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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
whereTij=?ij=
a21(t) =1
?0
By computation, we have
1
160 000.Inthiscase,correspondingtosystem(1.5),wehaveri(t)=80,a11(t)=a22(t)=100,a12(t)=
10,b12(t) = b21(t) =1
?0
20+ cos(t)/20.Also, we assume that
−Tij
Kij(s)ds = 1,
−?ij
fij(s)ds = 1, i,j = 1,2.
Mi=ru
i
al
ii
exp{ru
iTii} =
80
100exp
?
80 ·
1
160000
?
?80
10031/2000≈
80
100× 1.000549457?1,i = 1,2.
And so,
rl
i−
?
j∈Ji
au
ijMj−
?
j∈Ji
bu
ijMiMj>80 −1
5−
1
10>0,i = 1,2.
Aboveinequalityshowsthatcondition(H3)holds,thus,byTheorems2.2and2.3weknowthatsystem(5.1)ispermanent
and admits at least one positive 2?-periodic solution.
Now, we take ?i= 1>0,i = 1,2. By simple computation, corresponding to Corollary 3.1, we have
A∗
i = 1,2.
Therefore, from Corollary 3.1 we know that any positive solution of system (5.1) is globally attractive.
i?20>0,
Example 5.2. Consider the following system:
˙ x1(t) = x1(t)[4 − (1.5 + sin(10t))x1(t) − (1 + 0.5sin(10t))x2(t −1
˙ x2(t) = x2(t)[2 − (3 + 0.5cos(10t))x1(t −1
In this case, corresponding to system (4.13), we have
10) − (√2 + cos(10t))x1(t)x2(t −1
10)],
(5.2)
20) − (3.5 + 0.5sin(10t))x2(t) − 3x1(t −1
20)x2(t)].
r1(t) = 4,
r2(t) = 2,
b12(t) =
?12(t) = ?12(t) =1
By simple computation, we have
a11(t) = 1.5 + sin(10t),
a21(t) = 3 + 0.5cos(10t),
√2 + cos(10t),
10,
a12(t) = 1 + 0.5sin(10t),
a22(t) = 3.5 + 0.5sin(10t),
b21(t) = 3,
?21(t) = ?21(t) =1
20.
rl
1al
21= 4 × 2.5 = 10>au
1al
11ru
2= 2 × 2 = 4,
12= 2 × 1.5 = 3,
rl
22= 4 × 3 = 12?ru
1bl
Thus, condition (H7) is satisfied. From Theorem 4.3 it follows that limt→∞x2(t)=0 and limt→∞[x1(t)−x∗
where x∗
However, in this case, we have
22= (√2 + 1) × 3?au
and therefore, the third inequality in (H8) could not hold. Thus, our results improve the main results of Mahhuba [30].
Fig. 1 is the numeric simulation of the solution of system (5.2) with the initial condition (x1(?),x2(?)) = (3,4),? ∈
[−1
2au
rl
21= 4 × 3 = 12>6 = 2 × 3?ru
2bu
12= 2 × (√2 + 1).
1(t)]=0,
1(t) is the unique positive periodic solution of ˙ x1(t) = x1(t)[4 − (1.5 + sin(10t))x1(t)].
bu
12al
12bl
21= 1.5 × 3,
10,0].
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F. Chen et al. / Journal of Computational and Applied Mathematics 206 (2007) 733–754
753
0 0.51 1.522.533.544.55
0
0.5
1
1.5
2
2.5
3
3.5
4
time t
y(t)
Fig. 1. Numeric simulation of the solution of system (5.2) with initial condition (x1(?),x2(?)) = (3,4),? ∈ [−1
10,0] and t ∈ [0,5].
Acknowledgments
Theauthorsaregratefultoanonymousrefereesfortheirexcellentsuggestions,whichgreatlyimprovethepresentation
of the paper. Also, this work is supported by the National Natural Science Foundation of China (10501007), the
Foundation of Science and Technology of Fujian Province forYoung Scholars (2004J0002), the Foundation of Fujian
Education Bureau (JA04156).
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