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

Authors: Hui Fang
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 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:
(1.1)dN1(t)dt=N1(t)[r1-a11N1(t)-a12N2(t)-b1N1(t)N2(t)],dN2(t)dt=N2(t)[r2-a21N1(t)-a22N2(t)-b2N1(t)N2(t)],
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
(1.2)dN1(t)dt=N1(t)[r1(t)-a11(t)N1(t)-a12(t)N2(t)-b1(t)N1(t)N2(t)],dN2(t)dt=N2(t)[r2(t)-a21(t)N1(t)-a22(t)N2(t)-b2(t)N1(t)N2(t)],
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):
(1.3)N1(k+1)=N1(k)exp⁡{r1(k)-a11(k)N1(k)-a12(k)N2(k)-b1(k)N1(k)N2(k)},N2(k+1)=N2(k)exp⁡{r2(k)-a21(k)N1(k)-a22(k)N2(k)-b2(k)N1(k)N2(k)},
where ri,aij>0,bi>0(i,j=1,2) are ω-periodic, that is,(1.4)ri(k+ω)=ri(k),bi(k+ω)=bi(k),aij(k+ω)=aij(k),
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 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
(1.5)x1Δ(t)=r1(t)-a11(t)exp⁡{x1(t)}-a12(t)exp⁡{x2(t)}-b1(t)exp⁡{x1(t)}exp⁡{x2(t)},x2Δ(t)=r2(t)-a21(t)exp⁡{x1(t)}-a22(t)exp⁡{x2(t)}-b2(t)exp⁡{x1(t)}exp⁡{x2(t)},
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=ℝ (the set of all real numbers), then (1.5) reduces to (1.2). If 𝕋=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 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 [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 [9–11].Definition 2.1.
A time scale𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ.Letω>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 operatorσ:𝕋→𝕋, the backward jump operator ρ:𝕋→𝕋, and the graininess μ:𝕋→ℝ+=[0,∞) by
(2.1)σ(t)∶=inf⁡{s∈T:s>t},ρ(t)∶=sup⁡{s∈T:s<t},μ(t)=σ(t)-tfort∈T,
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.
Assumef:𝕋→ℝ is a function and let t∈𝕋. Then we define fΔ(t) to be the number (provided it exists) with the property that given any ɛ>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩𝕋 for some δ>0) such that
(2.2)|[f(σ(t))-f(s)]-fΔ(t)[σ(t)-s]|≤|σ(t)-s|∀s∈U.
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 t∈𝕋. A function F:𝕋→ℝ is called an antiderivative of f:𝕋→ℝ provided FΔ(t)=f(t) for all t∈𝕋. Then we define
(2.3)∫rsf(t)Δt=F(s)-f(r)forr,s∈T.Definition 2.4.
A functionf:𝕋→ℝ 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 f:𝕋→ℝ will be denoted by Crd(𝕋).Lemma 2.5.
Every rd-continuous function has an antiderivative.Lemma 2.6.
Ifa,b∈𝕋,α,β∈ℝ, and  f,g∈Crd(𝕋), then (a)
∫ab[αf(t)+βg(t)]Δt=α∫abf(t)Δt+β∫abg(t)Δt(b)
iff(t)≥0 for all a≤t<b, then ∫abf(t)Δt≥0(c)
if|f(t)|≤g(t) on [a,b)∶={t∈𝕋:a≤t<b}, then |∫abf(t)Δt|≤∫abg(t)Δt.For convenience, we now introduce some notation to be used throughout this paper.Let(2.4)κ=min⁡{[0,∞)∩T},Iω=[κ,κ+ω]∩T,gu=sup⁡t∈Iωg(t),gl=inf⁡t∈Iωg(t),g¯=1ω∫Iωg(s)Δs=1ω∫κκ+ωg(s)Δs,
where g∈Crd(𝕋) is an ω-periodic real function, that is, g(t+ω)=g(t) for all t∈𝕋.Lemma 2.7 (see [14]).
Lett1,t2∈Iω and t∈𝕋.  If g:𝕋→ℝ is  ω- periodic, then
(2.5)g(t)≤g(t1)+∫κκ+ω|g(s)|Δs,g(t)≥g(t2)-∫κκ+ω|g(s)|Δs.Lemma 2.8 (see [15]).
Assume that{fn}n∈N is a function on J such that(i)
{fn}n∈N is uniformly bounded on J,(ii)
{fnΔ}n∈N is uniformly bounded on J.Then there is a subsequence of {fn}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), we first make the following preparations [13].LetX, 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 ImL is closed in Z. If L is a Fredholm mapping of index zero, there then exist continuous projectors P:X→X and Q:Z→Z such that ImP=KerL, ImL=KerQ=Im(I-Q). If we define LP:domL∩KerP→ImL as the restriction L|domL∩KerPof Lto domL∩KerP, 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(I-Q)N:Ω¯→X is compact, that is, continuous and such that KP(I-Q)N(Ω¯) is relatively compact. Since ImQ is isomorphic to KerL, there exists isomorphism J:ImQ→KerL.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).
LetL be a Fredholm mapping of index zero and let  N:Ω¯→Z be  L-compact on  Ω¯. Suppose (a)
Lx≠λNx for every x∈domL∩∂Ω and every λ∈(0,1),(b)
QNx≠0 for every x∈∂Ω∩KerL, and Brouwer degree(3.1)deg⁡B(JQN,Ω∩Ker⁡L,0)≠0.
Then Lx=Nx has at least one solution in domL∩Ω¯.In the following, we shall use the notation(3.2)αij=a¯jib¯i-a¯iib¯j,αij′=a¯jib¯i-a¯iib¯je(R¯j+r¯j)ω,αij′′=(a¯jib¯ie(R¯j+r¯j)ω-a¯iib¯j)e(R¯i+r¯i)ω,βij=a¯iia¯jj+b¯ir¯j-a¯ija¯ji-b¯jr¯i,βij′=a¯iia¯jje(R¯j+r¯j)ω+b¯ir¯j-a¯ija¯jie(R¯i+r¯i)ω-b¯jr¯ie(R¯i+r¯i+R¯j+r¯j)ω,βij′′=a¯iia¯jje(R¯i+r¯i)ω+b¯ir¯je(R¯i+r¯i+R¯j+r¯j)ω-a¯ija¯jie(R¯j+r¯j)ω-b¯jr¯i,βij*=a¯iia¯jj+b¯ir¯j-a¯ija¯jie(R¯i+r¯i)ω-b¯jr¯ie(R¯i+r¯i+R¯j+r¯j)ω,γij=r¯ia¯jj-r¯ja¯ij,γij′=(r¯ia¯jje(R¯j+r¯j)ω-r¯ja¯ij)e(R¯i+r¯i)ω,γij′′=r¯ia¯jj-r¯ja¯ije(R¯j+r¯j)ω,γij*=r¯ia¯jj-r¯ja¯ije(R¯i+r¯i+R¯j+r¯j)ω,i≠j,i,j=1,2,N1(α,β,γ)=β-β2-4αγ2α,N2(α,β,γ)=β+β2-4αγ2α(α≠0,β2-4αγ>0).We make the following assumptions.(H1)
R¯i=(1/ω)∫κκ+ω|ri(t)|Δt≥(1/ω)∫κκ+ωri(t)Δt>0.(H2)
γij*=r¯ia¯jj-r¯ja¯ije(R¯i+r¯i+R¯j+r¯j)ω>0,i≠j,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:(3.3)a¯11N1+a¯12N2+b¯1N1N2=r¯1,a¯21N1+a¯22N2+b¯2N1N2=r¯2.
Assuming that (H1), (H2) hold, then the following conclusions hold.(i)
Ifα12>0, then (3.3) have two positive solutions:
(3.4)(Ni(α12,β12,γ12),N1(α21,β21,γ21)),i=1,2.(ii)
Ifα21>0, then (3.3) have two positive solutions:
(3.5)(N1(α12,β12,γ12),Ni(α21,β21,γ21)),i=1,2.Lemma 3.3.
Assume that (H1)–(H3) hold, then the following conclusions hold.(i)
β12>0,β122-4α12γ12>0,(ii)
β12*>0,β12*2-4α12γ12′>0.Proof.
The proof of (i) is the same as (i) of Lemma  3.5 in [7]. We omit it.(ii)
We have(3.6)β12*=(b¯1a¯11+a¯12r¯1e(R¯2+r¯2)ω)γ21*+(r¯1α12e(R¯1+r¯1+R¯2+r¯2)ωa¯11+a¯11γ12′r¯1e(R¯1+r¯1+R¯2+r¯2)ω)>0,β12*2-4α12γ12′=(b¯1a¯11+a¯12r¯1e(R¯2+r¯2)ω)2γ21*2+(r¯1α12e(R¯1+r¯1)ωa¯11-a¯11γ12′r¯1e(R¯1+r¯1)ω)2+2(b¯1a¯11+a¯12r¯1e(R¯2+r¯2)ω)(r¯1α12e(R¯1+r¯1+R¯2+r¯2)ωa¯11+a¯11γ12′r¯1e(R¯1+r¯1+R¯2+r¯2)ω)γ21*>0.Lemma 3.4.
Assume that (H1)–(H3) hold, then the following conclusions hold. 							(3.7)N1(α12,β12+m,γ12-n)<N1(α12,β12,γ12)<N1(α12,β12*,γ12′)<N2(α12,β12*,γ12′)<N2(α12,β12,γ12)<N2(α12,β12+m,γ12-n),
where
(3.8)m=a¯11a¯22(e(R¯1+r¯1)ω-1)+b¯1r¯2(e(R¯1+r¯1+R¯2+r¯2)ω-1)>0,n=a¯12r¯2(e(R¯2+r¯2)ω-1)>0.Proof.
Under the conditions thatα>0, β>0, γ>0, β2-4αγ>0, we have
(3.9)N1(α,β,γ)=2γβ+β2-4αγ,N2(α,β,γ)=β+β2-4αγ2α.
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
(3.10)α12′′>α12>α12′>0,γ12′>γ12>γ12′′>γ12*>0,β12>β12*.
So from (3.9), (3.10), and (H1)–(H3), we obtain that
(3.11)N1(α12,β12+m,γ12-n)<N1(α12,β12,γ12)<N1(α12,β12*,γ12′)<N2(α12,β12*,γ12′)<N2(α12,β12,γ12)<N2(α12,β12+m,γ12-n).Theorem 3.5.
Assume that (H1)–(H3) hold. Then system (1.5) has at least two ω-periodic solutions.Proof.
Take(3.12)X=Z={x=(x1,x2)T:xi∈C(T,R2),xi(t+ω)=xi(t)∀t∈T,i=1,2},‖x‖=[∑i=12(max⁡t∈Iω|xi(t)|)2]1/2,x∈X(orZ).
It is easy to verify that X and Z are both Banach spaces.
Define the following mappingsL:X→Z,N:X→Z, P:X→X and Q:Z→Z as follows:
(3.13)Nx=(r1(t)-∑j=12a1j(t)exp⁡{xj(t)}-b1(t)exp⁡{x1(t)}exp⁡{x2(t)}r2(t)-∑j=12a2j(t)exp⁡{xj(t)}-b2(t)exp⁡{x2(t)}exp⁡{x1(t)}),Lx=(x1Δx2Δ),Px=1ω∫κκ+ωx(t)Δt=Qx,x∈X(orZ).
We first show thatL 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:(x1(t),x2(t))T=(h1,h2)T∈ℝ2fort∈𝕋}, Im L={x∈X:∫κκ+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
(3.14)Im⁡P=Ker⁡L,Ker⁡Q=Im⁡L=Im⁡(I-Q).
On the other hand, Kp: ImL→domL∩KerP, the inverse to L, exists and is given by
(3.15)Kp(x)=∫κtx(s)Δs-1ω∫κκ+ω∫κηx(s)ΔsΔη.
Obviously, QN and Kp(I-Q)N are continuous. By Lemma 2.8, it is not difficult to show that Kp(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X. Moreover, QN(Ω¯) is bounded. Hence, N is L-compact on Ω¯ for any open bounded set Ω⊂X.
Corresponding to the operator equationLx=λNx,λ∈(0,1), one has
(3.16)xiΔ(t)=λ[ri(t)-∑j=12aij(t)exp⁡{xj(t)}-bi(t)exp⁡{xi(t)}exp⁡{xk(t)}],
where i,k=1,2,k≠i. Suppose x(t)=(x1(t),x2(t))T∈X is a solution of system (3.16) for some λ∈(0,1). Integrating (3.16) over the interval [κ,κ+ω], we have
(3.17)r¯iω=∑j=12∫κκ+ωaij(t)exp⁡{xj(t)}Δt+∫κκ+ωbi(t)exp⁡{xi(t)}exp⁡{xk(t)}Δt,
where i,k=1,2,k≠i.
It follows from (3.16) and (3.17) that
(3.18)∫κκ+ω|xiΔ(t)|Δt=λ∫κκ+ω|ri(t)-∑j=12aij(t)exp⁡{xj(t)}-bi(t)exp⁡{xi(t)}exp⁡{xk(t)}|Δt<∫κκ+ω|ri(t)|Δt+∑j=12∫κκ+ωaij(t)exp⁡{xj(t)}Δt+∫κκ+ωbi(t)exp⁡{xi(t)}exp⁡{xk(t)}Δt=(R¯i+r¯i)ω,i=1,2.
That is
(3.19)∫κκ+ω|xiΔ(t)|Δt<(R¯i+r¯i)ω,i=1,2.
Since x(t)∈X, there exist ξi,ηi, such that
(3.20)xi(ξi)=min⁡t∈[κ,κ+ω]xi(t),xi(ηi)=max⁡t∈[κ,κ+ω]xi(t),i=1,2.
From (3.17), (3.20), one obtains
(3.21)a¯11exp⁡{x1(η1)}+a¯12exp⁡{x2(η2)}+b¯1exp⁡{x1(η1)}exp⁡{x2(η2)}≥r¯1,(3.22)a¯21exp⁡{x1(ξ1)}+a¯22exp⁡{x2(ξ2)}+b¯2exp⁡{x1(ξ1)}exp⁡{x2(ξ2)}≤r¯2.
We can derive from (3.22) that
(3.23)x2(η2)≤x2(ξ2)+∫κκ+ω|x2Δ(t)|Δt<ln⁡r¯2-a¯21exp⁡{x1(ξ1)}a¯22+b¯2exp⁡{x1(ξ1)}+(R¯2+r¯2)ω,
which, together with (3.21), leads to
(3.24)exp⁡{x1(η1)}≥r¯1-a¯12exp⁡{x2(η2)}a¯11+b¯1exp⁡{x2(η2)}≥r¯1(a¯22+b¯2exp⁡{x1(ξ1)})-a¯12exp⁡{(R¯2+r¯2)ω}(r¯2-a¯21exp⁡{x1(ξ1)})a¯11(a¯22+b¯2exp⁡{x1(ξ1)})+b¯1exp⁡{(R¯2+r¯2)ω}(r¯2-a¯21exp⁡{x1(ξ1)}).
From (3.19), we have
(3.25)x1(ξ1)≥x1(η1)-∫κκ+ω|x1Δ(t)|Δt>x1(η1)-(R¯1+r¯1)ω.
That is
(3.26)exp⁡{x1(ξ1)}>exp⁡{x1(η1)}⋅exp⁡(-(R¯1+r¯1)ω),
which, together with (3.24), leads to
(3.27)exp⁡{(R¯1+r¯1)ω}exp⁡{x1(ξ1)}>r¯1(a¯22+b¯2exp⁡{x1(ξ1)})-a¯12exp⁡{(R¯2+r¯2)ω}(r¯2-a¯21exp⁡{x1(ξ1)})a¯11(a¯22+b¯2exp⁡{x1(ξ1)})+b¯1exp⁡{(R¯2+r¯2)ω}(r¯2-a¯21exp⁡{x1(ξ1)}).
Therefore, we have
(3.28)α12′′exp⁡(2x1(ξ1))-β12′′exp⁡{x1(ξ1)}+γ12′′<0.
So from (3.10), one obtains
(3.29)α12exp⁡(2x1(ξ1))-(β12+m)exp⁡{x1(ξ1)}+γ12-n<0,
where
(3.30)m=a¯11a¯22(e(R¯1+r¯1)ω-1)+b¯1r¯2(e(R¯1+r¯1+R¯2+r¯2)ω-1)>0,n=a¯12r¯2(e(R¯2+r¯2)ω-1)>0.
According to (i) of Lemma 3.3, we obtain
(3.31)N1(α12,β12+m,γ12-n)<exp⁡{x1(ξ1)}<N2(α12,β12+m,γ12-n).
In a similar way as the above proof, one can conclude from(3.32)a¯21exp⁡{x1(η1)}+a¯22exp⁡{x2(η2)}+b¯2exp⁡{x1(η1)}exp⁡{x2(η2)}≥r¯2,a¯11exp⁡{x1(ξ1)}+a¯12exp⁡{x2(ξ2)}+b¯1exp⁡{x1(ξ1)}exp⁡{x2(ξ2)}≤r¯1,
that
(3.33)α12′exp⁡(2x1(η1))-β12′exp⁡{x1(η1)}+γ12′>0.
Noticing that α12>α12′, β12′>β12*, one has
(3.34)α12exp⁡(2x1(η1))-β12*exp⁡{x1(η1)}+γ12′>0.
According to (ii) of Lemma 3.3, one has
(3.35)exp⁡{x1(η1)}>N2(α12,β12*,γ12′),orexp⁡{x1(η1)}<N1(α12,β12*,γ12′).
It follows from (3.19) and (3.31) that
(3.36)x1(η1)≤x1(ξ1)+∫κκ+ω|x1Δ(t)|Δt<ln⁡N2(α12,β12+m,γ12-n)+(R¯1+r¯1)ω∶=H.
On the other hand, it follows from (3.17) and (3.20) that
(3.37)a¯iiωexp⁡(xi(ξi))≤∫κκ+ωaii(t)exp⁡{xi(t)}Δt<r¯iω,i=1,2;
that is
(3.38)xi(ξi)<ln⁡r¯ia¯ii,i=1,2.
From (3.19) and (3.38), one obtains
(3.39)xi(t)≤xi(ξi)+∫κκ+ω|xiΔ(t)|Δt<ln⁡r¯ia¯ii+(R¯i+r¯i)ω,i=1,2.
It follows from (3.17) and (3.20) that
(3.40)r¯2ω=∑j=12∫κκ+ωa2j(t)exp⁡{xj(t)}Δt+∫κκ+ωb2(t)exp⁡{x2(t)}exp⁡{x1(t)}Δt≤∑j=12a¯2jωexp⁡{xj(ηj)}+b¯2ωexp⁡{x1(η1)}exp⁡{x2(η2)},
which implies that
(3.41)exp⁡{x2(η2)}≥r¯2-a¯21exp⁡{x1(η1)}a¯22+b¯2exp⁡{x1(η1)}.
From (3.39) and (3.41), we have
(3.42)x2(η2)≥ln⁡a¯11r¯2-a¯21r¯1exp⁡{(R¯1+r¯1)ω}a¯11a¯22+b¯2r¯1exp⁡{(R¯1+r¯1)ω}∶=M,
which leads to
(3.43)x2(t)≥x2(η2)-∫κκ+ω|x2Δ(t)|Δt>M-(R¯2+r¯2)ω.
By (3.39) and (3.43), we obtain that
(3.44)|x2(t)|<max⁡{|ln⁡r¯2a¯22+(R¯2+r¯2)ω|,|M-(R¯2+r¯2)ω|}:=A.
Now, let us considerQNx with x=(x1,x2)T∈ℝ2. Note that
(3.45)QN(x1,x2)=(r¯1-a¯11exp⁡(x1)-a¯12exp⁡(x2)-b¯1exp⁡(x1)exp⁡(x2)r¯2-a¯21exp⁡(x1)-a¯22exp⁡(x2)-b¯2exp⁡(x1)exp⁡(x2)).
According to Lemma 3.2, we can show that QNx=0 has two distinct solutions
(3.46)x̂i=(ln⁡Ni(α12,β12,γ12),ln⁡N1(α21,β21,γ21)),i=1,2.
Choose C>0 such that
(3.47)C>|ln⁡N1(α21,β21,γ21)|.
Let
(3.48)Ω1={x∈X|x1(t)∈(ln⁡N1(α12,β12+m,γ12-n),ln⁡N1(α12,β12*,γ12′)),|x2(t)|<A+C.},Ω2={x∈X|min⁡t∈Iωx1(t)∈(ln⁡N1(α12,β12+m,γ12-n),ln⁡N2(α12,β12+m,γ12-n)),max⁡t∈Iωx1(t)∈(ln⁡N2(α12,β12*,γ12′),H),|x2(t)|<A+C.},
Then both Ω1 and Ω2 are bounded open subsets of X. It follows from Lemma 3.2, Lemma 3.4, and (3.47) that x̂i∈Ωi, 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 Ω¯1∩Ω¯2=ϕ and Ωi satisfies the requirement (a) in Lemma 3.1 for i=1,2. Moreover, QNx≠0 for x∈∂Ωi∩KerL(i=1,2). A direct computation gives
(3.49)deg⁡B(JQN,Ωi∩Ker⁡L,0)≠0.
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 x̆i with x̆i∈DomL∩Ω¯i(i=1,2). Obviously x̆i(i=1,2) are different. The proof is complete.Example 3.6.
As an application of Theorem3.5, we consider the following system:
(3.50)x1Δ(t)=0.02+0.002cos⁡(0.4πt)-(1+0.001cos⁡(0.4πt))exp⁡{x1(t)}-(0.1+0.0137cos⁡(0.4πt))exp⁡{x2(t)}-(2+0.95cos⁡(0.4πt))exp⁡{x1(t)}exp⁡{x2(t)},x2Δ(t)=0.041+0.04cos⁡(0.4πt)-(1+0.038cos⁡(0.4πt))exp⁡{x1(t)}-(1+0.0379cos⁡(0.4πt))exp⁡{x2(t)}-(1+0.039cos⁡(0.4πt))exp⁡{x2(t)}exp⁡{x1(t)}.
If 𝕋=ℝ, then (3.50) reduces to the following system:
(3.51)dx1(t)dt=0.02+0.002cos⁡(0.4πt)-(1+0.001cos⁡(0.4πt))exp⁡{x1(t)}-(0.1+0.0137cos⁡(0.4πt))exp⁡{x2(t)}-(2+0.95cos⁡(0.4πt))exp⁡{x1(t)}exp⁡{x2(t)},dx2(t)dt=0.041+0.04cos⁡(0.4πt)-(1+0.038cos⁡(0.4πt))exp⁡{x1(t)}-(1+0.0379cos⁡(0.4πt))exp⁡{x2(t)}-(1+0.039cos⁡(0.4πt))exp⁡{x2(t)}exp⁡{x1(t)}.
A direct computation gives that
(3.52)R¯1=r¯1=0.02,R¯2=r¯2=0.041,a¯11=a¯22=1,a¯12=0.1,a¯21=1,b¯1=2,b¯2=1,ω=5,α12=a¯21b¯1-a¯11b¯2=1,γ12*=r¯1a¯22-r¯2a¯12e(R¯1+r¯1+R¯2+r¯2)ω=0.02-0.0041e0.61>0,γ21*=r¯2a¯11-r¯1a¯21e(R¯2+r¯2+R¯1+r¯1)ω=0.041-0.02e0.61>0.
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:
(3.53)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)].
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.Blue lines stand forx1*(t), x2*(t), red lines stand for x1**(t),x2**(t).
(a)(b)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.



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*Source: 602679-2012-04-30.xml*