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Journal of Computational andApplied Mathematics 206 (2007) 733–754

www.elsevier.com/locate/cam

Dynamic behaviors of a delay differential equation model of

plankton allelopathy

Fengde Chena,∗, Zhong Lia, Xiaoxing Chena, Jitka Laitochováb

aCollege of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China

bDepartment of Mathematics, Faculty of Education, Palacký University, Olomouc, Czech Republic

Received 5 February 2005; received in revised form 26 July 2006

Abstract

In this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive

and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system.

As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive

periodicsolutionofthesystem.Afterthat,bymeansofasuitableLyapunovfunctional,sufficientconditionsarederivedfortheglobal

attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components

will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility

of the main results.

© 2006 Elsevier B.V.All rights reserved.

MSC: 34C25; 92D25; 34D20; 34D40

Keywords: Competition; Toxicology; Lyapunov functional; Global attractivity; Permanence; Extinction

1. Introduction

Traditional Lotka–Volterra competitive system can be expressed as follows:

⎡

j=1

Themodelhasbeenstudiedextensively.Manyexcellentresultsconcernedwithpermanence,extinctionandtheexistence

of a globally attractive positive periodic solution (positive almost periodic solution) of system (1.1) are obtained (see

[1,2,5–7,9–11,13–23,26,28,33,35–42] and the references cited therein).

Ontheotherhand,aswaspointedoutbyChattopadhyay[4],theeffectsoftoxicsubstancesonecologicalcommunities

is an important problem from an environmental point of view. In [4], he had proposed the following two species

˙ xi(t) = xi(t)

⎣bi(t) −

n

?

aij(t)xj(t)

⎤

⎦,i = 1,2,...,n. (1.1)

∗Corresponding author. Tel./fax: +865913774501.

E-mail addresses: fdchen@fzu.edu.cn (F. Chen), lizhong04108@163.com (Z. Li), cxxing79@163.com (X. Chen), jitka.laitochova@upol.cz

(J. Laitochová).

0377-0427/$-see front matter © 2006 Elsevier B.V.All rights reserved.

doi:10.1016/j.cam.2006.08.020

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F. Chen et al. / Journal of Computational and Applied Mathematics 206 (2007) 733–754

competitive system:

˙ x1(t) = x1(t)[K1− ?1x1(t) − ?12x2(t) − ?1x1(t)x2(t)],

˙ x2(t) = x2(t)[K2− ?2x2(t) − ?21x2(t) − ?2x1(t)x2(t)],

wherex1(t)andx2(t)denotethepopulationdensityoftwocompetingspeciesattimetforacommonpoolofresources.

Theterms?1x1(t)x2(t)and?2x1(t)x2(t)denotetheeffectoftoxicsubstances.HereChattopadhyaymadetheassumption

that each species produces a substance toxic to the other, only when the other is present.

Noticing that the production of the toxic substance allelopathic to the competing species will not be instantaneous,

but delayed by different discrete time lags required for the maturity of both species, thus, Mukhopadhyay et al. [32]

modified system (1.2) to the following system:

(1.2)

˙ x1(t) = x1(t)[K1− ?1x1(t) − ?12x2(t) − ?1x1(t)x2(t − ?2)],

˙ x2(t) = x2(t)[K2− ?2x2(t) − ?21x2(t) − ?2x1(t − ?1)x2(t)],

where ?i>0, i = 1,2 are the time required for the maturity of the first species and second species, respectively.

Recently, Jin and Ma [24] argued that the environmental fluctuation is important in an ecosystem, and more realistic

models require the inclusion of the effect of environmental changes, especially environmental parameters which are

time-dependent and periodically changing (e.g., seasonal changes, food supplies, etc.). They also thought that the

distributed delay is more realistic, and proposed the following two-species competition model:

⎡

j=1

⎡

j=1

(1.3)

˙ x1(t) = x1(t)

⎣r1(t) −

2

?

a1j(t)

?0

−T1j

K1j(s)xj(t + s)ds − b1(t)x1(t)

?0

−?2

f2(s)x2(t + s)ds

⎤

⎦,

⎤

˙ x2(t) = x2(t)

⎣r2(t) −

2

?

a2j(t)

?0

−T2j

K1j(s)xj(t + s)ds − b2(t)x2(t)

?0

−?1

f1(s)x1(t + s)ds

⎦,(1.4)

where ri(t),aij(t)>0,bi(t)>0 (i,j =1,2) are continuous ?-periodic functions, Tij,?iare positive constants, Kij∈

C([−Tij,0],(0,+∞)) and?0

periodic solutions of system (1.4) are obtained. For additional works related to this topic, see [27,29–31,34].

Here, as far as system (1.4) is concerned, several issues are proposed:

(1) Is it possible to obtain a set of sufficient conditions which guarantee the permanence of the system?

(2) Is it possible to obtain a set of sufficient conditions which guarantee the global attractivity of the positive solution

of system (1.4)?

(3) As far as the two species Lotka–Volterra competitive system is concerned, the principle of exclusion is well known

(see [1]). However, seldom did scholars consider the final extinction of some of the species in the nonautonomous

system (1.2) (we call such a case the partial extinction, see [27,30]). To this day, to the best of our knowledge,

no scholar has investigated the partial extinction of system (1.4). Is it possible for us to obtain a set of sufficient

conditions which ensure one of the components in system (1.4) will be driven to extinction?

The aim of this paper is to give an affirmative answer to the above three issues. Also, since few things in nature

are truly periodic, unlike the consideration of [24], we feel that it is natural to consider the general nonautonomous,

nonperiodic system (1.4). Here we propose the following nonautonomous n-species competition system:

⎡

j=1

−TijKij(s)ds = 1,fi ∈ C([−?i,0],(0,+∞)) and?0

−?ifi(s)ds = 1 (i,j = 1,2). By

applying the coincidence degree theory, sufficient conditions which guarantee the existence of at least one positive

˙ xi(t) = xi(t)

⎣ri(t) −

n

?

aij(t)

?0

−Tij

Kij(s)xj(t + s)ds −

n

?

j=1,j?=i

bij(t)xi(t)

?0

−?ij

fij(s)xj(t + s)ds

⎤

⎦,

(1.5)

where i = 1,2,...,n and xi(t) (1?i?n) is the density of ith species at time t.

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F. Chen et al. / Journal of Computational and Applied Mathematics 206 (2007) 733–754

735

Throughout this paper, it is assumed that:

(H1) ri(t),aij(t),bij(t) (i ?= j), i,j=1,2,...,narecontinuousandboundedaboveandbelowbypositiveconstants

on [0,+∞);

(H2) Tij,?ijarepositiveconstants,Kij∈ C([−Tij,0],(0,+∞))and?0

We consider (1.5) together with the initial conditions

−TijKij(s)ds=1,fij∈ C([−?ij,0],(0,+∞))

(i ?= j) and?0

xi(?) = ?i(?)?0, ? ∈ [−?,0],

where ? = maxi,j{Tij,?ij}, ?iare continuous on [−?,0]. It is not difficult to see that solutions of (1.5)–(1.6) are well

defined for all t?0 and satisfy

−?ijfij(s)ds = 1 (i,j = 1,2,...,n).

?i(0)>0, (1.6)

xi(t)>0 for t?0, i = 1,2,...,n.

Throughout, we shall use the following notations:

• gl= min

where g is a continuous bounded function defined on [0,+∞);

• Ji= {1,...,i − 1,i + 1,...,n}.

t ?0g(t),gu= max

t ?0g(t),

We say a positive solution of system (1.5) is globally attractive if it attracts all other positive solution of the system.

Theorganizationofthispaperisasfollows:inthenextsection,byusingthedifferentialinequalitytheorem,sufficient

conditions are obtained for the permanence of system (1.5). As a corollary, for the periodic case, we obtain a set of

delay-dependent conditions which ensure the existence of positive periodic solutions of system (1.5). In Section 3, by

constructing a suitable Lyapunov functional, a set of easily verified sufficient conditions are obtained for the global

attractivity of positive solutions of system (1.5) with the initial conditions (1.6). In Section 4, we consider a two-

dimensional case of system (1.5), by further developing the analysis and technique of Li and Chen [27] and Montes

De Oca and Vivas [31], we obtain a set of sufficient conditions which ensure that one of the components will be

driven to extinction. We also compare our results with some previously known results. Some interesting relationship

among the results of [27,30] and our paper are discovered. In Section 5, some suitable examples are presented, which

show the feasibility of main results. For the works on the general nonautonomous ecosystem and the construction of

Lyapunov functional, one could refer to [12,8,25,39] and the references cited therein. For the works concerned with

partial extinction of the species in the ecosystem, one could refer to [1,27,30,31,36].

2. Permanence

This section is concerned with permanence of system (1.5).

Lemma 2.1. If a >0,b>0 and ˙ x(t)?(?)x(t)(b − ax(t)), x(t0)>0, we have

x(t)?b

a

lim sup

t→+∞

?

lim inf

t→+∞x(t)?b

a

?

.

Proof. From [8, Lemma 2.2], it follows that

?

Letting t → +∞ in the above inequality, it follows that

x(t)?b

a

x(t)?(?)b

a

1 +

?

b

ax(t0)− 1

?

e−b(t−t0)

?−1

. (2.1)

lim sup

t→+∞

?

lim inf

t→+∞x(t)?b

a

?

.

?

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F. Chen et al. / Journal of Computational and Applied Mathematics 206 (2007) 733–754

Theorem 2.1. Assume (H1)–(H2) hold, let x(t) = (x1(t),...,xn(t))Tbe any solution of system (1.5)–(1.6). Then

lim sup

t→+∞

where

Mi=ru

al

ii

xi(t)?Mi,i = 1,2,...,n, (2.2)

i

exp{ru

iTii},i = 1,2,...,n. (2.3)

Proof. Letx(t)=(x1(t),...,xn(t))Tbeanysolutionofsystem(1.5)–(1.6).Itfollowsfromthepositivityofthesolution

and the ith equation of system (1.5) that

˙ xi(t)?ru

ixi(t). (2.4)

For −Tii?s?0 and t?0, integrating the above differential inequality on the interval [t + s,t] leads to

xi(t + s)?xi(t)exp{ru

Owing to (2.5) and?0

˙ xi(t)?xi(t)

−Tii

?xi(t)[ru

By applying Lemma 2.1, it immediately follows that

is}?xi(t)exp{−ru

−TiiKii(s)ds = 1, again, from the positivity of the solution and ith equation of system (1.5)

?0

i− al

iTii},

0?s? − Tii. (2.5)

it follows that

?

ru

i− al

ii

Kii(s)xi(t + s)ds

?

iiexp{−ru

iTii}xi(t)].

lim sup

t→+∞

xi(t)?Mi,i = 1,2,...,n.

This ends the proof of Theorem 2.1.

?

Theorem 2.2. Assume (H1)–(H2) hold. Assume further that

(H3)

rl

i−

?

j∈Ji

au

ijMj−

?

j∈Ji

bu

ijMiMj>0,i = 1,2,...,n

holds; let x(t) = (x1(t),...,xn(t))Tbe any solution of system (1.5)–(1.6). Then

lim inf

i = 1,2,...,n,

where

i−?

and

⎧

⎩

Proof. Let x(t)=(x1(t),...,xn(t))Tbe any solution of system (1.5)–(1.6), for any positive constant ?>0, it follows

from Theorem 2.1 that there exists a T1>0 such that

t→+∞xi(t)?mi,

(2.6)

mi=

rl

j∈Jiau

ijMj−?

j∈Jibu

ijMiMj

au

ii?i

,i = 1,2,...,n,(2.7)

?i= exp

⎨

⎛

⎝rl

i+

n

?

j=1

au

ijMj+

?

j∈Ji

bu

ijMiMj

⎞

⎠Tii

⎫

⎬

⎭.(2.8)

xi(t)<Mi+ ?def

= M?

i

as t?T1.

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F. Chen et al. / Journal of Computational and Applied Mathematics 206 (2007) 733–754

737

Here, without loss of generality, from condition (H3) we may choose ? small enough such that

?

Thus, for t?T1+ ?,

⎛

j=1

For −Tii?s?0 and t?T1+ ?, integrating above differential inequality on the interval [t + s,t] leads to

⎧

⎩

where

⎧

⎩

−TiiKii(s)ds = 1, from the positivity of the solution and ith equation of system (1.5) it follows

that

⎡

j∈Ji

⎡

j∈Ji

By applying Lemma 2.1, it immediately follows that

i−?

i

Setting ? → 0, it follows that

lim inf

i = 1,2,...,n.

This ends the proof of Theorem 2.2.

?

rl

i−

j∈Ji

au

ijM?

j−

?

j∈Ji

bu

ijM?

iM?

j>0,i = 1,2,...,n.

˙ xi(t)?

⎝rl

i−

n

?

au

ijM?

j−

?

j∈Ji

bu

ijM?

iM?

j

⎞

⎠xi(t). (2.9)

xi(t + s)?xi(t)exp

⎨

⎛

⎝rl

i−

n

?

j=1

au

ijM?

j−

?

j∈Ji

bu

ijM?

iM?

j

⎞

⎠s

⎫

⎬

⎭?xi(t)??

i,

−Tii?s?0, (2.10)

??

i= exp

⎨

⎛

⎝rl

i−

n

?

j=1

au

ijM?

j−

?

j∈Ji

bu

ijM?

iM?

j

⎞

⎠Tii

⎫

⎬

⎭.

Owing to (2.10) and?0

˙ xi(t)?xi(t)

⎣rl

⎣rl

i−

?

?

au

ijM?

j−

?

?

j∈Ji

bu

ijM?

iM?

j− au

ii

?0

−Tii

Kii(s)xi(t + s)ds

⎤

⎤

⎦

?xi(t)

i−

au

ijM?

j−

j∈Ji

bu

ijM?

iM?

j− au

ii??

ixi(t)

⎦

for t?T1+ ?.

lim inf

t→+∞xi(t)?

rl

j∈Jiau

ijM?

j−?

j∈Jibu

ijM?

iM?

j

au

ii??

,i = 1,2,...,n.

t→+∞xi(t)?mi,

Noting that miand Miin Theorems 2.1 and 2.2 are independent of the solution of system (1.5)–(1.6), thus, as a

direct corollary of Theorems 2.1 and 2.2 we have:

Theorem 2.3. Assume that (H1)–(H3) hold, then system (1.5) with initial conditions (1.6) is permanent.

Now let us further assume that

(H4) ri(t),aij(t),bij(t) (i ?= j) are all continuous positive ?-periodic functions.

As a direct corollary of Theorem 2 in [35], from Theorem 2.3, we have:

Corollary 2.1. If (H1)–(H4) hold, then system (1.5) admits at least one positive ?-periodic solution.

Remark 2.1. Jin and Ma [24] showed that to ensure the existence of positive periodic solutions of system (1.4), it is

enough to make some restriction on aij(t) and ri(t), while bi(t), i=1,2 and delays Tij,?i, j =1,2 have no influence