Electronic Journal of Differential Equations, Vol. 2005(2005), No. 31, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) PERIODIC SOLUTIONS FOR A DELAYED PREDATOR-PREY SYSTEM WITH DISPERSAL AND IMPULSES QIMING LIU, SHIJIE DONG Abstract. A delayed predator-prey system with prey dispersal in n-patch environments and impulse effects is investigated. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions to the system. 1. Introduction An important and ubiquitous problem in mathematical ecology concerns the effect of environment change in the growth and diffusion of a species in a heterogenous habitat. There have been many studies in the literatures that investigate the population dynamics with diffusion process [5, 6, 10, 11]. In most of the models considered so far, it has been assumed that all biological and environmental parameters are constants in time. However, any biological or environmental parameters are naturally subject to fluctuation in time. The effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Thus, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (such as seasonal effects of weather, food supplies, mating habits and so forth); on the other hand, it is generally recognized that some kinds of time delays are inevitable in population interactions. Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the seasonality of the changing environment and the effect of time delays. There are still some other possible exterior effects under which the population densities change very rapidly. For example, impulse reduction of the population density of a given species is possible after its partial destruction by catching or by poisoning with chemical used at some transitory slots in fishing or agriculture [1, 7, 12]. Recently, many authors studies the existence of positive periodic solution in population models by using power and effective method of coincidence degree [8, 1991 Mathematics Subject Classification. 34K13, 34K45, 92D25. Key words and phrases. Predator-prey system; impulse; dispersion; periodic solution; coincidence degree. c 2005 Texas State University - San Marcos. Submitted November 1, 2004. Published March 18, 2005. 1 2 Q. LIU, S. DONG EJDE-2005/31 13], In the present paper, we are concerned with the study on the combined effects of dispersion, periodicity of environment, time delays and impulses on the dynamics of predator-prey system. To do so we are devoted to the study of the following delayed periodic predator-prey system with prey dispersal in n-patch environments and impulses ẋi (t) =xi (t)[ri (t) − aii (t)xi (t) − ai,n+1 (t)xn+1 (t)] + n X Dj (t)[xj (t) − xi (t)], i = 1, . . . , n, j=1,j6=i n h X ẋn+1 (t) =xn+1 (t) − rn+1 (t) + an+1,j (t)xj (t − σj ) (1.1) j=1 i − an+1,n+1 (t)xn+1 (t − σn+1 ) , ∆(xi (τk )) = −cik xi (τk ), t = τk , k ∈ Z+ , t 6= τk , k ∈ Z+ , i = 1, 2, . . . , n + 1. with initial conditions xi (s) = φi (s), s ∈ [−σ, 0], φi (0) > 0, i = 1, . . . , n + 1, (1.2) where xi (t) (i = 1, . . . , n) denote the densities of prey species in patch i, xn+1 denote the densities of all predator for all patches. ri (t) is the intrinsic growth rate of the prey, aii (t) is the density-dependent coefficient of the prey species. ai,n+1 (t) are the capturing rates of the predator and an+1,i /ai,n+1 are the conversion rates of nutrients into the reproduction of predator, rn+1 (t) are the death rates of the predator. Di (t) (i = 1, 2, . . . , n) is dispersal rate of prey species, σn+1 ≥ 0 denotes the delay due to negative feedback of the predator species σi (i = 1, 2, . . . , n) are the time delays due to gestation, that is, mature adult predators can only contribute to the production of predator biomass. aii (t), ai,n+1 (t), an+1,i (i, j = 1, . . . , n), ri (t) (i = 1, . . . , n + 1) and Di (t) (i = 1, 2, . . . , n) are continuously positive periodic functions with period ω > 0. cik are positive constants and 0 < cik < 1, Z+ is the set of all positive integers and there exists an integer p > 0 such that ci(k+p) = cik , τk+p = τk + ω. It is well known that by the fundamental theory of functional differential equations [4], system (1.1) has a unique solution x(t) = (x1 (t), . . . , xn+1 (t)) satisfying initial conditions (1.2). It is easy to verify that solutions of (1.1) corresponding to initial conditions (1.2) are defined on [0, +∞) and remain positive for all t ≥ 0. In this paper, the solution of system (1.1) satisfying initial conditions (1.2) is said to be positive. We shall use the following notation: Let J ⊂ R. Denote by P C(J, R) the set of function ψ : J → R which are continuous for t ∈ J, t 6= τk , are continuous from the left for t ∈ J and have discontinuities of the first kind at the points τk ∈ J. Denote the Banach space of ω-periodic functions by P Cω = {ψ ∈ P C[0, ω], R}|ψ(0) = ψ(ω)} and we denote Z 1 ω f¯ = f (t)dt, f L = min f (t), f M = max f (t), ω 0 t∈[0,ω] [0,ω] where f ∈ P Cω . EJDE-2005/31 PERIODIC SOLUTIONS 3 The organization of this paper is as follows. In the next section, by using Gaines and Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions are derived for the existence of positive periodic solutions of system (1.1) with initial conditions (1.2). 2. Main result In this section, by using Gaines and Mawhin’s continuation theorem of coincidence degree theory, we show the existence of positive ω-periodic solutions of (1.1)-(1.2). To this end, we first introduce the following notations. Let X, Z be real Banach spaces, let L: Dom L ⊂ X → Z be a linear mapping, and N : X → Z be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dim ker L = codim Im L < +∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X, and Q : Z → Z such that Im P = ker L, ker Q = Im L = Im(I − Q), then the restriction LP of L → Dom L ∩ ker P : (I − P )X → Im L is invertible. Denote the inverse of LP by KP . If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω̄ if QN (Ω̄) is bounded and KP (I − Q)N : Ω̄ → X is compact. Since Im Q is isomorphic to ker L, there exists isomorphism J : Im Q → ker L. For convenience, we introduce the continuation theorem of coincidence degree theory [3] and compactness criterion for set F ⊂ P Cω [2] as follows. Lemma 2.1. Let Ω ⊂ X be an open bounded set. Let L be a Fredholm mapping of index zero and N be L-compact on Ω̄. Assume (a) For each λ ∈ (0, 1), x ∈ ∂Ω ∩ Dom L, Lx 6= λN x (b) For each x ∈ ∂Ω ∩ ker L, QN x 6= 0 (c) deg{JQN, Ω ∩ ker L, 0} = 6 0. Then Lx = N x has at least one solution in Ω̄ ∩ Dom L. Lemma 2.2 (Compactness criterion). A set F ⊂ P Cω is relative compact if and only if (a) F is bounded, that is, kψk = sup{|ψ| : t ∈ J} ≤ M for each x ∈ F and some M > 0 (b) F is quasiequicontinuous in J. We are now in a position to state our main result on the existence of a positive periodic solution to system (1.1). Theorem 2.3. System (1.1) with initial conditions (1.2) has at least one strictly positive ω-periodic solution provided that (H1) n X n X aM n+1,j (rj − j=1 Dk − k=1,k6=j p p k=1 k=1 1 Y 1 1 Y 1 ln )/aM ln jj > rn+1 + ω 1 − cjk ω 1 − cn+1,k (H2) For i = 1, 2, . . . , n, ri − n X p Dj − j=1,j6=i > aM i,n+1 A 1 Y 1 ln ω 1 − cik k=1 Pn M j=1 an+1,j − rn+1 − 1 ω aL n+1,n+1 ln Qp 1 k=1 1−cn+1,k , 4 Q. LIU, S. DONG where A = max1≤i≤n {[(ri − Pn j=1,j6=i EJDE-2005/31 Pn Dj )M + j=1,j6=i DjM ]/aL ii }. Proof. Since solutions of (1.1)-(1.2) remain positive for all t ≥ 0, we let yi (t) = ln[xi (t)], i = 1, . . . , n + 1. (2.1) Substituting (2.1) into system (1.1), we derive ẏi (t) =ri (t) − + n X Dj (t) − aii (t)eyi (t) − ai,n+1 (t)eyn+1 (t) ] j=1,j6=i n X Dj (t)eyj (t)−yi (t) , i = 1, . . . , n, j=1,j6=i ẏn+1 (t) = − rn+1 (t) + n X (2.2) an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) , j=1 t 6= τk , k ∈ Z+ , ∆(yi (τk )) = ln(1 − cik ), t = τk , k ∈ Z+ , i = 1, 2, . . . , n + 1. ∗ It is easy to see that if (2.2) has one ω-periodic solution (y1∗ (t), . . . , yn+1 (t))T , ∗ ∗ ∗ T ∗ ∗ T then x (t) = (x1 (t), . . . , xn+1 (t)) = (exp[y1 (t)], . . . , exp[yn+1 (t)]) is a positive ω-periodic solution of system (1.1). Therefore, to complete the proof, it suffices to show that system (2.2) has one ω-periodic solution. Let X = {x ∈ P C(R, Rn+1 ) : y(t + ω) = y(t)}, Pn+1 with the norm kxk = supt∈[0,ω] i=1 |yi (t)|, where x = (y1 , y2 , . . . , yn+1 )T . Let Z = X × Rnp with the norm k(x, a1 , . . . , ap )k = (kxk2 + |a1 |2 + · · · + |ap |2 )1/2 . Then X and Z are Banach spaces. Let L : Dom L → Z L(x)(t) = (ẋ, ∆(y(τ1 ), . . . , ∆(y(τp )) where Dom L consist of functions x ∈ X such that x is continuous for t 6= τk , x is continuous from the left for t = τk , and ẋ(τk ) exists. Let N : X → Z, be defined as (N x)(t) = (f (t, x(t)), C1 , C2 , . . . , Cp ), where f (t, x) equals Pn Pn r1 (t) − j=2 Dj (t) − a11 (t)ey1 (t) − a1,n+1 (t)eyn+1 (t) ] + j=2 Dj (t)eyj (t)−y1 (t) .. . Pn−1 Pn−1 yn (t) yn+1 (t) yj (t)−yn (t) − an,n+1 (t)e ] + j=1 Dj (t)e rn (t) − j=1 Dj (t) − ann (t)e Pn −rn+1 (t) + j=1 an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) , and Ck = (ln(1 − c1k ), ln(1 − c2k ), . . . , ln(1 − cn+1,k ))T , k = 1, 2, . . . , p. Define two projectors P and Q as P : X → ker L, Q : Z → Z, Q(y, C1 , . . . , Cp) = Py = 1 Z ω 0 1 ω Z ω ydt, 0 ω ydt + p X k=1 0 o .. p Ck , . n 0 k=1 , EJDE-2005/31 PERIODIC SOLUTIONS 5 It is clear that ker L = {x : x ∈ X, x = h, h ∈ Rn+1 }, Z ω p X Im L = {z = (y, C1 , . . . , Cp ) ∈ Z, y(t)dt + Ck = 0} is closed in Z, 0 k=1 dim ker L = codim Im L = n + 1. Therefore, L is a Fredholm mapping of index zero. It is easy to show that P and Q are continuous projectors such that Im P = ker L, ker Q = Im L = Im(I − Q). Furthermore, the inverse KP of LP exists and is given by KP : Im Lto Dom L∩ker P , t Z y(s)ds − KP (z) = 0 1 ω Z ω Z t y(s)ds − 0 0 p X Ck . k=1 Then QN : X → Z and KP (I − Q)N : X → X read QN x = 1 Z ω ω 0 0 o p n X 1 p f (t, x)dt + Ck , ... , ω k=1 k=1 0 KP (I − Q)N x Z p 1 Z ωZ t X 1 X 1 t f (t, x)dt + Ck − f (t, x)ds + Ck = ω 0 ω t>τ ω 0 0 k=1 k Z ω p t 1 1 1X − ( − ) f (t, x)dt + Ck . ω 2 ω 0 ω k=1 Clearly, QN and KP (I − Q)N are continuous. To apply Lemma 2.1, we need to search for an appropriate open, bounded subset Ω. Corresponding to the operator equation Lx = λN x, λ ∈ (0, 1), we obtain n h i X ẏi (t) =λ ri (t) − Dj (t) − aii (t)eyi (t) − ai,n+1 (t)eyn+1 (t) j=1,j6=i + n X Dj (t)eyj (t)−yi (t) ], i = 1, . . . , n, j=1,j6=i h ẏn+1 (t) =λ − rn+1 (t) + n X (2.3) i an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) , j=1 t 6= τk , k ∈ Z+ , ∆(yi (τk )) = λ ln(1 − cik ), t = τk , k ∈ Z+ , i = 1, 2, . . . , n + 1. 6 Q. LIU, S. DONG EJDE-2005/31 Suppose that (y1 (t), . . . , yn+1 (t))T ∈ X is a solution of (2.3) for some λ ∈ (0, 1). Integrating system (2.3) over [0, ω], for i = 1, 2, . . . , n, we have Z ω aii (t)eyi (t) dt + ω Z 0 ai,n+1 (t)eyn+1 (t) dt + ln 0 Z n X ω (ri (t) − = 0 p Y k=1 Z n X Dj (t))dt + (2.4) ω yj (t)−yi (t) Dj (t)e dt, 0 j=1,j6=i j=1,j6=i 1 1 − cik and Z ω ω Z yn+1 (t−σn+1 ) rn+1 (t)dt + 0 = an+1,n+1 (t)e dt + ln 0 n Z ω X j=1 p Y k=1 yj (t−σj ) an+1,j (t)e 1 1 − cn+1,k (2.5) dt. 0 Multiplying the ith equation (2.3) by eyi (t) and integrating over [0, ω] gives Z ω aii (t)e2yi (t) dt 0 Z ≤ ω (ri (t) − 0 n X Z n X Dj (t))eyi (t) dt + j=1,j6=i j=1,j6=i ω Dj (t)eyj (t) dt + 0 p X k=1 in which ∆(eyi (τk ) ) = [(1 − cik )λ − 1]eyi (τk ) ≤ 0, then we have Z ω Z ω Z n n X X 2yi (t) M yi (t) M aL e dt ≤ [(r − D ) ] e dt + D i j ii j 0 0 j=1,j6=i ∆(eyi (τk ) ), ω eyj (t) dt . (2.6) 0 j=1,j6=i Using the inequality Z ω eyi (t) dt Z 2 ω e2yi (t) dt, ≤ω 0 0 it follows from (2.6) that Z Z Z ω n n X X ω yi (t) 1 L ω yi (t) 2 aii ( e dt) ≤ (ri − Dj )M e dt + DjM eyj (t) dt. ω 0 0 0 j=1,j6=i j=1,j6=i (2.7) Using the fact that if i = k, Z ω Z eyk (t) dt ≥ max 0 eyi (t) dt, i = 1, . . . , n 0 this , together with (2.7), leads to Z ω 1 L akk ( eyk (t) dt)2 ω 0 Z n X ≤ (rk − Dj )M j=1,j6=k ω 0 ω eyk (t) dt + ( n X j=1,j6=k DjM ) Z 0 ω eyk (t) dt, EJDE-2005/31 PERIODIC SOLUTIONS which implies Z ω Z yi (t) max{ e dt, i = 1, . . . , n} ≤ ω 0 7 eyk (t) dt Pn Pn [(rk − j=1,j6=k Dj )M + j=1,j6=k DjM ]ω 0 ≤ aL kk (2.8) . Set n (ri − Pn j=1,j6=i A = max Dj )M + Pn j=1,j6=i aL ii 1≤i≤n DjM o , we have from (2.8) that ω Z eyi (t) dt ≤ Aω, i = 1, . . . , n. (2.9) 0 Since yi ∈ P Cω , there exists ti , Ti ∈ [0, T ] ∪ {τ1+ , τ2+ , . . . , τp+ } such that yi (ti ) = min yi (t), yi (Ti ) = max yi (t), t∈[0,ω] i = 1, 2, . . . , n. t∈[0,ω] It follows from (2.9) that y(ti ) ≤ ln A, i = 1, 2, . . . , n. (2.10) We derive from (2.5) that Z ω an+1,n+1 (t)eyn+1 (t−σn+1 ) dt 0 ≤ n X aM n+1,j Z = dt − rn+1 ω − ln 0 aM n+1,j Z n X p Y k=1 p Y ω eyj (t) dt − rn+1 ω − ln 0 j=1 ≤ (A yj (t−σj ) e j=1 n X ω k=1 aM n+1,j − rn+1 )ω − ln j=1 p Y k=1 1 1 − cn+1,k (2.11) 1 1 − cn+1,k 1 , 1 − cn+1,k which yields yn+1 (tn+1 ) ≤ ln A Pn j=1 aM n+1,j − rn+1 − 1 ω ln Qp 1 k=1 1−cn+1,k an+1,n+1 , and Z ω eyn+1 (t) dt = 0 ≤ Z ω eyn+1 (t−σn+1 ) dt 0 Pn A j=1 aM n+1,j − rn+1 − 1 ω aL n+1,n+1 ln (2.12) Qp 1 k=1 1−cn+1,k ω. 8 Q. LIU, S. DONG EJDE-2005/31 It follows from (2.4), (2.5), (2.9) and (2.12) that ω Z |ẏn+1 (t)|dt 0 Z ω λ| − rn+1 (t) + = 0 Z n X an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) |dt j=1 n X ω ≤ [rn+1 (t) + 0 an+1,j (t)eyn+1 (t−σj ) + an+1,n+1 (t)eyn+1 (t−σn+1 ) ]dt j=1 Z ≤2 ω 0 < 2Aω n X an+1,j (t)eyj (t−σj ) dt − ln j=1 n X p Y k=1 (2.13) 1 1 − cn+1,k aM n+1,j := dn+1 , j=1 and Z ω n X ω Z |ẏi (t)|dt = λ|ri (t) − 0 0 Dj (t) − aii (t)eyi (t) − ai,n+1 (t)eyn+1 (t) j=1,j6=i n X + Dj (t)eyj (t)−yi (t) |dt j=1,j6=i ω Z ≤ [ri (t) + 0 n X Dj (t) + aii (t)eyi (t) + ai,n+1 (t)eyn+1 (t) j=1,j6=i n X + Dj (t)eyj (t)−yi (t) ]dt j=1,j6=i ω Z [aii (t)eyi (t) + ai,n+1 (t)eyn+1 (t) ]dt + ln ≤2 0 p Y k=1 ≤ 2 aM ii ω Z eyi (t) dt + aM i,n+1 0 Z ω 1 1 − cik p Y eyn+1 (t) dt + ln 0 k=1 n X p Y L aM )/a ω + ln n+1,j n+1,n+1 j=1 k=1 M < 2A aM ii + ai,n+1 ( 1 1 − cik 1 := di , 1 − cik (2.14) i = 1, . . . , n. From (2.10), (2.14) and (2.12), (2.13), we have yn+1 (t) Z ω ≤ yn+1 (tn+1 ) + p Y |ẏn+1 (t)|dt + ln 0 ≤ ln A Pn M j=1 an+1,j k=1 − rn+1 − 1 ω an+1,n+1 ln Qp 1 1 − cn+1,k 1 k=1 1−cn+1,k p Y + dn+1 + ln 1 , 1 − cn+1,k k=1 (2.15) EJDE-2005/31 PERIODIC SOLUTIONS 9 and Z ω yi (t) ≤yi (ti ) + p Y |ẏi (t)|dt + ln 0 k=1 p Y ≤ ln A + di + ln k=1 1 , 1 − cik 1 1 − cik (2.16) i = 1, 2, . . . , n. On the other hand, it follows from (2.4) and (2.9) that n X yi (Ti ) aM ω ≥ (ri − ii e Dj )ω − ln j=1,j6=i n X ≥ (ri − Dj )ω − ln j=1,j6=i p Y k=1 p Y k=1 1 − aM i,n+1 1 − cik Z ω eyn+1 (t) dt 0 1 1 − cik p n X 1 Y 1 M − aM A a − r − ln ω/aL n+1 i,n+1 n+1,j n+1,n+1 , ω 1 − c n+1,k j=1 k=1 which implies yi (Ti ) ≥ ln p n h 1 X 1 Y 1 r − D − ln i j ω 1 − cik aM ii j=1,j6=i k=1 − aM i,n+1 (A n X p aM n+1,j − rn+1 − j=1 i 1 Y 1 ln )/aL := ρi . n+1,n+1 ω 1 − cn+1,k k=1 (2.17) From (2.13) and (2.17) it follows that for i = 1, 2 . . . , n, Z ω yi (t) ≥ yi (Ti ) − |ẏi (t)|dt − | ln 0 k=1 Note that Z ω Z aii (t)eyi (t) dt + 0 p Y p Y 1 1 | ≥ ρi − di − | ln |, (2.18) 1 − cik 1 − cik k=1 ω ai,n+1 (t)eyn+1 (t) dt ≥ (ri − 0 n X j=1,j6=i Dj )ω − ln p Y k=1 1 , 1 − cik which, together with (2.5), leads to aM n+1,n+1 yn+1 (Tn+1 )ω Z ω n X ≥ aL eyj (t) dt − rn+1 ω n+1,j 0 j=1 ≥ n X aL n+1,j (rj − Pn k=1,k6=j Dk − j=1 − rn+1 ω − ln p Y k=1 1 1 − cn+1,k 1 ω ln Qp 1 k=1 1−cjk aM jj − aj,n+1 eyn +1(Tn+1 ) )ω 10 Q. LIU, S. DONG EJDE-2005/31 which yields yn+1 (Tn+1 ) ≥ aM n+1,n+1 n X 1 Pn M + ( j=1 aM n+1,j aj,n+1 )/ajj j=1 p n X aM n+1,j rj − × Dk − k=1,k6=j 1 Y 1 M ln /ajj ω 1 − cjk (2.19) k=1 p − rn+1 − 1 Y 1 ln := ρn+1 . ω 1 − cn+1,k k=1 From what has been discussed above, we finally derive that for i = 1, . . . , n, max |yi (t)| t∈[0,ω] p Y ≤ max | ln A| + di + | ln k=1 p Y 1 1 |, |ρi | + di + | ln | := Bi , 1 − cik 1 − cik k=1 and max |yn+1 (t) t∈[0,ω] ≤ | max{| ln A Pn j=1 1 aM n+1,j − rn+1 ω ln Qp 1 k=1 1−cn+1,k an+1,n+1 | p Y 1 1 |, |ρn+1 | + dn+1 + | ln | := Bn+1 . 1 − cn+1,k 1 − cn+1,k k=1 k=1 (2.20) Pn+1 Clearly, Bi (i = 1, 2, . . . , n + 1) are independent of λ. Denote B = i=1 Bi + B0 , ∗ here B0 is taken sufficiently large such that each solution (v1∗ , v2∗ , ..., vn∗ , vn+1 )T of the system of algebraic equations + dn+1 + | ln ri − n X Dj − j=2 p Y p n X 1 Y 1 ln − aii evi − ai,n+1 evn+1 + Dj evj −vi = 0, ω 1 − cik k=1 j=1,j6=i i = 1, 2, . . . , n, p −rn+1 − n X 1 Y 1 ln + an+1,j evj − an+1,n+1 evn+1 = 0, ω 1 − cn+1,k j=1 k=1 ∗ k(v1∗ , v2∗ , . . . , vn+1 )T k satisfies where for i = 1, 2, . . . , n, = Pn+1 ∗ i=1 |vi | < B (if it exists) and Pn+1 p n 1 X 1 Y 1 Ci = max | ln A0 |, ri − Dj − ln aii ω 1 − cik j=1,j6=i − ai,n+1 (A0 n X j=1 an+1,j k=1 − rn+1 )/an+1,n+1 i=1 (2.21) Ci < B, EJDE-2005/31 PERIODIC SOLUTIONS 11 and Cn+1 = max | ln A0 Pn j=1 an+1,j − rn+1 − 1 ω ln Qp 1 k=1 1−cn+1,k |, an+1,n+1 Pn Pn Qp 1 )/aii − rn+1 − i=1 an+1,i (ri − j=1,j6=i Dj − ω1 ln k=1 1−c ik Pn an+1,n+1 + ( j=1 an+1,j aj,n+1 )/ajj 1 ω ln Qp 1 k=1 1−cn+1,k (2.22) in which A0 = max {ri /aii }. 1≤i≤n Now, we take Ω = {y ∈ X : kyk < B}. Thus, the condition (a) of Lemma 2.1 is satisfied. When y ∈ ∂Ω ∩ ker L = ∂Ω ∩ Rn+1 , y = (y1 , y2 , ..., yn+1 )T is a constant vector in Rn+1 with kyk = B. If system (2.21) has solutions, then QN (y1 , . . . , yn , yn+1 )T Qp Pn Pn 1 r1 − j=2 Dj − ω1 ln k=1 1−c − a11 ey1 − a1,n+1 eyn+1 + j=2 Dj eyj −y1 1k .. . , = Pn−1 Qp Pn−1 1 1 yn yn+1 yj −yn e − a e + D e rn − j=1 Dj − ω lnQ k=1 1−cnk − aP nn n,n+1 j j=1 p n 1 + j=1 an+1,j eyj − an+1,n+1 eyn+1 −rn+1 − ω1 ln k=1 1−cn+1,k n 0 op .. 6= (0, . . . , 0, 0)T . . k=1 0 If system (2.21) does not have a solution, then we can directly derive y1 n 0 on+1 n 0 op .. QN ... 6= , ... . . yn+1 0 i=1 0 k=1 Thus, the condition (b) in Lemma 2.1 is satisfied. Finally, we will prove that the condition (c) in Lemma 2.1 is satisfied. To this end, we define φ : DomL × [0, 1] → X by φ(y1 , . . . , yn+1 , µ) Qp Pn 1 r1 − j=2 Dj − ω1 ln k=1 1−c − a11 ey1 1k .. . = Qp Pn−1 1 1 yn rn − j=1 Dj − ω ln k=1 1−cnk − ann e Qp Pn 1 1 −rn+1 − ω ln k=1 1−cn+1,k + j=1 an+1,j eyj − an+1,n+1 eyn+1 Pn −a1,n+1 eyn+1 + j=2 Dj eyj −y1 .. . + µ , P −ann eyn − an,n+1 eyn+1 + n−1 Dj eyj −yn j=1 0 where µ is a parameter. When (y1 , y2 , . . . , yn+1 )T ∈ ∂Ω ∩ Rn+1 , (y1 , y2 , . . . , yn+1 )T is a constant vector in Rn+1 with kyk = B. 12 Q. LIU, S. DONG EJDE-2005/31 We will show that when (y1 , y2 , . . . , yn+1 )T ∈ ∂Ω ∩ ker L, φ(y1 , y2 , . . . , yn+1 , µ) 6= 0. Otherwise, there is a constant vector (y1 , . . . , yn+1 )T ∈ Rn+1 with kyk = B satisfying φ(y1 , y2 , . . . , yn+1 , µ) = 0, that is ri − n X Dj − p n X 1 Y 1 ln − aii eyi − µai,n+1 eyn+1 + µ Dj eyj −yi = 0, ω 1 − cik k=1 j=1,j6=i j=1,j6=i i = 1, 2, . . . , n, p −rn+1 − n X 1 Y 1 an+1,j eyj − an+1,n+1 eyn+1 = 0. ln + ω 1 − cn+1,k j=1 k=1 By similar argument in (2.10), (2.12), (2.17) and (2.19), we have |yi | ≤ Ci , i = 1, . . . , n + 1, Pn+1 Pn+1 where Ci is defined by (2.22). Thus, we have i=1 |yi | ≤ i=1 Ci < B, which is leads to a contradiction. Using the property of topological degree and taking y1 y1 n 0 op .. .. J : Im Q → X, → ... , . , . k=1 yn+1 0 yn+1 we have deg(JQN (y1 , . . . , yn+1 )T , Ω ∩ ker L, (0, . . . , 0)T ) = deg(φ(y1 , . . . , yn+1 , 1), Ω ∩ ker L, (0, . . . , 0)T ) = deg(φ(y1 , . . . , yn+1 , 0), Ω ∩ ker L, (0, . . . , 0)T ) p n n X X 1 Y 1 = deg r1 − Dj − ln − a11 ey1 , . . . , rn − Dj ω 1 − c1k j=2 j=2 k=1 − + 1 ln ω n−1 X p Y k=1 p 1 1 Y 1 − ann eyn , −rn+1 − ln 1 − cnk ω 1 − cn+1,k k=1 an+1,j eyj − an+1,n+1 eyn+1 T , Ω ∩ ker L, (0, . . . , 0)T . j=1 Under assumption (H1), one can easily show that the system of algebraic equations ri − n X p Dj − j=1,j6=i 1 Y 1 ln − aii ui = 0, ω 1 − cik i = 1, 2, . . . , n, k=1 p (2.23) n X 1 Y 1 −rn+1 − ln + an+1,j uj − an+1,n+1 un+1 = 0, ω 1 − cn+1,k j=1 k=1 has a unique solution (u∗1 , . . . , u∗n+1 )T which satisfies Qp Pn ri − j=1,j6=i Dj − ω1 ln k=1 ∗ ui = aii 1 1−cik >0 EJDE-2005/31 PERIODIC SOLUTIONS 13 for i = 1, . . . , n, and u∗n+1 = ≥ n X 1 an+1,n+1 j=1 1 ln ω 1 n X − rn+1 − k=1 1 1 − cn+1,k aM n+1,j j=1 1 ln ω p Dj − n X rj − j=1,j6=i p Y k=1 1 1 − cn+1,k 1 Y 1 ln /(ajj ) ω 1 − cik k=1 j=1,j6=i p Y − rn+1 − an+1,n+1 n X an+1,j rj − p 1 Y 1 Dj − ln /(ajj ) ω 1 − cik k=1 > 0. A direct calculation shows that deg(JQN (y1 , y2 , . . . , yn+1 )T , Ω ∩ KerL, (0, 0, . . . , 0)T ) = sgn n+1 Y (−aii ) = (−1)n+1 6= 0. i=1 Finally, easily we show that the set {KP (I − Q)N u|u ∈ Ω̄} is equicontinuous and uniformly bounded. Using Lemma 2.2 and the Arzela-Ascoli Theorem, we see that KP (I − Q)N : Ω → X is compact. Moreover, QN (Ω̄) is bounded. Consequently, N is L−compact. We have proved that Ω satisfies all the requirements in Lemma 2.1. Hence, system (2.2) has at least one ω-periodic solution. Accordingly, system (1.1) has at least one positive ω-periodic solution. This completes the proof. Remark. In this paper, by borrowing Gaines and Mawhin’s continuation theorem of coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions to system (1.1) with initial conditions (1.2). We would like to mention here that it is interesting but challenging to discuss the existence of positive periodic solutions of (1.1) when we incorporate time delays to the self-regulated terms of the prey in n-patch environments. We leave this for our future work. References [1] G. Ballinger and X. Liu; Permanence of population growth models with impulse effects, Mathl. Comput. Modelling, 26 (1997) 59-72. [2] D. Bainov and P. Simeonov; Impulsive differential equations: periodic solutions and application, Pitman Monographs and Surveys in pure and applied Mathematics, 66 (1993), p. 25. [3] R. E. Gaines and J. L. Mawhin; Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977, p. 40. [4] J. Hale; Theory of Functional Differential Equations, Springer-Verlag, Heidelberg, 1977. [5] Y. Kuang and Y. Takeuchi; Predator-prey system dynamics in models of prey Dispersal in two-patch environment, Math.Biosci. 120 (1994) 77-98. [6] S. A. 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Mechanics in Medicine and Biology, 2 (2002) 207-206. [13] Z. Zhang and Z. Wang; Periodic solution for a two-species nonautonomous competition lotkavolterra patch system with time delay, J. Math. Anal. Appl. 265 (2002) 38-48. Qiming Liu Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China E-mail address: llqm2002@yahoo.com.cn Shijie Dong Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China E-mail address: j ds@sina.com