Stability Analysis of a Single Species Model with Allee Effect and
Density Dependent Birth Rate
FENGDE CHEN, QUN ZHU, QIANQIAN LI
College of Mathematics and Statistics
Fuzhou University
No. 2, wulongjiang Avenue, Minhou County, Fuzhou
CHINA
Abstract: Abstract: - A single species model with Allee effect and density-dependent birth rate
x
x
exd
cxb
a
x
dt
dx
)(
is proposed and studied in this paper, where , , , , and are all positive constants. Sufficient conditions
which ensure the system admits a unique globally stable positive equilibrium are obtained. Numeric simulations
show that with the increasing Alee effect, the system takes a much longer time to reach its stable steady-state
the solution, however, Allee effect has no influence on the final density of the species.
Key-Words: Single species model; Allee effect; Global stability
Received: June 25, 2022. Revised: February 19, 2023. Accepted: March 6, 2023. Published: March 15, 2023.
1 Introduction
The aim of this paper is to investigate the dynamic
behaviors of the following single-species model with
Allee effect and nonlinear birth rate
x
x
exd
cxb
a
x
dt
dx
)(
1.1
where
edcba ,,,,
and
are all positive constants.
)(tx is the densities of the species at time, here we
make the following assumptions:
(a) cxb
a
is the birth rate of the species, which is
density-dependent, the birth rate of the species is de-
clining as the density of the species is increasing;
(b) dis the death rate of the species, eis the density
dependent coefficients;
(c) We incor porate the Al lee effect x
x
x
)(
on the species, such an Allee effect describes the fact
of limitations in finding mates, which is also called
the weak Allee effect function. )(x
is the probabil-
ity of finding a mate where
is the indivi dual
searching efficiency ([4], [8], [11],[12], [13],
[14],[15], [16], [18]). The bigger
is the stronger
Allee effect.
Allee effect, which was first time observed by Al-
lee ([1]), describes a negative density dependence of
the species, become one of the main topics on the
ecosystem, many scholars have done works in this di-
rection, see [14 ], [16] and the references cited
therein.
󰇘Merdan [15] proposed the followin g
predator-prey system with Allee effect on prey spe-
cies

).(
,1
yxay
dt
dy
axy
x
x
xrx
dt
dx
2.1
where
is a positive constant, which describes the
intense of the Allee effect. 󰇘Merdan showed
that the sy stem subject to an Allee effect takes a
longer time to reach its steady-state solution and the
Allee effect reduces the population densities of both
predator and prey at the steady-state.
Stimulated by the work of Merdan, Guan, Liu and
Xie [8] argued that the higher the hie rarchy in the
food chain, the more likely it is to become extinct.
Hence, they proposed the following predator-prey
model with predator species subject to Allee effect:


,
,1
2yx
y
ay
dt
dy
axyxrx
dt
dx
3.1
where ar, are positive constants. They obtained a
set of sufficient conditions which ensure the exist-
ence of a unique globally asymptotically stable posi-
tive equilibrium. Their numeric si mulations showed
that the system subject to an Allee effect takes a much
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.30
Fengde Chen, Qun Zhu, Qianqian Li
E-ISSN: 2224-2678
282
Volume 22, 2023
longer time to reach its st able steady-state solution,
however, Allee effect has no influence on the final
density of the predator and prey species. Such a find-
ing is very different from that of 󰇘Merdan
[15].
During the past decade, the study of dynamic be-
haviors of mutualism or commensalism model be-
comes one of the main topics in the study of popula-
tion dynamics ([4]-[23]), among those works, some
of them ([4], [13], [18]) studied the influence of Allee
effect to the commensalism model.
Wu, Li, and Lin [18] proposed the follo wing two
species commensal symbiosis model with Holling
type functional response and Allee effect on the sec-
ond species
,)(
),
1
(
22
1
11
y
y
ybay
dt
dy
y
yc
xbax
dt
dx
p
p
4.1
where
,2,1,, piba ii and 1
care all positive con-
stants, .1pThey showed by numeric simulations
that with the increasing Allee effect, the system takes
much more time to reach its stable steady-state solu-
tion.
Chen [4] proposed the following two species com-
mensal symbiosis model involving Allee effect and
one party can not survive independently, which takes
the form:
,)(
),(
22
1
11
yu
y
ybay
dt
dy
yx
yc
xbax
dt
dx
5.1
where 22111 ,,,, bacba and uare all positive con-
stants, )(tx and )(ty are the densities of the first
and second species at time. The author investigated
the local and global stable properties of the boundary
equilibrium and the positive equilibrium.
Lin [13] investigated the dynamic behaviors of the
following two species co mmensal symbiosis model
incorporating Allee effect to the first species:
,
,)1(
cxbu
dt
du
axu
x
x
xrx
dt
dx
)6.1(
where 2,1,, iab iii
and 12
aare all positive con-
stants. He found that with the increase of Allee effect,
the final density of the species subject to Allee effect
is also increased. Such a phenomenon is the first time
observed, which is quite different from the known re-
sults ([4], [15]).
Lin [14] proposed a single species Logistic model
with Allee effect and feedback control
,
,)1(
cxbu
dt
du
axu
x
x
xrx
dt
dx
)7.1(
where bar ,,,
and care all positive constants. He
showed that for the s ystem without Allee effect, the
system admits a unique positive equilibrium which is
globally attractive, however, for the system with Al-
lee effect, depending on the intense of the Allee ef-
fect, the system could admit a unique positive equi-
librium which is locally asymptotically stable or the
species may be driven to extinction. Allee effect re-
duces the population density of the species.
It brings to our attention t hat in the system (1.2) -
(1.7), without the influenc e of the othe r species or
other factors, the species subject to Allee effect takes
the form
.)( 1x
x
exax
dt
dx
)8.1(
That is, all the works of [8], [13]-[15] are based on
the traditional single species Logistic equation
).( 1exax
dt
dx )9.1(
System (1.9) is very famous and is the cornerstone
of population biology. Noting that system (1.9) could
be revised as
).( exdax
dt
dx )10.1(
where ais the birth rate of the species and d is the
death rate of the species. Already, Brauer and Cas-
tillo-Chavez [3], Tang and Chen [17 ] and Berezan-
sky, Braverman, Idels [2] had shown that in so me
cases, the density dependent birth rate of the species
is more suitable. If we take the famous Beverton Holt
function ([2]) as the birth rate, then the system (1.10)
should be revised to
).( exd
cxb
a
x
dt
dx
)11.1(
If we further consider the influence of Allee effect to
the system (1.11), by adding the term x
x
to the
right hand side of the above sy stem, this will lead to
the system (1.1).
The paper is arranged as follows. In section 2, we
investigated the dy namic behaviors of the sy stem
(1.1); Section 3 presents some numerical simulations
Fengde Chen, Qun Zhu, Qianqian Li
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Volume 22, 2023
to show the feasibility of the main results. We end
this paper with a brief discussion.
2 Dynamic behaviors of the system
(1.1)
The equilibrium of system (1.1) is determined by the
equation
.0)(
x
x
exd
cxb
a
x
)1.2(
Assume that
d
b
a
)2.2(
holds, then system (1.1) admits a boundary equilib-
rium 0
1xand a positive equilibrium
x where
.
2
)(4)()( 2
ec
adbecebdcebdc
x
)3.2(
For the biol ogy meaning, we will focus our
attention on the stability of the positive equilibrium,
we have the following result.
Theorem 2.1. Assume that (2.2) holds, then the sys-
tem (2.1) admits a unique positive equilibrium
x
which is globally stable.
Proof. Obviously,
x satisfies the equation
.0
exd
cxb
a
)4.2(
Now let’s consider the Lyapunov function
.ln)(
x
x
xxxxV
)5.2(
One could easily see that the functionVis zero at the
positive equilibrium
xand is positive for all other
positive values of
x
. By applying (2.4), the tim e
derivative of V along the trajectories of (1.1) is
)(tVD
cxb
a
cxb
a
xx
x
x
exd
cxb
a
xx
)((
))((
x
x
exex
)
))((
)(
)((
cxbcxb
xxac
xx
)6.2(
x
x
xxe
))(
)
))((
(e
cxbcxb
ac
2
)(
xx
x
x
It then follows from (2.6) that 0)(
tVD strictly for
all 0xexcept the positive equilibrium
x, where
0)(
tD . Thus )(xV satisfies Lyapunov’s asymp-
totic stability theorem ([6]), and the positive
equilibrium
xof system (1.1) is globally asymptoti-
cally stable. This ends the proof of Theorem 2.1.
3 Numeric simulations
Now let’s consider the following example.
Example 3.1
.)1
1
2
(x
x
x
x
x
dt
dx
)1.3(
In this system, corresponding to system (1.1), we
take
,1,1,1,1,2
edcba
since
,12 dba
it follows from Theorem 2.1 that for
all
, system (3.1) always admits a unique positive
equilibrium ,4142.0
xwhich is globa lly asymp-
totically stable. Fig.1 is the case .1
Now let’s
take
2.0,0
and 0.5 a nd 1, respectively, togethe r
with the initi al condition 1.0)0( x, Fig. 2 show s
that with the increasing of the
(i. e., the increasing
of the Allee effect), the solution takes much time to
reach its steady state.
Dynamic behaviors of system (3.1) with beta=1
Fengde Chen, Qun Zhu, Qianqian Li
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Volume 22, 2023
Figure 1: Dynamic behaviors of the system (3.1) with
，1
the initial condition 5.1,1,4.0,1.0)0( x and
2, respectively.
4 Conclusion
Recently, many scholars investigated the dynam ic
behaviors of the ecosy stem subject to Allee effect,
see [4], [8], [11],[12], [13], [14],[15], [16], [18] and
the references cited therei n. By carefully checking
the models considered in [14], [15], we found that all
of them are based on the traditional Logistic model.
However, a more suitable model should consider the
influence of density on t he birth rate. Thus, we pro -
pose a single species model with Alle e effect and
density-dependent birth rate, i.e., system (1.1).
Our study shows that the conditions which ensure
Figure 2: Dynamic behaviors of the system (3.1) with
the initial condition 1.0)0( x, where a red curve is
the solution of
0
, green curve is the solution of
2.0
, the black curve is the solution of
5.0
and the blue curve is the solution of 1
, respec-
tively.
the existence of the positive equilibrium is enough to
ensure its global asymptotically stability, that is, once
the positive equilibrium exists, it is globally asymp-
totically stable. Numeric simulations show that the
Allee effect has no influence on the final density of
the species, however, with the increasing of the Allee
effect, the system takes more time to reach its steady
state. Such kind of property is similar to that of the
commensalism model ([4], [18]).
It seems interesting to consider the multi-species
system with both the no nlinear birth rate and Allee
effect, we leave this for future discussion.
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Contribution of individual authors to
the creation of a scientific article
(ghostwriting policy)
Qun Zhu wrote the draft.
Qianqian Li carried out the simulation.
Fengde Chen proposed the issue and revise the paper.
Sources of funding for research
presented in a scientific article or
scientific article itself
This work is supported by the Natural Science Foun-
dation of Fujian Province(2020J01499).
Creative Commons Attribution
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Fengde Chen, Qun Zhu, Qianqian Li
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Volume 22, 2023
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.