Research Article Existence of Almost-Periodic Solutions for Lotka-Volterra Cooperative Systems with Time Delay

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1 Applied Mathematics Volume 22, Article ID 274, 4 pages doi:55/22/274 Research Article Existence of Almost-Periodic Solutions for Lotka-Volterra Cooperative Systems with Time Delay Kaihong Zhao Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 6593, China Correspondence should be addressed to Kaihong Zhao, zhaokaihongs@26com Received 3 December 2; Accepted 29 February 22 Academic Editor: Hongyong Zhao Copyright q 22 Kaihong Zhao This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited This paper considers the existence of positive almost-periodic solutions for almost-periodic Lotka- Volterra cooperative system with time delay By using Mawhin s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost-periodic solutions are obtained An example and its simulation figure are given to illustrate the effectiveness of our results Introduction Lotka-Volterra system is one of the most celebrated models in mathematical biology and population dynamics In recent years, it has also been found with successful and interesting applications in epidemiology, physics, chemistry, economics, biological science, and other areas see 4 Moreover, in 5, it was shown that the continuous-time recurrent neural networks can be embedded into Lotka-Volterra models by changing coordinates, which suggests that the existing techniques in the analysis of Lotak-Volterra systems can also be applied to recurrent neural networks Owing to its theoretical and practical significance, Lotka-Volterra system have been studied extensively see 6 6 and the cites therein Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment eg, seasonal effects of weather, food supplies, mating habits, etc are considered as important selective forces on systems in a fluctuating environment Therefore, on the one hand, models should take into account both the periodically changing environment and the effects of time delays However, on the other hand, in fact, it is more

2 2 Applied Mathematics realistic and reasonable to study almost-periodic system than periodic system Recently, there are two main approaches to obtain sufficient conditions for the existence and stability of the almost-periodic solutions of biological models: one is using the fixed point theorem, Lyapunov functional method, and differential inequality techniques see 7 9 ; the other is using functional hull theory and Lyapunov functional method see 4 6 However, to the best of our knowledge, there are very few published letters considering the almostperiodic solutions for nonautonomous Lotka-Volterra cooperative system with time delay by applying the method of coincidence degree theory Motivated by this, in this letter, we apply the coincidence theory to study the existence of positive almost-periodic solutions for Lotka- Volterra cooperative system with time delay as follows: u i t u i t r i t b i t u i t τ i t j,j / i c ij t u j t τij t ), i, 2,,n, where u i t stands for the ith species population density at time t R, r i t is the natural reproduction rate, b i t represents the inner-specific competition, c ij t i / j stands for the interspecific cooperation, τ i t > andτ ij t > are all continuous almost-periodic functions on R Throughout this paper, we always assume that r i t, b i t, andc ij t are all nonegative almost periodic functions with respect to t R The initial condition of is of the form u i s φ i s, i, 2,,n, 2 where φ i s is positive bounded continuous function on τ, and τ max i,j n sup t R { τ ij t } The organization of the rest of this paper is as follows In Section 2, we introduce some preliminary results which are needed in later sections In Section 3, we establish our main results for the existence of almost-periodic solutions of Finally, an example and its simulation figure are given to illustrate the effectiveness of our results in Section 4 2 Preliminaries To obtain the existence of an almost-periodic solution of system, we first make the following preparations Definition 2 see 2 Letu t : R R be continuous in t u t is said to be almost-periodic on R, if, for any ɛ>, the set K u, ɛ {δ : u t δ u t <ɛ, for any t R} is relatively dense, that is, for any ɛ>, it is possible to find a real number l ɛ >, for any interval with length l ɛ, there exists a number δ δ ɛ in this interval such that u t δ u t <ɛ,for any t R Definition 22 A solution u t u t,u 2 t,,u n t T of is called an almost periodic solution if and only if for each i, 2,,n,u i t is almost periodic

3 Applied Mathematics 3 For convenience, we denote AP R the set of all real-valued, almost-periodic functions on R and for each j, 2,,n,let ) T } f j { λ R : lim f j s e i λs ds /, T T mod { ) N f j n i λi : n i Z, N N, λi ) } f j i 2 be the set of Fourier exponents and the module of f j, respectively, where f j is almost periodic Suppose f j t, φ j is almost periodic in t, uniformly with respect to φ j C τ,, R K j f j,ɛ,φ j denote the set of ɛ-almost periods for f j uniformly with respect to φ j C τ,, R l j ɛ denote the length of inclusion interval Let m f j /T T f j s ds be the mean value of f j on interval,t, where T>isaconstant clearly, m f j depends on T m f j lim T /T T f j s ds Lemma 23 see 2 Suppose that f and g are almost periodic Then the following statements are equivalent i mod f mod g, ii for any sequence {t n}, if lim n f t t n f t for each t R, then there exists a subsequence {t n } {t n} such that lim n g t t n f t for each t R Lemma 24 see 2 Let u AP R Then t u s ds is almost periodic t τ Let X and Z be Banach spaces A linear mapping L :dom L X Z is called Fredholm if its kernel, denoted by ker L {X dom L : Lx }, has finite dimension and its range, denoted by Im L {Lx : x dom L }, is closed and has finite codimension The index of L is defined by the integer dim K L codimdom L IfL is a Fredholm mapping with index, then there exist continuous projections P : X X and Q : Z Z such that Im P ker L and ker Q Im L Then L dom L ker P :Im L ker P Im L is bijective, and its inverse mapping is denoted by K P :Im L dom L ker P Since ker L is isomorphic to Im Q, there exists a bijection J :ker L Im Q LetΩ be a bounded open subset of X and let N : X Z be a continuous mapping If QN Ω is bounded and K P I Q N : Ω X is compact, then N is called L-compact on Ω, where I is the identity Let L be a Fredholm linear mapping with index and let N be a L-compact mapping on Ω Define mapping F :dom L Ω Z by F L NIfLx / Nx for all x dom L Ω, then by using P, Q, K P, J defined above, the coincidence degree of F in Ω with respect to L is defined by ) ) deg L F, Ω deg I P J Q K P I Q N, Ω,, 22 where deg g,γ,p is the Leray-Schauder degree of g at p relative to Γ Then The Mawhin s continuous theorem is given as follows

4 4 Applied Mathematics Lemma 25 see 22 Let Ω X be an open bounded set and let N : X Z be a continuous operator which is L-compact on Ω Assume a for each λ,, x Ω dom L,Lx/ λnx; b for each x Ω L, QNx / ; c deg JNQ,Ω ker L, / Then Lx Nx has at least one solution in Ω dom L In this paper, since we need some related properties of M-matrix we introduce them as follows In addition, A matrix A a ij means that each elements a ij Definition 26 see 23 If a real matrix A a ij n n satisfies the following conditions and 2 : a ii >, i, 2,,n, a ij, i/ j, i, j, 2,,n, 2 A is a positive-definite matrix, then A is called a M-matrix Lemma 27 see 23 If matrix A a ij n n is a M-matrix, then A exists and its every element is nonnegative Lemma 28 Suppose that matrix A a ij n n is a M-matrix, then AX B implies X A B Proof In fact, there exists a nonnegative positive vector ε ε,ε 2,,ε n T R n such that AX B ε,,, T which imply that X A B A ε,,, T According to Lemma 24, there exists at least one positive element in the every row of A, which imply A ε,,, T Thus, we obtain X A B 3 Main Result In this section, we state and prove our main results of our this paper By making the substitution u i t e y i t, i, 2,,n, 3 Equation can be reformulated as ẏ i t r i t b i t e y i t τ ii t c ij t e y j t τ ij t, i, 2,,n 32 j,i / j The initial condition 2 can be rewritten as follows: y i s ln φ i s : ψ i s, i, 2,,n 33

5 Applied Mathematics 5 Set X Z V V 2, where V V 2 { y t y t,y 2 t,,y n t ) T C R, R n : y i t AP R, mod y i t ) mod H i t, λ i y i t ) satisfies { y t h,h 2,,h n T R n}, λ i >β, i, 2,,n }, H i t r i t b i t e ψ i τ ii c ij t e ψ j τ ij j,i / j 34 and ψ is defined as 33, i, 2,,n, β > is a given constant For y y,y 2,,y n T Z, define y max i n sup t R y i t Lemma 3 Z is a Banach space equipped with the norm Proof If y {k} V and y {k} y {k},y {k} 2,,y {k} n T converges to y y, y 2,,y n T,that is, y {k} j y j,ask,j, 2,,n Then it is easy to show that y j AP R and mod y j mod H j For any λ j β, we have that therefore, lim T T lim T T T T y {k} j t e i λ j t dt, j, 2,,n; 35 y j t e i λ j t dt, j, 2,,n, 36 which implies y V Then it is not difficult to see that V is a Banach space equipped with the norm Thus, we can easily verify that X and Z are Banach spaces equipped with the norm The proof of Lemma 3 is complete Lemma 32 Let L : X Z, Ly ẏ,thenl is a Fredholm mapping of index Proof Clearly, L is a linear operator and ker L V 2 We claim that Im L V Firstly, we suppose that z t z t,z 2 t,,z n t T Im L Z Then there exist z {} t z {} t, z {} 2 t,,z {} n t T V and constant vector z {2} z {2},z {2} 2,,z {2} n T V 2 such that z t z {} t z {2}, 37 that is, z i t z {} i t z {2} i, i, 2,,n 38

6 6 Applied Mathematics From the definition of z i t and z {} i t, we can easily see that t t τ z i s ds and t t τ z{} i s ds are almost-periodic functions So we have z {2} i, i, 2,,n, then z {2},,, T, which implies z t V,thatisIm L V On the other hand, if u t u t,u 2 t,,u n t T V \{}, then we have t u j s ds AP R, j, 2,,nIf λ j /, then we obtain lim T T T t ) u j s ds e i λ j t dt lim i λ T j T T u j t e i λ j t dt, j, 2,,n 39 It follows that [ t t )] u j s ds m u j s ds u j t ), j, 2,,n, 3 hence t t ) u s ds m u s ds V X 3 Note that t u s ds m t u s ds is the primitive of u t in X, we have u t Im L, thatis, V Im L Therefore, Im L V Furthermore, one can easily show that Im L is closed in Z and dim ker L n co dim Im L ; 32 therefore, L is a Fredholm mapping of index The proof of Lemma 32 is complete Lemma 33 Let N : X Z, Ny G y,gy 2,,Gy n T,where G y i r i t b i t y i t τ i t c ij t e y j t τ ij t, i, 2,,n 33 j,i / j Set P : X Z, Py m y ),m y2 ),,m yn )) T, Q : Z Z, Qz m z,m z 2,,m z n T 34 Then N is L-compact on Ω,whereΩ is an open bounded subset of X Proof Obviously, P and Q are continuous projectors such that Im P ker L, Im L ker Q 35

7 Applied Mathematics 7 It is clear that I Q V 2 {,,, }, I Q V V Hence Im I Q V Im L 36 Then in view of Im P ker L, Im L ker Q Im I Q, 37 we obtain that the inverse K P :Im L ker P dom L of L P exists and is given by K P z t [ t ] z s ds m z s ds 38 Thus, QNy [ ] [ ] [ ]) m G y,m G y 2,,m G y T n K P I Q Ny f y ) Q f y )),f y2 ) Q f y2 )),,f yn ) Q f yn ))) T, 39 where f y i ) t [ ]) G y i m G y i ds, i, 2,,n 32 Clearly, QN and I Q N are continuous Now we will show that K P is also continuous By assumptions, for any < ɛ < and any compact set φ i C τ,, R, let l i ɛ i be the length of the inclusion interval of K i H i,ɛ i,φ i, i, 2,,n Suppose that {z k t } Im L V and z k t z k t,zk 2 t,,zk n t T uniformly converges to z t z t, z 2 t,,z n t T,thatisz k i z i,ask, i, 2,,n Because of t zk s ds Z, k, 2,,n, there exists σ i <σ i <ɛ i such that K i H i,σ i,φ i K i t zk i s ds, σ i,φ i, i, 2,,nLetl i σ i be the length of the inclusion interval of K i H i,σ i,φ i and l i max{l i ɛ i,l i σ i }, i, 2,,n 32 It is easy to see that l i is the length of the inclusion interval of K i H i,σ i,φ i and K i H i,ɛ i,φ i, i, 2,,n Hence, for any t/,l i, there exists ξ t K i H i,σ i,φ i K i t zk i s ds, σ i,φ i

8 8 Applied Mathematics such that t ξ t,l i, i, 2,,n Hence, by the definition of almost periodic function we have t t z k s ds max sup s ds z k i n t R i t t t ξt t ξt z k i s ds z k i s ds z k i s ds max i n sup t,l i 2max i n sup 2max i n t,l i li t z k i s ds max i n sup t/,l i z k i s ds max sup i n z k i s ds max i n ɛ i t/,l i t t ξt z k i s ds z k i s ds 322 From this inequality, we can conclude that t z s ds is continuous, where z t z t,z 2 t,,z n t T Im L Consequently, K P and K P I Q Ny are continuous From 322, we also have t z s ds and K P I Q Ny also are uniformly bounded in Ω Further, it is not difficult to verify that QN Ω is bounded and K P I Q Ny is equicontinuous in Ω By the Arzela-Ascoli theorem, we have immediately concluded that K P I Q N Ω is compact Thus N is L-compact on Ω The proof of Lemma 33 is complete Theorem 34 Assume that the following conditions H and H 2 hold: H e i : m r i t >, m b i t >, m c ij t >, i,j, 2,,n; H 2 D is a positive-definite matrix, where d ii m b i t, d ij m c ij t, i/ j, i, j, 2,,n, d d 2 d n d 2 d 22 d 2n D 323 d n d n2 d nn Then has at least one positive almost periodic solution Proof To use the continuation theorem of coincidence degree theorem to establish the existence of a solution of 32, we set Banach space X and Z the same as those in Lemma 3 and set mappings L, N, P, Q the same as those in Lemmas 32 and 33, respectively Then we can obtain that L is a Fredholm mapping of index and N is a continuous operator which is L-compact on Ω Now, we are in the position of searching for an appropriate open, bounded subset Ω for the application of the continuation theorem Corresponding to the operator equation Ly λny, λ,, 324

9 Applied Mathematics 9 we obtain ẏ i t λ r i t b i t e y i t τ i t c ij t e y j t τ ij t, i, 2,,n 325 j,i / j Assume that y t y t,y 2 t,,y n t T X is a solution of 325 for some λ, Denote M i sup t R {y i t }, M i inf t R {y i t } On the one hand, by 325, we derive ẏi t λ r i t b i t e y i t τ i t j,i / j c ij t e y j t τ ij t, i, 2,,n 326 Onthebothsidesof 326, integrating from to T and applying the mean value theorem of integral calculus, we have λ m r i t m b i t e y i ξ i τ i ξ i m c ij t ) e y j η ij τ ij η ij j,i / j 327 m ẏ i t ), i, 2,,n, where ξ i,t, η ij,t, i, j, 2,,n In the light of 327, wegetfori, 2,,n, [ ] λ m b i t e y i ξ i τ i ξ i λ m r i t j,i / j m c ij t ) e y j η ij τ ij η ij m ẏi t ) 328 Onthebothsidesof 328, taking the supremum with respect to ξ i,η ij and letting T, we obtain m b i t e M i m r i t m [ c ij t ] e M j, j,j / i 329 that is, d ii e M i d ij e M j e i, i, 2,,n 33 j,j / i

10 Applied Mathematics Equation 33 can be written by the following matrix form d d 2 d n d 2 d 22 d 2n d n d n2 d nn e M e M 2 e M n e e 2 e n 33 By Lemma 28 and assumption, we obtain e M e M 2 e M n e e 2 D e n : H H 2 H n, 332 which imply that M i ln H i, i, 2,,n 333 On the two sides of 328, taking the infimum with respect to ξ i, η ij, and letting T,we obtain M i ln H i, i, 2,,n 334 On the other hand, according to 325, we derive λr i t ẏ i t <λb i t e y i t τ i t, i, 2,,n 335 On the both sides of 335, integrating from to T and using the mean value theorem of integral calculus, we get λm r i t <m ẏ i t ) λm b i t e y i ζ i τ i ζ i, i, 2,,n, 336 where ζ i,t, i, 2,,n On the both sides of 336, take the supremum and infimum with respect to ζ i, respectively, and let T, then we have for i, 2,,n, m r i t <m b ii t e M i, m r i t <m b ii t e M i, 337 namely, e M i > m r i t m b i t e i d ii, e M i > m r i t m b i t e i d ii 338

11 Applied Mathematics which imply that M i > ln e i d ii, M i > ln e i d ii 339 Combining with 333, 334,and 339,wederiveforallt R, i, 2,,n, { min ln e } i ln e i <M i n d ii d i y i t M i ln H i ii { < max ln H i i n } 34 Denote M max{ min i n {ln e i /d ii }, max i n {ln H i } } Clearly, M is independent of λ Take Ω {y y,y 2,,y n ) T X : y <M } 34 It is clear that Ω satisfies the requirement a in Lemma 25 When y Ω ker L, y y,y 2,,y n T is a constant vector in R n with y M Then QNy m G,m G 2,,m G n T, y X, 342 where G i r i t b i t e y i c ij t e y j, j,j / i i, 2,,n, m G i m r i t m b ii t e y i m [ c ij t ] e y j j,j / i e i d ii e y i 343 d ij e y j, j,j / i i, 2,,n If QNy,,, T, then we have d d 2 d n d 2 d 22 d 2n d n d n2 d nn e y e y 2 e y n e e 2 e n, 344 which imply that y i ln H i, i, 2,,nThus,y y,y 2,,y n T Ω, this contradicts the fact that y Ω ker L Therefore, QNy /,,, T, which implies that the requirement b in Lemma 25 is satisfied If necessary, we can let M be greater such that y T QNy <, for

12 2 Applied Mathematics any y Ω ker L Furthermore, take the isomorphism J :Im Q ker L, Jz z and let Φ γ; y γy γ JQNy, then for any y Ω ker L, y T Φ γ; y <, we have deg{jqn,ω ker L, } deg { y, Ω ker L, } / 345 So, the requirement c in Lemma 25 is satisfied Hence, 32 has at least one almost-periodic solution in Ω, thatis, has at least one positive almost periodic solution The proof is complete 4 An Example and Simulation Consider the following two species cooperative system with time delay: ẋ t x t r t b t x t τ t c 2 t y t τ 2 t ), ẏ t y t r 2 t b 2 t y t τ 2 t c 2 t x t τ 2 t ), 4 where r t 2 sin 2t sin 3t, b t 2 sin 3t sin 5t, c 2 t 2 cos 2t cos 3t /2, τ t e sin 2t sin 5t, τ 2 t e sin t cos 2t, r 2 t 2 sin 2t sin 3t, b 2 t 2 sin 3t sin 5t, c 2 t 2 cos 2t cos 3t /2, τ 2 t e sin 2t cos 5t, τ 2 t e sin t cos 2t Since e m r t 2, d m b t 2, d 2 m c 2 t, e 2 m r t 2, d 22 m b t 2, d 2 m c 2 t, ) ) 2 2 D, det 3 >, D ) 2, then, the matrix D is positive definite, and ln e d ln e 2 d 22, ) H H 2 ) ) e 2 D, 2 e 2 ) ln H ln H 2 { M max{ min ln e } } i, i n d max { } ln H ii i n i ln 2, Ω {y ) T } y, y 2,,y n X : y < In 2 ) ln 2 ln 2, 43 Therefore, all conditions of Theorem 34 are satisfied By Theorem 34, system 4 has one positive almost-periodic solution The resulting numerical simulation is depicted in Figure

13 Applied Mathematics x y t Figure Acknowledgments This work is supported by the National Natural Sciences Foundation of People s Republic of China under Grant no 625, Yunnan Province natural scientific Research Fund Project no 2FZ58, and Yunnan Province education department scientific Research Fund Project no 2Z References T Cheon, Evolutionary stability of ecological hierarchy, Physical Review Letters, vol 9, no 25, Article ID 2585, 4 pages, 23 2 P Gao, Hamiltonian structure and first integrals for the Lotka-Volterra systems, Physics Letters A, vol 273, no -2, pp 85 96, 2 3 A Provata and G A Tsekouras, Spontaneous formation of dynamical patterns with fractal fronts in the cyclic lattice Lotka-Volterra model, Physical Review E, vol 67, no 5, part 2, Article ID 5662, 23 4 K Geisshirt, E Praestgaard, and S Toxvaerd, Oscillating chemical reactions and phase separation simulated by molecular dynamics, Chemical Physics, vol 7, no 22, pp , Y Moreau, S Louies, J Vandewalle, and L Brenig, Embedding recurrent neural networks into predator-prey models, Neural Networks, vol 2, no 2, pp , Y Li and Y Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems, Mathematical Analysis and Applications, vol 255, no, pp 26 28, 2 7 W Lin and T Chen, Positive periodic solutions of delayed periodic Lotka-Volterra systems, Physics Letters A, vol 334, no 4, pp , 25 8 X Yang, Global attractivity and positive almost periodic solution of a single species population model, Mathematical Analysis and Applications, vol 336, no, pp 26, 27 9 Y Li, Positive periodic solutions of discrete Lotka-Volterra competition systems with state dependent and distributed delays, Applied Mathematics and Computation, vol 9, no, pp , 27 Y Li, Positive periodic solutions of periodic neutral Lotka-Volterra system with state dependent delays, Mathematical Analysis and Applications, vol 33, no 2, pp , 27 Y Li, Positive periodic solutions of periodic neutral Lotka-Volterra system with distributed delays, Chaos, Solitons & Fractals, vol 37, no, pp , 28

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