Multiple periodic solutions for a discrete time model of plankton allelopathy

MULTIPLE PERIODIC SOLUTIONS FOR A DISCRETE TIME

MODEL OF PLANKTON ALLELOPATHY

JIANBAO ZHANG AND HUI FANG

Received 19 May 2005; Revised 25 September 2005; Accepted 27 September 2005

We study a discrete time model of the growth of two species of plankton with compet-itive and allelopathic effects on each otherN1(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)}. A set of sufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin’s continuation theorem of coincidence degree theory as well as some a priori estimates. Some new results are obtained.

Copyright © 2006 J. Zhang and H. 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.

1. Introduction

Many researchers have noted that the increased population of one species of phytoplank-ton might affect the growth of one or several other species by the production of allelo-pathic toxins or stimulators, influencing bloom, pulses, and seasonal succession. The study of allelopathic interactions in the phytoplanktonic world has become an impor-tant subject in aquatic ecology. For detailed studies, we refer to [1,2,7,9–11,13] and references cited therein.

Maynard-Smith [9] and Chattopadhyay [2] proposed the following two species Lotka-Volterra competition system, which describes the changes of size and density of phyto-plankton:

dN1 dt =N1

r1−a11N1(t)−a12N2(t)−b1N1(t)N2(t), dN2

dt =N1

r2−a21N1(t)−a22N2(t)−b2N1(t)N2(t)

,

(1.1)

whereb1andb2are the rates of toxic inhibition of the first species by the second and vice versa, respectively.

Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 90479, Pages1–14

Naturally, more realistic models require the inclusion of the periodic changing of envi-ronment (e.g., seasonal effects of weather, food supplies, etc). For such systems, as pointed out by Freedman and Wu [5] and Kuang [8], it would be of interest to study the existence of periodic solutions. This motivates us to modify system (1.1) to the form

dN1

dt =N1(t)

r1(t)−a11(t)N1(t)−a12(t)N2(t)−b1(t)N1(t)N2(t)

,

dN2

dt =N1(t)

r2(t)−a21(t)N1(t)−a22(t)N2(t)−b2(t)N1(t)N2(t),

(1.2)

whereri(t),ai j(t)>0,bi(t)>0 (i,j=1, 2) are continuousω-periodic functions.

The main purpose of this paper is to propose a discrete analogue of system (1.2) and to obtain sufficient conditions for the existence of multiple positive periodic solutions by employing the coincidence degree theory. To our knowledge, no work has been done for the existence of multiple positive periodic solutions for this model using this way.

The paper is organized as follows. InSection 2, we propose a discrete analogue of sys-tem (1.2). InSection 3, motivated by the recent work of Fan and Wang [4] and Chen [3], we study the existence of multiple positive periodic solutions of the difference equations derived inSection 2.

2. Discrete analogue of system (1.2)

Assume that the average growth rates in (1.2) change at equally spaced time intervals and estimates of the population size are made at equally spaced time intervals, then we can incorporate this aspect in (1.2) and obtain the following system:

dN1(t) dt

1 N1(t)=r1

[t]−a11[t]N1[t]−a12[t]N2[t]−b1[t]N1[t]N2[t],

dN2(t) dt

1 N2(t)=

r2

[t]−a21

[t]N1

[t]−a22

[t]N2

[t]−b2

[t]N1

[t]N2

[t], (2.1)

wheret=0, 1, 2,. . ., [t] denotes the integer part oft,t∈(0, +∞).

By a solution of (2.1), we mean a functionx=(x1,x2)T, which is defined for t∈ [0, +∞), and possesses the following properties.

(1)xis continuous on [0, +∞).

(2) The derivativesdx1(t)/dt,dx2(t)/dtexist at each pointt∈[0, +∞) with the pos-sible exception of the pointst∈ {0, 1, 2,. . .}, where left-sided derivatives exist. The equations in (2.1) are satisfied on each interval [k,k+ 1) withk=0, 1, 2,. . . . Fork≤t < k+ 1,k=0, 1, 2,. . ., integrating (2.1) fromktot, we obtain

N1(t)=N1(k) exp

r1(k)−a11(k)N1(k)−a12(k)N2(k)−b1(k)N1(k)N2(k)

(t−k), N2(t)=N2(k) exp

r2(k)−a21(k)N1(k)−a22(k)N2(k)−b2(k)N1(k)N2(k)

Lettingt→k+ 1, we have

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) expr2(k)−a21(k)N1(k)−a22(k)N2(k)−b2(k)N1(k)N2(k), (2.3)

fork=0, 1, 2,. . . .Equation (2.3) is a discrete analogue of system (1.2). Notice that the periodicity of parameters of (2.1) is sufficient, but not necessary for the periodicity of coefficients in (2.3).

In system (2.3), we always assume thatri,ai j>0,bi>0 (i,j=1, 2) areω-periodic, that

is,

ri(k+ω)=ri(k), bi(k+ω)=bi(k), ai j(k+ω)=ai j(k), (2.4)

for anyk∈Z(the set of all integers),i,j=1, 2, whereω, a fixed positive integer, denotes the prescribed common period of the parameters in (2.3).

3. Existence of multiple positive periodic solutions

In this section, in order to obtain the existence of multiple positive periodic solutions of (2.3), we first make the following preparations.

LetX andY be normed vector spaces. LetL: DomL⊂X→Y be a linear mapping and N:X→Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim kerL=codim ImL <∞ and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors P:X→X andQ:Y→Y such that ImP=kerLand ImL=kerQ=Im(I−Q). It follows thatL| DomL∩kerP: (I−P)X→ImLis invertible and its inverse is denoted byKp. IfΩ is

a bounded open subset ofX, the mapping N is calledL-compact onΩif (QN)(Ω) is bounded andKp(I−Q)N:Ω→Xis compact. Because ImQis isomorphic to kerL, there

exists an isomorphismJ: ImQ→kerL.

For convenience, we introduce Mawhin’s continuation theorem as follows.

Lemma3.1 [6, page 40] (Continuation theorem). LetLbe a Fredholm mapping of index zero and letN: ¯Ω→ZbeL-compact onΩ¯. Suppose

(a)Lx=λNxfor everyx∈domL∩∂Ωand everyλ∈(0, 1); (b)QNx=0for everyx∈∂Ω∩KerL, and Brouwer degree

deg_BJQN,Ω∩KerL, 0)=0. (3.1)

ThenLx=Nxhas at least one solution indomL∩_Ω¯_.

Suppose{g(k)}is anω-periodic (ω∈Z+_{) sequence of real numbers defined fork∈Z.} Throughout this paper, we will use the following notation:

Iω=

0, 1,. . .,ω−1, g=_ω1

ω−1

k=0

g(k), R¯i=_ω1 ω−1

k=0

_{ri(k) ,}

αi j=a¯jib¯i−a¯iib¯j, αi j=a¯jib¯i−a¯iib¯jeR¯jω, α i j=

¯

ajib¯ieR¯jω−a¯iib¯jeR¯iω,

βi j=a¯iia¯j j+ ¯bir¯j−a¯i ja¯ji−b¯jr¯i,

β_{i j}=a¯iia¯j jeR¯jω+ ¯bir¯j−a¯i ja¯jieR¯iω−b¯jr¯ie( ¯Ri+ ¯Rj)ω,

β _{i j}=a¯iia¯j jeR¯iω+ ¯bir¯je( ¯Ri+ ¯Rj)ω−a¯i ja¯jieR¯jω−b¯jr¯i,

γi j=r¯ia¯j j−r¯ja¯i j, γi j=

¯

ria¯j jeR¯jω−r¯ja¯i jeR¯iω,

γ_{i j} =r¯ia¯j j−r¯ja¯i jeR¯jω, i,j=1, 2,i=j,

N1(α,β,γ)=

β−β2_−4αγ

2α , N2(α,β,γ)=

β+β2_−4αγ

2α (α=0,β

2_{−4αγ >0.} (3.2)

Define

l2=

x=x(k):x(k)∈R2_{,k∈Z. (3.3)}

Fora=(a1,a2)T ∈R2, define|a| =max{|a1|,|a2|}. Letlω⊂l2 denote the subspace of allω-periodic sequences equipped with the usual supremum norm · , that is, forx= {x(k) :k∈Z} ∈lω_{,x =max}

k∈Iω|x(k)|. It is not difficult to show that lω is a

finite-dimensional Banach space.

Let the linear operatorS:lω_→R2_{be defined by}

S(x)= 1 ω

ω−1

k=0

x(k), x=x(k) :k∈Z∈lω_{. (3.4)}

Then we obtain two subspaceslω0 andlωc oflωdefined by

lω

0 =

x=x(k)∈lω_:S(x)=0,

lω c =

x=x(k)∈lω_{:x(k)≡β, for someβ∈R}2_and∀k∈Z, (3.5)

respectively. Denote byL:lω_→lω_{the difference operator given byLx= {(Lx)(k)}with}

(Lx)(k)=x(k+ 1)−x(k), forx∈lω_{andk∈Z. (3.6)}

Let a linear operatorK:lω_→lω

c be defined byKx= {(Kx)(k)}with

(Kx)(k)≡S(x), forx∈lω_{andk∈Z. (3.7)}

Lemma3.2 [12, Lemma 2.1]. (i)Bothlω0 andlωc are closed linear subspaces oflωandlω=

lω0⊕lcω,dimlωc =2.

(ii)Lis a bounded linear operator withkerL=lω

c andImL=l0ω.

(iii)Kis a bounded linear operator withker(L+K)= {0}andIm(L+K)=lω_.

Lemma3.3. Letg,r:Z→Rbeω-periodic, that is,g(k+ω)=g(k),r(k+ω)=r(k). Assume that for anyk∈Z,

g(k+ 1)−g(k)≤ r(k) . (3.8)

Then for any fixedk1,k2∈Iω, and anyk∈Z, one has

g(k)≤gk1

+

ω−1

s=0

_{r(s) ,}

g(k)≥gk2

−

ω−1

s=0

_{r(s) .} (3.9)

Proof. It is only necessary to prove that the inequalities hold for anyk∈Iω. For the first

inequality, it is easy to see the first inequality holds ifk=k1. Ifk > k1, then

g(k)−gk1

=

k−1

s=k1

g(s+ 1)−g(s)≤

k−1

s=k1

_{r(s) ≤}ω−1

s=0

_{r(s) , (3.10)}

and hence,g(k)≤g(k1) +

ω−1

s=0 |r(s)|. Ifk < k1, thenk+ω > k1. Therefore,

g(k)−gk1

=g(k+ω)−gk1

=

k+ω−1

s=k1

g(s+ 1)−g(s)

≤

k+ω−1

s=k1

_{r(s) ≤}k1+ω−1

s=k1

_{r(s) =}ω−1

s=0

_{r(s) ,} (3.11)

equivalently, g(k)≤g(k1) +_sω₌−01|r(s)|. Now we can claim that the first inequality is

valid.

Similar to the above proof, we can prove that the second inequality is valid. In the following, we make the following assumptions.

(H1) ¯Ri=(1/ω)

ω−1

k=0|ri(k)| ≥(1/ω)

ω−1

k=0ri(k)>0.

(H2)γ i j=r¯ia¯j j−r¯ja¯i jeR¯jω>0,i=j,i,j=1, 2.

(H3)α12>0.

(H4)β12/α12> β12/α12.

Lemma3.4 [13, Lemma 3.2]. Consider the following algebraic equations: ¯

a11N1+ ¯a12N2+ ¯b1N1N2=r¯1, ¯

a21N1+ ¯a22N2+ ¯b2N1N2=r¯2.

(3.12)

(i)Ifα12>0, then (3.12) have two positive solutions:

Niα12,β12,γ12,N1α21,β21,γ21, i=1, 2. (3.13)

(ii)Ifα21>0, then (3.12) have two positive solutions:

Lemma3.5. Assume that(H1)–(H3)hold, then the following conclusions hold. (i)β12>0,β122 −4α12γ12>0;

ThusN1(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

α 12> α12> α12>0, γ12> γ12> γ 12>0, (3.20)

we have

γ12 α12

>γ12

α12. (3.21)

So from (3.19), (3.20), (3.21) and (H1)–(H4), we obtain that

N1

α12,β12+m,γ12−n

< N1

α12,β12,γ12

=N1

1,β12 α12,

γ12 α12

< N1

1,β12 α12

,γ12 α12

=N1

α12,β12,γ12

< N2

α12,β12,γ12

=N2

1,β12 α12

,γ12 α12

< N2

1,β12 α12,

γ12 α12

=N2

α12,β12,γ12

< N2

α12,β12+m,γ12−n

.

(3.22)

Theorem3.7. In addition to(H1)–(H3), assume further that system (2.3) satisfies (H5)N1(α12,β12,γ12)< N1(α12,β12,γ12)< N2(α12,β12,γ12).

Then system (2.3) has at least two positiveω-periodic solutions.

Proof. Since we are concerned with positive solutions of (2.3), we make the change of variables,

Ni(k)=exp

xi(k)

, i=1, 2. (3.23)

Then (2.3) is rewritten as

xi(k+ 1)−xi(k)=ri(k)−aii(k) expxi(k)−ai j(k) expxj(k)

−bi(k) exp

xi(k)

expxj(k)

, (3.24)

wherei,j=1, 2,i=j. TakeX=Y=lω_{, (Lx)(k)=x(k+ 1)−x(k), and denote}

(ᏺx)(k)=

r1(k)−a11(k) expx1(k)−a12(k) expx2(k)−b1(k) expx1(k)expx2(k) r2(k)−a22(k) exp

x2(k)

−a21(k) exp

x1(k)

−b2(k) exp

x2(k)

expx1(k)

,

(3.25)

for anyx∈Xandk∈Z. It follows fromLemma 3.2thatLis a bounded linear operator and

kerL=lωc, ImL=lω0, dim kerL=2=codim ImL, (3.26)

Define

Px= 1 ω

ω−1

s=0

x(s), x∈X, Qy= 1 ω

ω−1

s=0

y(s), y∈Y. (3.27)

It is not difficult to show thatPandQare two continuous projectors such that

ImP=kerL, ImL=kerQ=Im(I−Q). (3.28)

Furthermore, the generalized inverse (ofL)Kp: ImL→kerP∩DomLexists and is given

by

Kp(z)= k−1

s=0

z(s)−_ω1

ω−1

s=0

(ω−s)z(s). (3.29)

Notice thatQᏺ,Kp(I−Q)ᏺare continuous andXis a finite-dimensional Banach space,

it is not difficult to show thatKp(I−Q)ᏺ(Ω) is compact for any open bounded setΩ⊂X.

Moreover, Qᏺ(Ω) is bounded. Thus, ᏺ isL-compact on with any open bounded set Ω⊂X.

Corresponding to the operator equationLx=λᏺx,λ∈(0, 1), we have

xi(k+ 1)−xi(k)=λ

ri(k)−aii(k) exp

xi(k)

−ai j(k) exp

xj(k)

−bi(k) exp

xi(k)

expxj(k)

, (3.30)

wherei,j=1, 2,i=j. Suppose thatx=(x1(k),x2(k))T ∈Xis a solution of (3.30) for a certainλ∈(0, 1). Summing on both sides of (3.30) from 0 toω−1 aboutk, we get

0=

ω−1

k=0

xi(k+ 1)−xi(k)

=λ

ω−1

k=0

ri(k)−aii(k) exp

xi(k)

−ai j(k) exp

xj(k)

−bi(k) exp

xi(k)

expxj(k)

,

(3.31)

that is

¯ riω=

ω−1

k=0

aii(k) exp

xi(k)

+ai j(k) exp

xj(k)

+bi(k) exp

xi(k)

expxj(k)

, (3.32)

wherei,j=1, 2,i=j. It follows from (3.30) that

xi(k+ 1)−xi(k)< ri(k) , k∈Z,i=1, 2. (3.33)

Sincex(t)∈X, there existξi,ηi∈Iωsuch that

xi

ξi

=min

k∈Iω

xi(k)

, xi

ηi

=max

k∈Iω

xi(k)

From (3.32), (3.34), one obtains

We can derive fromLemma 3.3, (3.33) and (3.36) that

x2η2≤x2ξ2+ ¯R2ω≤lnr¯2−a¯21exp

which, together with (3.35), leads to

expx1

FromLemma 3.3and (3.33), we have

x1

which, together with (3.38), leads to

expR¯1ω

So from (3.20), one obtains

α12exp2x1ξ1−β12+mexpx1ξ1+γ12−n <0, (3.43)

According to (i) ofLemma 3.5, we obtain

Therefore, the equation

α12x2−

β12+m

x+γ12−n=0 (3.46)

has two positive solutions

Ni

α12,β12+m,γ12−n

, i=1, 2. (3.47)

Thus, we have

N1

α12,β12+m,γ12−n

<expx1

ξ1

< N2

α12,β12+m,γ12−n

. (3.48)

In a similar way as the above proof, we can conclude from

¯ a21exp

x1

η1

+ ¯a22exp

x2

η2

+ ¯b2exp

x1

η1

expx2

η2

≥r¯2, ¯

a11expx1ξ1+ ¯a12expx2ξ2+ ¯b1expx1ξ1expx2ξ2≤r¯1,

(3.49)

that

α12exp

2x1

η1

−β12exp

x1

η1

+γ12>0. (3.50)

According to (ii) ofLemma 3.5, one has

β122−4α12γ12>0. (3.51)

Therefore, the equation

α12x2−β12x+γ12=0 (3.52)

has two positive solutions

Ni

α12,β12,γ12

, i=1, 2. (3.53)

Thus, we have

expx1η1> N2α12,β12,γ12

, or expx1η1< N1α12,β12,γ12

. (3.54)

It follows fromLemma 3.3, (3.33) and (3.48) that

x1

η1

≤x1

ξ1

+ ¯R1ω <lnN2

α12,β12+m,γ12−n

+ ¯R1ω:=H.

(3.55)

On the other hand, it follows from (3.32) and (3.34) that

¯ aiiωexp

xi

ξi

≤

ω−1

k=0

aii(k) exp

xi(k)

that is

xiξi<ln r¯i

¯

aii, i=1, 2. (3.57)

FromLemma 3.3, (3.33) and (3.57), one obtains

xi(k)≤xi

It follows from (3.32) and (3.34) that

¯

which implies that

expx2

From (3.58) and (3.60), we have

x2

which, together withLemma 3.3, leads to

x2(k)≥x2

η2

−_R¯_{2ω > M−R}¯_{2ω. (3.62)}

By (3.58) and (3.62), we obtain that

_{x2(k) <max ln} r¯2

According toLemma 3.4, we can show thatQᏺx=0 has two distinct solutions

ChooseC >0 such that

C > lnN1

Let

Ω1=

⎧ ⎨ ⎩x∈X

x1(k)∈

lnN1α12,β12+m,γ12−n, lnN1α12,β12,γ12

,

_{x2(k) < A+C.}

⎫ ⎬ ⎭,

Ω2=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

x∈X

mink∈Iωx1(k)∈(lnN1

α12,β12+m,γ12−n, lnN2α12,β12+m,γ12−n, maxk∈Iωx1(k)∈

minlnN2

α12,β12,γ12

, lnN2

α12,β12,γ12

−δ,H,

_{x2(k) < A+C.}

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

,

(3.67)

whereδis a constant such that

minlnN2

α12,β12,γ12

, lnN2

α12,β12,γ12

−lnN1

α12,β12,γ12

> δ >0. (3.68)

Then bothΩ1 andΩ2 are bounded open subsets ofX. It follows fromLemma 3.4and (3.66) thatxi_∈Ω

i,i=1, 2. With the help of (3.48), (3.54), (3.55), (3.63) and (H5), it is easy to see that ¯Ω1Ω2¯ =φandΩisatisfies the requirement (a) inLemma 3.1fori=1, 2.

Moreover,Qᏺx=0 forx∈∂ΩiKerL,i=1, 2. A direct computation gives

deg_BJQᏺ,Ωi∩KerL, 0=0. (3.69)

HereJis taken as the identity mapping since ImQ=KerL. So far we have proved thatΩi

satisfies all the assumptions inLemma 3.1. Hence (3.24) has at least twoω-periodic solu-tions ˘xi_{with ˘x}i_∈DomLΩ¯

i(i=1, 2). Obviously ˘xi(i=1, 2) are different. Let ˘Nij(k)=

exp( ˘xi_j(k)),i,j=1, 2. Then ˘Ni_{=( ˘N}i

1, ˘N2i) (i=1, 2) are two different positiveω-periodic

solutions of (2.3). The proof is complete.

With the help ofLemma 3.6andTheorem 3.7, we have the following.

Corollary3.8. Under Assumptions(H1)–(H4), system (2.3) has at least two positiveω -periodic solutions.

Example 3.9. As an application ofCorollary 3.8, we consider the following system

N1(k+ 1)=N1(k) exp

0.0002 + 0.0002 cosπk/50

−0.0001 + 0.00005 cosπk/50N1(k) −1000 + cosπk/50N2(k)

−20000 + cosπk/50N1(k)N2(k)

, N2(k+ 1)=N1(k) exp

0.00041 + 0.00041 cosπk/50

−0.0002 + 0.0001 cosπk/50N1(k) −10000 + cosπk/50N2(k) −20000 + cosπk/50N1(k)N2(k)

.

A direct computation gives that

¯

r1=0.0002=R¯1, r¯2=0.00041=R¯2, a¯11=0.0001, a¯12=1000, ¯

a21=0.0002, a¯22=10000, b¯1=b¯2=20000, ω=100, α12>1.91629, γ 12>1.57284, γ 21>1.9194×10−10,

α12β12−α12β12>0.00904.

(3.71)

So according to Corollary 3.8, the above system has at least two positive 100-periodic solutions.

Similar to the proof ofTheorem 3.7, we can prove the following results.

Theorem3.10. In addition to(H1)and(H2), assume further that system (2.3) satisfies (H3)α21>0.

(H5)N1(α21,β21,γ21)< N1(α21,β21,γ21)< N2(α21,β21,γ21). Then system (2.3) has at least two positiveω-periodic solutions.

Corollary3.11. In addition to(H1),(H2)and(H3), assume further that system (2.3) satisfies

(H4)β21/α21> β21/α21.

Then system (2.3) has at least two positiveω-periodic solutions.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (No. 10161007, 10561004).

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Jianbao Zhang: Center for Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

E-mail address:[email protected]

Hui Fang: Center for Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, China