Research Article Existence of at Least Two Periodic Solutions for a Competition System of Plankton Allelopathy on Time Scales

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1 Applied Mathematics Volume 2012, Article ID , 14 pages doi: /2012/ Research Article Existence of at Least Two Periodic Solutions for a Competition System of Plankton Allelopathy on Time Scales Hui Fang Department of Mathematics, Kunming University of Science and Technology, Yunnan , China Correspondence should be addressed to Hui Fang, kmustfanghui@hotmail.com Received 3 October 2011; Accepted 15 February 2012 Academic Editor: Meng Fan Copyright q 2012 Hui Fang. 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. We study a competition system of the growth of two species of plankton with competitive and allelopathic effects on each other on time scales. With the help of Mawhin s continuation theorem of coincidence degree theory, a set of easily verifiable criteria is obtained for the existence of at least two periodic solutions for this model. Some new existence results are obtained. An example and numerical simulation are given to illustrate the validity of our results. 1. Introduction The allelopathic interactions in the phytoplanktonic world have been studied by many researchers. For instance, see 1 4 and references cited therein. Maynard-Smith 2 and Chattopadhyay 3 proposed the following two-species Lotka-Volterra competition system, which describes the changes of size and density of phytoplankton: dn 1 t dn 2 t N 1 tr 1 a 11 N 1 t a 12 N 2 t b 1 N 1 tn 2 t, N 2 tr 2 a 21 N 1 t a 22 N 2 t b 2 N 1 tn 2 t, 1.1 where b 1 and b 2 are the rates of toxic inhibition of the first species by the second and vice versa, respectively. Naturally, more realistic models require the inclusion of the periodic changing of environment caused by seasonal effects of weather, food supplies, and so forth. For such

2 2 Applied Mathematics systems, as pointed out by Freedman and Wu 5 and Kuang 6, it would be of interest to study the existence of periodic solutions. This motivates us to modify system 1.1 to the form dn 1 t dn 2 t N 1 tr 1 t a 11 tn 1 t a 12 tn 2 t b 1 tn 1 tn 2 t, N 2 tr 2 t a 21 tn 1 t a 22 tn 2 t b 2 tn 1 tn 2 t, 1.2 where r i t,a ij t > 0,b i t > 0 i, j 1, 2 are continuous ω-periodic functions. If the estimates of the population size and all coefficients in 1.2 are made at equally spaced time intervals, then we can incorporate this aspect in 1.2 and obtain the following discrete analogue of system 1.2: N 1 k 1 N 1 k exp{r 1 k a 11 kn 1 k a 12 kn 2 k b 1 kn 1 kn 2 k}, N 2 k 1 N 2 k exp{r 2 k a 21 kn 1 k a 22 kn 2 k b 2 kn 1 kn 2 k}, 1.3 where r i,a ij > 0,b i > 0 i, j 1, 2 are ω-periodic, that is, r i k ω r i k, b i k ω b i k, a ij k ω a ij k, 1.4 for any Z the set of all integers, ω is a fixed positive integer. System 1.3 was considered by Zhang and Fang 7. However, dynamics in each equally spaced time interval may vary continuously. So, it may be more realistic to assume that the population dynamics involves the hybrid discrete-continuous processes. For example, Gamarra and Solé pointed out that such hybrid processes appear in the population dynamics of certain species that feature nonoverlapping generations: the change in population from one generation to the next is discrete and so is modelled by a difference equation, while within-generation dynamics vary continuously due to mortality rates, resource consumption, predation, interaction, etc. and thus are described by a differential equation 8, page 619. The theory of calculus on time scales see 9, 10 and references cited therein was initiated by Hilger in his Ph.D. thesis in in order to unify continuous and discrete analysis, and it has become an effective approach to the study of mathematical models involving the hybrid discrete-continuous processes. This motivates us to unify systems 1.2 and 1.3 to a competition system on time scales T of the form x Δ 1 t r 1t a 11 t exp{x 1 t} a 12 t exp{x 2 t} b 1 t exp{x 1 t} exp{x 2 t}, x Δ 2 t r 2t a 21 t exp{x 1 t} a 22 t exp{x 2 t} b 2 t exp{x 1 t} exp{x 2 t}, 1.5 where r i t,a ij t > 0,b i t > 0 i, j 1, 2 are rd-continuous ω-periodic functions. In 1.5, letn i t exp{x i t}, i 1, 2. If T R the set of all real numbers, then 1.5 reduces to 1.2. IfT Z the set of all integers, then 1.5 reduces to 1.3. To our knowledge, few papers have been published on the existence of multiple periodic solutions for this model. Motivated by the work of Chen 12, we study the existence of multiple periodic solutions of 1.5 by applying Mawhin s continuation theorem

3 Applied Mathematics 3 of coincidence degree theory 13. Some new results are obtained. Even in the special case when T Z, our conditions are also easier to verify than that of Preliminaries on Time Scales In this section, we briefly present some foundational definitions and results from the calculus on time scales so that the paper is self-contained. For more details, one can see Definition 2.1. AtimescaleT is an arbitrary nonempty closed subset of the real numbers R. Let ω>0. Throughout this paper, the time scale T is assumed to be ω-periodic, that is, t T implies t ω T. In particular, the time scale T under consideration is unbounded above and below. Definition 2.2. We define the forward jump operator σ : T T, the backward jump operator ρ : T T, and the graininess μ : T R 0, by σt : inf{s T : s>t}, ρt : sup{s T : s<t}, μt σt t for t T, 2.1 respectively. If σt t, then t is called right-dense otherwise: right-scattered,andifρt t, then t is called left-dense otherwise: left-scattered. Definition 2.3. Assume f : T R is a function and let t T. Then we define f Δ t to be the number provided it exists with the property that given any ε>0, there is a neighborhood U of t i.e., U t δ, t δ T for some δ>0such that [ fσt fs ] f Δ tσt s σt s s U. 2.2 In this case, f Δ t is called the delta or Hilger derivative of f at t. Moreover, f is said to be delta or Hilger differentiable on T if f Δ t exists for all t T. AfunctionF : T R is called an antiderivative of f : T R provided F Δ t ft for all t T. Then we define s r ftδt Fs fr for r, s T. 2.3 Definition 2.4. A function f : T R is said to be rd-continuous if it is continuous at rightdense points in T and its left-sided limits exist finite at left-dense points in T. Thesetof rd-continuous functions f : T R will be denoted by C rd T. Lemma 2.5. Every rd-continuous function has an antiderivative. Lemma 2.6. If a, b T,α,β R, and f, g C rd T,then a b a αftβgtδt α b a ftδt β b a gtδt b if ft 0 for all a t<b,then b a ftδt 0 c if ft gt on a, b : {t T : a t<b},then b a ftδt b a gtδt.

4 4 Applied Mathematics For convenience, we now introduce some notation to be used throughout this paper. Let min{0, T}, I ω, ω T, g u supgt, t I ω g 1 gsδs 1 gsδs, ω I ω ω g l inf t I ω gt, 2.4 where g C rd T is an ω-periodic real function, that is, gt ω gt for all t T. Lemma 2.7 see 14. Let t 1,t 2 I ω and t T.Ifg : T R is ω- periodic, then gt gt 1 gsδs, gt gt2 gsδs. 2.5 Lemma 2.8 see 15. Assume that {f n } n N is a function on J such that i {f n } n N is uniformly bounded on J, ii {fn Δ } n N is uniformly bounded on J. Then there is a subsequence of {f n } n N which converges uniformly on J. 3. Existence of Multiple Periodic Solutions In this section, in order to obtain the existence of multiple periodic solutions of 1.5,wefirst make the following preparations 13. Let X, Z be normed vector spaces, let L :doml X Z be a linear mapping, and let N: X Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimkerl codimiml < and Im L is closed in Z.IfLis a Fredholm mapping of index zero, there then exist continuous projectors P : X X and Q : Z Z such that Im P Ker L, ImL Ker Q ImI Q. If we define L P :doml Ker P Im L as the restriction L dom L Ker P of L to dom L Ker P, then L P is invertible. We denote the inverse of that map by K P.IfΩis an open bounded subset of X, the mapping N will be called L-compact on Ω if QNΩ is bounded and K P I QN : Ω X is compact, that is, continuous and such that K P I QNΩ is relatively compact. Since Im Q is isomorphic to Ker L, there exists isomorphism J :ImQ Ker L. Mawhin s continuation theorem of coincidence degree theory is a very powerful tool to deal with the existence of periodic solutions of differential equations, difference equations and dynamic equations on time scales. For convenience, we introduce Mawhin s continuation theorem 13, page 40 as follows. Lemma 3.1 continuation theorem. Let L be a Fredholm mapping of index zero and let N : Ω Z be L-compact on Ω. Suppose a Lx / λnx for every x dom L Ω and every λ 0, 1,

5 Applied Mathematics 5 b QNx / 0 for every x Ω Ker L, and Brouwer degree deg B JQN,Ω Ker L, 0 / Then Lx Nx has at least one solution in dom L Ω. In the following, we shall use the notation α ij a ji b i a ii b j, α ij a jib i a ii b j e R jr j ω, α ij a ji b i e R jr j ω a ii b j e R ir i ω, β ij a ii a jj b i r j a ij a ji b j r i, β ij a iia jj e R jr j ω b i r j a ij a ji e R ir i ω b j r i e R ir i R j r j ω, β ij a iia jj e R ir i ω b i r j e R ir i R j r j ω a ij a ji e R jr j ω b j r i, β ij a iia jj b i r j a ij a ji e R ir i ω b j r i e R ir i R j r j ω, γ ij r i a jj r j a ij, γ ij r i a jj e R jr j ω r j a ij e R ir i ω, 3.2 γ ij r ia jj r j a ij e R jr j ω, γ ij r ia jj r j a ij e R ir i R j r j ω, i/ j, i, j 1, 2, β β 2 4αγ β β 2 4αγ N 1 α, β, γ, N 2 α, β, γ α / 0, β 2 4αγ > 0. 2α 2α We make the following assumptions. H 1 R i 1/ω r i t Δt 1/ω r i tδt >0. H 2 γ ij r ia jj r j a ij e R ir i R j r j ω > 0, i/ j, i, j 1, 2. H 3 α 12 > 0. Next, we introduce some lemmas. Lemma 3.2 see 16, Lemma 3.2. Consider the following algebraic equations: a 11 N 1 a 12 N 2 b 1 N 1 N 2 r 1, a 21 N 1 a 22 N 2 b 2 N 1 N 2 r Assuming that H 1, H 2 hold, then the following conclusions hold. i If α 12 > 0, then3.3 have two positive solutions: Ni α12,β 12,γ 12,N1 α21,β 21,γ 21, i 1,

6 6 Applied Mathematics ii If α 21 > 0, then3.3 have two positive solutions: N1 α12,β 12,γ 12,Ni α21,β 21,γ 21, i 1, Lemma 3.3. Assume that H 1 H 3 hold, then the following conclusions hold. i β 12 > 0, β α 12γ 12 > 0, ii β 12 > 0, β α 12γ 12 > 0. Proof. The proof of i is the same as i of Lemma 3.5 in 7. We omit it. ii We have β 12 b1 a 11 β α 12γ 12 b1 a 11 2 b1 a 12 γ r 1 e R 2r 2 ω 21 r1 α 12 e R 1r 1 R 2 r 2 ω a 11 γ 12 > 0, a 11 r 1 e R 1r 1 R 2 r 2 ω a 12 r 1 e R 2r 2 ω 2 γ 2 21 r1 α 12 e R 1r 1 ω a 11 a 11 γ 12 r 1 e R 1r 1 ω a 12 r1 α 12 e R 1r 1 R 2 r 2 ω a 11 γ 12 γ a 11 r 1 e R 2r 2 ω a 11 r 1 e R 1r 1 R 2 r 2 ω 21 > Lemma 3.4. Assume that H 1 H 3 hold, then the following conclusions hold. N 1 α12,β 12 m, γ 12 n <N 1 α12,β 12,γ 12 <N1 α12,β 12,γ 12 <N 2 α12,β 12 12,γ <N2 α12,β 12,γ 12 <N2 α12,β 12 m, γ 12 n, 3.7 where m a 11 a 22 e R 1r 1 ω 1 b 1 r 2 e R 1r 1 R 2 r 2 ω 1 > 0, n a 12 r 2 e R 2r 2 ω 1 > Proof. Under the conditions that α>0, β>0, γ>0, β 2 4αγ > 0, we have N 1 α, β, γ β 2γ β 2 4αγ β, N 2 α, β, γ β 2 4αγ 2α. 3.9 Thus N 1 α, β, γ N 2 α, β, γ is increasing decreasing in the first variable, decreasing increasing in the second variable, increasing decreasing in the third variable. Notice that α 12 >α 12 >α 12 > 0, γ 12 >γ 12 >γ 12 >γ 12 > 0, β 12 >β

7 Applied Mathematics 7 So from 3.9, 3.10, andh 1 H 3,weobtainthat N 1 α12,β 12 m, γ 12 n <N 1 α12,β 12,γ 12 <N1 α12,β 12,γ 12 <N 2 α12,β 12 12,γ <N2 α12,β 12,γ 12 <N2 α12,β 12 m, γ 12 n Theorem 3.5. Assume that H 1 H 3 hold. Then system 1.5 has at least two ω-periodic solutions. Proof. Take X Z { x x 1,x 2 T : x i C T, R 2 },x i t ω x i t t T,i 1, 2, [ 2 ] 1/2 x max x i t, x X or Z. t I ω i It is easy to verify that X and Z are both Banach spaces. Define the following mappings L : X Z, N : X Z, P : X X and Q : Z Z as follows: r 1 t Nx r 2 t Lx x Δ 1 x Δ 2, a 1j t exp { x j t } b 1 t exp{x 1 t} exp{x 2 t} a 2j t exp { x j t }, b 2 t exp{x 2 t} exp{x 1 t} j1 j Px 1 ω xtδt Qx, x XorZ. We first show that L is a Fredholm mapping of index zero and N is L-compact on Ω for any open bounded set Ω X. The argument is quite standard. For example, one can see 14, 17, 18. But for the sake of completeness, we give the details here. It is easy to see that KerL{x X : x 1 t,x 2 t T h 1,h 2 T R 2 for t T},ImL{x X : xtδt 0} is closed in Z, and dimkerl codimiml 2. Therefore, L is a Fredholm mapping of index zero. Clearly, P and Q are continuous projectors such that Im P Ker L, Ker Q Im L ImI Q On the other hand, K p :ImL dom L Ker P, the inverse to L, exists and is given by K p x t xsδs 1 ω η xsδsδη. 3.15

8 8 Applied Mathematics Obviously, QN and K p I QN are continuous. By Lemma 2.8,itisnotdifficult to show that K p I QNΩ is compact for any open bounded set Ω X. Moreover, QNΩ is bounded. Hence, N is L-compact on Ω for any open bounded set Ω X. Corresponding to the operator equation Lx λnx, λ 0, 1, one has x Δ i t λ r i t a ij t exp { x j t } b i t exp{x i t} exp{x k t}, 3.16 j1 where i, k 1, 2, k/ i. Suppose xt x 1 t,x 2 t T X is a solution of system 3.16 for some λ 0, 1. Integrating 3.16 over the interval, ω, we have r i ω a ij t exp { x j t } Δt j1 b i t exp{x i t} exp{x k t}δt, 3.17 where i, k 1, 2, k/ i. It follows from 3.16 and 3.17 that x Δ Δt i t λ r it a ij t exp { x j t } b i t exp{x i t} exp{x k t} Δt j1 < r i t Δt j1 a ij t exp { x j t } Δt b i t exp{x i t} exp{x k t}δt R i r i ω, i 1, That is x Δ Δt i t < R i r i ω, i 1, Since xt X, there exist ξ i,η i, such that x i ξ i min t,ω x it, x i ηi max t,ω x it, i 1, From 3.17, 3.20, oneobtains a 11 exp { x 1 } a12 exp { x 2 η2 } b1 exp { x 1 } exp { x2 η2 } r1, a 21 exp{x 1 ξ 1 } a 22 exp{x 2 ξ 2 } b 2 exp{x 1 ξ 1 } exp{x 2 ξ 2 } r

9 Applied Mathematics 9 We can derive from 3.22 that ω x 2 η2 x2 ξ 2 x Δ Δt 2 t r 2 a 21 exp{x 1 ξ 1 } <ln a 22 b 2 exp{x 1 ξ 1 } which, together with 3.21,leadsto exp { x 1 } r 1 a 12 exp { x 2 η2 } R 2 r 2 ω, 3.23 a 11 b 1 exp { } x 2 η2 } r2 r 1 a 22 b 2 exp{x 1 ξ 1 } a 12 exp {R 2 r 2 ω a 21 exp{x 1 ξ 1 } } r2 a 11 a 22 b 2 exp{x 1 ξ 1 } b 1 exp {R 2 r 2 ω a 21 exp{x 1 ξ 1 } From 3.19, we have ω x 1 ξ 1 x 1 x Δ Δt 1 t >x1 R 1 r 1 ω That is exp{x 1 ξ 1 } > exp { } x 1 exp R 1 r 1 ω, 3.26 which, together with 3.24,leadsto } exp {R 1 r 1 ω exp{x 1 ξ 1 } } r2 r 1 a 22 b 2 exp{x 1 ξ 1 } a 12 exp {R 2 r 2 ω a 21 exp{x 1 ξ 1 } 3.27 > } r2 a 11 a 22 b 2 exp{x 1 ξ 1 } b 1 exp {R 2 r 2 ω a 21 exp{x 1 ξ 1 }. Therefore, we have So from 3.10, oneobtains α 12 exp2x 1ξ 1 β 12 exp{x 1ξ 1 } γ 12 < α 12 exp2x 1 ξ 1 β 12 m exp{x 1 ξ 1 } γ 12 n<0, 3.29 where m a 11 a 22 e R 1r 1 ω 1 b 1 r 2 e R 1r 1 R 2 r 2 ω 1 > 0, n a 12 r 2 e R 2r 2 ω 1 >

10 10 Applied Mathematics According to i of Lemma 3.3, weobtain N 1 α12,β 12 m, γ 12 n < exp{x 1 ξ 1 } <N 2 α12,β 12 m, γ 12 n In a similar way as the above proof, one can conclude from a 21 exp { x 1 } a22 exp { x 2 η2 } b2 exp { x 1 } exp { x2 η2 } r2, a 11 exp{x 1 ξ 1 } a 12 exp{x 2 ξ 2 } b 1 exp{x 1 ξ 1 } exp{x 2 ξ 2 } r 1, 3.32 that α 12 exp 2x 1 β 12 exp{ } x 1 γ 12 > Noticing that α 12 >α 12, β 12 >β 12, one has α 12 exp 2x 1 β 12 exp{ } x 1 γ 12 > According to ii of Lemma 3.3, one has exp { x 1 } >N2 α12,β 12,γ 12, or exp { x1 } <N1 α12,β 12,γ It follows from 3.19 and 3.31 that x 1 x1 ξ 1 x Δ Δt 1 t < ln N 2 α12,β 12 m, γ 12 n R 1 r 1 ω : H On the other hand, it follows from 3.17 and 3.20 that a ii ω expx i ξ i a ii t exp{x i t}δt <r i ω, i 1, 2; 3.37 that is x i ξ i < ln r i a ii, i 1, From 3.19 and 3.38, oneobtains x i t x i ξ i x Δ Δt i t r i <ln a ii R i r i ω, i 1,

11 Applied Mathematics 11 It follows from 3.17 and 3.20 that r 2 ω j1 a 2j t exp { x j t } Δt b 2 t exp{x 2 t} exp{x 1 t}δt a 2j ω exp { } x j ηj b2 ω exp { } { } x 1 exp x2 η2, j which implies that exp { x 2 η2 } r 2 a 21 exp { x 1 } a 22 b 2 exp { x 1 } From 3.39 and 3.41, we have } a 11 r 2 a 21 r 1 exp {R 1 r 1 ω x 2 η2 ln } : M, 3.42 a 11 a 22 b 2 r 1 exp {R 1 r 1 ω which leads to ω x 2 t x 2 η2 x Δ Δt 2 t >M R 2 r 2 ω By 3.39 and 3.43, weobtainthat { x 2 t < max ln r 2 } R 2 r 2 ω M a, R 2 r 2 ω 22 : A Now, let us consider QNx with x x 1,x 2 T R 2.Notethat QNx 1,x 2 r 1 a 11 expx 1 a 12 expx 2 b 1 expx 1 expx r 2 a 21 expx 1 a 22 expx 2 b 2 expx 1 expx 2 According to Lemma 3.2, we can show that QNx 0 has two distinct solutions x i ln N i α12,β 12,γ 12, ln N1 α21,β 21,γ 21, i 1, Choose C>0 such that C> ln N 1 α21,β 21,γ

12 12 Applied Mathematics Let { x 1 t ln N 1 α12,β 12 m, γ 12 n, ln N 1 α12,β } Ω 1 x X,γ,, x 2 t <AC. Ω 2 x X min x 1 t ln N 1 α12,β 12 m, γ 12 n, ln N 2 α12,β 12 m, γ 12 n, t I ω max x 1 t ln N 2 α12,β t I 12,γ 12,H, x2 t <AC., ω 3.48 Then both Ω 1 and Ω 2 are bounded open subsets of X. It follows from Lemma 3.2, Lemma 3.4, and 3.47 that x i Ω i, i 1, 2. With the help of 3.31, 3.35, 3.36, 3.44, andlemma 3.4, it is easy to see that Ω 1 Ω 2 φ and Ω i satisfies the requirement a in Lemma 3.1 for i 1, 2. Moreover, QNx / 0forx Ω i Ker L i 1, 2. A direct computation gives deg B JQN,Ω i Ker L, 0 / Here J is taken as the identity mapping since Im Q Ker L. So far we have proved that Ω i satisfies all the assumptions in Lemma 3.1. Hence 1.5 has at least two ω-periodic solutions x i with x i Dom L Ω i i 1, 2. Obviously x i i 1, 2 are different. The proof is complete. Example 3.6. As an application of Theorem 3.5, we consider the following system: x Δ 1 t cos0.4πt cos0.4πt exp{x 1t} cos0.4πt exp{x 2 t} cos0.4πt exp{x 1 t} exp{x 2 t}, x Δ 2 t cos0.4πt cos0.4πt exp{x 1t} cos0.4πt exp{x 2 t} cos0.4πt exp{x 2 t} exp{x 1 t}. If T R, then 3.50 reduces to the following system: dx 1 t cos0.4πt cos0.4πt exp{x 1 t} cos0.4πt exp{x 2 t} dx 2 t cos0.4πt exp{x 1 t} exp{x 2 t}, cos0.4πt cos0.4πt exp{x 1 t} cos0.4πt exp{x 2 t} cos0.4πt exp{x 2 t} exp{x 1 t}.

13 Applied Mathematics 13 x1t t x2t t x 1 t x 1 t a x 2 t x 2 t b Figure 1: Blue lines stand for x 1 t, x 2 t, red lines stand for x 1 t, x 2 t. A direct computation gives that R 1 r , R 2 r , a 11 a 22 1, a , a 21 1, b 1 2, b 2 1, ω 5, α 12 a 21 b 1 a 11 b 2 1, 3.52 γ 12 r 1a 22 r 2 a 12 e R 1r 1 R 2 r 2 ω e 0.61 > 0, γ 21 r 2a 11 r 1 a 21 e R 2r 2 R 1 r 1 ω e 0.61 > 0. So according to Theorem 3.5, System3.51 has at least two 5-periodic solutions x 1 t,x 2 t and x 1 t,x 2 t. The simulation results given in Figure 1 verify the above conclusion. Set N i t exp{x i t} i 1, 2, then 3.51 can be changed into the following system: dn 1 t dn 2 t N 1 t cos0.4πt cos0.4πtn 1 t cos0.4πtn 2 t cos0.4πtn 1 tn 2 t, N 2 t cos0.4πt cos0.4πtn 1 t cos0.4πtn 2 t cos0.4πtn 2 tn 1 t. 3.53

14 14 Applied Mathematics Therefore, System 3.53 has at least two positive 5-periodic solutions. Similar to the proof of Theorem 3.5, we can prove the following result. Theorem 3.7. In addition to H 1 and H 2, assume further that system 1.5 satisfies H 3 α 21 > 0. Then system 1.5 has at least two ω-periodic solutions. Acknowledgment This research is supported by the National Natural Science Foundation of China Grant nos , References 1 E. L. Rice, Allelopathy, Academic Press, New York, NY, USA, J. Maynard-Smith, Models in Ecology, Cambridge University, Cambridge, UK, J. Chattopadhyay, Effect of toxic substances on a two-species competitive system, Ecological Modelling, vol. 84, no. 1 3, pp , A. Mukhopadhyay, J. Chattopadhyay, and P. K. Tapaswi, A delay differential equations model of plankton allelopathy, Mathematical Biosciences, vol. 149, no. 2, pp , H. I. Freedman and J. H. Wu, Periodic solutions of single-species models with periodic delay, SIAM Journal on Mathematical Analysis, vol. 23, no. 3, pp , Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, J. Zhang and H. Fang, Multiple periodic solutions for a discrete time model of plankton allelopathy, Advances in Difference Equations, vol. 2006, Article ID 90479, J. G. P. Gamarra and R. V. Solé, Complex discrete dynamics from simple continuous population models, Bulletin of Mathematical Biology, vol. 64, no. 3, pp , M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, S. Hilger, Analysis on measure chains a unified approach to continuous and discrete calculus, Results in Mathematics, Resultate der Mathematik, vol. 18, no. 1-2, pp , Y. Chen, Multiple periodic solutions of delayed predator-prey systems with type IV functional responses, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 5, no. 1, pp , R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics, Vol. 568, Springer, Berlin, Germany, M. Bohner, M. Fan, and J. Zhang, Existence of periodic solutions in predator-prey and competition dynamic systems, Nonlinear Analysis: Real World Applications, vol. 7, no. 5, pp , Y. Xing, M. Han, and G. Zheng, Initial value problem for first-order integro-differential equation of Volterra type on time scales, Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 3, pp , Z. Jin and Z. Ma, Periodic solutions for delay differential equations model of plankton allelopathy, Computers & Mathematics with Applications, vol. 44, no. 3-4, pp , M. Fazly and M. Hesaaraki, Periodic solutions for predator-prey systems with Beddington- DeAngelis functional response on time scales, Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp , W. Zhang, P. Bi, and D. Zhu, Periodicity in a ratio-dependent predator-prey system with stagestructured predator on time scales, Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp , 2008.

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