Merdan-type Allee Effect on a Lotka-Volterra Commensal Symbiosis
Model with Density-dependent Birth Rate
Abstract: - A Lotka-Volterra commensal symbiosis model with a density dependent birth rate and a Merdan-type
Allee effect on the second species has been proposed and examined. The global attractivity of system’s equilibria
is ensured by using the differential inequality theory. Our results show that the Allee effect has no effect on the
existence or stability of the system’s equilibrium point. However, both species take longer to approach extinction
or a stable equilibrium state as the Allee effect increases.
Key-Words: -Commensalism; Global attractivity; Allee effect
1 Introduction
The objective of this study is to analyze the dynamic
patterns of a commensalism model that incorporates a
density-dependent birth rate and a Merdan-type Allee
effect. The model can be expressed as follows:
dx
dt =xb11
b12 +b13x−b14 −a11x+a12y,
dy
dt =yb21
b22 +b23y−b24 −a22yy
β+y,
(1)
The given equation involves positive constants bij for
i= 1,2and j= 1,2,3,4, as well as a11,a12,a22,
and β. The variables x(t)and y(t)represent the den-
sity of the first and second species, respectively, at a
given time t. It is assumed that the second species is
subject to the Allee effect, which is mathematically
represented as y
β+y.
One of the fundamental relationships between
species is commensalism, in which one species uses
the resources of the other without causing harm or
gaining anything in return. Commensalism occurs
frequently in nature, but theoretical study of it has
only just started in the last 20 years ([1]-[24]). Re-
garding this particular direction, the study’s initial
framework is based on the well-known Lotka-Volterra
type commensalism model ([6, 7, 8, 9, 12]). After-
ward, numerous scholars ([1, 2, 3, 4, 5, 10] argued that
the functional response function must be included to
describe how intense the commensalism between the
species is. and they primarily concentrate on the sys-
tem’s stability property. Recently, researchers started
looking into the Allee effect’s impact on the commen-
salism model ([4], [11],[14]) and the nonlinear density
dependent birth rate ([19],[21], [22]).
It is well known that food and other resources, and
space are always limited. With the increase in popula-
tion density, intraspecific competition intensifies and
the number of natural enemies increases continuously.
leading to a reduction in the birth rate of the popula-
tion, This suggests that it is more realistic to consider
density-dependent birth rates in population models.
Recently, Chen et al [19] argued that the Beverton-
Holt function provides a suitable framework for un-
derstanding this phenomenon. The model they con-
sidered takes the form:
dx
dt =xb11
b12 +b13x−b14 −a11x+a12y,
dy
dt =yb21
b22 +b23y−b24 −a22y,
(2)
The given constants bij , i = 1,2, j = 1,2,3,4,and
a11, a12 and a22 are all positive. These values play
an important role in determining the density of the
first and second species at time t, which is repre-
sented by x(t)and y(t)respectively. The system can
have four nonnegative equilibria. The authors demon-
strated that by using appropriate Lyapunov functions,
all four equilibria can be globally asymptotically sta-
ble, given certain assumptions. This distinguishes
this system from commensalism systems with con-
stant birth rates and is a special characteristic.
On the other hand, with the overexploitation of
natural resources by humans, more and more species
have become endangered, and their populations have
declined sharply, which will make it difficult for them
to find mates and cooperate effectively, which will ex-
acerbate further reductions in populations. The phe-
nomenon is called the Allee effect. Merdan [13] pro-
posed a predator-prey system described below:
dx
dt =rx(1 −x)x
β+x−axy,
dy
dt =ay(x−y),
(3)
FENGDE CHEN, XIAQING HE, ZHONG LI, TINGTING YAN
College of Mathematics and Statistics
Fuzhou University
No. 2, wulongjiang Avenue, Minhou County, Fuzhou
CHINA
Received: May 17, 2022. Revised: August 21, 2023. Accepted: September 16, 2023. Published: October 5, 2023.
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where x
β+xrepresents the Allee effect term, and
βindicates the strength of the Allee effect. Merdan
demonstrated the detrimental effects of the Allee ef-
fect on the species, the Allee effect delays the system’s
arrival at its steady-state solution and reduces the final
densities of both species. Since then, many academics
have suggested the ecosystem incorporating the Allee
effect of the Merdan type ([4],[14, 15, 16, 17, 18]).
Lin[4] introduced a Lotka-Volterra commensalism
model incorporating the Allee effect in relation to the
first species.
dx
dt =x(b1−a11x)x
β+x+a12xy,
dy
dt =y(b2−a22y).
(4)
He demonstrated that the Allee effect increases the
species’ final densities. Such a finding is very dif-
ferent from Merdan’s findings.
In her study, Wu[4] presented the following com-
mensal symbiosis model:
dx
dt =xa1−b1x+c1yp
1 + yp,
dy
dt =y(a2−b2y)y
u+y,
(5)
The findings of her study indicate that the system
exhibits permanence and possesses a singular, glob-
ally stable positive equilibrium. Furthermore, the
Allee effect does not appear to exert any discernible
influence on the ultimate density of the species.
As we can see from systems (3)-(5), the Merdan
type Allee effect affects each ecosystem in a different
way. To the best of our knowledge, the Allee effect
has not yet been included in the system (2). This
inspired us to suggest the system (1).
Finding out the impact of the nonlinear density
birth rate is a fascinating topic with regard to the
system (1). Is it possible that the system (1) admits
some dynamic behaviors that are comparable
to those of the system (2)? Could we provide a
positive response regarding the Allee effect?
The remainder of the paper is organized as fol-
lows. The purpose of this essay is to determine the
resolution to the above problems. Section 2 discusses
the existence of the equilibria; Section 3 presents the
paper’s main conclusion; and Section 4 establishes a
new lemma and uses it and the differential inequality
theory to strictly prove the paper’s main conclusions.
A brief discussion follows this essay’s conclusion.
2 Equilibria
The equilibria of system (1) fulfill the equation
xb11
b12 +b13x−b14 −a11x+a12y= 0,
yb21
b22 +b23y−b24 −a22yy
β+y= 0.
(6)
There is always a boundary equilibrium A1(0,0) in
the system (1). If
b11
b12
> b14 (7)
holds, then the system represented by equation (1) ex-
hibits a nonnegative boundary equilibrium denoted by
A2(x∗,0), where
x∗=−(b14b13 +a11b12) + √∆1
2a11b13
.(8)
here
∆1= (b14b13 +a11b12)2−4a11b13(b14b12 −b11).
Assume that b21
b22
> b24 (9)
holds, then it can be observed that the system rep-
resented by equation (1) possesses a boundary equi-
librium denoted as A3(0, y∗), which is nonnegative,
where
y∗=−(b24b23 +a22b22) + √∆2
2a22b23
.(10)
here
∆2= (b24b23 +a22b22)2−4a22b23(b24b22 −b21).
Assume that (9) and
b11
b12
+a12y1> b14 (11)
holds, the system represented by equation (1) pos-
sesses a unique positive equilibrium denoted as
A4(x1, y1), where
y1=−(b24b23 +a22b22) + √∆3
2a22b23
,(12)
here
∆3= (b24b23 +a22b22)2−4a22b23(b24b22 −b21).
the variable x1denotes the unique positive root of the
following equation:
b11
b12 +b13x−b14 −a11x+a12y1= 0.(13)
Remark 2.1. The observation of four equilibria in-
dicates that the Allee effect does not exert any dis-
cernible impact on the existence of the equilibrium.
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3 Main results
The preceding section entailed a discourse on the
presence of equilibria. With respect to the stability
of those equilibrium points, the following outcomes
have been observed:
Theorem 3.1
(1) Assume that
b11
b12
< b14 (14)
and
b21
b22
< b24 (15)
holds, the global attractivity of the boundary equilib-
rium A1(0,0) holds true;
(2) Assume that
b11
b12
> b14 (16)
and
b21
b22
< b24 (17)
holds, then the boundary equilibrium A2(x∗,0) is
globally attractive;
(3) Assume that
b21
b22
> b24 (18)
and
b11
b12
+a12y∗< b14 (19)
holds, it can be inferred that the boundary equilib-
rium denoted as A3(0, y∗)exhibits global attractivity,
where y∗is defined by (10);
(4) Assume that
b21
b22
> b24 (20)
and
b11
b12
+a12y1> b14 (21)
then the system (1) exists a unique positive equilib-
rium A4(x1, y1),which is globally attractive, where
y1is defined by (12).
Remark 3.1. Equations (20) and (21) represent the
necessary conditions for the presence of a positive
equilibrium point. The global attractivity of the posi-
tive equilibrium point is guaranteed if and only if (20)
and (21) holds.
4 Proof of the main result
The following Lemma is useful in the demonstration
of Theorem 3.1.
Lemma 4.1 Consider the following equation
dy
dt =ya
b+cy −d−eyy
β+y.(22)
Assuming positive constants a,b,c,d, and e, and a
non-negative constant β, we can conclude the follow-
ing:
(a) If the inequality a > bd is satisfied, then the sys-
tem represented by equation (22) possesses a single
positive equilibrium y∗that is globally asymptotically
stable.
(b) Conversely, if a < bd, then the equilibrium y= 0
of the same system is globally asymptotically stable.
Proof. The approach employed to prove Lemma 4.1
has a similarity to that of the proof of Lemma 2.1 as
presented in Chen et al.’s work [19]. For the sake of
brevity, we shall not analyze into the specifics of these
proofs in this context.
Proof of Theorem 4.1.
(1) For enough small ε > 0, without loss of generality,
one many assume that
ε < 1
2(b14 −b11
b12
),
then b11
b12 −b14 +ε < −ε. (23)
holds. Consider the equation
dy
dt =yb21
b22 +b23y−b24 −a22yy
β+y,(24)
the global attractiveness of the equilibrium y= 0 in
system (24) can be deduced from (14) and Lemma
4.1.
lim
t→+∞
y(t) = 0.(25)
From (25), there exists a T1>0such that
y(t)<ε
a12
for all t≥T1.(26)
Subsequently, it can be deduced from the first equa-
tion of system (1) and (23) that
dx
dt =xb11
b12 +b13x−b14 −a11x+a12y
≤xb11
b12 +b13x−b14 −a11x+ε
≤xb11
b12 −b14 +ε
≤ −εx,
(27)
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consequently
x(t)≤x(T1)exp n−ε(t−T1)o→0as t→+∞.
(28)
(25) and (28) show that A1(0,0) is globally attractive.
(2) Similarly to the analysis of (24)-(26), under the
assumption (17) holds, from the second equation of
system (1) it can be inferred that
lim
t→+∞
y(t) = 0.(29)
From (29), there exists a T2>0such that for ε > 0
small enough, the following inequality holds.
y(t)<ε
a12
for all t≥T2.(30)
Subsequently, it can be deduced from the first equa-
tion of the system (1) that
dx
dt ≤xb11
b12 +b13x−b14 −a11x+ε.(31)
Consider the equation
du
dt =ub11
b12 +b13u−b14 −a11u+ε.(32)
Obviously, condition (16) implies that
b11
b12
> b14 −ε. (33)
It follows from Lemma 4.1 that system (32) admit a
unique positive equilibrium u1(ε), which is globally
attractive, indeed,
u1(ε) = −A2+qA2
2−4A1A3
2A1
,(34)
here
A1=a11b13,
A2=a11b12 +b13b14 −b13ε,
A3=b12b14 −b12ε−b11.
(35)
Now, from (31) and (32), by using comparison theo-
rem of differential equation, we have
lim sup
t→+∞
x(t)≤u1(ε) + ε. (36)
At the same time, it can be inferred from the first equa-
tion of system (1) that
dx
dt ≥xb11
b12 +b13x−b14 −a11x(37)
Consider the equation
dv
dt =vb11
b12 +b13v−b14 −a11v(38)
It follows from Lemma 4.1 that under the assumption
(16) holds, system (38) admit a unique positive equi-
librium x∗, where
x∗=−(b14b13 +a11b12) + √∆1
2a11b13
,(39)
which is globally attractive, here
∆1= (b14b13 +a11b12)2−4a11b13(b14b12 −b11).
Now, from (37) and (38), by using comparison theo-
rem of differential equation, we have
lim inf
t→+∞
x(t)≥x∗−ε. (40)
(36) together with (40) leads to
x∗−ε≤lim inf
t→+∞
x(t)≤lim sup
t→+∞
x(t)≤u1(ε) + ε.
(41)
Since εare any small positive constants, and noting
that u1(ε)→x∗as ε→0.setting ε→0in (41)
leads to
x∗≤lim inf
t→+∞
x(t)≤lim sup
t→+∞
x(t)≤x∗.(42)
Therefore,
lim
t→+∞
x(t) = x∗.(43)
(29) and (43) show that A2(x∗,0) is globally attrac-
tive.
(3) For enough small ε > 0, from (19), one could see
that b11
b12
+a12(y∗+ε)< b14 −ε. (44)
holds. Consider the equation
dy
dt =yb21
b22 +b23y−b24 −a22yy
β+y,(45)
it follows from (18) and Lemma 4.1 that the equilib-
rium y=y∗of system (45) is globally attractive, i.
e.,
lim
t→+∞
y(t) = y∗.(46)
where
y∗=−(b24b23 +a22b22) + √∆2
2a22b23
,(47)
here
∆2= (b24b23 +a22b22)2−4a22b23(b24b22 −b21).
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Hence, for above ε > 0, there exists a T3>0such
that
y∗−ε < y(t)< y∗+εfor all t≥T3.(48)
For t>T3, from the first equation of system (1) and
(48), we have
dx
dt ≤xb11
b12 +b13x−b14 −a11x+a12(y∗+ε)
≤xb11
b12 +b13x−b14 +a12(y∗+ε)
≤ −εx.
(49)
Hence
x(t)≤x(T3)exp n−ε(t−T3)o→0as t→+∞.
(50)
(46) and (50) shows that A3(0, y∗)is globally attrac-
tive.
(4) For enough small ε > 0, from (21) one could see
that b11
b12
+a12(y1+ε)> b14 (51)
holds. Consider the equation
dy
dt =yb21
b22 +b23y−b24 −a22y,(52)
it follows from (20) and Lemma 4.1 that the positive
equilibrium y=y1of system (52) is globally attrac-
tive, i. e.,
lim
t→+∞
y(t) = y1.(53)
where
y1=−(b24b23 +a22b22) + √∆3
2a22b23
,(54)
here
∆3= (b24b23 +a22b22)2−4a22b23(b24b22 −b21).
Hence, for above ε > 0, there exists a T4>0such
that
y1−ε < y(t)< y1+εfor all t≥T4.(55)
For t>T4, from the first equation of system (1) and
(55), we have
dx
dt ≤xb11
b12 +b13x−b14 −a11x+a12(y1+ε).
(56)
Now let’s consider the equation
du
dt =ub11
b12 +b13u−b14 −a11u+a12(y1+ε).
(57)
Noting that from (51) we have
b11
b12
> b14 −a12(y1+ε).(58)
Therefore, it can be deduced from Lemma 4.1 that the
system represented by equation (58) possesses a glob-
ally attractive positive equilibrium denoted as u∗(ε).
Indeed,
u∗(ε) = −B2+qB2
2−4B1B3
2B1
,(59)
here
B1=a11b13,
B2=a11b12 +b13b14 −b13a12(y1+ε),
B3=b12b14 −b12a12(y1+ε)−b11.
(60)
Now, from (56) and (57), by using comparison theo-
rem of differential equation, we have
lim sup
t→+∞
x(t)≤u∗(ε) + ε. (61)
On the other hand, for enough small ε > 0, from (??)
one could see that
b11
b12
+a12(y1−ε)> b14 (62)
holds. For t>T4, again, by utilizing the first equa-
tion of system (1) in conjunction with (55), it follows
that
dx
dt ≥xb11
b12 +b13x−b14 −a11x+a12(y1−ε).
(63)
Consider the equation
dv
dt =vb11
b12 +b13x−b14 −a11v+a12(y1−ε).
(64)
Noting that from (62) we have
b11
b12
> b14 −a12(y1−ε).(65)
Therefore, it can be deduced from Lemma 4.1 that the
system represented by equation (64) possesses a glob-
ally attractive positive equilibrium, denoted as v∗(ε).
Indeed,
v∗(ε) = −C2+qC2
2−4C1C3
2C1
,(66)
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here
C1=a11b13,
C2=a11b12 +b13b14 −b13a12(y1−ε),
C3=b12b14 −b12a12(y1−ε)−b11.
(67)
Now, from (63) and (64), by using comparison theo-
rem of differential equation, we have
lim inf
t→+∞
x(t)≥v∗(ε)−ε. (68)
(61) together with (68) leads to
v∗(ε)−ε≤lim inf
t→+∞
x(t)≤lim sup
t→+∞
x(t)≤u∗(ε) + ε.
(69)
Since εare any small positive constants, and noting
that u∗(ε)→x1as ε→0, v∗(ε)→x1as ε→0,
setting ε→0in (69) leads to
x1≤lim inf
t→+∞
x(t)≤lim sup
t→+∞
x(t)≤x1.(70)
Therefore,
lim
t→+∞
x(t) = x1.(71)
(53) and (71) provide evidence that the equilibrium
A4(x1, y1)exhibits global attractivity. The demon-
stration of Theorem 4.1 has been concluded.
5 Numeric simulations
As was shown in Section 2, 3 and 4, it has been deter-
mined that the Allee effect does not exert any impact
on the presence or stability of the equilibria. It seems
interesting to give some more insight to the influence
of Allee effect, let’s consider some numeric simula-
tions. Following we will use Maple 2021 to draw the
numeric simulations.
Example 5.1 In system (1), let’s choose b13 =b14 =
a11 =a12 =b24 =a22 =b23 = 1.
(A) Take b11 =b21 = 1, b12 =b22 = 2, than the
global asymptotic stability of the boundary equilib-
rium A1(0,0) can be deduced from Theorem 4.1.
Figures 1 and 2 present numerical simulations of
x(t)and y(t), respectively, for values of βequal to
0, 2, 5, and 10. Figures 1 and 2 demonstrate that as
the Allee effect of the second species increases, both
species xand yrequire a longer time to dying out.
Such a phenomenon is quite different to the knowl-
edge about the Allee effect, since generally speaking,
the species suffer to Allee effect become more endan-
gered and the chance for it to be driven to extinction
is increasing. However, numeric simulations shows
that it is possible for human being to take some suit-
able method to avoid the extinction of the species. In
a sense, it seems that the Allee effect is an effective
way to slow down the rate of population extinction.
(b) Take b11 =b21 = 2, b12 =b22 = 1. Accord-
ing to Theorem 4.1, it can be deduced that the pos-
itive equilibrium A4(0.6364,0.4142) exhibits global
asymptotic stability. Figures 3 and 4 demonstrate that
as the Allee effect of the second species increases,
both species xand yrequire a longer time to reach
their final density.
Figure 1: The dynamic behaviors exhibited by
the first species in Case (A), the initial condition
(x(0), y(0)) = (1,1),β= 0,2,5and 10, respec-
tively.
6 Summary and discussion
Our research shows that: Allee effect will not affect
the existence and stability of the equilibrium point of
the system, but with the increase of the Allee effect,
the solution of the system needs more time to tend
to the boundary equilibrium point or positive equilib-
rium point. That is, it takes more time to go extinct or
to stabilize population densities.
As we all know, when the amount of species is
small or the generations are distinct, it is more ap-
propriate to use the difference equation to describe
the dynamic behavior of the population. Recently,
scholars such as Zhou et al [25] discussed the dis-
crete amensalism system with Merdan-type Allee ef-
fect. Their research shows that, the Allee effect will
substantive change the dynamic behaviors of the sys-
tem, and the system will produce various bifurcation
phenomena. An interesting question: what about the
dynamic behavior of the discrete system correspond-
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Figure 2: The dynamic behaviors exhibited by
the second species in Case (A), the initial con-
dition (x(0), y(0)) = (1,1), corresponding to
β= 0,2,5and 10, respectively.
Figure 3: The dynamic behaviors exhib-
ited by the first species in Case (B), the
initial condition (x(0), y(0)) = (1,1),
corresponding to β= 0,2,5and 10, re-
spectively.
Figure 4: The dynamic behaviors exhib-
ited by the second species in Case (B), the
initial condition (x(0), y(0)) = (1,1),
corresponding to β= 0,2,5and 10, re-
spectively.
ing to the system (1.1)? We will conduct further re-
search in subsequent articles.
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WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2023.20.10
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Contribution of individual authors to
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This work is supported by the Natural Science Foun-
dation of Fujian Province(2020J01499).
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WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE
DOI: 10.37394/23208.2023.20.10
Fengde Chen, Xiaqing He, Zhong Li, Tingting Yan
E-ISSN: 2224-2902
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Volume 20, 2023
