Existence and global stability of positive periodic solutions of a predator–prey system with delays

Existence and global stability of positive periodic solutions of a predator–prey system with delays

Applied Mathematics and Computation 146 (2003) 167–185 www.elsevier.com/locate/amc Existence and global stability of positive periodic solutions of a...

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Applied Mathematics and Computation 146 (2003) 167–185 www.elsevier.com/locate/amc

Existence and global stability of positive periodic solutions of a predator–prey system with delays Lin-Lin Wang, Wan-Tong Li

*

Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China

Abstract This paper studies the existence, global stability and uniform persistence of positive periodic solutions of a periodic predator–prey system with Holling type III functional response. By using the continuation theorem of coincidence degree theory and Liapunov functional, some sufficient conditions are obtained. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Positive solution; Global stability; Uniform persistence; Predator–prey system; MawhinÕs continuous theorem

1. Introduction Predator–prey models play a crucial role in bioeconomics, that is the management of renewable resources. Renewable resources management is complicated and constructing accurate mathematical models about the effect of many factors we would take into account is even more complicated. But it is obvious that a perfect model cannot be achieved because even if we could put all possible factors in a model, the model could never predict ecological catastrophes or mother nature caprice. Therefore, the best we can do is to look for analyzable models that describe as well as possible the reality on populations. Massive work has been done on this issue. Worthy of mentioning, time

*

Corresponding author. E-mail address: [email protected] (W.-T. Li).

0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00534-9

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delays of one type or another have been incorporated into biological models by many researchers, we refer to the monographs of Cushing [4], Gopalsamy [8], Kuang [17] and MacDonald [20] for general delayed biological systems and to Beretta and Kuang [2], Gopalsamy [9,10], Hastings [11], May [21], Ruan [24] and the references cited therein for studies on delayed predator–prey systems. There are many different kinds of delayed predator–prey models in the literature, for more details we can refer to [3] and [6]. Holling [12] proposed three types of functional response functions according to different kinds of species on the foundation of experiments in 1965. Systems with Holling type functional response has been investigated by many authors, see, for example, Hsu and Huang [15], Rosenzweig [22,23]. They studied the stability of the equilibria, existence of Hopf bifurcation, limit cycles, homoclinic loops, and even catastrophe. Huo et al. [14] considered the delayed diffusive predator–prey models with Holling type II functional response, and sufficient conditions for the existence and global stability of positive periodic solutions are obtained. Jia [16] studied the persistence and periodic solution for the nonautonomous predator– prey systems with Holling type III functional response without delays. In fact, it would be of interest to study the existence and globally asymptotically stability of positive solutions for systems with periodic coefficients [17,25]. Recently, some excellent results have been obtained by using the coincidence degree method, such as [5,13,18,19]. Now we consider the existence and global stability of periodic solutions for predator–prey delayed systems with Holling type III functional response. In the present paper, we consider the following system   dN1 ðtÞ a1 ðtÞN1 ðtÞ ¼ N1 ðtÞ b1 ðtÞ  a1 ðtÞN1 ðt  s1 ðtÞÞ  N ðt  rðtÞÞ ; 2 dt 1 þ mN12 ðtÞ   dN2 ðtÞ a2 ðtÞN12 ðt  s2 ðtÞÞ ¼ N2 ðtÞ  b2 ðtÞ  a2 ðtÞN2 ðtÞ þ ; dt 1 þ mN12 ðt  s2 ðtÞÞ ð1Þ where N1 ðtÞ; N2 ðtÞ are the densities of the prey population and predator population at time t, bi : R ! RR; ai ; si ; r; ai : R ! ½0; þ1Þ ði ¼ 1; 2Þ are continuous T functions of period T and 0 bi ðtÞ dt > 0, ai ðtÞ 6¼ 0, m is a nonnegative constant. In view of the actual applications of system (1), we consider the initial value problem Ni ðsÞ ¼ Ui ðsÞ;

s 2 ½s; 0;

where s ¼ max fs1 ðtÞ; s2 ðtÞ; rðtÞg t2½0;T 

Ui ð0Þ > 0; i ¼ 1; 2;

L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185

169

and denote Z 1 T  bi ¼ bi ðtÞ dt; T 0 Z 1 T ai ¼ ai ðtÞ dt; T 0

Bi ¼

1 T

Z

T

jbi ðtÞj dt;

0

f L ¼ min f ðtÞ t2½0;T 

and

ai ¼

1 T

Z

T

ai ðtÞ dt;

0

f M ¼ max f ðtÞ: t2½0;T 

2. Existence of positive periodic solution In order to obtain the existence of positive periodic solution of (1), for the readerÕs convenience, we shall summarize in the following a few concepts and results from [7] that will be basic for this section. Let X,Z be normed Banach spaces, L : DomL  X ! Z be a linear mapping, N : X ! Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL ¼ CodimImL < þ1 and ImL is closed in Z. If L is a Fredholm mapping of index zero there exist continuous projects P : X ! X and Q : Z ! Z such that ImP ¼ KerL; ImL ¼ KerQ ¼ ImðI  QÞ. It follows that LjDomL \ KerP : ðI  P ÞX ! ImL is invertible. We denote the inverse of that map by KP . If X be an open bounded subset of X, the mapping N will be called L-compact on X if QN ðXÞ is bounded and KP ðI  QÞN : X ! X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ ! KerL. In the proof of our main theorems below, we will use the continuation theorem of Mawhin and Gaines [7]. Lemma 2.1 (Continuation theorem). Let L be a Fredholm mapping of index zero and let N be L-compact on X. Suppose further (a) For each k 2 ð0; 1Þ, every solution x of Lx ¼ Nx satisfies x 62 oX; (b) QNx ¼ 6 0 for each x 2 oX \ KerL and degfJQN ; X \ KerL; 0g 6¼ 0: Then the equation Lx ¼ Nx has at least one solution lying in DomL \ X. For convenience, we shall introduce the notation: Z 1 T g¼ gðtÞ dt; T 0 where g is an T-periodic function. Now we state our first theorem for the existence of a positive T-periodic solution.

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Theorem 2.1. Assume that 1  b1  pffiffiffiffi H21 a1 > 0; 2 m expf2H12 g b2 > 0 a2   1 þ m expf2H12 g and that (C): the algebraic equations 8 a1 u1 u2 > > b1   a1 u1  ¼ 0; < 1 þ mu2 1

> > a2 u2  b2 þ  :

ð2Þ

a2 u21 ¼ 0; 1 þ mu21 T

T

has a unique solution ðu1 ; u2 Þ 2 intR2þ ¼ fðu1 ; u2 Þ jui > 0ði ¼ 1; 2Þg, where ( ) 1 b2 a   m 2 b2 ÞT H21 ¼ ln þ ðB2 þ   a2 and ( H12 ¼ ln

 a1 b1  2p1 ffiffimffi expfH21 g  a1

)  ðB1 þ b1 ÞT :

Then (1) has at least one positive T-periodic solution. Proof. Since N1 ðtÞ ¼ N1 ð0Þ exp

Z t 

b1 ðsÞ  a1 ðsÞN1 ðs  s1 ðsÞÞ   a1 ðsÞN1 ðsÞ N2 ðs  rðsÞÞ ds ;  1 þ mN12 ðsÞ Z t    a2 ðsÞN12 ðs  s2 ðsÞÞ N2 ðtÞ ¼ N2 ð0Þ exp  b2 ðsÞ  a2 ðsÞN2 ðsÞ þ ds ; 1 þ mN12 ðs  s2 ðsÞÞ 0 0

the solution of system (1) remains positive for t P 0. Make the change of variables N1 ðtÞ ¼ expfx1 ðtÞg

and

then (1) can be reformulated as

N2 ðtÞ ¼ expfx2 ðtÞg;

L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185

dx1 ðtÞ ¼ b1 ðtÞ  a1 ðtÞ expfx1 ðt  s1 ðtÞÞg dt a1 ðtÞ expfx1 ðtÞg expfx2 ðt  rðtÞÞg ;  1 þ m expf2x1 ðtÞg dx2 ðtÞ a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg ¼ b2 ðtÞ  a2 ðtÞ expfx2 ðtÞg þ : dt 1 þ m expf2x1 ðt  s2 ðtÞÞg

171

ð3Þ

Define T

X ¼ Z ¼ fxðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ 2 CðR; R2 Þ : xi ðT þ tÞ ¼ xi ðtÞ; i ¼ 1; 2g and kxk ¼ kðx1 ðtÞ; x2 ðtÞÞT k ¼ max jx1 ðtÞj þ max jx2 ðtÞj t2½0;T 

t2½0;T 

for any x 2 X (or Z). Then X and Z are Banach spaces with the norm k:k. Let " # 1 ðtÞg expfx2 ðtrðtÞÞg b1 ðtÞ  a1 ðtÞ expfx1 ðt  s1 ðtÞÞg  a1 ðtÞ expfx 1þm expf2x1 ðtÞg Nx ¼ 2 ðtÞ expf2x1 ðts2 ðtÞÞg b2 ðtÞ  a2 ðtÞ expfx2 ðtÞg þ a1þm expf2x1 ðts2 ðtÞÞg   D1 ðtÞ  ; D2 ðtÞ RT RT Lx ¼ dxðtÞ=dt; Px ¼ ð1=T Þ 0 xðtÞdt; x 2 XR; Qz ¼ ð1=T Þ 0 zðtÞdt; z 2 Z. Then it T follows that KerL ¼ R2 ; Im L ¼ fz 2 Z : 0 zðtÞdt ¼ 0g is closed in Z, dimKerL ¼ 2 ¼ codimImL and P, Q are continuous projectors such that ImP ¼ KerL

and

KerQ ¼ ImL ¼ ImðI  QÞ:

Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse(to L) KP : ImL ! KerP \ DomL exists and can be read as Z Z t Z T 1 T KP ðzÞ ¼ zðsÞds  zðsÞds dt: T 0 0 0 Thus 2 RT h 1

i 3 1 ðsÞþx2 ðsrðsÞÞg b1 ðsÞ  a1 ðsÞ expfx1 ðs  s1 ðsÞÞg  a1 ðsÞ expfx ds 1þm expf2x1 ðsÞg 5 i QNx ¼ 4 R T h a2 ðsÞ expf2x1 ðss2 ðsÞÞg 1  b ðsÞ  a ðsÞ expfx ðsÞg þ ds 2 2 2 T 0 1þm expf2x1 ðss2 ðsÞÞg T

0

172

and

L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185

"R

# " R R # T t 1 D1 ðsÞds D1 ðsÞds dt 0 0 T  1 RT Rt KP ðI  QÞNx ¼ D2 ðsÞds D ðsÞds dt 0 2 T 0 " # R T ðT  1Þ D1 ðsÞds  Tt 21 R0T : ð t  2Þ 0 D2 ðsÞds t R0t 0

Obviously, QN and KP ðI  QÞN are continuous. It is not difficult to show that KP ðI  QÞN ðXÞ is compact for any open bounded X  X by using Arzela– Ascoli theorem. Moreover, QN ðXÞ is clearly bounded. Thus, N is L-compact on X with any open bounded set X  X . Now we reach the position to search for an appropriate open bounded set X for the application of the continuation theorem. Corresponding to the operator equation Lx ¼ kNx; k 2 ð0; 1Þ; we have  dx1 ðtÞ ¼ k b1 ðtÞ  a1 ðtÞ expfx1 ðt  s1 ðtÞÞg dt  a1 ðtÞ expfx1 ðtÞg expfx2 ðt  rðtÞÞg  ; 1 þ m expf2x1 ðtÞg   dx2 ðtÞ a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg ¼ k  b2 ðtÞ  a2 ðtÞ expfx2 ðtÞg þ : dt 1 þ m expf2x1 ðt  s2 ðtÞÞg ð4Þ Assume that xðtÞ 2 X is a solution of (4) for a certain k 2 ð0; 1Þ. Integrating (4) over the interval ½0; T , we obtain Z T ½b1 ðtÞ  a1 ðtÞ expfx1 ðt  s1 ðtÞÞgdt 0 Z T a1 ðtÞ expfx1 ðtÞg expfx2 ðt  rðtÞÞg dt ¼ 0 ð5Þ  1 þ m expf2x1 ðtÞg 0 and

Z

T



0

 a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg  b2 ðtÞ  a2 ðtÞ expfx2 ðtÞg þ dt ¼ 0: 1 þ m expf2x1 ðt  s2 ðtÞÞg

ð6Þ

Thus Z

T

0

  a1 ðtÞ expfx1 ðtÞ þ x2 ðt  rðtÞÞg a1 ðtÞ expfx1 ðt  s1 ðtÞÞg þ dt ¼ b1 T 1 þ m expf2x1 ðtÞg ð7Þ

and Z 0

T

  a2 ðtÞ expfx2 ðtÞg þ

 a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg dt ¼ b2 T : 1 þ m expf2x1 ðt  s2 ðtÞÞg

ð8Þ

L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185

173

From (4), (5), (7) and (8), we obtain Z

T

jx01 ðtÞjdt ¼ k

Z

T

jb1 ðtÞ  a1 ðtÞ expfx1 ðt  s1 ðtÞÞg  a1 ðtÞ expfx1 ðtÞg expfx2 ðt  rðtÞÞg   dt 1 þ m expf2x1 ðtÞg Z T Z T < jb1 ðtÞjdt þ a1 ðtÞ expfx1 ðt  s1 ðtÞÞg 0 0  a1 ðtÞ expfx1 ðtÞg expfx2 ðt  rðtÞÞg þ dt 1 þ m expf2x1 ðtÞg b1 ÞT ¼ ðB1 þ 

0

0

ð9Þ ð10Þ

and Z

T

jx02 ðtÞjdt ¼ k

Z

T

j  b2 ðtÞ  a2 ðtÞ expfx2 ðtÞg  a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg  dt < ðB2 þ b2 ÞT : þ 1 þ m expf2x1 ðt  s2 ðtÞÞg 

0

0

ð11Þ

In view of (6) we get Z

Z T b2 ðtÞdt þ a2 ðtÞ expfx2 ðtÞgdt 0 0 Z T Z T a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg a2 ðtÞ dt dt 6 ¼ 1 þ m expf2x m ðt  s ðtÞÞg 1 2 0 0 T

and so Z

T

a2 ðtÞ expfx2 ðtÞgdt 6

0

1 b2 T : a2 T   m T

By mean value theorem and note that xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ 2 X , we know there exists n2 2 ½0; T  such that expfx2 ðn2 Þg

Z

T

a2 ðtÞdt 6

0

so we have ( x2 ðn2 Þ 6 ln

 b2  a2

1 a m 2

) :

1 b2 T ; a2 T   m

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L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185

Therefore x2 ðtÞ 6 x2 ðn2 Þ þ

Z

(

T

jx02 ðtÞjdt 6

ln

 b2  a2

1 a m 2

0

) þ ðB2 þ b2 ÞT ,H21 :

ð12Þ

From (7) we get Z T  b1 T P a1 ðtÞ expfx1 ðt  s1 ðtÞÞgdt: 0

By mean value theorem, we know there exists n1 2 ½0; T  such that Z T expfx1 ðn1 Þg a1 ðtÞdt 6  b1 T ; 0

which yields ( x1 ðn1 Þ 6 ln

 b1  a1

) :

Hence x1 ðtÞ 6 x1 ðn1 Þ þ

Z

(

T

jx01 ðtÞjdt 6 0

From (5) we obtain Z T Z b1 ðtÞdt  0

Z

¼

ln

 b1  a1

) þ ðB1 þ b1 ÞT ,H11 :

T

a1 ðtÞ expfx1 ðt  s1 ðtÞÞgdt

0 T

0

a1 ðtÞ expfx1 ðtÞg expfx2 ðt  rðtÞÞg dt: 1 þ m expf2x1 ðtÞg

Then Z

T

a1 ðtÞ expfx1 ðt  s1 ðtÞÞgdt

0

Z

T

Z

T

a1 ðtÞ expfx1 ðtÞg expfx2 ðt  rðtÞÞg dt 1 þ m expf2x1 ðtÞg 0 0 Z T 1 1 P b1 T  pffiffiffiffi expfH21 g a1 ðtÞdt ¼ b1 T  pffiffiffiffi expfH21 g a1 T : 2 m 2 m 0 ¼

b1 ðtÞdt 

By mean value theorem, there exists g0 2 ½0; T  such that Z T 1 expfx1 ðg0  s1 ðg0 ÞÞg a1 ðtÞdt P  b1 T  pffiffiffiffi expfH21 g a1 T : 2 m 0

ð13Þ

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175

Denote g0  s1 ðg0 Þ ¼ g1 þ kT , g1 2 ½0; T , k is an integer, then ( )  a1 b1  2p1 ffiffimffi expfH21 g x1 ðg1 Þ P ln  a1 and x1 ðtÞ P x1 ðg1 Þ 

Z

(

T

jx01 ðtÞjdt P

ln

 a1 b1  2p1 ffiffimffi expfH21 g

)

a1

0

 ðB1 þ  b1 ÞT ,H12 :

ð14Þ

Thus, in view of (13) and (14) we obtain max jx1 ðtÞj 6 maxfjH11 j; jH12 jg :¼ H1 :

t2½0;T 

Furthermore, from (6) we also have Z T Z T Z b2 ðtÞdt þ a2 ðtÞ expfx2 ðtÞgdt ¼ 0

0

T

0

a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg dt: 1 þ m expf2x1 ðt  s2 ðtÞÞg

2

By the monotonicity of the function u =ð1 þ u2 Þðu > 0Þ and (14) we get Z T Z T a2 ðtÞ expf2x1 ðt  s2 ðtÞÞg dt a2 ðtÞ expfx2 ðtÞgdt ¼ 1 þ m expf2x1 ðt  s2 ðtÞÞg 0 0 Z T expf2H12 g  b2 ðtÞdt P a2 T  b2 T : 1 þ m expf2H12 g 0 Then the mean value theorem implies that there exists g2 2 ½0; T  such that Z T expf2H12 g expfx2 ðg2 Þg a2 ðtÞdt P a2 T  b2 T : 1 þ m expf2H12 g 0 Direct calculation yields ( expf2H12 g x2 ðg2 Þ P ln

1þm expf2H12 g

 a2

b2 a2  

) :¼ H22 ;

thus x2 ðtÞ P x2 ðg2 Þ 

Z

T

jx02 ðtÞjdt P H22  ðB2 þ b2 ÞT :

ð15Þ

0

From (12) and (15) we have b2 ÞT jg :¼ H2 : max jx2 ðtÞj 6 maxfjH21 j; jH22  ðB2 þ 

t2½0;T 

ð16Þ

Obviously, Hi ; Hij ði; j ¼ 1; 2Þ are independent on the choice of k. Under the assumptions in Theorem 2.1, it is easy to show that the algebraic equation (2)

176

L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185 T

T

has a unique solution ðu1 ; u2 Þ 2 intR2þ ¼ fðu1 ; u2 Þ jui > 0ði ¼ 1; 2Þg since condition (C) holds. Let H ¼ H1 þ H2 þ H3 , where H3 > 0 is large enough such that kðlnfu1 g; lnfu2 gÞT k ¼ j lnfu1 gj þ j lnfu2 gj < H3 : Define T

X ¼ fxðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞ 2 X : kxk < H g: Thus X satisfies (a) in Lemma 2.1. When x 2 oX \ KerL ¼ oX \ R2 ; x is a constant vector in R2 with kxk ¼ H . Then 2 3 1 expfx1 þx2 g  b1   a1 expfx1 g  a1þm expf2x1 g 5 6¼ 0: QNx ¼ 4 a2 expf2x1 g  b2   a2 expfx2 g þ 1þm expf2x1 g Since Im P ¼ KerL, J can be the identity mapping, in view of the assumptions in Theorem 2.1, direct calculation produces degfJQN ; X \ KerL; 0g 2 1 þx2 gð1m expf2x1 gÞ  a1 expfx1 g  a1 expfxð1þm expf2x1 gÞ2 ¼ sgn det 4 2 a2 expf2x1 g ð1þm expf2x1 gÞ2

¼ sgn  a1  a2 expfx1 þ x2 g þ þ

2 a2 a1 expf3x1 þ x2 g

a2 expfx2 g

3 5

a2 expfx1 þ 2x2 gð1  m expf2x1 gÞ a1 

!

ð1 þ m expf2x1 gÞ3

1 expfx1 þx2 g  a1þm expf2x1 g

ð1 þ m expf2x1 gÞ2

6¼ 0:

By now we have proved that X satisfies all the conditions in Lemma 2.1. Hence T (3) has at least one solution ðx1 ðtÞ; x2 ðtÞÞ in DomL \ X. Set N1 ðtÞ ¼ expfx1 ðtÞg; N2 ðtÞ ¼ expfx2 ðtÞg; then N  ðtÞ ¼ ðN1 ðtÞ; N2 ðtÞÞ completes the proof. 

T

is a positive T-periodic solution of (1). This

3. Global asymptotic stability In this section, we assume s1 ðtÞ ¼ 0;

s2 ðtÞ ¼ s2 ;

rðtÞ ¼ r

ð17Þ

f M ¼ max f ðtÞ:

ð18Þ

and denote f L ¼ min f ðtÞ; t2½0;T 

t2½0;T 

L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185

177

First we give two Lemmas without proof which will be used in the proof of our main result. Lemma 3.1 (BarbalatÕs Lemma, see [1]). Let f be a nonnegative function defined on ½0; þ1Þ such that f is integrable and uniformly continuous on ½0; þ1Þ. Then limt!1 f ðtÞ ¼ 0. Lemma 3.2. The domain R2þ ¼ fðy; uÞjy P 0; u P 0g is invariant with respect to system (1). The proof is simple and will be omitted here. Thus we only consider the nonnegative solution of (1) which satisfies the initial conditions. For convenience, we set w¼

2 H11 : 2 1 þ mH11

Theorem 3.1. Assume that (17) and the conditions in Theorem 2.1 hold. Assume further that L (C1) ðw þ 2p1 ffiffimffiÞaM 1 < ðb2 þ a2  wa2 Þ . M L (C2) a2 ½expfH21 g þ 1 < a1 expfH22 g. M L (C3) bM 1 þ a1 expfH21 g < a1 .

Then system (1) has a unique T-periodic solution which is globally asymptotically stable. Proof. By Theorem 2.1, there exists a positive periodic solution ðN1 ðtÞ; N2 ðtÞÞ of (1). From the conditions listed above, we can choose positive constants k1 ; k2 such that   1 L < k1 < ðb2 þ a2  wa2 Þ w þ pffiffiffiffi aM 2 m 1 and L aM 2 ð1 þ expfH21 gÞ < k2 < a1 expfH22 g:

Consider a Liapunov functional V ðtÞ defined by V ðtÞ ¼ j lnðN1 ðtÞÞ  lnðN1 ðtÞÞj þ j lnðN2 ðtÞÞ  lnðN2 ðtÞÞj Z t   þ jN1 ðtÞ  N1 ðtÞj þ jN2 ðtÞ  N2 ðtÞj þ k1 jN2 ðsÞ  N2 ðsÞj ds tr  Z t  N12 ðsÞ   N12 ðsÞ  þ k2  ds: 2 1 þ mN12 ðsÞ  ts2  1 þ mN1 ðsÞ

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L.-L. Wang, W.-T. Li / Appl. Math. Comput. 146 (2003) 167–185

Denote s1 ¼

jN1 ðtÞ  N1 ðtÞj N1 ðtÞ  N1 ðtÞ

and

s2 ¼

jN2 ðtÞ  N2 ðtÞj : N2 ðtÞ  N2 ðtÞ

Then it is easy to see that j lnðN1 ðtÞÞ  lnðN1 ðtÞÞj ¼ s1 ; lnðN1 ðtÞÞ  lnðN1 ðtÞÞ

j lnðN2 ðtÞÞ  lnðN2 ðtÞÞj ¼ s2 lnðN2 ðtÞÞ  lnðN2 ðtÞÞ

and    N12 ðsÞ N12 ðsÞ  N12 ðsÞ N12 ðsÞ     1 þ mN 2 ðsÞ 1 þ mN 2 ðsÞ  1 þ mN 2 ðsÞ 1 þ mN 2 ðsÞ ¼ s1 : 1

1

1

1

Calculating the right derivative Dþ V ðtÞ of V ðtÞ along the solutions of (1) leads to Dþ V ðtÞ ¼ s1 a1 ðtÞ½N1 ðtÞ  N1 ðtÞ  s1 a1 ðtÞ½N12 ðtÞ  N12 ðtÞ " # N1 ðtÞ N1 ðtÞ  N2 ðt  rÞ  N ðt  rÞ  s1 a1 ðtÞ 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 2  N12 ðt  s2 Þ   s2 a2 ðtÞ½N2 ðtÞ  N2 ðtÞ þ s2 a2 ðtÞ 1 þ mN12 ðt  s2 Þ  N12 ðt  s2 Þ  þ s1 b1 ðtÞ½N1 ðtÞ  N1 ðtÞ 1 þ mN12 ðt  s2 Þ   N12 ðtÞ N12 ðtÞ  N2 ðt  rÞ  N ðt  rÞ  s1 a1 ðtÞ 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 2  s2 b2 ðtÞ½N2 ðtÞ  N2 ðtÞ  s2 a2 ðtÞ½N22 ðtÞ  N22 ðtÞ   N12 ðt  s2 Þ N12 ðt  s2 Þ  N2 ðtÞ  N ðtÞ þ s2 a2 ðtÞ 1 þ mN12 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ 2 þ s2 k1 ½N2 ðtÞ  N2 ðtÞ  s2 k1 ½N2 ðt  rÞ  N2 ðt  rÞ    N12 ðtÞ N12 ðtÞ N12 ðt  s2 Þ k þ s1 k2   s 1 2 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 1 þ mN12 ðt  s2 Þ  N12 ðt  s2 Þ  1 þ mN12 ðt  s2 Þ ¼ a1 ðtÞjN1 ðtÞ  N1 ðtÞj  a1 ðtÞjN12 ðtÞ  N12 ðtÞj   N1 ðtÞ N1 ðtÞ  N N ðt  rÞ  ðt  rÞ  s1 a1 ðtÞ 2 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 2

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179



N12 ðt  s2 Þ  a2 ðtÞjN2 ðtÞ  N2 ðtÞj þ s2 a2 ðtÞ 1 þ mN12 ðt  s2 Þ  N12 ðt  s2 Þ  þ b1 ðtÞjN1 ðtÞ  N1 ðtÞj 1 þ mN12 ðt  s2 Þ   N12 ðtÞ N12 ðtÞ   s1 a1 ðtÞ ðt  rÞ  ðt  rÞ N N 2 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 2  b2 ðtÞjN2 ðtÞ  N2 ðtÞj  a2 ðtÞjN22 ðtÞ  N22 ðtÞj   N12 ðt  s2 Þ N12 ðt  s2 Þ  ðtÞ  ðtÞ þ s2 a2 ðtÞ N N 2 1 þ mN12 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ 2   N12 ðtÞ   þ k1 jN2 ðtÞ  N2 ðtÞj  k1 jN2 ðt  rÞ  N2 ðt  rÞj þ k2  1 þ mN12 ðtÞ    2 2 2  N1 ðt  s2 Þ N1 ðtÞ  N1 ðt  s2 Þ    k2   ;  1 þ mN12 ðtÞ  1 þ mN12 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ 

where 

 N1 ðtÞ N1 ðtÞ  N N ðt  rÞ  ðt  rÞ 2 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 2  N1 ðtÞ N1 ðtÞ ¼ s1 a1 ðtÞ N2 ðt  rÞ  N  ðt  rÞ 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 2  N1 ðtÞ N1 ðtÞ   þ N N ðt  rÞ  ðt  rÞ 1 þ mN12 ðtÞ 2 1 þ mN12 ðtÞ 2

 s1 a1 ðtÞ

N1 ðtÞ ½N2 ðt  rÞ  N2 ðt  rÞ 1 þ mN12 ðtÞ   N1 ðtÞ N1 ðtÞ   s1 a1 ðtÞN2 ðt  rÞ  1 þ mN12 ðtÞ 1 þ mN12 ðtÞ   N1 ðtÞ N1 ðtÞ  < s1 a1 ðtÞN2 ðt  rÞ  1 þ mN12 ðtÞ 1 þ mN12 ðtÞ ¼ s1 a1 ðtÞ

a1 ðtÞ þ pffiffiffiffi jN2 ðt  rÞ  N2 ðt  rÞj 2 m a1 ðtÞ < s1 a1 ðtÞN2 ðt  rÞ½N1 ðtÞ  N1 ðtÞ þ pffiffiffiffi jN2 ðt  rÞ  N2 ðt  rÞj 2 m a1 ðtÞ < a1 ðtÞ expfH21 gjN1 ðtÞ  N1 ðtÞj þ pffiffiffiffi jN2 ðt  rÞ  N2 ðt  rÞj 2 m

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and 

 N12 ðt  s2 Þ N12 ðt  s2 Þ  N N ðtÞ  ðtÞ 2 1 þ mN12 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ 2 N12 ðt  s2 Þ ¼ s2 a2 ðtÞ ½N2 ðtÞ  N2 ðtÞ 1 þ mN12 ðt  s2 Þ   N12 ðt  s2 Þ N12 ðt  s2 Þ þ s2 a2 ðtÞ  N  ðtÞ 1 þ mN12 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ 2   N12 ðt  s2 Þ   < s2 a2 ðtÞwjN2 ðtÞ  N2 ðtÞj þ s2 a2 ðtÞN2 ðtÞ 1 þ mN12 ðt  s2 Þ  2 N1 ðt  s2 Þ  ¼ a2 ðtÞwjN2 ðtÞ  1 þ mN12 ðt  s2 Þ     N12 ðt  s2 Þ N12 ðt  s2 Þ      :  N2 ðtÞj þ a2 ðtÞN2 ðtÞ 1 þ mN12 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ 

s2 a2 ðtÞ

Futhermore, 

 N12 ðtÞ N12 ðtÞ  N N ðt  rÞ  ðt  rÞ 2 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ 2  N12 ðtÞ N12 ðtÞ N2 ðt  rÞ  N  ðt  rÞ ¼ s1 a1 ðtÞ 2 1 þ mN1 ðtÞ 1 þ mN12 ðtÞ 2  N12 ðtÞ N12 ðtÞ   N ðt  rÞ  N ðt  rÞ þ 1 þ mN12 ðtÞ 2 1 þ mN12 ðtÞ 2   N12 ðtÞ  s1 a1 ðtÞN2 ðt  rÞ ¼ s1 a1 ðtÞ N2 ðt  rÞ  N2 ðt  rÞ 1 þ mN12 ðtÞ   N12 ðtÞ N12 ðtÞ   6 wa1 ðtÞjN2 ðt  rÞ  N2 ðt  rÞj 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ    N12 ðtÞ N12 ðtÞ     :  a1 ðtÞN2 ðt  rÞ 1 þ mN 2 ðtÞ 1 þ mN 2 ðtÞ 

s1 a1 ðtÞ

1

1

Hence we get Dþ V ðtÞ < a1 ðtÞjN1 ðtÞ  N1 ðtÞj  a1 ðtÞjN12 ðtÞ  N12 ðtÞja1 ðtÞ a1 ðtÞ  expfH21 gjN1 ðtÞ  N1 ðtÞj þ pffiffiffiffi jN2 ðt  rÞ  N2 ðt  rÞj 2 m  N12 ðt  s2 Þ  a2 ðtÞjN2 ðtÞ  N2 ðtÞj þ s2 a2 ðtÞ 1 þ mN12 ðt  s2 Þ  N12 ðt  s2 Þ  þ b1 ðtÞjN1 ðtÞ  N1 ðtÞjwa1 ðtÞjN2 ðt  rÞ 1 þ mN12 ðt  s2 Þ

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     N2 ðt  rÞj  a1 ðtÞN2 ðt  rÞ

181

 N12 ðtÞ N12 ðtÞ    b2 ðtÞjN2 ðtÞ 1 þ mN 2 ðtÞ 1 þ mN 2 ðtÞ  1

1

 N2 ðtÞj  a2 ðtÞjN22 ðtÞ  N22 ðtÞj þ wa1 ðtÞjN2 ðt  rÞ  N2 ðt  rÞj    N12 ðtÞ N12 ðtÞ   a1 ðtÞN2 ðt  rÞ  þ k1 jN2 ðtÞ  N2 ðtÞj 1 þ mN12 ðtÞ 1 þ mN12 ðtÞ     N12 ðtÞ N12 ðtÞ     k1 jN2 ðt  rÞ  N2 ðt  rÞj þ k2   1 þ mN12 ðtÞ 1 þ mN12 ðtÞ     N12 ðt  s2 Þ N12 ðt  s2 Þ   k2   2 1 þ mN1 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ  < ½a1 ðtÞ  b1 ðtÞ  a1 ðtÞ expfH21 gjN1 ðtÞ  N1 ðtÞj  ½b2 ðtÞ þ a2 ðtÞ   a1 ðtÞ  wa2 ðtÞ  k1 jN2 ðtÞ  N2 ðtÞj þ pffiffiffiffi þ wa1 ðtÞ  k1 jN2 ðt  rÞ 2 m    N12 ðtÞ N12 ðtÞ      N2 ðt  rÞj þ ½k2  a1 ðtÞN2 ðt  rÞ  1 þ mN12 ðtÞ 1 þ mN12 ðtÞ     N12 ðt  s2 Þ N12 ðt  s2 Þ    þ ½a2 ðtÞ þ a2 ðtÞN2 ðtÞ  k2   : 1 þ mN12 ðt  s2 Þ 1 þ mN12 ðt  s2 Þ 

If we choose L aM 2 ðexpfH21 g þ 1Þ < k2 < a1 expfH22 g

and k1 >

  1 w þ pffiffiffiffi aM ; 2 m 1

then Dþ V ðtÞ < ½a1 ðtÞ  b1 ðtÞ  a1 ðtÞ expfH21 gjN1 ðtÞ  N1 ðtÞj  ½b2 ðtÞ þ a2 ðtÞ  wa2 ðtÞ  k1 jN2 ðtÞ  N2 ðtÞj:

ð19Þ

By (C1)–(C3), there exists a positive constant c such that   1 L þ c; ðb2 þ a2  wa2 Þ > w þ pffiffiffiffi aM 2 m 1 aL1 expfH22 g > aM 2 ðexpfH21 g þ 1Þ þ c and M aL1 > bM 1 þ a1 expfH21 g þ c:

Hence Dþ V ðtÞ < cðjN1 ðtÞ  N1 ðtÞj þ jN2 ðtÞ  N2 ðtÞjÞ:

ð20Þ

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On the other hand, we have V ðtÞ P jN1 ðtÞ  N1 ðtÞj þ jN2 ðtÞ  N2 ðtÞj

ð21Þ

and V ð0Þ 6 j lnðN1 ð0ÞÞ  lnðN1 ð0ÞÞj þ j lnðN2 ð0ÞÞ  lnðN2 ð0ÞÞj þ jN1 ð0Þ  N1 ð0Þj þ jN2 ð0Þ  N2 ð0Þj þ k1 r sup jU2 ðtÞ  N2 ðtÞj þ k2 s2 t2½r;0

   U21 ðtÞ N12 ðtÞ  ;  sup   1 þ mU2 ðtÞ 1 þ mN 2 ðtÞ  t2½s2 ;0

ð22Þ

1

1

which, by (20), implies that V ðtÞ 6 V ð0Þ < þ1;

tP0

ð23Þ

and V ðtÞ is nonincreasing for t P 0. From (21) and (23), we know that jN1 ðtÞ  N1 ðtÞj þ jN2 ðtÞ  N2 ðtÞj is bounded. Combine (21) and (23) with (22), we know that the periodic solution ðN1 ðtÞ; N2 ðtÞÞ is stable. Moreover, we have from (20) that Z t V ðtÞ þ c ðjN1 ðsÞ  N1 ðsÞj þ jN2 ðsÞ  N2 ðsÞjÞds 6 V ð0Þ; t P 0; ð24Þ 0

which leads to ðjN1 ðsÞ  N1 ðsÞj þ jN2 ðsÞ  N2 ðsÞjÞ 2 L1 ½0; 1Þ:

ð25Þ

Since ðN1 ðtÞ; N2 ðtÞÞ and ðN1 ðtÞ; N2 ðtÞÞ are bounded for t P 0 with bounded derivations (from the Eq. (1) satisfied by them), it will follow that jN1 ðtÞ  N1 ðtÞj and jN2 ðtÞ  N2 ðtÞj are uniformly continuous on ½0; 1Þ. By Lemma 3.1, we obtain lim jN1 ðtÞ  N1 ðtÞj ¼ 0;

t!1

Thus we complete the proof.

lim jN2 ðtÞ  N2 ðtÞj ¼ 0:

t!1



4. Uniform persistence In this section, we will consider the uniform persistence of the system (1). And assume that s1 ðtÞ ¼ 0;

ð26Þ

throughout this section. First we give a lemma, which can be found in [17].

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183

Lemma 4.1. Consider the following equation u0 ðtÞ ¼ uðtÞ½d1  d2 uðtÞ

ðd2 > 0Þ:

If d1 > 0; then lim uðtÞ ¼

t!þ1

d1 : d2

Theorem 4.1. Assume that L aM 2 > mb2 ;

ð27Þ

aM 1

bL1 > pffiffiffiffi b1 2 m

ð28Þ

b22 bM > 2L ; 2 a2 1 þ mb2

ð29Þ

and

where b1 ¼

L aM 2  mb2 ; L ma2

b2 ¼

pffiffiffiffi 2 mbL1  aM b pffiffiffiffi 1 1 : 2 m aM 1

Then system (1) is uniformly persistence. Proof. Since dN1 ðtÞ L 6 N1 ðtÞ½bM 1  a1 N1 ðtÞ; dt we consider the following equation L u0 ðtÞ ¼ uðtÞ½bM 1  a1 uðtÞ:

By Lemma 4.1, we have lim uðtÞ ¼

t!þ1

bM 1 : aL1

Then by comparison theorem, there exists T1 > 0 such that N1 ðtÞ 6

bM 1 ; aL1

for all t P T1

and also   dN2 ðtÞ aM 6 N2 ðtÞ  bL2  aL2 N2 ðtÞ þ 2 ; dt m

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which and (27) imply that there exists T2 > 0 such that N2 ðtÞ 6 b1 ;

for all t P T2 :

Similarly,   dN1 ðtÞ aM 1 p ffiffiffi ffi P N1 ðtÞ bL1  aM N ðtÞ  b 1 1 dt 2 m 1 and (28) imply there exists T3 > 0 such that N1 ðtÞ P b2 ;

for all t P T3 :

Moreover, " # 2 dN2 ðtÞ aL2 ðb2 Þ M M P N2 ðtÞ  b2  a2 N2 ðtÞ þ 2 dt 1 þ mðb2 Þ and (29) mean that there exists T4 > 0 such that " # 1 aL2 ðb2 Þ2 N2 ðtÞ P M  bM :¼ b3 ; for all t P T4 ; 2 a2 1 þ mðb2 Þ2 here we use the monotonicity of the function u2 =ð1 þ u2 Þ for u > 0. Hence we choose T ¼ maxfTi g;

i ¼ 1; 2; 3; 4;

then for any t > T we have b2 6 N1 ðtÞ 6 b0

and

b3 6 N2 6 b1 :

Therefore D ¼ fðN1 ; N2 Þjb2 6 N1 ðtÞ 6 b0 ; b3 6 N2 6 b1 g is the invariant set of (1), hence the system is uniformly persistence. The proof is complete. 

Acknowledgements This work is supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), the Foundation for University Key Teacher by the Ministry of Education of China and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.

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