Global Attractivity of a Single Species Model with both Infinite Delay
Merdan Type Allee effect
XIAORAN LIa，b, QIN YUEb, FENGDE CHENc
a Faculty of Economics and Business, Universiti Malaysia Sarawak Kota Samarahan, 94300,
Sarawak, MALAYSIA
b College of Finance and Mathematics, West Anhui University Yueliang Dao, West of Yunlu Bridge,
Lu An, An Hui, CHINA
c College of Mathematics and Statistics, Fuzhou University No. 2, wulongjiang Avenue,
Minhou County, Fuzhou, CHINA
Abstract: - This paper proposes and investigates a single-species model with infinite delay and Merdan- type
Allee effect. The model takes the form
,)( x
x
dssxstKcbxax
dt
dx t
where a, b, c, and β are all positive constants. The differential inequality theory and iterative method are used to
obtain sufficient conditions that ensure the global attractivity of the system's positive equilibrium. Our research
shows that the Allee effect does not affect the final density of the species; however, the numerical simulations
show that with the increase of the Allee effect, the system takes much time to the stale state.
Key-Words: Dynamical Systems, Systems Theory, Single species; Infinite delay; Global attractivity; Allee effect
Received: April 17, 2022. Revised: November 3, 2022. Accepted: November 22, 2022. Published: December 31, 2022.
1 Introduction
The paper aims to investigate the dynamic behav-
iors of the following single-species model with both
infinite delay and the Allee effect
bxa
x
x
x
dt
dx
(
tdssxstKc ),)()(
)1.1(
where
cba ,,
and
are all positive constants.
)(tx
is
the density of the d species at the time
t
. We shall
consider (1.1) together with the initial conditions
)2.1(
Where
BC
i
and
0)0(:)),0[],0,(({
CBC
, and
bounded
}
.
The delay kernel
),0(),0[: K
is a con-
tinuous function such that

0.1)( dssK
)3.1(
According to the fundamental theory of functional
differential equations [39], system (1.1) has a unique
solution
)(tx
satisfying the initial condition (1.2). We
can easily demonstrate
0)( tx
for all
0t
.
Many scholars have studied the dynamic be-
haviors of population models over the last few
decades ([1]-[37]). Among those works, the model
with infinite delays has been studied by many schol-
ars ([1]-[18]). For example, Chen, Xie, and Wang [7]
proposed a competition model of plankton allelopa-
thy with infinite delay,
)()[()( 11111 txKtxtx
dsstxsK
t
)()( 21212
],)()()( 21211 dsstxsftx t
)()[()( 22222 txKtxtx
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dsstxsK
t)()( 12121
tdsstxsftx ],)()()( 12122
)4.1(
They showed that for this system, delay and toxic
substances are harmless for the stability of the inte-
rior equilibrium point.
On the other hand, Merdan [19] proposed the fol-
lowing predator-prey system with the Allee effect on
prey species:
),(
,)1(
yxay
dt
dy
axy
x
x
xrs
dt
dx
)5.1(
where β is a positive constant, which describes the
intensity of the Allee effect. Merdan showed that the
Allee effect harms the species. Since then, many
scholars have proposed an ecosystem incorporating a
Merdan-type Allee effect ([19]-[34]). The Merdan-
type Allee effect appears to have a different im-
pact on different population systems; for example,
Lin[21] proposed a Lotka-Volterra commensal sym-
biosis model with the first species subject to the Allee
effect
),(
,)(
222
12111
yaby
dt
dy
xya
x
x
xabx
dt
dx
)6.1(
he showed that the final density of the species subject
to the Allee effect is also increased.
It draws our attention to the fact that all of
those works ([19]-[34]) did not consider the influ-
ence of the delay, which is one of the most important
factors to determine the dynamic behaviors of the
species. This motivated us to propose the system
(1.1), which is the most simple single species model
with infinite delay and the Allee effect.
As far as the system (1.1) is concerned, one inter-
esting issue is finding out the influence of the Allee
effect. One may think it is possible to investigate the
dynamic behaviors of the system (1.1) by construct-
ing some suitable Lyapunov function similar to [5, 6,
8]. i.e.,
xtxtV ln)(ln)(
.)()(
0dsdtxxskc t
st

However, by simple computation, with the Allee ef-
fect term, it seems impossible to obtain conditions to
ensure the negative of the D+V (t), hence, one could
not apply the method of [5, 6, 8] to investigate the
dynamic behaviors of the system (1.1).
The paper aims to find out the influence of the Al-
lee effect on the system (1.1), the rest of the paper is
organized as follows. In Section 2, by using the iter-
ative method, we obtain a set of sufficient conditions
that ensure the system (1.1) admits a unique globally
attractive positive equilibrium. In Section 3, we give
a numeric simulation to show the feasibility of the
main result. We end this paper with a brief discus-
sion.
2 Main result
Concerned with the stability of the system (1.1),
we have the following result.
Theorem 2.1.
Assume that
cb
)1.2(
then the system (1.1) admits a unique positive equi-
librium
cb
a
x
, which is globally attractive.
Before we begin to prove the main result, we need
several Lemmas.
Lemma 2.1. [27] Consider the following equation:
,)( 22 yu
y
ybay
dt
dy
)2.2(
the unique positive equilibrium
2
2
b
a
y
is global
stable.
Following Lemma 2.2 is Lemma 3 of Francisco
Montes de Oca and Miguel Vivas[9].
Lemma 2.2. Let
RRx :
be a bounded nonnega-
tive continuous function, and let
),0(),0[: k
be a continuous kernel such
that
01)( dssk
. Then
t
tx )(inflim
dssxstk
t
t)()(inflim

).(suplim
)()(suplim
tx
dssxstk
t
t
t


Lemma2.3[27] System (1.1) has a unique positive
equilibrium
cb
a
x
.
Proof The equilibria of system (1.1) satisfy the
equation
.0))((
x
x
xdsstKcbxax t
)3.2(
Noting that
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,1)(
dsstK
t
the positive solution of the system (1.1) satisfies the
equation
.0 cxbxa
That is, system (1.1) allows for a unique positive
equilibrium
cb
a
x
.
This ends the proof of Lemma 2.3.
Lemma 2.4. Let
)(tx
be any solution of
)2.1()1.1(
, then
0)( tx
for all
.0t
Proof. We note that in (1.1), one has
),()(
)( txtP
dt
tdx
)4.2(
where
bxa
x
x
tP
()(
).)()( dssxstKc t
)5.2(
Hence,
.0})(exp{)0()( 0 dssPxtx t
)6.2(
This ends the proof of Lemma 2.4.
Following, we will develop the method of Chen,
Xie, and Wang [6] and Wu, Gao, and Chen [16] to
prove the main result.
Proof of Theorem 2.1. From (1.7), it follows that
there exists a
0
sufficiently small such that
.
)(1
)(
,
)(1
)(
2
1
1
1
1
2
2
1
2
2
2
2
1
1
a
a
r
a
r
b
r
a
a
r
a
r
b
r
)7.2(
Let
)(tx
be any positive solution of system (1.1) with
initial condition (1.1). From system (1.1) it follows
that
.)( x
x
bxax
dt
dx
)8.2(
Now let’s consider the auxiliary equation
.)( u
u
buau
dt
du
)9.2(
It follows from Lemma 2.1 that the unique positive
equilibrium
b
a
u
is globally stable. That is,
,)(lim b
a
tu
t

As a result of employing the differential inequality
theory, one has
,)(suplim b
a
tx
t

)10.2(
It follows from Lemma 2.2 that

t
t
dssxstK )()(suplim
,
)(suplim
b
a
tx
t

)11.2(
Hence, for sufficiently small
0
, which satisfies
(2.7), it follows from (2.10) and (2.11) that there ex-
ists a
0
1T
such that for all
1
Tt
.)( )1(
M
b
a
tx
def
)12.2(
.)()( )1(
M
b
a
dssxstK
def
t
)13.2(
For
1
Tt
, it follows from the system (1.1) and (2.13)
That
.][
)( )1(
x
x
cMbuax
dt
tdx
)14.2(
Now let’s consider the auxiliary equation
.)( )1(
u
u
cMbuau
dt
du
)15.2(
It follows from Lemma 2.1 that the unique positive
equilibrium
b
cMa
u
)1(
is globally stable. That
is,
.)(lim
)1(
b
cMa
tu
t

As a result of employing the differential inequality
theory, one has
,)(inflim
)1(
b
cMa
tx
t

)16.2(
It follows from Lemma 2.2 and (2.16) that
dssxstK
t
t)()(inflim

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,
)(inflim
)1(
b
cMa
tx
t

)17.2(
Hence, for enough small
0
, which satisfies (2.7),
it follows from (2.7) and (2.17) that there exists a
0
2T
such that for all
2
Tt
,
.)( )1(
)1(
m
b
cMa
tx
def
)18.2(
tdssxstK )()(
.
)1(
)1(
m
b
cMadef
)19.2(
For
2
Tt
, it follows from the system (1.1) and (2.19)
that
.][
)( )1(
x
x
cmbxax
dt
tdx
)20.2(
Now let’s consider the auxiliary equation
.][ )1(
u
u
cmbuau
dt
du
)21.2(
It follows from Lemma 2.1 that the unique positive
equilibrium
b
cma
u
)1(
is globally stable. That is,
.)(lim
)1(
b
cma
tu
t

Thus, by using the differential inequality theory, one
has
,)(suplim
)1(
b
cma
tx
t

)22.2(
It follows from Lemma 2.2 that

t
t
dssxstK )()(suplim
.
)(suplim
)1(
b
cma
tx
t
)23.2(
Hence, for
0
which satisfies (3.1), it follows from
(3.15)-(3.16) that there exists a
0
3T
such that for
all
3
Tt
,
.
2
)( )2(
)1(
1M
b
cma
ty
def
)24.2(
tdssxstK )()(
.
2
)2(
)1(
M
b
cmadef
)25.2(
For
3
Tt
, it follows from the system (1.1) and (2.25)
That
.])[(
)( )2(
x
x
cMbxatx
dt
tdx
)26.2(
Now let’s consider the auxiliary equation
.][ )2(
u
u
cMbuau
dt
du
)27.2(
It follows from Lemma 2.1 that the unique positive
equilibrium
b
cMa
u
)2(
b is globally stable. That
is,
.)(lim
)2(
b
cMa
tu
t

Thus, by using the differential inequality theory, one
has
,)(inflim
)2(
b
cMa
tx
t

)28.2(
It follows from Lemma 2.3 that
dssxstK
t
t)()(inflim

.
)(inflim
)2(
b
cMa
tx
t

)29.2(
Hence, for
0
which satisfies (3.1), it follows from
(2.28) and (2.29) that there exists a
0
4T
such that
for all
4
Tt
,
.)( )2(
)2(
m
b
cMa
tx
def
)30.2(
tdssxstK )()(
.
)2(
)1(
m
b
cMadef
)31.2(
One could easily see that
2
)1(
)2(
b
cma
M
;
)1(
M
b
a
2
)2(
)2(
b
cMa
m
.
)1(
)1(
m
b
cMa
)32.2(
Repeating the above procedure, we get four se-
quences
,,2,1,, )()( nmM nn
such that for
2n
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；
nb
cma
M
n
n
)1(
)(
.
)(
)(
nb
cMa
m
n
n
)33.2(
Obviously,
,)( )()( nn Mtxm
for
.
2n
Tt
)34.2(
By induction, similar to the analysis of Chen, Xie,
and Wang[7], we could prove that sequences
)(n
M
are non-increasing, and sequences
)(n
m
are non-de-
creasing. Therefore,
.lim,lim )()( xmxM n
t
n
t 
)35.2(
Letting
n
in (2.33), we obtain
.
,
xcaxb
xcaxb
)36.2(
It follows from (2.36) that
.0))(( xxcb
Hence,
.xx
)37.2(
Again, substituting (2.37) with (2.36) leads to
.
x
cb
a
xx
that is
.)(lim
xtx
t
Thus, the unique interior equilibrium
x
is globally
attractive. This completes the proof of Theorem 2.1.
3 Numeric simulations
Now let us consider the following examples.
Example 3.1
,))(26( )(
tst
x
x
dssxexx
dt
dx
)1.3(
where
is a positive constant.
Let
,)(
)(
tst dssxey
then system (3.1) is equiv-
alent to the system
,)26( x
x
yxx
dt
dx
.yx
dt
dy
)2.3(
It follows from Theorem 2.1 that system (3.1) admits
a unique globally attractive positive equilibrium
2
x
. Consequently, system (3.2) admits a unique
globally attractive positive equilibrium
)2,2(),(
yx
. Fig.1 supports this assertion.
Now let’s choose
= 2, 5, and 20, respectively. Fig.
2 shows that as
(i.e., Allee effect) increases, the
solution takes longer to reach its steady state.
Dynamic behaviors of the system (3.2)
Figure 1: Dynamic behaviors of the system (3.2), the
initial condition
)4,1(),3,1.0(),1.0,4())0(),0(( yx
,
)1,4(
and
)4,4(
, respectively.
4 Discussion
Merdan [19] proposed a predator-prey system with
the Allee effect on prey species, see system (1.5), his
study showed that this harms the species. The final
density of the species will decrease as the Allee effect
increases. Since then, many scholars ([19]-[34]) have
investigated the dynamic behaviors of the population
system with the Merdan-type Allee effect, and for
different systems, the Allee effect has different influ-
ences.
In this paper, we further incorporate the Merdan-
type Allee effect into the single-species delayed sys-
tem. As demonstrated in Theorem 2.1, system (1.1)
admits a unique globally attractive positive equilib-
rium
x
under the assumption
cb
. Noting that this
condition is also necessary to ensure the system
tdssxstKcbxax
dt
dx ),)()((
)1.4(
admits the unique globally attractive positive equilib-
rium. Also, noting that
x
is independent of β. Hence,
we can conclude: the Allee effect does not influence
the existence and stability of the positive equilibrium.
However, Fig. 2 shows that with the increasing Allee
effect, the system should take more time to approach
its steady state. It is in this sense that the Allee effect
harms the stability property of the system.
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Figure 2: Numeric simulations of
)(tx
, with
= 2, 5,
20, and
)1,1())0(),0(( yx
, respectively, where the
black curve is the solution of
= 2, the red curve is
the solution of
= 5, and the blue curve is the solu-
tion of
= 20.
We would like to mention at the end of the paper
that in [40], one of our recent works, we discovered
that for the discrete commensalism model with
Merdan type Allee effect, the Allee effect increased
the stability property of the system in the sense that,
without Allee effect, the system may be chaos, how-
ever, only with the increasing of Allee effect, if the
Allee effect is enough large, the system may become
stable. The influence of the Merdan-type Allee
effect appears to be quite different for the contin-
uous and discrete systems. To this day, no scholar
has proposed a discrete model with infinite delay and
a Merdan-type Allee effect, we will try to do work in
this direction in the future.
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Xiaoran Li carried out the computation and wrote the
draft.
Qin Yue carried out the simulation.
Fengde Chen was responsible for the proposal of
the problem.
Sources of funding for research
presented in a scientific article or
scientific article itself
Social Science Project of Anhui Provincial Depart-
ment of Education (2022AH051657).
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.34
Xiaoran Li, Qin Yue, Fengde Chen
E-ISSN: 2224-266X
322
Volume 21, 2022
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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