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Journal of Applied Mathematics
Volume 2012, Article ID 602679, 14 pages
http://dx.doi.org/10.1155/2012/602679
Research Article

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

Department of Mathematics, Kunming University of Science and Technology, Yunnan 650093, China

Received 3 October 2011; Accepted 15 February 2012

Academic Editor: Meng Fan

Copyright © 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.

Abstract

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 [14] 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: 𝑑𝑁1(𝑡)𝑑𝑡=𝑁1𝑟(𝑡)1𝑎11𝑁1(𝑡)𝑎12𝑁2(𝑡)𝑏1𝑁1(𝑡)𝑁2,(𝑡)𝑑𝑁2(𝑡)𝑑𝑡=𝑁2𝑟(𝑡)2𝑎21𝑁1(𝑡)𝑎22𝑁2(𝑡)𝑏2𝑁1(𝑡)𝑁2,(𝑡)(1.1) where 𝑏1 and 𝑏2 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 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 𝑑𝑁1(𝑡)𝑑𝑡=𝑁1𝑟(𝑡)1(𝑡)𝑎11(𝑡)𝑁1(𝑡)𝑎12(𝑡)𝑁2(𝑡)𝑏1(𝑡)𝑁1(𝑡)𝑁2,(𝑡)𝑑𝑁2(𝑡)𝑑𝑡=𝑁2𝑟(𝑡)2(𝑡)𝑎21(𝑡)𝑁1(𝑡)𝑎22(𝑡)𝑁2(𝑡)𝑏2(𝑡)𝑁1(𝑡)𝑁2,(𝑡)(1.2) where 𝑟𝑖(𝑡),𝑎𝑖𝑗(𝑡)>0,𝑏𝑖(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): 𝑁1(𝑘+1)=𝑁1𝑟(𝑘)exp1(𝑘)𝑎11(𝑘)𝑁1(𝑘)𝑎12(𝑘)𝑁2(𝑘)𝑏1(𝑘)𝑁1(𝑘)𝑁2,𝑁(𝑘)2(𝑘+1)=𝑁2(𝑟𝑘)exp2(𝑘)𝑎21(𝑘)𝑁1(𝑘)𝑎22(𝑘)𝑁2(𝑘)𝑏2(𝑘)𝑁1(𝑘)𝑁2(,𝑘)(1.3) where 𝑟𝑖,𝑎𝑖𝑗>0,𝑏𝑖>0(𝑖,𝑗=1,2) are 𝜔-periodic, that is,𝑟𝑖(𝑘+𝜔)=𝑟𝑖(𝑘),𝑏𝑖(𝑘+𝜔)=𝑏𝑖(𝑘),𝑎𝑖𝑗(𝑘+𝜔)=𝑎𝑖𝑗(𝑘),(1.4) for any (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 1988 [11] 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 𝑥Δ1(𝑡)=𝑟1(𝑡)𝑎11(𝑥𝑡)exp1(𝑡)𝑎12(𝑥𝑡)exp2(𝑡)𝑏1(𝑥𝑡)exp1(𝑥𝑡)exp2(,𝑥𝑡)Δ2(𝑡)=𝑟2(𝑡)𝑎21(𝑥𝑡)exp1(𝑡)𝑎22(𝑥𝑡)exp2(𝑡)𝑏2(𝑥𝑡)exp1(𝑥𝑡)exp2(,𝑡)(1.5) where 𝑟𝑖(𝑡),𝑎𝑖𝑗(𝑡)>0,𝑏𝑖(𝑡)>0(𝑖,𝑗=1,2)are rd-continuous 𝜔-periodic functions.

In (1.5), let 𝑁𝑖(𝑡)=exp{𝑥𝑖(𝑡)}, 𝑖=1,2. If 𝕋= (the set of all real numbers), then (1.5) reduces to (1.2). If 𝕋= (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 of coincidence degree theory [13]. Some new results are obtained. Even in the special case when 𝕋=, our conditions are also easier to verify than that of [7].

2. 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 [911].

Definition 2.1. A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers .

Let 𝜔>0. Throughout this paper, the time scale 𝕋 is assumed to be 𝜔-periodic, that is, 𝑡𝕋 implies 𝑡+𝜔𝕋. In particular, the time scale 𝕋 under consideration is unbounded above and below.

Definition 2.2. We define the forward jump operator 𝜎𝕋𝕋, the backward jump operator 𝜌𝕋𝕋, and the graininess 𝜇𝕋+=[0,) by 𝜎(𝑡)=inf{𝑠𝕋𝑠>𝑡},𝜌(𝑡)=sup{𝑠𝕋𝑠<𝑡},𝜇(𝑡)=𝜎(𝑡)𝑡for𝑡𝕋,(2.1) respectively. If 𝜎(𝑡)=𝑡, then 𝑡 is called right-dense (otherwise: right-scattered), and if 𝜌(𝑡)=𝑡, then 𝑡 is called left-dense (otherwise: left-scattered).

Definition 2.3. Assume 𝑓𝕋 is a function and let 𝑡𝕋. Then we define 𝑓Δ(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=(𝑡𝛿,𝑡+𝛿)𝕋 for some 𝛿>0) such that ||[]𝑓(𝜎(𝑡))𝑓(𝑠)𝑓Δ[]||||||(𝑡)𝜎(𝑡)𝑠𝜎(𝑡)𝑠𝑠𝑈.(2.2) In this case, 𝑓Δ(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 𝑓Δ(𝑡) exists for all 𝑡𝕋. A function 𝐹𝕋 is called an antiderivative of 𝑓𝕋 provided 𝐹Δ(𝑡)=𝑓(𝑡) for all 𝑡𝕋. Then we define 𝑠𝑟𝑓(𝑡)Δ𝑡=𝐹(𝑠)𝑓(𝑟)for𝑟,𝑠𝕋.(2.3)

Definition 2.4. A function 𝑓𝕋 is said to be rd-continuous if it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋. The set of rd-continuous functions 𝑓𝕋 will be denoted by 𝐶rd(𝕋).

Lemma 2.5. Every rd-continuous function has an antiderivative.

Lemma 2.6. If 𝑎,𝑏𝕋,𝛼,𝛽, and 𝑓,𝑔𝐶rd(𝕋), then (a)𝑏𝑎[𝛼𝑓(𝑡)+𝛽𝑔(𝑡)]Δ𝑡=𝛼𝑏𝑎𝑓(𝑡)Δ𝑡+𝛽𝑏𝑎𝑔(𝑡)Δ𝑡(b)if 𝑓(𝑡)0 for all 𝑎𝑡<𝑏, then 𝑏𝑎𝑓(𝑡)Δ𝑡0(c)if |𝑓(𝑡)|𝑔(𝑡) on [𝑎,𝑏)={𝑡𝕋𝑎𝑡<𝑏}, then |𝑏𝑎𝑓(𝑡)Δ𝑡|𝑏𝑎𝑔(𝑡)Δ𝑡.

For convenience, we now introduce some notation to be used throughout this paper.

Let [𝜅=min{0,)𝕋},𝐼𝜔=[]𝜅,𝜅+𝜔𝕋,𝑔𝑢=sup𝑡𝐼𝜔𝑔(𝑡),𝑔𝑙=inf𝑡𝐼𝜔𝑔(𝑡),1𝑔=𝜔𝐼𝜔𝑔1(𝑠)Δ𝑠=𝜔𝜅𝜅+𝜔𝑔(𝑠)Δ𝑠,(2.4) where 𝑔𝐶rd(𝕋) is an 𝜔-periodic real function, that is, 𝑔(𝑡+𝜔)=𝑔(𝑡) for all 𝑡𝕋.

Lemma 2.7 (see [14]). Let 𝑡1,𝑡2𝐼𝜔 and 𝑡𝕋. If 𝑔𝕋 is 𝜔- periodic, then 𝑔𝑡(𝑡)𝑔1+𝜅𝜅+𝜔||𝑔||𝑡(𝑠)Δ𝑠,𝑔(𝑡)𝑔2𝜅𝜅+𝜔||𝑔||(𝑠)Δ𝑠.(2.5)

Lemma 2.8 (see [15]). Assume that {𝑓𝑛}𝑛𝑁 is a function on 𝐽 such that(i){𝑓𝑛}𝑛𝑁 is uniformly bounded on 𝐽,(ii){𝑓Δ𝑛}𝑛𝑁 is uniformly bounded on 𝐽.Then there is a subsequence of {𝑓𝑛}𝑛𝑁 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), we first make the following preparations [13].

Let 𝑋, 𝑍 be normed vector spaces, let 𝐿dom𝐿𝑋𝑍 be a linear mapping, and let 𝑁: 𝑋𝑍 be a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿<+ and Im𝐿 is closed in 𝑍. If 𝐿 is a Fredholm mapping of index zero, there then exist continuous projectors 𝑃𝑋𝑋 and 𝑄𝑍𝑍 such that Im𝑃=Ker𝐿, Im𝐿=Ker𝑄=Im(𝐼𝑄). If we define 𝐿𝑃dom𝐿Ker𝑃Im𝐿 as the restriction 𝐿|dom𝐿Ker𝑃of 𝐿to dom𝐿Ker𝑃, then 𝐿𝑃 is invertible. We denote the inverse of that map by 𝐾𝑃. 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 𝐾𝑃(𝐼𝑄)𝑁Ω𝑋 is compact, that is, continuous and such that 𝐾𝑃(𝐼𝑄)𝑁(Ω) is relatively compact. Since Im𝑄 is isomorphic to Ker𝐿, there exists isomorphism 𝐽Im𝑄Ker𝐿.

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 𝑁Ω𝑍 be 𝐿-compact on Ω. Suppose (a)𝐿𝑥𝜆𝑁𝑥 for every 𝑥dom𝐿𝜕Ω and every 𝜆(0,1),(b)𝑄𝑁𝑥0 for every 𝑥𝜕ΩKer𝐿, and Brouwer degreedegB(𝐽𝑄𝑁,ΩKer𝐿,0)0.(3.1) Then 𝐿𝑥=𝑁𝑥 has at least one solution in dom𝐿Ω.

In the following, we shall use the notation 𝛼𝑖𝑗=𝑎𝑗𝑖𝑏𝑖𝑎𝑖𝑖𝑏𝑗,𝛼𝑖𝑗=𝑎𝑗𝑖𝑏𝑖𝑎𝑖𝑖𝑏𝑗𝑒(𝑅𝑗+𝑟𝑗)𝜔,𝛼𝑖𝑗=𝑎𝑗𝑖𝑏𝑖𝑒(𝑅𝑗+𝑟𝑗)𝜔𝑎𝑖𝑖𝑏𝑗𝑒(𝑅𝑖+𝑟𝑖)𝜔,𝛽𝑖𝑗=𝑎𝑖𝑖𝑎𝑗𝑗+𝑏𝑖𝑟𝑗𝑎𝑖𝑗𝑎𝑗𝑖𝑏𝑗𝑟𝑖,𝛽𝑖𝑗=𝑎𝑖𝑖𝑎𝑗𝑗𝑒(𝑅𝑗+𝑟𝑗)𝜔+𝑏𝑖𝑟𝑗𝑎𝑖𝑗𝑎𝑗𝑖𝑒(𝑅𝑖+𝑟𝑖)𝜔𝑏𝑗𝑟𝑖𝑒(𝑅𝑖+𝑟𝑖+𝑅𝑗+𝑟𝑗)𝜔,𝛽𝑖𝑗=𝑎𝑖𝑖𝑎𝑗𝑗𝑒(𝑅𝑖+𝑟𝑖)𝜔+𝑏𝑖𝑟𝑗𝑒(𝑅𝑖+𝑟𝑖+𝑅𝑗+𝑟𝑗)𝜔𝑎𝑖𝑗𝑎𝑗𝑖𝑒(𝑅𝑗+𝑟𝑗)𝜔𝑏𝑗𝑟𝑖,𝛽𝑖𝑗=𝑎𝑖𝑖𝑎𝑗𝑗+𝑏𝑖𝑟𝑗𝑎𝑖𝑗𝑎𝑗𝑖𝑒(𝑅𝑖+𝑟𝑖)𝜔𝑏𝑗𝑟𝑖𝑒(𝑅𝑖+𝑟𝑖+𝑅𝑗+𝑟𝑗)𝜔,𝛾𝑖𝑗=𝑟𝑖𝑎𝑗𝑗𝑟𝑗𝑎𝑖𝑗,𝛾𝑖𝑗=𝑟𝑖𝑎𝑗𝑗𝑒(𝑅𝑗+𝑟𝑗)𝜔𝑟𝑗𝑎𝑖𝑗𝑒(𝑅𝑖+𝑟𝑖)𝜔,𝛾𝑖𝑗=𝑟𝑖𝑎𝑗𝑗𝑟𝑗𝑎𝑖𝑗𝑒(𝑅𝑗+𝑟𝑗)𝜔,𝛾𝑖𝑗=𝑟𝑖𝑎𝑗𝑗𝑟𝑗𝑎𝑖𝑗𝑒(𝑅𝑖+𝑟𝑖+𝑅𝑗+𝑟𝑗)𝜔𝑁,𝑖𝑗,𝑖,𝑗=1,2,1(𝛼,𝛽,𝛾)=𝛽𝛽24𝛼𝛾2𝛼,𝑁2(𝛼,𝛽,𝛾)=𝛽+𝛽24𝛼𝛾2𝛼𝛼0,𝛽2.4𝛼𝛾>0(3.2)

We make the following assumptions.(H1)𝑅𝑖=(1/𝜔)𝜅𝜅+𝜔|𝑟𝑖(𝑡)|Δ𝑡(1/𝜔)𝜅𝜅+𝜔𝑟𝑖(𝑡)Δ𝑡>0. (H2)𝛾𝑖𝑗=𝑟𝑖𝑎𝑗𝑗𝑟𝑗𝑎𝑖𝑗𝑒(𝑅𝑖+𝑟𝑖+𝑅𝑗+𝑟𝑗)𝜔>0,𝑖𝑗,𝑖,𝑗=1,2. (H3)𝛼12>0.

Next, we introduce some lemmas.

Lemma 3.2 (see [16, Lemma 3.2]). Consider the following algebraic equations: 𝑎11𝑁1+𝑎12𝑁2+𝑏1𝑁1𝑁2=𝑟1,𝑎21𝑁1+𝑎22𝑁2+𝑏2𝑁1𝑁2=𝑟2.(3.3) Assuming that (H1), (H2) hold, then the following conclusions hold.(i)If 𝛼12>0, then (3.3) have two positive solutions: 𝑁𝑖𝛼12,𝛽12,𝛾12,𝑁1𝛼21,𝛽21,𝛾21,𝑖=1,2.(3.4)(ii)If 𝛼21>0, then (3.3) have two positive solutions: 𝑁1𝛼12,𝛽12,𝛾12,𝑁𝑖𝛼21,𝛽21,𝛾21,𝑖=1,2.(3.5)

Lemma 3.3. Assume that (H1)–(H3) hold, then the following conclusions hold.(i)𝛽12>0,𝛽2124𝛼12𝛾12>0, (ii)𝛽12>0,𝛽2124𝛼12𝛾12>0.

Proof. The proof of (i) is the same as (i) of Lemma  3.5 in [7]. We omit it.(ii)We have𝛽12=𝑏1𝑎11+𝑎12𝑟1𝑒(𝑅2+𝑟2)𝜔𝛾21+𝑟1𝛼12𝑒(𝑅1+𝑟1+𝑅2+𝑟2)𝜔𝑎11+𝑎11𝛾12𝑟1𝑒(𝑅1+𝑟1+𝑅2+𝑟2)𝜔𝛽>0,2124𝛼12𝛾12=𝑏1𝑎11+𝑎12𝑟1𝑒(𝑅2+𝑟2)𝜔2𝛾221+𝑟1𝛼12𝑒(𝑅1+𝑟1)𝜔𝑎11𝑎11𝛾12𝑟1𝑒(𝑅1+𝑟1)𝜔2+2𝑏1𝑎11+𝑎12𝑟1𝑒(𝑅2+𝑟2)𝜔𝑟1𝛼12𝑒(𝑅1+𝑟1+𝑅2+𝑟2)𝜔𝑎11+𝑎11𝛾12𝑟1𝑒(𝑅1+𝑟1+𝑅2+𝑟2)𝜔𝛾21>0.(3.6)

Lemma 3.4. Assume that (H1)–(H3) hold, then the following conclusions hold. 𝑁1𝛼12,𝛽12+𝑚,𝛾12𝑛<𝑁1𝛼12,𝛽12,𝛾12<𝑁1𝛼12,𝛽12,𝛾12<𝑁2𝛼12,𝛽12,𝛾12<𝑁2𝛼12,𝛽12,𝛾12<𝑁2𝛼12,𝛽12+𝑚,𝛾12,𝑛(3.7) where 𝑚=𝑎11𝑎22𝑒(𝑅1+𝑟1)𝜔+1𝑏1𝑟2𝑒(𝑅1+𝑟1+𝑅2+𝑟2)𝜔1>0,𝑛=𝑎12𝑟2𝑒(𝑅2+𝑟2)𝜔1>0.(3.8)

Proof. Under the conditions that 𝛼>0, 𝛽>0, 𝛾>0, 𝛽24𝛼𝛾>0, we have 𝑁1(𝛼,𝛽,𝛾)=2𝛾𝛽+𝛽24𝛼𝛾,𝑁2(𝛼,𝛽,𝛾)=𝛽+𝛽24𝛼𝛾2𝛼.(3.9) Thus 𝑁1(𝛼,𝛽,𝛾)(𝑁2(𝛼,𝛽,𝛾)) is increasing (decreasing) in the first variable, decreasing (increasing) in the second variable, increasing (decreasing) in the third variable. Notice that 𝛼12>𝛼12>𝛼12>0,𝛾12>𝛾12>𝛾12>𝛾12>0,𝛽12>𝛽12.(3.10) So from (3.9), (3.10), and (H1)–(H3), we obtain that 𝑁1𝛼12,𝛽12+𝑚,𝛾12𝑛<𝑁1𝛼12,𝛽12,𝛾12<𝑁1𝛼12,𝛽12,𝛾12<𝑁2𝛼12,𝛽12,𝛾12<𝑁2𝛼12,𝛽12,𝛾12<𝑁2𝛼12,𝛽12+𝑚,𝛾12.𝑛(3.11)

Theorem 3.5. Assume that (H1)–(H3) hold. Then system (1.5) has at least two 𝜔-periodic solutions.

Proof. Take 𝑥𝑋=𝑍=𝑥=1,𝑥2𝑇𝑥𝑖𝐶𝕋,2,𝑥𝑖(𝑡+𝜔)=𝑥𝑖,(𝑡)𝑡𝕋,𝑖=1,2𝑥=2𝑖=1max𝑡𝐼𝜔||𝑥𝑖||(𝑡)21/2,𝑥𝑋(or𝑍).(3.12) It is easy to verify that 𝑋 and 𝑍 are both Banach spaces.
Define the following mappings 𝐿𝑋𝑍,𝑁𝑋𝑍, 𝑃𝑋𝑋 and 𝑄𝑍𝑍 as follows: 𝑟𝑁𝑥=1(𝑡)2𝑗=1𝑎1𝑗𝑥(𝑡)exp𝑗(𝑡)𝑏1𝑥(𝑡)exp1𝑥(𝑡)exp2𝑟(𝑡)2(𝑡)2𝑗=1𝑎2𝑗𝑥(𝑡)exp𝑗(𝑡)𝑏2𝑥(𝑡)exp2𝑥(𝑡)exp1,𝑥(𝑡)𝐿𝑥=Δ1𝑥Δ2,1𝑃𝑥=𝜔𝜅𝜅+𝜔𝑥(𝑡)Δ𝑡=𝑄𝑥,𝑥𝑋(or𝑍).(3.13)
We first show that 𝐿 is a Fredholm mapping of index zero and 𝑁 is 𝐿-compact on Ω for any open bounded set Ω𝑋. 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={𝑥𝑋(𝑥1(𝑡),𝑥2(𝑡))𝑇=(1,2)𝑇2for𝑡𝕋}, Im 𝐿={𝑥𝑋𝜅𝜅+𝜔𝑥(𝑡)Δ𝑡=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 Im𝑃=Ker𝐿,Ker𝑄=Im𝐿=Im(𝐼𝑄).(3.14) On the other hand, 𝐾𝑝: Im𝐿dom𝐿Ker𝑃, the inverse to 𝐿, exists and is given by 𝐾𝑝(𝑥)=𝑡𝜅1𝑥(𝑠)Δ𝑠𝜔𝜅𝜅+𝜔𝜂𝜅𝑥(𝑠)Δ𝑠Δ𝜂.(3.15) Obviously, 𝑄𝑁 and 𝐾𝑝(𝐼𝑄)𝑁 are continuous. By Lemma 2.8, it is not difficult to show that 𝐾𝑝(𝐼𝑄)𝑁(Ω) is compact for any open bounded set Ω𝑋. Moreover, 𝑄𝑁(Ω) is bounded. Hence, 𝑁 is 𝐿-compact on Ω for any open bounded set Ω𝑋.
Corresponding to the operator equation 𝐿𝑥=𝜆𝑁𝑥,𝜆(0,1), one has 𝑥Δ𝑖𝑟(𝑡)=𝜆𝑖(𝑡)2𝑗=1𝑎𝑖𝑗𝑥(𝑡)exp𝑗(𝑡)𝑏𝑖𝑥(𝑡)exp𝑖𝑥(𝑡)exp𝑘(𝑡),(3.16) where 𝑖,𝑘=1,2,𝑘𝑖. Suppose 𝑥(𝑡)=(𝑥1(𝑡),𝑥2(𝑡))𝑇𝑋 is a solution of system (3.16) for some 𝜆(0,1). Integrating (3.16) over the interval [𝜅,𝜅+𝜔], we have 𝑟𝑖𝜔=2𝑗=1𝜅𝜅+𝜔𝑎𝑖𝑗𝑥(𝑡)exp𝑗(𝑡)Δ𝑡+𝜅𝜅+𝜔𝑏𝑖𝑥(𝑡)exp𝑖𝑥(𝑡)exp𝑘(𝑡)Δ𝑡,(3.17) where 𝑖,𝑘=1,2,𝑘𝑖.
It follows from (3.16) and (3.17) that 𝜅𝜅+𝜔||𝑥Δ𝑖||(𝑡)Δ𝑡=𝜆𝜅𝜅+𝜔|||||𝑟𝑖(𝑡)2𝑗=1𝑎𝑖𝑗𝑥(𝑡)exp𝑗(𝑡)𝑏𝑖𝑥(𝑡)exp𝑖𝑥(𝑡)exp𝑘|||||<(𝑡)Δ𝑡𝜅𝜅+𝜔||𝑟𝑖||(𝑡)Δ𝑡+2𝑗=1𝜅𝜅+𝜔𝑎𝑖𝑗𝑥(𝑡)exp𝑗+(𝑡)Δ𝑡𝜅𝜅+𝜔𝑏𝑖𝑥(𝑡)exp𝑖𝑥(𝑡)exp𝑘(𝑡)Δ𝑡=𝑅𝑖+𝑟𝑖𝜔,𝑖=1,2.(3.18) That is 𝜅𝜅+𝜔||𝑥Δ𝑖||(𝑡)Δ𝑡<𝑅𝑖+𝑟𝑖𝜔,𝑖=1,2.(3.19) Since 𝑥(𝑡)𝑋, there exist 𝜉𝑖,𝜂𝑖, such that 𝑥𝑖𝜉𝑖=min[]𝑡𝜅,𝜅+𝜔𝑥𝑖(𝑡),𝑥𝑖𝜂𝑖=max[]𝑡𝜅,𝜅+𝜔𝑥𝑖(𝑡),𝑖=1,2.(3.20) From (3.17), (3.20), one obtains 𝑎11𝑥exp1𝜂1+𝑎12𝑥exp2𝜂2+𝑏1𝑥exp1𝜂1𝑥exp2𝜂2𝑟1,(3.21)𝑎21𝑥exp1𝜉1+𝑎22𝑥exp2𝜉2+𝑏2𝑥exp1𝜉1𝑥exp2𝜉2𝑟2.(3.22) We can derive from (3.22) that 𝑥2𝜂2𝑥2𝜉2+𝜅𝜅+𝜔||𝑥Δ2||(𝑡)Δ𝑡<ln𝑟2𝑎21𝑥exp1𝜉1𝑎22+𝑏2𝑥exp1𝜉1+𝑅2+𝑟2𝜔,(3.23) which, together with (3.21), leads to 𝑥exp1𝜂1𝑟1𝑎12𝑥exp2𝜂2𝑎11+𝑏1𝑥exp2𝜂2𝑟1𝑎22+𝑏2𝑥exp1𝜉1𝑎12exp𝑅2+𝑟2𝜔𝑟2𝑎21𝑥exp1𝜉1𝑎11𝑎22+𝑏2𝑥exp1𝜉1+𝑏1exp𝑅2+𝑟2𝜔𝑟2𝑎21𝑥exp1𝜉1.(3.24) From (3.19), we have 𝑥1𝜉1𝑥1𝜂1𝜅𝜅+𝜔||𝑥Δ1||(𝑡)Δ𝑡>𝑥1𝜂1𝑅1+𝑟1𝜔.(3.25) That is 𝑥exp1𝜉1𝑥>exp1𝜂1exp𝑅1+𝑟1𝜔,(3.26) which, together with (3.24), leads to exp𝑅1+𝑟1𝜔𝑥exp1𝜉1>𝑟1𝑎22+𝑏2𝑥exp1𝜉1𝑎12exp𝑅2+𝑟2𝜔𝑟2𝑎21𝑥exp1𝜉1𝑎11𝑎22+𝑏2𝑥exp1𝜉1+𝑏1exp𝑅2+𝑟2𝜔𝑟2𝑎21𝑥exp1𝜉1.(3.27) Therefore, we have 𝛼12exp2𝑥1𝜉1𝛽12𝑥exp1𝜉1+𝛾12<0.(3.28) So from (3.10), one obtains 𝛼12exp2𝑥1𝜉1𝛽12𝑥+𝑚exp1𝜉1+𝛾12𝑛<0,(3.29) where 𝑚=𝑎11𝑎22𝑒(𝑅1+𝑟1)𝜔+1𝑏1𝑟2𝑒(𝑅1+𝑟1+𝑅2+𝑟2)𝜔1>0,𝑛=𝑎12𝑟2𝑒(𝑅2+𝑟2)𝜔1>0.(3.30) According to (i) of Lemma 3.3, we obtain 𝑁1𝛼12,𝛽12+𝑚,𝛾12𝑥𝑛<exp1𝜉1<𝑁2𝛼12,𝛽12+𝑚,𝛾12𝑛.(3.31)
In a similar way as the above proof, one can conclude from 𝑎21𝑥exp1𝜂1+𝑎22𝑥exp2𝜂2+𝑏2𝑥exp1𝜂1𝑥exp2𝜂2𝑟2,𝑎11𝑥exp1𝜉1+𝑎12𝑥exp2𝜉2+𝑏1𝑥exp1𝜉1𝑥exp2𝜉2𝑟1,(3.32) that 𝛼12exp2𝑥1𝜂1𝛽12𝑥exp1𝜂1+𝛾12>0.(3.33) Noticing that 𝛼12>𝛼12, 𝛽12>𝛽12, one has 𝛼12exp2𝑥1𝜂1𝛽12𝑥exp1𝜂1+𝛾12>0.(3.34) According to (ii) of Lemma 3.3, one has 𝑥exp1𝜂1>𝑁2𝛼12,𝛽12,𝛾12𝑥,orexp1𝜂1<𝑁1𝛼12,𝛽12,𝛾12.(3.35)
It follows from (3.19) and (3.31) that 𝑥1𝜂1𝑥1𝜉1+𝜅𝜅+𝜔||𝑥Δ1||(𝑡)Δ𝑡<ln𝑁2𝛼12,𝛽12+𝑚,𝛾12+𝑛𝑅1+𝑟1𝜔=𝐻.(3.36)
On the other hand, it follows from (3.17) and (3.20) that 𝑎𝑖𝑖𝑥𝜔exp𝑖𝜉𝑖𝜅𝜅+𝜔𝑎𝑖𝑖𝑥(𝑡)exp𝑖(𝑡)Δ𝑡<𝑟𝑖𝜔,𝑖=1,2;(3.37) that is 𝑥𝑖𝜉𝑖<ln𝑟𝑖𝑎𝑖𝑖,𝑖=1,2.(3.38) From (3.19) and (3.38), one obtains 𝑥𝑖(𝑡)𝑥𝑖𝜉𝑖+𝜅𝜅+𝜔||𝑥Δ𝑖||(𝑡)Δ𝑡<ln𝑟𝑖𝑎𝑖𝑖+𝑅𝑖+𝑟𝑖𝜔,𝑖=1,2.(3.39) It follows from (3.17) and (3.20) that 𝑟2𝜔=2𝑗=1𝜅𝜅+𝜔𝑎2𝑗𝑥(𝑡)exp𝑗+(𝑡)Δ𝑡𝜅𝜅+𝜔𝑏2𝑥(𝑡)exp2𝑥(𝑡)exp1(𝑡)Δ𝑡2𝑗=1𝑎2𝑗𝑥𝜔exp𝑗𝜂𝑗+𝑏2𝑥𝜔exp1𝜂1𝑥exp2𝜂2,(3.40) which implies that 𝑥exp2𝜂2𝑟2𝑎21𝑥exp1𝜂1𝑎22+𝑏2𝑥exp1𝜂1.(3.41) From (3.39) and (3.41), we have 𝑥2𝜂2ln𝑎11𝑟2𝑎21𝑟1exp𝑅1+𝑟1𝜔𝑎11𝑎22+𝑏2𝑟1exp𝑅1+𝑟1𝜔=𝑀,(3.42) which leads to 𝑥2(𝑡)𝑥2𝜂2𝜅𝜅+𝜔||𝑥Δ2||(𝑡)Δ𝑡>𝑀𝑅2+𝑟2𝜔.(3.43) By (3.39) and (3.43), we obtain that ||𝑥2||||||(𝑡)<maxln𝑟2𝑎22+𝑅2+𝑟2𝜔||||,|||𝑀𝑅2+𝑟2𝜔|||=𝐴.(3.44)
Now, let us consider 𝑄𝑁𝑥 with 𝑥=(𝑥1,𝑥2)𝑇2. Note that 𝑥𝑄𝑁1,𝑥2=𝑟1𝑎11𝑥exp1𝑎12𝑥exp2𝑏1𝑥exp1𝑥exp2𝑟2𝑎21𝑥exp1𝑎22𝑥exp2𝑏2𝑥exp1𝑥exp2.(3.45) According to Lemma 3.2, we can show that 𝑄𝑁𝑥=0 has two distinct solutions ̂𝑥𝑖=ln𝑁𝑖𝛼12,𝛽12,𝛾12,ln𝑁1𝛼21,𝛽21,𝛾21,𝑖=1,2.(3.46) Choose 𝐶>0 such that ||𝐶>ln𝑁1𝛼21,𝛽21,𝛾21||.(3.47) Let Ω1=||||||𝑥𝑥𝑋1(𝑡)ln𝑁1𝛼12,𝛽12+𝑚,𝛾12𝑛,ln𝑁1𝛼12,𝛽12,𝛾12,||𝑥2||,Ω(𝑡)<𝐴+𝐶.2=|||||||𝑥𝑋min𝑡𝐼𝜔𝑥1(𝑡)ln𝑁1𝛼12,𝛽12+𝑚,𝛾12𝑛,ln𝑁2𝛼12,𝛽12+𝑚,𝛾12,𝑛max𝑡𝐼𝜔𝑥1(𝑡)ln𝑁2𝛼12,𝛽12,𝛾12,||𝑥,𝐻2||,(𝑡)<𝐴+𝐶.(3.48) Then both Ω1 and Ω2 are bounded open subsets of 𝑋. It follows from Lemma 3.2, Lemma 3.4, and (3.47) that ̂𝑥𝑖Ω𝑖, 𝑖=1,2. With the help of (3.31), (3.35), (3.36), (3.44), and Lemma 3.4, it is easy to see that Ω1Ω2=𝜙 and Ω𝑖 satisfies the requirement (a) in Lemma 3.1 for 𝑖=1,2. Moreover, 𝑄𝑁𝑥0 for 𝑥𝜕Ω𝑖Ker𝐿(𝑖=1,2). A direct computation gives deg𝐵𝐽𝑄𝑁,Ω𝑖Ker𝐿,00.(3.49) Here 𝐽 is taken as the identity mapping since Im𝑄=Ker𝐿. So far we have proved that Ω𝑖 satisfies all the assumptions in Lemma 3.1. Hence (1.5) has at least two 𝜔-periodic solutions ̆𝑥𝑖 with ̆𝑥𝑖Dom𝐿Ω𝑖(𝑖=1,2). Obviously ̆𝑥𝑖(𝑖=1,2) are different. The proof is complete.

Example 3.6. As an application of Theorem 3.5, we consider the following system: 𝑥Δ1(𝑥𝑡)=0.02+0.002cos(0.4𝜋𝑡)(1+0.001cos(0.4𝜋𝑡))exp1(𝑥𝑡)(0.1+0.0137cos(0.4𝜋𝑡))exp2𝑥(𝑡)(2+0.95cos(0.4𝜋𝑡))exp1(𝑥𝑡)exp2(,𝑥𝑡)Δ2(𝑥𝑡)=0.041+0.04cos(0.4𝜋𝑡)(1+0.038cos(0.4𝜋𝑡))exp1(𝑥𝑡)(1+0.0379cos(0.4𝜋𝑡))exp2𝑥(𝑡)(1+0.039cos(0.4𝜋𝑡))exp2(𝑥𝑡)exp1(.𝑡)(3.50) If 𝕋=, then (3.50) reduces to the following system: 𝑑𝑥1(𝑡)𝑥𝑑𝑡=0.02+0.002cos(0.4𝜋𝑡)(1+0.001cos(0.4𝜋𝑡))exp1𝑥(𝑡)(0.1+0.0137cos(0.4𝜋𝑡))exp2𝑥(𝑡)(2+0.95cos(0.4𝜋𝑡))exp1𝑥(𝑡)exp2,(𝑡)𝑑𝑥2(𝑡)𝑥𝑑𝑡=0.041+0.04cos(0.4𝜋𝑡)(1+0.038cos(0.4𝜋𝑡))exp1𝑥(𝑡)(1+0.0379cos(0.4𝜋𝑡))exp2𝑥(𝑡)(1+0.039cos(0.4𝜋𝑡))exp2𝑥(𝑡)exp1.(𝑡)(3.51) A direct computation gives that 𝑅1=𝑟1=0.02,𝑅2=𝑟2=0.041,𝑎11=𝑎22=1,𝑎12=0.1,𝑎21=1,𝑏1=2,𝑏2𝛼=1,𝜔=5,12=𝑎21𝑏1𝑎11𝑏2𝛾=1,12=𝑟1𝑎22𝑟2𝑎12𝑒(𝑅1+𝑟1+𝑅2+𝑟2)𝜔=0.020.0041𝑒0.61𝛾>0,21=𝑟2𝑎11𝑟1𝑎21𝑒(𝑅2+𝑟2+𝑅1+𝑟1)𝜔=0.0410.02𝑒0.61>0.(3.52) So according to Theorem 3.5, System (3.51) has at least two 5-periodic solutions (𝑥1(𝑡),𝑥2(𝑡)) and (𝑥1(𝑡),𝑥2(𝑡)). The simulation results given in Figure 1 verify the above conclusion. Set 𝑁𝑖(𝑡)=exp{𝑥𝑖(𝑡)}(𝑖=1,2), then (3.51) can be changed into the following system: 𝑑𝑁1(𝑡)𝑑𝑡=𝑁1(𝑡)0.02+0.002cos(0.4𝜋𝑡)(1+0.001cos(0.4𝜋𝑡))𝑁1(𝑡)(0.1+0.0137cos(0.4𝜋𝑡))𝑁2(𝑡)(2+0.95cos(0.4𝜋𝑡))𝑁1(𝑡)𝑁2,(𝑡)𝑑𝑁2(𝑡)𝑑𝑡=𝑁2(𝑡)0.041+0.04cos(0.4𝜋𝑡)(1+0.038cos(0.4𝜋𝑡))𝑁1(𝑡)(1+0.0379cos(0.4𝜋𝑡))𝑁2(𝑡)(1+0.039cos(0.4𝜋𝑡))𝑁2(𝑡)𝑁1.(𝑡)(3.53) 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.

fig1
Figure 1: Blue lines stand for 𝑥1(𝑡), 𝑥2(𝑡), red lines stand for 𝑥1(𝑡),𝑥2(𝑡).

Theorem 3.7. In addition to (H1) and (H2), assume further that system (1.5) satisfies(H3)'𝛼21>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. 10971085, 11061016).

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