Periodic solutions for a two-species nonautonomous competition system with diffusion and impulses

Chaos, Solitons and Fractals 32 (2007) 1916–1926 www.elsevier.com/locate/chaos

Periodic solutions for a two-species nonautonomous competition system with diffusion and impulses Lingzhen Dong b

a,*

, Lansun Chen b, Peilin Shi

a

a Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China Department of Applied Mathematics, Dalian University of Technology, Dalian 116023, China

Accepted 28 December 2005

Communicated by A. Helal

Abstract Rx By re-estimating the upper bound of 0 eui ðtÞ dt (i = 1, 2), we generalize a result about the existence of a positive periodic solution for a two-species nonautonomous patchy competition system with time delay. Based on that system, we consider the impulsive harvesting and stocking, and establish a two-species nonautonomous competition Lotka–Volterra system with diffusion and impulsive effects. With the continuation theorem of coincidence degree theory, we obtain the existence of a positive periodic solution for such a system. At last, two examples are given to demonstrate our results.  2006 Elsevier Ltd. All rights reserved.

1. Introduction One of the most interesting questions in mathematical biology concerns the existence of positive periodic solutions for population dynamical systems. For the continuous Lotka–Volterra systems, such a problem has been investigated extensively, and many skills and techniques have been developed. The existence of positive periodic solutions for such systems can be obtained by Brouwer fixed point theorem [2], by standard techniques of bifurcation theory [3], or by theory of topological degree [1]. In fact, these methods have been widely applied to various L–V systems [4–6]. However, in population dynamics, many evolutionary processes experience short-time rapid changes after undergoing relatively long smooth variation. For examples, the harvesting and stocking occur at fixed moments, and some species usually immigrate at the same time every year, etc. If we still thought of the population dynamical systems with these phenomena as continuous systems, it would be unreasonable or incorrect. We should establish systems with impulsive effects. Recently, theories for impulsive differential equations have been introduced into population dynamics [7–10,12,13]. To the authors knowledge, the population dynamical systems with diffusion and impulsive effects are seldom discussed. *

Corresponding author. E-mail address: [email protected] (L. Dong).

0960-0779/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.01.003

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

In this paper, we mainly study the following impulsive system: 8 9 x_ ðtÞ ¼ x1 ðtÞðr1 ðtÞ  a1 ðtÞx1 ðtÞ  b1 ðtÞyðtÞÞ þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ; > > > > > 1 = > > x_ 2 ðtÞ ¼ x2 ðtÞðr2 ðtÞ  a2 ðtÞx2 ðtÞÞ þ D2 ðtÞðx1 ðtÞ  x2 ðtÞÞ; > > >   > > > < y_ ðtÞ ¼ yðtÞ r3 ðtÞ  a3 ðtÞx1 ðtÞ  b3 ðtÞyðtÞ  bðtÞ R 0 KðsÞyðt þ sÞds ; > ; s 9 > > Dx1 ðtk Þ ¼ ck x1 ðtk Þ; > > = > > > > Dx2 ðtk Þ ¼ d k x2 ðtk Þ; > > > > ; : Dyðtk Þ ¼ ek yðtk Þ;

1917

t 6¼ tk ; ð1:1Þ

with period where ri(t), ai(t) (i = 1, 2, 3), Di(t) (i = 1, 2), bi(t) (i = 1, 3), b(t) are all positive periodic continuous functions R0 x > 0, s is a nonnegative constant, K(s) is a continuous nonnegative function on [s, 0] such that s KðsÞds ¼ 1, and there exists a number q > 0 such that ck+q = ck, dk+q = dk, ek+q = ek, tk+q = tk + x (k 2 Z+). From the viewpoint of biology, we assume 1 + ck, 1 + dk, 1 + ek are all positive. System (1.1) shows that species X can diffuse between patch 1 and 2, while species Y is restricted to patch 1, and in patch 1 species X competes with Y for space or food. Furthermore, species X in two patches and species Y are harvested and stocked periodically at fixed moments, which occur when 1 + ck, 1 + dk, 1 + ek are smaller than 1 or larger than 1. These human activities occur frequently and therefore the study of their effects on the population behaviors is of great practical value. This paper R x is arranged as follows. In the next section, Theorem 2.1 in [1] is generalized by re-estimating the upper bound of 0 eui ðtÞ dt (i = 1, 2). And we introduce the continuation theorem of coincidence degree theory, which is used to prove the existence of a positive periodic solution of (1.1). Moreover, some notations are introduced. In Section 3, we study the existence of a positive periodic solution of (1.1). At last, two examples are worked out.

2. Generalization Rx We estimate the upper bound of 0 eui ðtÞ dt (i = 1, 2) in a easier way, and Theorem 2.1 in [1] can be generalized. In fact, let v(t) = max{u1(t), u2(t)}. We know that v(t) is a x-periodic continuous function and (1) If u1(t) > u2(t), or u1(t) = u2(t) but u_ 1 ðtÞ P u_ 2 ðtÞ, then vðtÞ ¼ u1 ðtÞ and

u_ 1 ðtÞ 6 k½r1 ðtÞ  a1 ðtÞeu1 ðtÞ  6 k½rm1  al1 eu1 ðtÞ .

(2) If u2(t) > u1(t), or u2(t) = u1(t) but u_ 2 ðtÞ P u_ 1 ðtÞ, then vðtÞ ¼ u2 ðtÞ and

u_ 2 ðtÞ 6 k½r2 ðtÞ  a2 ðtÞeu2 ðtÞ  6 k½rm2  al2 eu2 ðtÞ .

Consequently, Dþ vðtÞ 6 k½r  aevðtÞ ; r ¼ maxfrm1 ; rm2 g, Rwhere x vðtÞ e dt 6 rx . Further, 0 a Z 0

x

eui ðtÞ dt 6

rx a

a¼

ð2:1Þ minfal1 ; al2 g.

Integrating (2.1) over [0, x], we have 0 6 k½rx  a

Rx 0

vðtÞ

e

dt, which implies that

ði ¼ 1; 2Þ.

Thus, we can replace (H5) of Theorem 2.1 in [1] with ðH 05 Þ : ar3 > am3 r, and obtain Theorem 2.1 0 . If (H3): ri(t) > Di(t) (i = 1, 2), r3 , and (H4): bl3 ðr1  D1 Þ > bm 1 (H 05 ): ar3 > am r, are satisfied, then system (1.1) in [1] has at least one x-periodic solution. 3 In order to obtain the existence of positive x-periodic solutions for (1.1), we shall introduce the continuation theorem of coincidence degree theory [11]. Let X, Y be real Banach spaces, L : Dom L  X ! Y be a Fredholm mapping of index zero, and P : X ! X, Q : Y ! Y be continuous projectors such that Im P = Ker L, Ker Q = Im L, so that X = Ker L  Ker P, Y = Im L  Im Q. Denote by LP the restriction of L to Dom L \ Ker P and by KP : Im L ! Dom L \ Ker P the inverse to LP. Let J : Im Q ! Ker L be an isomorphism of Im Q onto Ker L. Then the continuation theorem can be described as follows.

1918

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

Lemma 2.1. Let X  X be an open bounded set and let N : X ! Y be a continuous operator which is L-compact on X (i.e. QN : X ! Y and K P ðI  QÞN : X ! Y are compact). Assume that (a) for each k 2 (0, 1), x 2 oX \ Dom L, Lx 5 kNx, (b) for each x 2 oX \ Dom L, QNx 5 0, and (c) deg{JQN, X \ Ker L, 0} 5 0. Then Lx = Nx has at least one solution in X \ Ker L. For convenience, we use the following notations in the rest parts of the paper. (1) (2) (3) (4)

PC(R+, R) = {u(t)ju : R+ ! R, lims!tu(s) = u(t) if t 5 tk, limt!tk uðtÞ ¼ uðtk Þ, limt!tþk uðtÞ exists, k 2 Z+}, PC 0 (R+, R) = {u(t)ju : R+ ! R, u 0 (t) 2 PC(R+, R)}, PCx = {u 2 PC(R+, R)ju(t)R= u(t + x)}, PC 0x ¼ fu 2 PC 0 ðRþ ; RÞjuðtÞ ¼ uðt þ xÞg, x For f 2 PCx, denote f ¼ x1 0 f ðtÞdt, fl = mint2[0,x]jf(t)j, fm = maxt2[0,x]jf(t)j.

3. Positive x-periodic solutions In this section, we demonstrate the existence of a positive x-periodic solution for (1.1). We have Theorem 3.1. If system (1.1) satisfies P lnð1 þ ck Þ > D1 ðtÞ and r2 ðtÞ þ qk¼1 lnð1 þ d k Þ > D2 ðtÞ, P P (2) ððr1  D1 Þx þ qk¼1 lnð1 þ ck ÞÞbl3 > bm1 ðr3 x þ qk¼1 lnð1 þ ek ÞÞ, and Pq P (3) aðr3 x þ k¼1 lnð1 þ ek ÞÞ > am3 ðrx þ qk¼1 lnð1 þ Bk ÞÞ,

(1) r1 ðtÞ þ

Pq

k¼1

where r ¼ maxfrm1 ; rm2 g, Bk = max{ck, dk}, a ¼ minfal1 ; al2 g, then system (1.1) has at least one positive x-periodic solution. Proof. Let x1(t) = exp(u1(t)), x2(t) = exp(u2(t)), y(t) = exp(u3(t)). We obtain 8 9 u_ 1 ðtÞ ¼ r1 ðtÞ  D1 ðtÞ  a1 ðtÞeu1 ðtÞ  b1 ðtÞeu3 ðtÞ þ D1 ðtÞeu2 ðtÞu1 ðtÞ ; > > > = > > > u_ 2 ðtÞ ¼ r2 ðtÞ  D2 ðtÞ  a2 ðtÞeu2 ðtÞ þ D2 ðtÞeu1 ðtÞu2 ðtÞ ; > t 6¼ tk ; > > > R0 > ; < u_ 3 ðtÞ ¼ r3 ðtÞ  a3 ðtÞeu1 ðtÞ  b3 ðtÞeu3 ðtÞ  bðtÞ s KðsÞeu3 ðtþsÞ ds; 9 > Du1 ðtk Þ ¼ lnð1 þ ck Þ; > > = > > > > Du2 ðtk Þ ¼ lnð1 þ d k Þ; > > > > : ; Du3 ðtk Þ ¼ lnð1 þ ek Þ.

ð3:1Þ

T It is obvious that if system (3.1) has an x-periodic solution ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ , then 





ðeu1 ðtÞ ; eu2 ðtÞ ; eu3 ðtÞ ÞT ¼ ðx1 ðtÞ; x2 ðtÞ; y  ðtÞÞT is a positive x-periodic solution of system (1.1). So in the following we discuss the existence of x-periodic solutions of system (3.1). T To apply P Lemma 2.1, we denote X = {(u1(t), u2(t), u3(t)) jui(t) 2 PCx, i = 1, 2, 3} and denote kðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT k ¼ 3i¼1 supt2½0;x jui ðtÞj. Then (X, k Æ k) is a real Banach space. Moreover, let 820 9 1 0 1 0 13  ui ðtÞ 2 PC x ði ¼ 1; 2; 3Þ; ðmk ; nk ; p ÞT ¼ ðDU 1 ðtk ÞDU 2 ðtk Þ; DU 3 ðtk ÞÞT ; > mq m1 u1 ðtÞ > k  < = 6B C B C B C7 Y ¼ 4@ u2 ðtÞ A; @ n1 A; . . . ; @ nq A5 a constant vector, k ¼ 1; 2; . . . ; q;  > > : ;  U ðtÞ; a primitive function of u ðtÞ; i ¼ 1; 2; 3. pq u3 ðtÞ p1 i i . , yq] 2 Y, where y1(t) = (u1(t), u2(t), u3(t))T and yk = (mk, nk, pk)T (k = 1, 2, . . . , q), and define Assume Pthat y = [y1(t), y1, . . P kyk ¼ 3i¼1 supt2½0;x jui ðtÞj þ qk¼1 ky k k. Then (Y, k Æ k) is also a Banach space.

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

1919

Set L : Dom L  X ! Y, 2 3 20 1 0 1 0 13 Du1 ðtq Þ u_ 1 ðtÞ Du1 ðt1 Þ u1 ðtÞ 6 7 6B C B C B C7 L4 u2 ðtÞ 5 ¼ 4@ u_ 2 ðtÞ A; @ Du2 ðt1 Þ A; . . . ; @ Du2 ðtq Þ A5; Du3 ðtq Þ u3 ðtÞ u_ 3 ðtÞ Du3 ðt1 Þ where Dom L ¼ fðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT 2 X ju_ i ðtÞ 2 PC x g ¼ fðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT 2 X jui ðtÞ 2 PC 0x ði ¼ 1; 2; 3Þg. At the same time, we denote f1 ðtÞ ¼ r1 ðtÞ  D1 ðtÞ  a1 ðtÞeu1 ðtÞ  b1 ðtÞeu3 ðtÞ þ D1 ðtÞeu2 ðtÞu1 ðtÞ ; f2 ðtÞ ¼ r2 ðtÞ  D2 ðtÞ  a2 ðtÞeu2 ðtÞ þ D2 ðtÞeu1 ðtÞu2 ðtÞ ; Z 0 KðsÞeu3 ðtþsÞ ds. f3 ðtÞ ¼ r3 ðtÞ  a3 ðtÞeu1 ðtÞ  b3 ðtÞeu3 ðtÞ  bðtÞ s

Then we may define N : X ! Y, 2

3 20 10 1 0 13 lnð1 þ cq Þ lnð1 þ c1 Þ f1 ðtÞ u1 ðtÞ 6 7 6B CB C B C7 N 4 u2 ðtÞ 5 ¼ 4@ f2 ðtÞ A@ lnð1 þ d 1 Þ A; . . . ; @ lnð1 þ d q Þ A5; lnð1 þ eq Þ u3 ðtÞ f3 ðtÞ lnð1 þ e1 Þ and define two projectors P and Q as P : X ! X, 0Rx 1 1 u1 u1 ðtÞdt 0 C B C 1 BRx P @ u2 A ¼ @ 0 u2 ðtÞdt A. x Rx u3 u3 ðtÞdt 0 0

Q : Y ! Y, 1 2 0R 3 q P x u m 1 ðtÞdt þ k 0 C 0 1 1 0 1 20 0 13 6 B 0 17 k¼1 C 0 6 B mq m1 u1 ðtÞ 0 7 C 61 B R 7 q P x C B C B n C7 6 B u ðtÞdt þ 6B C B C B 7 nk C; @ 0 A; . . . ; @ 0 C Q4@ u2 ðtÞ A; @ n1 A; . . . ; @ q A5 ¼ 6 B 0 2 A7. C 6x B 7 k¼1 C 0 6 B pq u3 ðtÞ p1 0 7 q A 4 @Rx 5 P u3 ðtÞdt þ pk 0 k¼1

3

Obviously, Ker L = R , and  8 9 q P Rx > >  0 u1 ðtÞdt þ > > m ¼ 0 k > > > > 20 1 0 1 0 1 3 > > k¼1  > > mq m1 u1 ðtÞ > > R < = q P 6B C B C B n C7 x u ðtÞdt þ nk ¼ 0 Im L ¼ 4@ u2 ðtÞ A; @ n1 A; . . . ; @ q A5 0 2  > > k¼1 > >  > > pq p1 u3 ðtÞ > > q  > > R P > > x > >  : ; u ðtÞdt þ p ¼ 0 3 k  0 k¼1

is closed in Y. Noting that Im P = Ker L, Ker Q = Im L and dimKer L = codimIm L = 3, we know that L is a Fredholm mapping of index zero. Moreover, by computation, the inverse KP of LP has the form KP : Im L ! Ker P \ Dom L, 1 Z Z q q t X X 1 x t 1 X u1 ðsÞds þ mk  u1 ðsÞds dt  mk þ mk t k C B x 0 0 x k¼1 C tk
1920

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

Since 1 2 0R 3 q P x f1 ðtÞdt þ lnð1 þ ck Þ 0 C 0 1 0 1 6 B 0 17 k¼1 C 0 6 B u1 ðtÞ 0 7 C 61 B R 7 q P B C 6 B x f ðtÞdt þ C B C B 7 lnð1 þ d k Þ C; @ 0 A; . . . ; @ 0 C QN @ u2 ðtÞ A ¼ 6 B 0 2 A7; C 6x B 7 k¼1 C 0 6 B u3 ðtÞ 0 7 q A 4 @Rx 5 P f ðtÞdt þ lnð1 þ ek Þ 0 3 k¼1

and Kp(I  Q)N : X ! X,

1 lnð1 þ c Þ k B C 1 0 k¼1 tk
P

f ðsÞds þ 0 1

lnð1 þ ck Þ

0R

1

x f ðsÞds 0 1

þ

q P

k¼1

0 R R 1 q q P P x t 1 1 f ðsÞds dt  lnð1 þ c Þ þ lnð1 þ c Þt 1 k k k x Bx 0 0 C k¼1 k¼1 B C B R R C q q P P B 1 x t f ðsÞds dt  1 lnð1 þ d k Þ þ x lnð1 þ d k Þtk C  Bx 0 0 2 C; B C k¼1 k¼1 B C q q @ R xR t A P P 1 1 f ðsÞds dt  lnð1 þ e Þ þ lnð1 þ e Þt 3 k k k x 0 0 x k¼1

k¼1

by the Lebesque convergence theorem, QN and KP(I  Q)N are continuous. Moreover, from the Arzela–Ascoli theorem, QN ðXÞ, K P ðI  QÞN ðXÞ are relatively compact for any open bounded set X  X. Therefore N is L-compact on X. In the following, the operator equation, Lx = kNx, k 2 (0, 1), is considered, which can be rewritten as: 9 8 u_1 ðtÞ ¼ k½r1 ðtÞ  D1 ðtÞ  a1 ðtÞeu1 ðtÞ  b1 ðtÞeu3 ðtÞ þ D1 ðtÞeu2 ðtÞu1 ðtÞ ; > > > > = > > u2 ðtÞ u1 ðtÞu2 ðtÞ > _ u ðtÞ ¼ k½r ðtÞ  D ðtÞ  a ðtÞe þ D ðtÞe ; > t 6¼ tk ; 2 2 2 2 2 > > > > > ; < u_ ðtÞ ¼ k½r ðtÞ  a ðtÞeu1 ðtÞ  b ðtÞeu3 ðtÞ  bðtÞ R 0 KðsÞeu3 ðtþsÞ ds > 3 3 3 3 s ð3:2Þ 9 > > Du ðt Þ ¼ k lnð1 þ c Þ; k > 1 k > > = > > > > Du2 ðtk Þ ¼ k lnð1 þ d k Þ; > > > : ; Du3 ðtk Þ ¼ k lnð1 þ ek Þ. Suppose (u1(t), u2(t), u3(t))T 2 X is a solution of system (3.2) for some k 2 (0, 1). Integrating (3.2) over [0, x], it can be obtained that Z x q X    lnð1 þ ck Þ ¼ r1 ðtÞ  D1 ðtÞ  a1 ðtÞeu1 ðtÞ  b1 ðtÞeu3 ðtÞ þ D1 ðtÞeu2 ðtÞu1 ðtÞ dt; 0

k¼1



q X

lnð1 þ d k Þ ¼

k¼1



q X

lnð1 þ ek Þ ¼

Z Z

a1 ðtÞeu1 ðtÞ dt þ

0

Z

0

x



 r2 ðtÞ  D2 ðtÞ  a2 ðtÞeu2 ðtÞ þ D2 ðtÞeu1 ðtÞu2 ðtÞ dt;



u1 ðtÞ

r3 ðtÞ  a3 ðtÞe

x

a2 ðtÞeu2 ðtÞ dt 

Z

x

b1 ðtÞeu3 ðtÞ dt 

b3 ðtÞeu3 ðtÞ dt þ

 b3 ðtÞe

q X

Z

lnð1 þ d k Þ ¼

lnð1 þ ck Þ ¼

 bðtÞ

Z

x

Z

Z

0 u3 ðtþsÞ

KðsÞe

x

ðr1 ðtÞ  D1 ðtÞÞdt þ

x

ðr2 ðtÞ  D2 ðtÞÞdt þ

0

s

Z 0

0

bðtÞ 0

q X k¼1

k¼1

x

u3 ðtÞ

 ds dt.

s

0

0

Z

0

0

k¼1

Therefore, Z x

x

KðsÞeu3 ðtþsÞ ds dt þ

Z 0

Z

Z

x

D1 ðtÞeu2 ðtÞu1 ðtÞ dt;

ð3:3Þ

0 x

D2 ðtÞeu1 ðtÞu2 ðtÞ dt;

0 x

a3 ðtÞeu1 ðtÞ dt 

q X k¼1

lnð1 þ ek Þ ¼

ð3:4Þ Z

x

r3 ðtÞdt. 0

ð3:5Þ

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

Consequently, Z Z x ju_ 1 ðtÞjdt 6 2 0

Z

a1 ðtÞeu1 ðtÞ dt þ

Z

0 x

ju_ 2 ðtÞjdt 6 2

Z

0

Z

x

ju_ 3 ðtÞjdt 6 2

a2 ðtÞeu2 ðtÞ dt 

Z

0

q X

k¼1

lnð1 þ d k Þ;

k¼1 x

r3 ðtÞdt þ

0

q X

lnð1 þ ek Þ ¼ 2r3 x þ

k¼1

q X

D

lnð1 þ ek Þ ¼ d 3 .

k¼1

From Eq. (3.5), we have Z Z x q X b3 ðtÞeu3 ðtÞ dt  lnð1 þ ek Þ 6 0

X q b1 ðtÞeu3 ðtÞ dt  lnð1 þ ck Þ;

0

x

0 x

x

1921

x

r3 ðtÞdt

0

k¼1

and therefore, P Z x r3 x þ qk¼1 lnð1 þ ek Þ eu3 ðtÞ dt 6 . bl3 0

ð3:6Þ

Let v(t) = max{u1(t), u2(t)}. Then v(t) 2 PCx. Moreover, (1) If u1(t) > u2(t), or u1(t) = u2(t) but u_ 1 ðtÞ P u_ 2 ðtÞ, then vðtÞ ¼ u1 ðtÞ and

u_ 1 ðtÞ 6 k½r1 ðtÞ  a1 ðtÞeu1 ðtÞ  6 k½rm1  al1 eu1 ðtÞ .

(2) If u2(t) > u1(t), or u2(t) = u1(t) but u_ 2 ðtÞ P u_ 1 ðtÞ, then vðtÞ ¼ u2 ðtÞ and

u_ 2 ðtÞ 6 k½r2 ðtÞ  a2 ðtÞeu2 ðtÞ  6 k½rm2  al2 eu2 ðtÞ .

Denote r ¼ maxfrm1 ; rm2 g and a ¼ minfal1 ; al2 g. Then (

Dþ vðtÞ 6 k½r  aevðtÞ ; Dvðtk Þ 6 k lnð1 þ Bk Þ;

ð3:7Þ

where Bk = max{ck, dk} (k 2 Z+). Integrating (3.7) over [0, x], we have Z x q X lnð1 þ Bk Þ 6 rx  a evðtÞ dt;  0

k¼1

Rx

Pq

which leads to 0 evðtÞ dt 6 ½rx þ k¼1 lnð1 þ Bk Þ=a. Therefore, P Z x rx þ qk¼1 lnð1 þ Bk Þ ði ¼ 1; 2Þ. eui ðtÞ dt 6 a 0

ð3:8Þ

So, Z

x

ju_ 1 ðtÞjdt 6 2 0



! ! P P am1 r bm1 r3 am q lnð1 þ Bk Þ bm1 qk¼1 lnð1 þ ek Þ þ l x þ 2 1 k¼1 þ a a b3 bl3 q X k¼1

Z

x 0

D

ð3:9Þ

lnð1 þ ck Þ ¼ d 1 > 0;

am r am ju_ 2 ðtÞjdt 6 2 2 x þ 2 2 a

Pq

k¼1

lnð1 þ Bk Þ X D lnð1 þ d k Þ ¼ d 2 > 0.  a k¼1

From Eq. (3.4), we know that P Z x ðr2  D2 Þx þ qk¼1 lnð1 þ d k Þ eu2 ðtÞ dt > . am2 0

q

ð3:10Þ

1922

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

From Eq. (3.3), we have Z x Z q X a1 ðtÞeu1 ðtÞ dt  lnð1 þ ck Þ P ðr1  D1 Þx  0

x

b1 ðtÞeu3 ðtÞ dt

0

k¼1

and consequently Z

x 0

ðr1  D1 Þbl3  bm1 r3 eu1 ðtÞ dt P x am1 bl3

bm1

Pq

k¼1

q P

lnð1 þ ek Þ  bl3

lnð1 þ ck Þ

k¼1

ð3:11Þ

> 0.

am1 bl3

With equality (3.5), we know that     Z 0 b3 þ b KðsÞds exp max u3 ðtÞ s

P r3 x 

Z

t2½0;x

x

a3 ðtÞeu1 ðtÞ dt þ

0

P

q X

lnð1 þ ek Þ

k¼1

P P ar3  am3 r a qk¼1 lnð1 þ ek Þ  am3 qk¼1 lnð1 þ Bk Þ xþ > 0. a a

ð3:12Þ

Inequalities (3.6) and (3.8) show that there exist three points si ði ¼ 1; 2; 3Þ such that P rx þ qk¼1 lnð1 þ Bk Þ ði ¼ 1; 2Þ; ui ðsi Þ 6 ln Pqax r3 x þ k¼1 lnð1 þ ek Þ u3 ðs3 Þ 6 ln . bl3 x

ð3:13Þ ð3:14Þ

Consequently, there are three positive numbers di (i = 1, 2, 3) such that ui ðsi Þ < di

ði ¼ 1; 2; 3Þ.

ð3:15Þ

Similarly, inequalities (3.10)–(3.12) show that there exist three points (i = 1, 2, 3) such that ui ðni Þ > qi

ni

ði ¼ 1; 2; 3Þ and three positive numbers qi

ði ¼ 1; 2; 3Þ.

ð3:16Þ

Since for an arbitrary t 2 [0, x], Z t Z X u_ 1 ðsÞds þ k lnð1 þ ck Þ 6 u1 ðs1 Þ þ u1 ðtÞ ¼ u1 ðs1 Þ þ s1

u2 ðtÞ ¼ u2 ðs2 Þ þ u3 ðtÞ ¼ u3 ðs3 Þ þ

Z

s2

Z

u1 ðtÞ P u1 ðn1 Þ 

X

u_ 2 ðsÞds þ k

X

u_ 3 ðsÞds þ k

s3

Z Z

x

ju_ 1 ðsÞjds 

Z

0

X

Z Z

j lnð1 þ ck Þj;

0
ju_ 2 ðsÞjds þ 0

0

X X

j lnð1 þ d k Þj;

0
ju_ 3 ðsÞjds þ

X

j lnð1 þ ek Þj;

0
j lnð1 þ ck Þj;

0
ju_ 2 ðsÞjds 

0

u3 ðtÞ P u3 ðn3 Þ 

lnð1 þ ek Þ 6 u3 ðs3 Þ þ

s3
0

u2 ðtÞ P u2 ðn2 Þ 

lnð1 þ d k Þ 6 u2 ðs2 Þ þ

s2
t

ju_ 1 ðsÞjds þ

0

s1
t

x

X

j lnð1 þ d k Þj;

0
ju_ 3 ðsÞjds 

X

j lnð1 þ ek Þj

0
and inequalities (3.15) and (3.16), we have ðqi þ d i þ ri Þ 6 ui ðtÞ 6 di þ d i þ ri ði ¼ 1; 2; 3Þ; P P P where r1 ¼ 0
jui ðtÞj 6 maxfqi þ d i þ ri ; di þ d i þ ri g ¼ Ri

ði ¼ 1; 2; 3Þ.

Obviously, Ri (i = 1, 2, 3) are independent of k. Now, we are in the position of proving all conditions of Lemma 2.1 are satisfied.

ð3:17Þ

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

1923

Denote M = R1 + R2 + R3 + R0, where R0 is sufficiently large such that solution (a*, b*, c*)T of the system 8 q P > x yx  z > r a ¼  lnð1 þ ck Þ; 1  D1   1 e  b1 e þ D1 e > > > k¼1 > > < q P r2  D2  a2 ey þ D2 exy ¼  lnð1 þ d k Þ; ð3:18Þ > k¼1 > > >   q > R > >  0 KðsÞds ez ¼  P lnð1 þ ek Þ; : r3  a3 ex  b3 þ b s k¼1

k(a*, b*, c*)Tk

ja*j

jb*j

jc*j

= + + < M, provided that system (3.18) has a solution or a number of solutions. Let satisfies X = {(u1(t), u2(t), u3(t))T 2 X, k(u1(t), u2(t), u3(t))Tk < M}. Then condition (a) of Lemma 2.1 is satisfied. T 3 T 3 P3For each (u1, u2, u3) 2 oX \ Ker L = oX \ R , we know that (u1(t), u2(t), u3(t)) is a constant vector in R with i¼1 jui j ¼ M. If system (3.18) has a solution or a number of solutions, then 20 1 3 q P u1 u2 u1  u3 r a þ lnð1 þ c 1  D1   1 e  b1 e þ D1 e kÞ C0 1 0 1 6B 0 17 20 1 0 1 0 13 k¼1 C 0 6B u1 ðtÞ 0 7 0 0 0 C 6B 7 q P B C 6B CB C B C7 6B C B C B C7 a2 eu2 þ D2 eu1 u2 þ lnð1 þ d k Þ QN @ u2 ðtÞ A ¼ 6B r2  D2   C; @ 0 A; . .. ; @ 0 A7 6¼ 4@ 0 A; @ 0 A;. . .; @ 0 A5. C 6B 7 k¼1 C 0 6B u3 ðtÞ 0 7 0 0 0   q 4@ A 5 R P  0 KðsÞds eu3 þ lnð1 þ ek Þ r3   a 3 eu 1   b3 þ b s k¼1

If system (3.18) does not have any solution, naturally, 0 1 20 1 0 1 0 13 0 0 0 u1 ðtÞ B C 6B C B C B C7 QN @ u2 ðtÞ A 6¼ 4@ 0 A; @ 0 A; . . . ; @ 0 A5; u3 ðtÞ 0 0 0 which shows that condition (b) in Lemma 2.1 is satisfied. Finally, we prove that condition (c) in Lemma 2.1 is satisfied. Define U : Dom L · [0, 1] ! X, 2 3 q P u1 u3   r  D  a e  b e þ lnð1 þ c Þ 1 1 1 k 6 1 7 2 3 k¼1 6 7 D1 eu2 u1 6 7 q P 6 7 6 7 lnð1 þ d k Þ Uðu1 ; u2 ; u3 ; lÞ ¼ 6 r2  D2  a2 eu2 þ 7 þ l4 D2 eu1 u2 5; 6 7 k¼1 6 7 0   q 4 5 R0 P u1 u3   r3  a3 e  b3 þ b s KðsÞds e þ lnð1 þ ek Þ k¼1

0 for (u1, u2, u3)T 2 oX \ Ker L. Assuming where l 2 [0, 1] is a parameter. With the mapping U, we have U(u1, u2, u3, l) P5 3 T that this is not true, then there exists a constant vector (u1, u2, u3) with i¼1 jui j ¼ M satisfying U(u1, u2, u3, l) = 0, i.e. r1  D1  a1 eu1  b1 eu3 þ lD1 eu2 u1 þ

q X

lnð1 þ ck Þ ¼ 0;

k¼1

r2  D2  a2 eu2 þ lD2 eu1 u2 þ  Z  r3  a3 eu1  b3 þ b

q X

lnð1 þ d k Þ ¼ 0;

k¼1

0

s

 q X KðsÞds eu3 þ lnð1 þ ek Þ ¼ 0. k¼1

Following the discussion of (3.13)–(3.18), we know that juij < max{di, qi}. Therefore P 3 i¼1 jui j ¼ M. Using the property of topological degree and taking J : Im Q ! Ker L by 3 2 0R 1 1 0R q q P P x x u m u m 1 ðtÞdt þ k 1 ðtÞdt þ k 0 0 6 B C 0 1 C B 0 17 k¼1 k¼1 6 B C 0 C B 0 7 7 1 BR 61 B R C C q q P P x 7 6 B x u ðtÞdt þ C B C C B B nk C; @ 0 A; . . . ; @ 0 A7 ¼ B 0 u2 ðtÞdt þ nk C J6 B 0 2 C; 7 xB 6x B C C k¼1 k¼1 7 6 B C C B 0 0 q q 5 4 @Rx A A @Rx P P u ðtÞdt þ p u ðtÞdt þ p 3 3 k k 0 0 k¼1

k¼1

P3

i¼1 jui j

< M, which contradicts

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L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

we have degðJQN ðu1 ; u2 ; u3 ÞT ; X \ Ker L; ð0; 0; 0ÞT Þ ¼ degðUðu1 ; u2 ; u3 ; 1Þ; X \ Ker L; ð0; 0; 0ÞT Þ ¼ degðUðu1 ; u2 ; u3 ; 0Þ; X \ Ker L; ð0; 0; 0ÞT Þ ¼ deg

r1  D1  a1 eu1  b1 eu3 þ

q X

lnð1 þ ck Þ; r2  D2   a2 eu2 þ

k¼1

 Z  r3  a3 e  b3 þ b

0

u1

q X

lnð1 þ d k Þ;

k¼1

1 !T  q X TA u3 KðsÞds e þ lnð1 þ ek Þ ; X \ Ker L; ð0; 0; 0Þ .

s

k¼1

By the hypotheses of the theorem, the algebraic system 8 q P > > lnð1 þ ck Þ ¼ 0; > r1  D1  a1 x1  b1 y þ > > k¼1 > > > < q P r2  D2  a2 x2 þ lnð1 þ d k Þ ¼ 0; > k¼1 > > > >   q > R >  0 KðsÞds y þ P lnð1 þ ek Þ ¼ 0; > : r3  a3 x1  b3 þ b s

k¼1

T has a unique solution ðx1 ; x2 ; y  Þ , where     P P a1 r3 þ qk¼1 lnð1 þ ek Þ  a3 r1  D1 þ qk¼1 lnð1 þ ck Þ    > 0; x1 ¼ R  0 KðsÞds  a3 b1 a1 b3 þ b s P r2  D2 þ qk¼1 lnð1 þ d k Þ > 0; x2 ¼ a2      R P P  0 KðsÞds   r1  D1 þ qk¼1 lnð1 þ ck Þ b3 þ b b1 r3 þ qk¼1 lnð1 þ ek Þ s    > 0. y ¼ R  0 KðsÞds   a1 b3 þ b b1 a3  s

Consequently, we have degðJQN ðu1 ; u2 ; u3 ÞT ; X \ Ker L; ð0; 0; 0ÞT Þ ¼ sign

  Z   b1   b3 þ b a2  a2  a3  a1 

0

KðsÞds

s

 x1 x2 y  ¼ 1.

And the proof is completed. h In fact, let ck = dk = ek = 0, then the above system (1.1) is the system (1.1) in [1], and conditions (1), (2) and (3) of Theorem 3.1 are identical with (H3), (H4) and ðH 05 Þ of Theorem 2.1 0 . Example 1. Consider the system 8   x_ ðtÞ ¼ x1 ðtÞðð2 þ sin tÞ  ð2 þ sin tÞx1 ðtÞ  yðtÞÞ þ 54 þ sin t ðx2 ðtÞ  x1 ðtÞÞ; > > < 1   x_2 ðtÞ ¼ x2 ðtÞðð2 þ cos tÞ  ð2 þ sin tÞx2 ðtÞÞ þ 54 þ cos t ðx1 ðtÞ  x2 ðtÞÞ; >   > : yðtÞ _ ¼ yðtÞ 4þsin t  1 x ðtÞ  ð2 þ sin tÞyðtÞ . 2p

ð3:19Þ

2p 1

It is easily proved that this system satisfies (H3), (H4) and ðH 05 Þ, and has at least one positive 2p-periodic solution (Fig. 1). Example 2. For the system 8 9   > x_1 ðtÞ ¼ x1 ðtÞðð2 þ sin tÞ  ð2 þ sin tÞx1 ðtÞ  yðtÞÞ þ 54 þ sin t ðx2 ðtÞ  x1 ðtÞÞ; > > > > = > > < x_2 ðtÞ ¼ x2 ðtÞðð2 þ cos tÞ  ð2 þ sin tÞx2 ðtÞÞ þ 5 þ cos tðx1 ðtÞ  x2 ðtÞÞ; 4 >   > ; t 1 >  2p x1 ðtÞ  ð2 þ sin tÞyðtÞ ; > y_ ðtÞ ¼ yðtÞ 4þsin > 2p > > : Dyðtk Þ ¼ 0:05yðtk Þ;

t 6¼ tk ;

ð3:20Þ

where tk = 2kp, k 2 Z+. It can be verified that all the conditions of Theorem 3.1 hold, so there exists at least one positive 2p-periodic solution for this impulsive differential system (Fig. 2).

L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

1925

Fig. 1. Positive periodic solution for continuous system (3.19). (a) Phase portrait. (b) and (c) Time-series of species X in patch 1, 2, respectively. (d) Time-series of species Y in patch 1.

Fig. 2. Positive periodic solution for impulsive differential system (3.20). (a) Phase portrait. (b) and (c) Time-series of species X in patch 1, 2 respectively. (d) Time-series of species Y in patch 1.

4. Conclusion In this paper, we study a two-species periodic competition system with diffusion and impulses.RIn fact, system (1.1) x with ck = dk = ek = 0 (k 2 N) was discussed in paper [1]. Here, we re-estimate the upper bounds of 0 eui ðtÞ dt (i = 1, 2) by constructing the special function v(t) = max{u1(t), u2(t)}, then the conditions of Theorem 2.1 in [1], which guarantee the existence of a positive periodic solution, are simplified. Further, the impulsive effects are considered besides the diffusion of species X between two patches, and a periodic competition system with diffusion and impulses is established. With the theory of topological degree, the existence of a positive periodic solution for such an impulsive differential system is

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L. Dong et al. / Chaos, Solitons and Fractals 32 (2007) 1916–1926

obtained. Moreover, some examples are given to illustrate our results. How to prove the stability of positive periodic solutions will be our future work.

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