þ þ c2 x2 ð r0 ÞÞ þ x2 ð0Þðb2 þ c2 L1 Þ þ 1 ð r0 Þ 4 2 2 4 i 11 sin2 x dx þ h:o:t: þ c2 x11 2 ð r0 ÞÞ x2 ð0Þðb2 þ c2 L1 Þ 2 For determining the Floquet exponent we transform Eq. (A.14) into the Poincare normal form [6,10] n_ ¼ imk n þ c1 ð0Þnjnj2 þ Oðjnj5 Þ;
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L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
where " # i jg02 j2 g21 2 2jg11 j þ : c1 ð0Þ ¼ g20 g11 3 2 2mk Moreover r r0 ¼ r2 e2 þ Oðe3 Þ; bðeÞ ¼ b2 e2 þ Oðe3 Þ;
r2 ¼
Re c1 ð0Þ ; a0 ðr0 Þ
b2 ¼ Re c1 ð0Þ:
Note that here b2 is the coefficient of the main part of the Floquet exponent and r2 determines the bifurcation direction, since g11 ¼ 0, that is 1 Re c1 ð0Þ ¼ Re g21 : 2 In the general case, we can use a numerical program to compute the value of g21 , and for a special kind of parameters which satisfy the property ðSÞ, we can determine the sign of Re c1 ð0Þ directly. Recall the Lemma 5.6 in Section 5.1, we have a1 ¼ b1 ¼ 1=ðb1 þ b2 Þ;
L1 ¼ L 1 ¼ h1 ¼ 1:
Direct calculations yield that p 16ðb1 þ b2 Þ ; S1 ¼ p 1 þ i ; C~1 ¼ 2 3pð4 þ p2 Þ 4ðb1 þ b2 Þð2i pÞ ; C4 ¼ 3pð4 þ p2 Þ
8ðb1 þ b2 Þ C~2 ¼ ; 3ð4 þ p2 Þ
C3 ¼
4ðb1 þ b2 Þð2i þ pÞ ; 3pð4 þ p2 Þ
and E11 ðkÞ ¼ E21 ðkÞ ¼
ðb1 þ b2 Þð2 þ iÞ þ h:o:t: 10
Substituting C3 ; C4 and E1 into (5.41), (5.42) we get x20 ð0Þ ¼
b1 þ b2 8ið2i þ pÞ 8ið2i pÞ 2 i 1 sin x þ Oð1Þ þ þ 1 10 k 1 3pð4 þ p2 Þ 9pð4 þ p2 Þ
and x20 ðr0 Þ ¼
b1 þ b2 8ð2i þ pÞ 8ð2i pÞ 2i 1 sin x þ Oð1Þ: 1 k 1 3pð4 þ p2 Þ 9pð4 þ p2 Þ 10
20 Noting that x11 ¼ 0; x20 1 ¼ x2 we have Z p 4 1 20 ð r Þ þ ix ð0ÞÞ sin2 x dx þ h:o:t: g21 ¼ ðb1 þ b2 Þðx20 0 1 1 S1 0 2 Z p 20 2 4ð2 piÞ ðb þ b Þ x1 ð r0 Þ þ ix20 ¼ 1 2 1 ð0Þ sin x dx þ h:o:t: pð4 þ p2 Þ 0
Denote D¼
Z
p 2 20 ½x20 1 ðr0 Þ þ ix1 ð0Þ sin x dx ¼ 0
b1 þ b2 16ð2i pÞ ð2 iÞði 1Þ 4 þ þ Oð1Þ: k1 9pð4 þ p2 Þ 10 3
Since Reð2 piÞð2i pÞ ¼ 0, we obtain Reðg21 Þ ¼
8ðb1 þ b2 Þ2 ð3p þ 2Þ þ Oð1Þ: 15pð4 þ p2 Þðk 1Þ
L. Zhou et al. / Chaos, Solitons and Fractals 14 (2002) 1201–1225
1225
Therefore Reðg21 Þ < 0;
0 < k 1 1:
It means that the bifurcation periodic solutions are stable.
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