Existence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1

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1 Advances in Dynamical Systems and Applications. ISSN Volume 1 Number 2 (26), pp c Research India Publications Existence of Positive Periodic Solutions of Mutualism Systems with Several Delays 1 Yonghui Xia College of Mathematics and Computer Science, Fuzhou University, Fuzhou 352, P.R.China yhxia@fzu.edu.cn; xiayonhui@163.com Abstract In this paper, by using Mawhin coincidence degree, we study the global existence of positive periodic solutions for a class of mutualism systems with several delays and obtain a new and interesting criterion, which is much different from those of Yang etc. [8]. Moreover, our arguments for obtaining bounds of solutions to the operator equation Lx = λn x are much different from those used in [8]. So some new arguments are employed for the first time. AMS subject classification: 34C25, 92D25, 92B5. Keywords: Periodic solutions, coincidence degree, mutualism model. 1. Introduction Traditional Lotka Volterra type predator-prey model or competitive model has received great attention from both theoretical and mathematical biologists and has been studied extensively (for example, see [1 5, 9, 11 22]). But there are few papers considering the mutualism system. Goh [6] discussed the stability in models of mutualism of the form ẋ1 (t) = x 1 (t) [ r 1 (t) a 11 (t)x 1 (t) + a (t)x 2 (t) ], ẋ 2 (t) = x 2 (t) [ r 2 (t) + a 21 (t)x 1 (t) a 22 (t)x 2 (t) ]. (1.1) 1 This work was supported by the Foundation of Developing Science and Technology of Fuzhou University under the grant 24-XY- and the National Natural Science Foundation of China under Grant (No.16717) and Shanghai Shuguang Genzong Project (6SGG7). Received March 11, 26; Accepted June 9, 26

2 21 Yonghui Xia Then Gopalsamy and He [7] studied the following system with discrete delay ẋ1 (t) = x 1 (t) [ r 1 (t) a 11 (t)x 1 (t τ) + a (t)x 2 (t τ) ], ẋ 2 (t) = x 2 (t) [ r 2 (t) + a 21 (t)x 1 (t τ) a 22 (t)x 2 (t τ) ]. (1.2) Some sufficient conditions were obtained for the persistence and global attractivity of system (1.2). Then Yang etc. [8] obtained some sufficient conditions for the existence of positive periodic solutions of the system ẏ1 (t) = y 1 (t) [ r 1 (t) a 11 (t)y 1 (t τ 1 (t)) + a (t)y 2 (t τ 2 (t)) ], ẏ 2 (t) = y 2 (t) [ r 2 (t) + a 21 (t)y 1 (t τ 1 (t)) a 22 (t)y 2 (t τ 2 (t)) ] (1.3). Now we generalize the system to ẏ1 (t) = y 1 (t) [ r 1 (t) a 11 (t)y 1 (t τ 1 (t)) + a (t)y 2 (t τ 2 (t)) ], ẏ 2 (t) = y 2 (t) [ r 2 (t) + a 21 (t)y 1 (t σ 1 (t)) a 22 (t)y 2 (t σ 2 (t)) ], (1.4) where a ij (t), i, j = 1, 2, τ i (t), σ i (t), i = 1,..., n are positive periodic functions with period ω > and r i (t)dt >, i = 1, 2. The assumption of periodicity of the parameters a ij (t) is a way of incorporating the periodicity of the environment (e.g., seasonal effects of weather condition, food supplies, temperature, mating habits, harvesting etc.). The growth functions r i (t) are unnecessary to remain positive, since the environment fluctuates randomly, in bad condition, r i (t) may be negative. Obviously, system (1.3) is a special case of system (1.4). Unlike system (1.3), σ i (t) τ i (t). Thus the existing argument in Yang etc. [8] for obtaining bounds of solutions to the operator equation Lx = λnx are not applicable to our case and some new arguments are employed for the first time. The initial conditions for system (1.4) take the form of ẏ i (s) = ψ i (s), τ = max sup τ i (t), σ i (t), i = 1, 2} t [,ω] ψ i (t) C([ τ, ], R + ), ψ i () >, i = 1, 2. Throughout this paper, we will use the following natation: f = 1 ω where f(t) is an ω-periodic function. f(t)dt, f ι = min f(t), f µ = max f(t), t [,ω] t [,ω] 2. Existence of Positive Periodic Solutions (1.5) In order to obtain the existence of positive periodic solutions of (1.4), for convenience, we shall summarize in the following a few concepts and results from [1] that will be prerequisite for this section.

3 Positive Periodic Solutions of Mutualism Systems with Several Delays 211 Let X, Y be real Banach spaces, let L : DomL X Y be a linear mapping, and N : X Y be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dimkerl = codim ImL < + and ImL is closed in Y. If L is a Fredholm mapping of index zero and there exist continuous projections P : X X, and Q : Y Y such that ImP = KerL, KerQ = ImL = Im(I Q). It follows that L doml KerP : (I P )X ImL is invertible. We denote the inverse of that map by K p. If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω if QN(Ω) is bounded and K p (I Q)N : Ω X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ KerL. Lemma 2.1. [1] Let Ω X be an open bounded set. Let L be a Fredholm mapping of index zero and N be L-compact on Ω. Assume (a) for each λ (, 1), x Ω DomL, Lx λnx; (b) for each x Ω KerL, QNx ; (c) degjqn, Ω KerL, } =. Then Lx = Nx has at least one solution in Ω DomL. Lemma 2.2. Assume that f, g are continuous nonnegative functions defined on the interval [a, b]. Then there exists ξ [a, b] such that Theorem 2.3. Let b a f(t)g(t)dt = f(ξ) b a g(t)dt. b ι 11 = min t [,ω] b ι 22 = min t [,ω] a 11 (t) 1 τ 1 (t), bµ = max t [,ω] a 22 (t) 1 σ 2 (t), bµ 21 = max t [,ω] a (t) 1 τ 2 (t), a 21 (t) 1 σ 1 (t). If b ι 11b ι 22 > b µ 21b µ and τ i (t) < 1, σ i (t) < 1, (i = 1, 2) hold, then system (1.4) has at least one positive ω-periodic solution. Proof. We make the change of variables x i (t) = ln y i (t), i = 1, 2. (2.1) Then (1.4) is rewritten as ẋ1 (t) = r 1 (t) a 11 (t) expx 1 (t τ 1 (t))} + a (t) expx 2 (t τ 2 (t))}, ẋ 2 (t) = r 2 (t) + a 21 (t) expx 1 (t σ 1 (t))} a 22 (t) expx 2 (t σ 2 (t))}. (2.2)

4 2 Yonghui Xia Take and define X = Y = x = (x 1, x 2 ) T C(R, R 2 ) : x(t + ω) = x(t)} x = max t [,ω] x 1(t) + max t [,ω] x 2(t), x = (x 1, x 2 ) T X or Y (here denotes the Euclidean norm). Then X and Y are Banach spaces with the norm. For any x = (x 1, x 2 ) T X, because of the periodicity, we can easily check that r 1 (t) a 11 (t) expx 1 (t τ 1 (t))} + a (t) expx 2 (t τ 2 (t))} := 1 (x, t) C(R, R), r 2 (t) + a 21 (t) expx 1 (t σ 1 (t))} a 22 (t) expx 2 (t σ 2 (t))} := 2 (x, t) C(R, R), are ω-periodic. Set ( dx1 (t) L : DomL X, L(x 1 (t), x 2 (t)) =, dx ) 2(t), dt dt where DomL = (x 1 (t), x 2 (t)) C 1 (R, R 2 )} and N : X X, ( ) ( ) x1 1 (x, t) N =. 2 (x, t) x 2 Define P ( x1 x 2 ) ( x1 = Q x 2 ) = 1 ω ω 1 ω x 1 (t)dt x 2 (t)dt, ( x1 x 2 ) X = Y. It is not difficult to show that KerL = x x X, x = C, C R 2 }, } ImL = y y Y, y(t)dt = is closed in Y, dim KerL = codim ImL = 2, and P and Q are continuous projections such that ImP = KerL, KerQ = ImL = Im(I Q). It follows that L is a Fredholm mapping of index zero. Furthermore, the inverse K p of L p exists and has the form K p : ImL DomL KerP K p (y) = t y(s)ds 1 ω t y(s)dsdt.

5 Positive Periodic Solutions of Mutualism Systems with Several Delays 213 Then QN : X Y and K p (I Q)N : X X read QNx = 1 ω ω 1 ω K p (I Q)Nx = 1 (x, t)dt 2 (x, t)dt t Nx(s)ds 1 ω t ( t Nx(s)dsdt ω 1 2) Nx(s)ds. Clearly, QN and K p (I Q)N are continuous. By using the Arzelà Ascoli Theorem, it is not difficult to prove that K p (I Q)N(Ω) is compact for any open bounded set Ω X. Moreover, QN(Ω) is bounded. Therefore, N is L-compact on Ω for any open bounded set Ω X. Now we reach the position to search for an appropriate open bounded subset Ω for the application of Lemma 2.1. Corresponding to the operator equation Lx = λnx, λ (, 1), we have ẋ1 (t) = λ [ r 1 (t) a 11 (t) expx 1 (t τ 1 (t))} + a (t) expx 2 (t τ 2 (t))} ], ẋ 2 (t) = λ [ r 2 (t) + a 21 (t) expx 1 (t σ 1 (t))} a 22 (t) expx 2 (t σ 2 (t))} ]. (2.3) Suppose x = (x 1, x 2 ) T X is a solution of (2.3) for a certain λ (, 1). Integrating (2.3) over the interval [, ω], we obtain Hence, r 1 ω + r 2 ω + [ r1 (t) a 11 (t) expx 1 (t τ 1 (t))} + a (t) expx 2 (t τ 2 (t))} ] dt =, [ r2 (t) + a 21 (t) expx 1 (t σ 1 (t))} a 22 (t) expx 2 (t σ 2 (t))} ] dt =. a (t) expx 2 (t τ 2 (t))}dt = a 21 (t) expx 1 (t σ 1 (t))}dt = a 11 (t) expx 1 (t τ 1 (t))}dt, (2.4) a 22 (t) expx 2 (t σ 2 (t))}dt. (2.5) Let s = t τ 1 (t). We note that τ 1 (t) < 1, which implies ds dt = 1 τ 1(t) >, ds = (1 τ 1 (t))dt. Therefore, the function s = t τ 1 (t) has the continuous inverse function t = τ 1 (s), s [ τ 1 (), ω τ 1 (ω)]. So we have a 11 (t) expx 1 (t τ 1 (t))}dt = τ1 (ω) τ 1 () a 11 (τ 1 (s)) 1 τ 1 (τ 1 (s)) expx 1(s)}ds.

6 214 Yonghui Xia By Lemma 2.2, there exists ξ [ τ 1 (), ω τ 1 (ω)] = [ τ 1 (), ω τ 1 ()] such that a 11 (t) expx 1 (t τ 1 (t))}dt = = a 11 (τ1 (ξ)) 1 τ 1 (τ1 (ξ)) a 11 (η 1 ) 1 τ 1 (η 1 ) τ1 (ω) τ 1 () expx 1 (s)}ds expx 1 (t)}dt, (η 1 = τ 1 (ξ)). For the equations (2.3) and (2.4), similar to the above discussion, there exist η i [, ω], (i = 1, 2, 3, 4) such that We denote r 1 ω + a (η 2 ) ω expx 2 (t)}dt = a 11(η 1 ) ω expx 1 (t)}dt, 1 τ 2 (η 2 ) 1 τ 1 (η 1 ) (2.6) r 2 ω + a 21(η 3 ) ω expx 1 (t)}dt = a 22(η 4 ) ω expx 2 (t)}dt. 1 σ 1 (η 3 ) 1 σ 2 (η 4 ) (2.7) b 11 = a 11(η 1 ) 1 τ 1 (η 1 ), b = a (η 2 ) 1 τ 2 (η 2 ), b 21 = a 21(η 2 ) 1 σ 1 (η 2 ), b 22 = a 22(η 4 ) 1 σ 2 (η 4 ). It follows from (2.6) (2.7) that r 1 ω + b expx 2 (t)}dt = b 11 expx 1 (t)}dt, (2.8) r 2 ω + b 21 expx 1 (t)}dt = b 22 expx 2 (t)}dt. (2.9) Under the assumption of Theorem 2.3, it is not difficult to derive that and expx 1 (t)}dt = b r 2 + b 22 r 1 ω bµ r 2 + b µ 22 r 1 b 11 b 22 b 21 b b ι 11b ι 22 b µ 21b µ ω (2.1) expx 2 (t)}dt = b 21 r 1 + b 11 r 2 ω bµ 21 r 1 + b µ 11 r 2 b 11 b 22 b 21 b b ι 11b ι 22 b µ 21b µ ω. (2.11) Therefore, there exists t i [, ω], i = 1, 2 such that x 1 (t 1 ) ln bµ r 2 + b µ 22 r 1 b ι 11b ι 22 b µ 21b µ, x 2 (t 2 ) ln bµ 21 r 1 + b µ 11 r 2 b ι 11b ι 22 b µ 21b µ. Obviously, there exists positive constants M i >, i = 1, 2 such that x i (t i ) M i, i = 1, 2. (2.)

7 Positive Periodic Solutions of Mutualism Systems with Several Delays 215 On the other hand, it follows from (2.3), (2.4) and (2.1) that ẋ 1 (t) dt = λ r1 (t) a 11 (t) expx 1 (t τ 1 (t))} +a (t) expx 2 (t τ 2 (t))} dt, < r 1 ω + = 2 + 2b µ 11 a (t) expx 2 (t τ 2 (t))}dt a 11 (t) expx 1 (t τ 1 (t))}dt a 11 (t) expx 1 (t τ 1 (t))}dt Similarly, it follows from (2.3), (2.5) and (2.11) that It follows from (2.) (2.14) that expx 1 (t)}dt 2b µ b µ r 2 + b µ 22 r 1 11 b ι 11b ι 22 b µ 21b µ ω := H 1. (2.13) ẋ 2 (t) dt < 2b µ b µ 21 r 1 + b µ 11 r 2 11 b ι 11b ι 22 b µ 21b µ ω := H 2. (2.14) x i (t) x i (t i ) + ẋ i (t) dt < M i + H i. Clearly, M i, H i, i = 1, 2 are independent of the choice of λ. We note that ā 11 = 1 ω a 11 (t)dt = 1 ω τ1 () τ 1 () a 11 (τ 1 (s)) 1 τ 1 (τ 1 (s)) ds = a 11(τ 1 (s 1 )) 1 τ 1 (τ 1 (s 1 )), s 1 [ τ 1 (), ω τ 1 ()], (2.15) where t = τ 1 (s) is the inverse function of s = t τ 1 (t). Similar to the discussion of (2.15), ā = a (τ 2 (s 2 )) 1 τ 2 (τ 2 (s 2 )), ā 21 = a 21(σ 1(s 3 )) 1 σ 1 (σ 1(s 3 )), ā 22 = a 22(σ 2(s 4 )) 1 σ 2 (σ 2(s 4 )). (2.16) Thus, under the assumptions in Theorem 2.3, it follows from (2.15) (2.16) that ā 11 ā 22 ā ā 21 > b ι 11b ι 22 b µ b µ 21 >. (2.17) Now it is easy to show that the system of algebraic equations r i ā ii v i + ā ij v j =, i, j = 1, 2.

8 216 Yonghui Xia has a unique solution (v 1, v 2) T R 2 + with x i >. Take H = maxm i + H i + C, M i + H i +C}, where C > is taken sufficiently large such that (ln v 1, ln v 2) T < C. Define Ω = x(t) X : x < H}. It is clear that Ω satisfies condition (a) of Lemma 2.1. Let x Ω KerL = Ω IR 2, x be a constant vector in IR 2 with x = H. Then QNx = [ r i ā ii expx i } + ā ij expx j } = ] 2 1. Furthermore, let J : ImQ KerL. In view of assumptions in Theorem 2.3 and (2.15) (2.17), it is easy to see that [ ] ā11 ā degjqn, Ω KerL, } = sgn det sgn(b ι ā 21 ā 22 11b ι 22 b µ b µ 21), where deg( ) is the Brouwer degree and the J is the identity mapping since ImQ = KerL. By now, we have shown that Ω verifies all requirements of Lemma 2.1. Then it follows that Lx = N x has at least one solution in DomL Ω. By (2.1), we derive that (1.4) has at least one positive ω-periodic solution. The proof is completed. Remark 2.4. Our results and the method used in the proof are much different from those in the known literature. We remark that we generalize system (1.3) in [8] to system (1.4), where τ i (t) σ i (t), i.e., all the delays are different. It is in this aspect that we generalize the results in [8]. Acknowledgement The authors would like to express their gratitude to the reviewers and editor for their valuable comments and careful reading. References [1] L. Chen. Mathematical Models and Methods in Ecology, Science Press, Bejing, 1998, (in Chinese). [2] Y. Kuang. Delay Differential Equations With Application Dynamic, New York, Academic Press, [3] M. Fan and K. Wang. Positive periodic solutions of a periodic integro-differential competition system with infinite delays, Z. Angew. Math. Mech., 81(3):197 23, 21. [4] M. Fan and K. Wang. Periodicity in a delayed ratio-dependent pedator-prey system, J. Math. Anal. Appl., 262:179 19, 21. [5] B.S. Goh. Management and Analysis of Biological Populations, Elsevier Scientific, The Netherlands, 198. [6] B.S. Goh. Stability in models of mutualism, Amer. Natural, 113: , 1979.

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