EXISTENCE AND GLOBAL ATTRACTIVITY OF POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE

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1 Electronic Journal of Differential Equations, Vol. 28 (28), No. 9, pp. 7. ISSN: URL: or EXISTENCE AND GLOBAL ATTRACTIVITY OF POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL WITH CROWLEY-MARTIN FUNCTIONAL RESPONSE XIAOWAN LI, XIAOJIE LIN, JIANG LIU Abstract. In this article, we consider a predator-prey system with mutual interference and Crowley-Martin functional response. We obtain positive solutions for the system by using the comparison principle. The existence of periodic solutions is established by applying coincidence degree theory. In addition, we obtain that the system has only one positive periodic solution which is a global attractor by constructing a proper Lyapunov function.. Introduction Predator-prey model is one of the dominant themes in both ecology and mathematical ecology because of its universal existence and importance with many concerned biological studies []. In recent years, classical predator-prey models have been extensively studied; see [2, 3, 4, 5] and the references cited therein. Hassell [6] discussed the following model with the mutual interference between predator and prey, ẋ = xg(x) ψ(x)y m, ẏ = y( d kψ(x)y m (.) q(y)), where the mutual interference constant m (, ] is a real number. During his research of the capturing behavior between hosts and parasites, he found that the hosts or parasites had the tendency to leave from each other when they met, which interfered with the hosts capturing effects. It is obvious that the mutual interference will be stronger while the size of the parasite becomes larger. Mathematicians and ecologists have explored the dynamical behavior of predatorprey models with the Holling type II [7, 8] and Holling type III functional responses. Lv [9, ] investigated the existence and globally attractivity of the positive periodic solutions of the predator-prey model with mutual interference and Holling type III, ẋ = x(r b x) c x 2 k 2 x 2 ym, ẏ = y( r 2 b 2 y) c 2x 2 k 2 x 2 ym. 2 Mathematics Subject Classification. 92D25, 34C25. Key words and phrases. Predator-prey model; Crowley-Martin functional response; periodic solution; permanence; global attractor; Lyapunov function. c 28 Texas State University. Submitted April, 27. Published November 26, 28. (.2)

2 2 X. LI, X. LIN, J. LIU EJDE-28/9 Later, the permanence and existence of a unique globally attractive positive almost periodic solution of model (.2) were considered by Zhang et al. []. There are many other types of functional response such as Beddington-DeAngelis which has been discussed [2, 3, 4, 5, 6, 7]. For example, Lin and Chen [6] studied an almost periodic Volterra model with mutual interference and Beddington-DeAngelis functional response as follows k x ẋ = x(r b x) a dx cy ym, k 2 x ẏ = y( r 2 b 2 y) a dx cy ym. (.3) Guo and Chen [7] investigated a special case of system (.3) and proved the existence and global attractivity of positive periodic solutions for the model. System (.3) with Beddington-DeAngelis functional response adopts that handing and interference are exclusive activities. Crowley-Martin [8] assumed that predator s predation will decrease due to high predator density (interference among the predator individuals) even when prey density is high (presence of handing or searching of prey by predator individual) [9]. There are very few literature available on predator-prey model with Crowley- Martin functional response [2, 2, 22]. The Crowley-Martin functional response is predator dependent. The per capita fedding rate for predator y in this formulation is η(x, y) = bx ( a x)( b y), where b, a, b are positive parameters that are used for effects of capture rate, handling time and magnitude of interference among predators, respectively, on the fedding rate. If we consider a x b y along with absence of mutual interference among predators at high prey density (i.e. when a b xy becomes too small) then the food supply (prey population) will be superabundant i.e. the increase in prey density (x) will not increase the feeding rate per predator η(x, y). In this case, the predation rate per unit of predator η(x, y) becomes constant and η(x, y) = b a. As y, the limiting value of η(x, y) becomes a function of x only and when y, η(x, y). One can easily observe that η(x, y) varies inversely with respect to y. Crowley-Martin response function represents classical response function for a =, b = while Michaelis-Menten (Holling type II) functional response for a >, b = [, 23]. Tripathi et al. [24] investigated the globally stability of the predator-prey model with Crowley-Martin response function with time delay of the form dx CY = X(A BX dt dy F = Y ( D EY dt A B X C Y B C XY ), X A B X C Y B C XY ). (.4) Egami and Hirano [25] considered the almost periodic solution and global attractivity on the basis of system (.4).

3 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL 3 Motivated by the above results, in this article we consider a predator-prey model with Crowley-Martin functional response ẋ = x[r a x a 2 x(t τ)] Cxy m A B x C y B C xy, ẏ = y[ r 2 b y b 2 y(t τ)] Dxy m A B x C y B C xy, (.5) where < m, x and y represent prey and predator densities at time t, respectively, r is intrinsic growth rate of the prey in the absence of the predator and r 2 is the death rate of the predator, a and b are decay rates of the prey and the predator in competition among their own populations, a 2 and b 2 are decay rates of the prey and the predator effected by harmful environmental for a period of past time. r i, a i, b i (i =, 2), A, B, C, C, D are positive ω-periodic function, t [, ), delay τ >, n 2. This article is organized as follow. In section 2, by using the comparison principle in ordinary differential equation and some analytical techniques, we study the permanence of positive solutions of delayed predator-prey model (.5). In section 3, we prove the existence of positive periodic solutions to systems (.5) by applying the coincidence degree theory. Section 4 is devoted to the global attractivity by constructing a suitable Lyapunov function. 2. Permanence For the sake of convenience and simplicity, we introduce some notation as follows: f L = min t [,ω] f, ˆf = ω f M = max t [,ω] f, f dt, f = ω f = max { f }, t [,ω] fdt, where f is a continuous ω-periodic function. To obtain the permanence of positive solutions of system (.5), we state some lemmas. Lemma 2. ([26]). If a >, b >, ż ( )z(b az) and z() >, then, for any small constant ε >, there exists a positive constant T, such that z b a ε, ( b ε), for t T. (2.) a Lemma 2.2 ([26]). If a >, b >, ż ( )z m (b az m ) and z() >, then, for any small constant ε >, there exists a positive constant T, such that z ( b a ) m ε, ( ( b a ) m ε), for t T. (2.2) Theorem 2.3. System (.5) is permanent; which means that for any positive solution (x, y) T of (.5), there exist positive constants K i, i =, 2, 3, 4, and T > such that K 3 x K, K 4 y K 2, for t T.

4 4 X. LI, X. LIN, J. LIU EJDE-28/9 Proof. Assume (x, y) T is an arbitrary positive solution of system (.5), then the first equation in (.5) yields ẋ x(r M a L x). From Lemma 2., for any small constant ε >, there exists a positive constant T, such that x ( r M ) ε a L := K, for t T. (2.3) Similarly, from the second equation in (.5), we obtain ( D ẏ y m M K ) r2 L y m. A L By Lemma 2.2, for the ε chosen above, there exists a positive constant T 2, such that ( D M K ) m y ε := K 2, for t T 2. (2.4) A L rl 2 From (2.3) and (2.4), for any small enough positive constant ɛ, there exists a positive number T 3 such that x K ɛ and y K 2 ɛ for all t T 3. From the first equation in (.5), one has Denoting ẋ x rl CM (K 2 ɛ) m A L (a a 2 ) M (K ɛ). δ(ɛ) = r L CM (K 2 ɛ) m A L (a a 2 ) M (K ɛ), and integrating above inequality from t τ to t, we obtain x(t τ) e δ(ɛ)τ x. From the first equation in (.5), we obtain that ẋ x(r L CM (K 2 ɛ) m (a M a M 2 e δ(ɛ)τ )x). (2.5) A L According to Lemma 2., for any positive constant ε (r L CM (K 2 ɛ) m A L )/(a M a M 2 e δ(ɛ)τ ), when ɛ, there exists a positive constant T 4, such that x (r L CM K m 2 A L )/(a M a M 2 e δτ ) ε := K 3, for t T 4. (2.6) From the second equation in (.5), we obtain ( ẏ y m D L K 3 A M BM (K ɛ) C M (K 2 ɛ) C M (K 2 ɛ)b M (K ɛ) ) (b M b M 2 )(K 2 ɛ) 2 m r2 M y m. Then by Lemma 2.2, letting ɛ and for any positive constant ε 2, we have D L K 3 ε 2 ( A M BM K C M K 2 C M K 2B M K (b M b M 2 )K 2 m 2 )/r M 2,

5 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL 5 and there exists a positive constant T 5, such that [( D L K 3 y A M BM K C M K 2 C M K 2B M K ) ] (b M b M 2 )K2 2 m /r2 M m ε 2 := K 4, for t T 5. Let T = max{t, T 2, T 3, T 4, T 5 }, and chose ε min{ε, ε 2 }, then we have K 3 x K, K 4 y K 2. (2.7) 3. Existence of positive periodic solutions To understand the sufficient conditions for guaranteeing the existence of positive periodic solutions, we will introduce the coincidence degree briefly as follows. Definition 3. ([27]). Let X and Y be two Banach spaces, L : Dom L X Y be a linear map. If the following conditions are satisfied (a) Im L is a closed subspace of Y ; (b) dim ker L = co dim Im L <, then we call the operator L is a Fredholm operator of index zero. If L is a Fredholm operator with index zero and there exists continuous projections P : X X and Q : Y Y such that X = ker L ker P, Y = Im L Im Q, Im P = ker L and Im L = ker Q = Im(I Q), then L Dom L ker P : (I P )X Im L has an inverse function, and we set it as K p, then Kp : Im L Dom L ker P. Definition 3.2 ([27]). Let N : X Y be a continuous map and Ω [, ] X is an open set. If QN(Ω [, ]) is bounded and K p (I Q)N(Ω [, ]) X is relatively compact, then we say that N(Ω [, ]) is L-compact. Lemma 3.3 ([27]). Let both X and Y be Banach spaces, L : Dom L X Y be a Fredholm operator with index zero, Ω X be an open bounded set, and N : Ω Y be L-compact on Ω. If all the following conditions hold: () Lx λn(x), for x Ω Dom L, λ [, ]; (2) Nx / Im L, x Ω ker L; (3) deg{jqn, Ω ker L, } =, where J : Im Q ker L is an isomorphism, then the equation Lx = N(x) has at least one solution on Ω Dom L. Suppose (x, y) T is an arbitrary positive solution of (.5), let u = ln x and v = ln y, then system (.5) can be changed into u = r a e u a 2 e u(t τ) Ce mv A B e u C e v B e u C e v, v = r 2 b e v b 2 e v(t τ) De u e (m )v A B e u C e v B e u C e v. (3.)

6 6 X. LI, X. LIN, J. LIU EJDE-28/9 Denoting the right terms of first equation and second equation in (3.) by F (t, u, v) and F 2 (t, u, v) respectively and considering system where λ (, ]. u = λf (t, u, v), v = λf 2 (t, u, v), (3.2) Lemma 3.4. Suppose (u, v) T is a ω-periodic solution of (3.2), then there exists a positive number R, such that u v R, where R will be calculated as in the proof. Proof. Since (u, v) T is periodic, the following discussion will be restricted to t [, ω]. Integrating the first equation of (3.2) from to ω and in view of udt =, we obtain Thus, r dt = (a e u a 2 e u(t τ) Ce mv )dt. A B e u C e v B e u C ev u dt = λ F dt Note that z = (u, v) T X, there exist ξ, η, ξ, η, such that u(ξ) = min t [,ω] u, v(η) = min t [,ω] v, (3.3) 2r dt = 2 r ω. (3.4) u(ξ) = max u, t [,ω] v(η) = max v. (3.5) t [,ω] So u(ξ) = u(ξ) = v(η) = v(η) =. From (3.3) and (3.5), we obtain Hence we have i.e., r dt e u(ξ) From (3.4) and (3.7), we have u u(ξ) (a e u a 2 e u(t τ) )dt (a a 2 )e u(ξ) dt = ω(ā ā 2 )e u(ξ). ω(ā ā 2 ) u(ξ) ln u dt ln r dt = Letting t = η in the second equation of (3.2), one has r 2 (η) r ā ā 2, (3.6) r ā ā 2. (3.7) r ā ā 2 2 r ω := U. (3.8) D(η)e u(η) e (m )v(η) A (η) B (η)e u(η) C (η)e v(η) B (η)e u(η) C (η)e v(η) D(η)e(m )v(η). B (η)

7 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL 7 So, we obtain v(η) m ln[r 2B D ]L := H. (3.9) Meanwhile, we know that b (η)e v(η) D(η)e u(η) e (m )v(η) A (η) B (η)e u(η) C (η)e v(η) B (η)e u(η) C (η)e v(η) D(η)e(m )v(η), B (η) and v(η) m 2 ln [ b B ] L := H2. (3.) D Combining inequalities (3.9) with (3.), we obtain On the other hand, in view of (3.2), one has r (ξ) = a (ξ)e u(ξ) a 2 (ξ)e u(ξ τ) v max{h, H 2 } := H 3. (3.) C(ξ)e mv(ξ) A (ξ) B (ξ)e u(ξ) C (ξ)e v(ξ) B (ξ)e u(ξ) C (ξ)e, (3.2) v(ξ) r 2 (η) = b (η)e v(η) b 2 (η)e v(η τ) By (3.2), we have So i.e. u(ξ) ln S := S. Then D(η)e u(η) e (m )v(η) A (η) B (η)e u(η) C (η)e v(η) B (η)e u((η) C (η)e. (3.3) v(η) r (ξ) (a (ξ) a 2 (ξ))e u(ξ) C(ξ)emv(ξ). A (ξ) e u(ξ) [ A (ξ)r (ξ) C(ξ)e mh3 (a (ξ) a 2 (ξ))a (ξ) ] := S, u u(ξ) u dt S 2 r ω := U 2. (3.4) Next we estimate the lower bound of v. If v(η), then the lower bound of v is. If v(η) <, then e ( m)v(η) e (2 m)v(η) and e nv(η) <. Combined with (3.3), we obtain (r 2 (η) b (η) b 2 (η))e ( m)v(η) D(η)e u(η) A (η) B (η)e u(η) C (η)e v(η) B (η)e u((η) C (η)e v(η) D(η)e U2 A (η) B (η)e U C (η)e H3 B (η)e U C (η)e H3.

8 8 X. LI, X. LIN, J. LIU EJDE-28/9 So v(η) ( ( m ln D(η)e U2 / (r 2 (η) b (η) b 2 (η))a (η) )) B (η)e U C (η)e H3 B (η)e U C (η)e H3 := S 2. Donating S 3 = min{, S 2 }, we obtain U 2 u U and S 3 v H 3. Thus u = max{ U, U 2 } := U, v = max{ S 3, H 3 } := H. So u v U H := R. Suppose (u, v) T is a constant solution of system (3.), then r a e u a 2 e u Ce mv A B e u C e u B e u C e v =, r 2 b e v b 2 e v De u e (m )v A B e u C e v B e u C e v =. (3.5) (3.6) Integrating two side of above equations on [, ω] and applying integral mean theorem, we obtain r ā e u ā 2 e u C(t )e mv A (t ) B (t )e u C (t )e u B (t )e u C (t )e v =, r 2 b e v b 2 e v D(t 2 )e u e (m )v A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v =, where t, t 2 [, ω]. Next we consider the equations r ā e u ā 2 e u C(t )e mv µ A (t ) B (t )e u C (t )e u B (t )e u C (t )e v =, µ r 2 b e v b 2 e v where µ is a parameter. D(t 2 )e u e (m )v A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v =, (3.7) (3.8) Lemma 3.5. Suppose (u, v) T is a solution of (3.8), then there exists a positive number R 2, such that u v R 2, where R 2 will be calculated as in the following proof. Proof. From the first equation in (3.8) we obtain r (ā ā 2 )e u. Then u ln From the second equation in (3.8), one has ( b b 2 )e v r ā ā 2 := W. (3.9) D(t 2 )e u e (m )v A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v.

9 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL 9 Then v 2 m ln D(t 2 ) B ( b (t 2 ) b 2 (t 2 )) := V. (3.2) If u, then is the low bound of u. If u <, from the first equation of (3.8), we obtain r (ā ā 2 )e u C(t )e mv A (t ). Thus u ln{( r C(t )e mv )/(ā ā 2 )} := W 2. A (t ) Letting W 3 = min{, W 2 }, we obtain u W 3. (3.2) If v, then is a lower bound of v. If v <, from the second equation of (3.8), one has r 2 ( b b 2 )e v D(t 2 )e u e (m )v A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v. Then, in view of ( m)v (2 m)v, and e mv <, we obtain and ( r 2 b b 2 )e ( m)v D(t 2 )e u A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v D(t 2 )e W3 A (t 2 ) B (t 2 )e W C (t 2 )e V B (t 2 )e W C (t 2 )e V, v ( ( m ln D(t 2 )e W3 / (A (t 2 ) B (t 2 )e W C (t 2 )e V B (t 2 )e W C (t 2 )e V )( r 2 b b )) 2 ) := V 2. Letting V 3 = min{, V 2 }, we have From (3.9), (3.2), (3.2) and (3.22), we know that Denoting v V 3. (3.22) W 3 u W, V 3 v V. (3.23) u = max{ W, W 3 } = O, v = max{ V, V 3 } = O 2. We have u v O O 2 := R 2. Theorem 3.6. System (.5) has at least one positive ω-periodic solution. Proof. Suppose that (x, y) T is an arbitrary positive solution of (.5) and let u = ln x, v = ln y, then (.5) is changed into (3.). Let X = Y = {z z = (u, v) T C(R, R 2 ) : z(t ω) = z}, be equipped with the norm z = max t [,ω] { z }.

10 X. LI, X. LIN, J. LIU EJDE-28/9 Then X and Y are both Banach spaces with the norm. Take z X and define operators L, P and Q as follows L : Dom L X Y, Lz = dz dt, P (z) = ω zdt, Q(z) = ω where Dom L = {z X : z C (R, R 2 )}. Define N:X Y by N(z) = ( r a e u a 2 e u(t τ) C r 2 b e v b 2 e v(t τ) D ) where C = D = Ce mv A B e u C e u B e u C e v De u e (m )v A B e u C e v B e u C e v Then ker L = R 2, dim ker L = codim Im L = 2, and { } Im L = z Y : zdt =, is closed in Y, and P, Q are both continuous projections satisfying Im P = ker L, Im L = ker Q = Im(I Q). zdt, So L is a Fredholm operation of index zero, which implies that L has a unique inverse. We denote by K p : Im L ker P Dom L the inverse of L. By a straightforward calculation, we obtain K p (z) = For any z X, we obtain and where W i = t z(s)ds ω t z(s) ds dt. QN(z) = Q(F (t, u, v), F 2 (t, u, v)) T ( ω ω ) T = F (t, u, v)dt, F 2 (t, u, v)dt) ω ω = ( F, F 2 ) T, F i (s, u(s), v(s))ds ω K p (I Q)Nz = (W, W 2 ) T, t F i (s, u(s), v(s))dsdt F i t ω 2 F. Obviously, it is undemanding to check by the Lebesgue convergence theorem that both QN and K p (I Q)N are continuous. Moreover, by using the Arzela-Ascoli Theorem, the operator K p (I Q)N(Ω) is compact and QN(Ω) is bounded for any open set Ω X. So N is L-compact on Ω respect to any bounded open set Ω X. Particularly, we take Ω = {z z = (u, v) T X, z R}, where R = R R 2 ε (ε > ), R and R 2 are defined in Lemma 3.4 and Lemma 3.5. Now, we check the three conditions in Lemma 3.3.

11 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL (i) For each λ (, ), z Ω Dom L, we have Lx λn(x). Otherwise, z is a ω-periodic solution of (3.2) and then z R, will be derived by Lemma 3.4. It is impossible because z = R > R for z Ω Dom L. dz dt (ii) When z Ω ker L, =, i.e., z is a constant vector (u, v) T with (u, v) T = R R 2 ε. If QN(u, v) T =, then (u, v) T is a solution of (3.8) for µ =. By Lemma 3.5, we have (u, v) T R 2 which contradicts to (u, v) T = R R 2 ε. Thus, for each z Im Q. When z Ω ker L, QNz. (iii) Choose J : Im Q ker L such that J(z) = z for each z Im Q. When z Ω ker L, z = (u, v) T is a constant vector and satisfies JQN(u, v) T = JQ(F (t, u, v), F 2 (t, u, v)) T = ( F (t, u, v)dt, ω ω ( r (ā ā 2 )e u = r 2 ( b b 2 )e v F 2 (t, u, v)dt)) T C(t )e mv A (t )B (t )e u C (t )e u B (t )e u C (t )e v D(t 2)e u e (m )v A (t 2)B (t 2)e u C (t 2)e v B (t 2)e u C (t 2)e v where t, t 2 were defined as in (3.7). We define ϕ : z Ω ker L [, ] X as follows ( ) r (ā ā 2 )e u ϕ(u, v, µ) = ( b b 2 )e v D(t 2)e u e (m )v A (t 2)B (t 2)e u C (t 2)e v B (t 2)e u C (t 2)e ( ) v C(t )e mv µ A (t )B (t )e u C (t )e u B (t )e u C (t )e v. r 2 Then JQN(u, v) T = ϕ(u, v, ). By Lemma 3.4, we see ϕ(u, v, ) (, ) T. Hence, using the homotopy invariance theorem of topological degree, we obtain deg{jqn(u, v) T, Ω ker L, (, ) T } = deg{ϕ(u, v, ), Ω ker L, (, ) T } = deg{ϕ(u, v, ), Ω ker L, (, ) T } { = deg ( r (ā ā 2 )e u, ( b b 2 )e v Denote D(t 2 )e u e (m )v A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v )T, Ω ker L, (, ) T }. ψ 2 (u, v) := ( b b 2 )e v and consider the algebraic equations ψ (u, v) := r (ā ā 2 )e u, D(t 2 )e u e (m )v A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v, ), ψ (u, v) =, ψ 2 (u, v) =. (3.24)

12 2 X. LI, X. LIN, J. LIU EJDE-28/9 From the first equation of (3.24) we obtain its unique u = ln r ā ā 2. Substituting it into second equation of (3.24), we obtain ( b b 2 )e v D(t 2 )e u e (m )v A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v =, which is easy checked to have a unique solution v on R. So equations (3.24) has unique solution (u, v ) T on Ω ker L. For convenience, we denote p(u, v) = A (t 2 ) B (t 2 )e u C (t 2 )e v B (t 2 )e u C (t 2 )e v. Then ψ u = (ā ā 2 )e u, ψ v =, ψ 2 u = D(t 2)e u e (m )v p(u, v) D(t 2 )e u e (m )v [B (t 2 )e u B (t 2 )e u C (t 2 )e v ] p 2, (u, v) ψ 2 v = M(u, v) ( b b 2 )e v, where ( M(u, v) = (m )D(t 2 )e u e (m )v p(u, v) D(t 2 )e u e (m )v [C (t 2 )e v ) B (t 2 )e u C (t 2 )e v ] /p 2 (u, v). Hence, we have deg{jqn(u, v) T, Ω ker L, (, ) T } ψ ψ = sgn u v ψ 2 u ψ 2 v (u,v ) = sgn[ (ā ā 2 )e u (M(u, v) ( b b 2 )e v )] =. All the conditions in Lemma 3.3 have been checked. This implies that (3.) has at least one ω-periodic solution. Further system (.5) has at least one ω-periodic solution. 4. Global attractivity Definition 4.. Suppose ( x, ỹ) T is a positive ω-periodic solution of (.5), (x, y) T is arbitrary positive solution of (.5), with lim x x = and lim y ỹ =, t t then ( x, ỹ) T is called a global attractor. Lemma 4.2 ([28]). If function f is nonnegative, integrable and uniformly continuous on [, ), then lim t f =. From Theorem 2.3, we know that for any positive ε enough small, there exists T, such that, when t T, an arbitrary positive solution (x, y) T of system (.5) satisfies K 3 ε x K ε, K 4 ε y K 2 ε. (4.)

13 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL 3 For convenience, we denote where γ = min{k 3, K 4, e U2, e S3 }, Γ = max{k, K 2, e U, e H3 }, (4.2) g = g(t, γ, γ), G = g(t, Γ, Γ), (4.3) g(t, x, y) = A B x C y B C xy. Theorem 4.3. Suppose system (.5) satisfies σ 2 σ = min { a CB t [,ω] g 2 γ m CB C g 2 γ m A 2 (m 2)DC G 2 Γ m } >, = min t [,ω] { ( mca g 2 CB C g 2 b mda G 2 (m 2)DC G 2 γ m mcb g 2 γ m mcb C g 2 γ m ) Γ m (m )DB G 2 Γ m γ m (m )CC g 2 γ m Γ m (m 2)DC B } G 2 Γ m B 2 >, (4.4) (4.5) then system (.5) has only one positive ω-periodic solution which is a global attractor. Proof. Suppose that (x, y) T is an arbitrary periodic solution of (.5). Theorem 3.6 indicates that (.5) has at least one positive ω-periodic solution ( x, ỹ) T satisfying e U2 x e U, e S3 ỹ e H3. We choose the Lyapunov function V = V V 2, where Then D V (.5) V = ln x ln x A 2 t t τ V 2 = ln y ln ỹ B 2 t t τ x(s) x(s) ds, y(s) ỹ(s) ds. = sgn(x x)(ẋ x x x ) A 2 x x A 2 x(t τ) x = sgn(x x)[ a (x x) a 2 (x(t τ) x(t τ)) ( Cym g(t, x, y) Cỹm g(t, x, ỹ) )] A 2 x x A 2 x(t τ) x(t τ). Since Cy m g(t, x, y) Cỹm g(t, x, ỹ) (4.6)

14 4 X. LI, X. LIN, J. LIU EJDE-28/9 = C { A [y m ỹ m ] B [ xy m g(t, x, y)g(t, x, ỹ) xy m xy m xỹ m ] C yỹ[y m ỹ m ] B C [ xỹy m xyy m xyy m } xyỹ m xyỹ m xyỹ m ] C { = A (y m ỹ m ) B ( x x)y m g(t, x, y)g(t, x, ỹ) B x(y m ỹ m ) C yỹ(y m ỹ m ) B C [ xy m (ỹ y) xy(y m ỹ m ) } yỹ m ( x x)]. Substituting this inequality in (4.6), we obtain D V (.5) C { a x x A y m ỹ m g(t, x, y)g(t, x, ỹ) B [y m x x x y m ỹ m ] C yỹ y m ỹ m B C [ xy m ỹ y } xy y m ỹ m yỹ m x x ] A 2 x x. (4.7) Meanwhile, D V 2 (.5) = sgn(y ỹ)(ẏ y ỹ ỹ ) B 2 y ỹ B 2 y(t τ) ỹ(t τ) = sgn(y ỹ)[ b (y ỹ) b 2 (y(t τ) ỹ(t τ)) Dxym g(t, x, y) B 2 y(t τ) ỹ(t τ). D xỹm ] B 2 y ỹ g(t, x, ỹ) (4.8) Since Dxy m g(t, x, y) = D g(t, x, y)g(t, x, ỹ) D xỹm g(t, x, ỹ) { A [x(y m ỹ m ) ỹ m (x x)] B xxy m ỹ m C yỹ[y m 2 (x x) } x(y m 2 ỹ m 2 )] B C x xyỹ(y m 2 ỹ m 2 ). (4.9)

15 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL 5 Substituting the above equality in (4.8), we obtain D V 2 (.5) b y ỹ D g(t, x, y)g(t, x, ỹ) (A [x y m ỹ m ỹ m x x ] B xx y m ỹ m C yỹ[y m 2 x x x y m 2 ỹ m 2 ] B C x xyỹ y m 2 ỹ m 2 B 2 y ỹ. For any x, x 2 [a, b] (, ), we have α b α x x 2 x α x α 2 α a α x x 2, for α <, αa α x x 2 x α x α 2 α b α x x 2, for α >. (4.) Then, from (4.7) and (4.), in view of (4.), (4.2) and (4.3), letting ε, we obtain D V (.5) and [ a CB g 2 [ mca g 2 CB C g 2 γ m CB C g 2 γ m A 2 ] x x γ m mcb g 2 D V 2 (.5) [ b mda G 2 γ m (m )CC g 2 γ m γ m mcb C g 2 γ m ] y ỹ, (m 2)DC G 2 Γ m (m )DB G 2 Γ m Γ m (m 2)DC B G 2 Γ m B 2 ] y ỹ [ (m 2)DC G 2 Γ m A Γ m ] x x. Summing (4.), (4.2), (4.4) and (4.5), we obtain, for t > T >, that (4.) (4.2) D V (.5) = D V (.5) D V 2 (.5) σ x x σ 2 y ỹ. Integrating the two sides of above inequality, we have T T V σ x(s) x(s) ds σ 2 y(s) ỹ(s) ds V (T ) <. From Lemma 4.2, we obtain t lim x x =, lim t t y ỹ =. t This proves that any positive ω-periodic solution of (.5) is a global attractor. Next we prove the uniqueness of the positive ω-periodic solution ( x(s), ỹ). Suppose there is another positive ω-periodic solution ( x (s), ỹ ). Otherwise, there exists a ξ [, ω] such that x(ξ) x (ξ) or ỹ(ξ) ỹ (ξ), then ε >. However ε = lim t x(ξ nω) x (ξ nω) = lim t x x =.

16 6 X. LI, X. LIN, J. LIU EJDE-28/9 This is a contradiction. Hence, the positive ω-periodic solution ( x(s), ỹ) is unique. Acknowledgements. This work was partly supported by grants 7785, 8725, and 6776 from the National Natural Science Foundation of China. References [] A. Berryman; The origin and evolution of predator-prey theory, Ecol. Soc. Am. 73 (992), [2] S. Ahmad; On the nonautonomous Volterra-Lotka competition equation, Proc. Amer. Math. Soc. 7() (993), [3] X. Chen, Z. Du; Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse, Qual. Theory Dyn. Syst., 7 (28), [4] Z. Du, Z. Feng; Existence and asymptotic behavior of traveling waves in a modified vectordisease model, Commun. Pure Appl. Anal., 7(5) (28), [5] B. Lisena; Global attractive periodic models of predator-prey type, Nonlinear Anal. Real world Appl. 6 (25), [6] M. Hassel; Mutual interference between searching insect parasites, J. Anim. Ecol., 4 (97), [7] A. Nindjin, M. Aziz-Alaoui, M. Cadivel; Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real world Appl., 7 (26), 4-8. [8] Y. Zhu, K. Wang; Existence and global attractivity of positive periodic solution for a predatorprey model with modified Leslie-Gower Holling-Type II schemes, J. Math. Anal. Appl., 384 (2), [9] Y. Lv, Z. Du; Existence and global attractivity of positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response, Nonlinear Anal. Real world Appl., 2 (2) [] X. Wang, Z. Du, J. Liang; Existence and global attractivity of positive periodic solution to a Lotka-Volterra model, Nonlinear Anal. Real world Appl., (2), [] C. Zhang, N. Huang, D. O Regan; Almost periodic solutions for a Volterra model with mutual interference and Holling III type functional response, Appl. Math. Comput., 225 (23), [2] Z. Du, Z. Feng; Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays, J. Comput. Appl. Math., 258 (24), [3] X. Lin, Z. Du, Y. Lv; Periodic solution to a multispecies predator- prey competition dyanmic sysytem with Beddington-DeAngelis functional response and time delay, Appl. Math., 58(6) (23), [4] L. Haiyin, Y. Takeuchi; Dynamics of the density dependent predator-prey system with Beedington-DeAngelis functional response, J. Math. Anal. Appl., 374 (2) [5] J. Tripathi, S. Abbas, M. Thakur; Dynamics analysis of a prey-predator model with Beedington-DeAngelis type function response incorporating a prey refuge, Nonlinear. Dyn., 8 (25), [6] X. Lin, F. Chen; Almost periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response, Appl. Math. Comput., 24 (29), [7] H. Guo, X. Chen; Existence and global attractivity of positive periodic solution for a Volterra model with mutual interference and Beddington-DeAngelis functional response, Appl. Math. Comput., 27 (2), [8] P. Crowley, E. Martin; Functional responses and interference within and between year classes of a dragonfly population, J. North Am. Benth. Soc., 8 (989), [9] G. Sklaski, J. Gilliam; Functional responses with predator interference: viable alternative to Holling type II model, Ecology., 82 () (2), [2] R. Upadhyay, R. Naji; Dynamics of three species food chain model with Crowley-Martin type functional response, Chao Solitions Fractals., 42 (29),

17 EJDE-28/9 POSITIVE PERIODIC SOLUTIONS FOR A PREDATOR-PREY MODEL 7 [2] R. Upadhyay, R. Raw, V. Rai; Dynamic complexities in a tri-trophic food chain model with Holling type II and Crowley-Martin type functional response, Nonlinear Anal. Model. Control., 5 (2), [22] Q. Dong, W. Ma, M. Sun; The asymptotic behaviour of a chemostar model with Crowley- Martin type functional response and time delay, J. Math. Chem., 5 (23), [23] S. Ruan, D. Xiao; Global analysis in a predator-prey system with non-monotonic functional response, SIAM J. Appl. Math., 6 (4) (2), [24] J. Tripathi, S. Tyagi, S. Abbas; Global analysis of a delayed density dependent predator-prey model with Crowley-Martin functional response, Commun. Nonlinear Sci. Numer. Simul., 3 (26), [25] C. Egami, N. Hirano; Periodic solution in a class of periodic delay predator-prey systems, Yokohama Math. J., 5 (24), [26] K. Wang; Permanence and global asymptotical stability of a predator-prey model with mutual interference, Nonlinear Anal. Real world Appl., 2 (2), [27] R. Gaines, J. Mawhin; Coincidence Degree and Nonlinear Differential Equations, Springer- Verlag, Berlin, 977. [28] Y. Zhu, K. Wang; Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes, J. Math. Anal. Appl., 384 (2), Xiaowan Li School of Mathematics, Jilin University, Changchun 32, China address: xiaowan27@63.com Xiaojie Lin School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 226, China address: linxiaojie973@63.com Jiang Liu School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 226, China address: jiangliu@jsnu.edu.cn