Global stability of a commensal symbiosis model with Holling II
functional response and feedback controls
FENGDE CHEN, YANBO CHONG, SHANGMING CHEN
College of Mathematics and Statistics
Fuzhou University
No. 2, wulongjiang Avenue, Minhou County, Fuzhou
CHINA
Abstract: - A commensal symbiosis model with Holling II functional response and feedback controls is proposed
and studied in this paper. The system admits four equilibria, and three boundary equilibria are unstable, only
positive equilibrium is locally asymptotically stable. By applying the comparison theorem of differential equation,
we show that the unique positive equilibrium is globally attractive. Numeric simulations show the feasibility of
the main result.
Key-Words: -Commensalism; Feedback controls; Holling II functional response; Comparison theorem; Global
attractivity
Received: June 18, 2021. Revised: April 19, 2022. Accepted: May 20, 2022. Published: June 16, 2022.
1 Introduction
The aim of this paper is to investigate the global sta-
bility property of the following commensal symbio-
sis model with Holling type functional response and
feedback controls :
˙x=xb1−a11x+a12y
a13 +a14y−α1u1,
˙y=y(b2−a22y−α2u2),
˙u1=−η1u1+a1x,
˙u2=−η2u2+a2y,
(1)
where x(t)and y(t)denote the density of the first
and second species at time t.u1and u2are feedback
control variables. All parameters used in this model
are positive.
During the lase decade, many scholars inves-
tigated the dynamic behaviors of the mutualism
model or commensalism model ([1]-[30]). Also,
due to it is importance, many scholars ([31]-[41])
investigated the dynamic behaviors of the ecosystem
with feedback controls. however, it is very strange
that to this day, only one paper[20] considered the
influence of feedback controls to the commensalism
models. In [20], Han and Chen proposed and studied
the following Lotka-volterra commensal symbiosis
model with feedback controls:
˙x=x(b1−a11x+a12y−α1u1),
˙y=y(b2−a22y−α2u2),
˙u1=−η1u1+a1x,
˙u2=−η2u2+a2y.
(2)
By constructing a suitable Lyapunov function, the au-
thors showed that the positive equilibrium of the sys-
tem is globally stable.
On the other hand, there were also several schol-
ars ([7], [16], [17],[22], [23],[26]) argued that the the
relationship of two commensalism model should be
described by the suitable functional response, for ex-
ample, Li, Lin and Chen[17] studied the positive peri-
odic solution of a discrete commensalism model with
Holling II functional response. The system takes the
form
x1(k+ 1) = x1(k)exp a1(k)−b1(k)x1(k)
+c1(k)x2(k)
e1(k) + f1(k)x2(k),
x2(k+ 1) = x2(k)exp {a2(k)−b2(k)x2(k)},
(3)
Wu [7] argued that between two species nonlinear
type of relationship between two species is more fea-
sible, and she established the following two species
commensal symbiosis model
dx
dt =xa1−b1x+c1yp
1 + yp,
dy
dt =y(a2−b2y),
(4)
where ai, bi, i = 1,2pand c1are all positive con-
stants, p≥1. The results of [7] is then generalized
by Lei [23] and Wu, Li and Lin[16] to the commen-
salism model with Allee effect.
Stimulated by the above works, we propose the
system (1). As far as system (1) is concerned, since
it seems that the system is similar to system (2),
only with the cooperation term a12xy in system (2)
changed to the term with Holling type functional
response a12xy
a13 +a14yin system (1). One may expect
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the analysis method used in Han and Chen[20] could
be applied to system (1), however, this is impossible.
In their paper, Han and Chen could deal with the
stability property of the system (1) by constructing
suitable Lyapunov function, by using this method,
one could always obtain some interesting result about
the linear system. When it come to the nonlinear
case, it is very difficult to deal with the nonlinear
term to ensure the negative definite of the Lyapunov
function; We mention here that in [7], Wu investigat-
ed the global stability of the equilibrium of system
(4) by using the Dulac criterion, which could only be
applied to the two dimensional system, and could not
be applied to the higher dimensional system.
The aim of this paper, is to investigated the stabil-
ity property of the system (1). To deal with this, we
need to develop some new analysis technique, more
precisely, we will combine the analysis technique of
Han and Chen [20], Wu[7] and Yue [42], to overcome
the difficulty of nonlinearity.
The paper is arranged as follows. We will investi-
gate the existence and locally stability property of the
equilibria of system (1) in section 2. In section 3, we
first establish a global stability result of single species
feedback control system via the Lyapunov function,
after that, by developing the analysis technique of
Yue[32], more precisely, by using the differential
inequality theory and the comparison theorem, we
investigate the global attractivity property of the
positive equilibrium of system (1). In section 4,
we present some numerical simulations to show the
feasibility of the main result. We end this paper by a
briefly discussion.
2 Existence and local stability of
Equilibria
This section we will focus our attention to investigate
the existence and local stability property of the system
(1).
The equilibria of system (1) is determined by the
following system
xb1−a11x+a12y
a13 +a14y−α1u1= 0,
y(b2−a22y−α2u2) = 0,
−η1u1+a1x= 0,
−η2u2+a2y= 0.
(5)
The system always admits three boundary equilibria:
A1(0,0,0,0),
A2β1b1
a1α1+a11η1
,0,a1b1
a1α1+a11η1
,0,
A30,b2η2
a2α2+a22η2
,0,a2b2
a2α2+a22η2.
Also, system (1) admits a unique positive equilibrium
A4x∗, y∗, u∗
1, u∗
2, where
x∗=
b1+a12y∗
a13 +a14y∗
a11 +α1a1
η1
,
y∗=b2
a22 +α2a2
η2
,
u∗
1=a1x∗
η1
,
u∗
2=a2y∗
η2
.
(6)
Obviously, x∗, y∗,u∗
1and u∗
2satisfy the equations
b1−a11x∗+a12y∗
a13 +a14y∗−α1u∗
1= 0,
b2−a22y∗−α2u∗
2= 0,
−η1u∗
1+a1x∗= 0,
−η2u∗
2+a2y∗= 0.
(7)
We shall now investigate the local stability prop-
erty of the above equilibria.
The variational matrix of system (1) is
J(x, y, u1, u2)
=
A11 A12 −α1x0
0A22 0−yα2
a10−η10
0a20−η2
,
(8)
where
A11 =b1−2a11x+a12y
a14y+a13
−α1u1,
A12 =xa12
a14y+a13
−a12ya14
(a14y+a13)2,
A22 =−2a22y−α2u2+b2.
Theorem 2.1 A1(0,0,0,0) is unstable.
Proof. From (8) we could see that the Jacobian
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matrix of the system about the equilibrium point
A1(0,0,0,0) is given by
b10 0 0
0b20 0
a10−η10
0a20−eta2
.(9)
The characteristic equation of above matrix is
(λ−b1)(λ−b2)(λ+η1)(λ+η2) = 0.(10)
Hence, it has two positive characteristic root
λ1=b1, λ2=b2, consequently, A1(0,0,0,0) is
unstable. This ends the proof of Theorem 2.1.
Theorem 2.2 A2β1b1
a1α1+a11η1
,0,a1b1
a1α1+a11η1
,0
is unstable.
Proof. From (8) we could see that the Jacobian
matrix of the system about the equilibrium point
A2β1b1
a1α1+a11η1
,0,a1b1
a1α1+a11η1
,0is given by
B11 B12 B13 0
0b20 0
a10−η10
0a20−eta2
.(11)
where
B11 =b1−2a11η1b1
a1α1+a11η1
−α1a1b1
a1α1+a11η1
,
B12 =η1b1a12
(a1α1+a11η1)a13
,
B13 =−η1b1α1
a1α1+a11η1
.
The characteristic equation of above matrix is
(λ−b2)(λ+η2)C1λ2+C2λ+C3= 0,(12)
where
C1=a1α1+a11η1,
C2=a1α1η1+a11η1b1+a11η2
1,
C3=a1η1b1α1+a11b1η2
1.
Hence, it has a positive characteristic root λ1=b2,
consequently, A2is unstable. This ends the proof of
Theorem 2.2.
Theorem 2.3 A30,b2η2
a2α2+a22η2
,0,a2b2
a2α2+a22η2
is unstable.
Proof. From (8) we could see that the Jacobian
matrix of the system about the equilibrium point
A30,b2η2
a2α2+a22η2
,0,a2b2
a2α2+a22η2is given by
D11 0 0 0
0D22 0D24
a10−η10
0a20−eta2
,(13)
where
D11 =b1+a12b2η2
(a2α2+a22η2) ∆1
,
D22 =−2a22b2η2
a2α2+a22η2
−α2a2b2
a2α2+a22η2
+b2,
D24 =−b2η2α2
a2α2+a22η2
,
∆1=a14b2η2
a2α2+a22η2
+a13.
The characteristic equation of above matrix is
(λ−D11)(λ+η1)E1λ2+E2λ+E3= 0,(14)
where
E1=a2α2+a22η2,
E2=a2α2η2+a22η2b2+a22η2
2,
E3=a2η2b2α2+a22b2η2
2.
Hence, it has a positive characteristic root λ1=D11,
consequently, A3is unstable. This ends the proof of
Theorem 2.3.
Theorem 2.4 A4x∗, y∗, u∗
1, u∗
2is locally asymptot-
ically stable.
Proof. From (8) we could see that the Jacobian
matrix of the system about the equilibrium point
A4x∗, y∗, u∗
1, u∗
2is given by
−a11x∗m−α1x∗0
0−a22y∗0−α2y∗
a10−η10
0a20−eta2
,
(15)
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where
m=x∗a12
a14y∗+a13
−a12y∗a14
(a14y∗+a13)2.
The characteristic equation of above matrix is
(λ2+F1λ+F2)λ2+G1λ+G2= 0,(16)
where
F1=a22y∗+η2,
F2=a2α2y∗+a22η2y∗,
G1=a11x∗+η1,
G2=a1α1x∗+a11η1x∗.
From F1, F2, G1, G2are all positive constants, one
could easily see that four roots of equation (16)
is negative. Consequently, A4x∗, y∗, u∗
1, u∗
2is
locally asymptotically stable. This ends the proof of
Theorem 2.4.
3 Global attractivity
We had showed in the previous section system (1) ad-
mits four equilibria, however, only the positive equi-
librium A4x∗, y∗, u∗
1, u∗
2is locally asymptotically
stable, while the other three equilibria are all unsta-
ble. Now, one interesting issue proposed: Is it possi-
ble for us to find out the suitable conditions to ensure
the positive equilibrium A4x∗, y∗, u∗
1, u∗
2be glob-
ally asymptotically stable?
We will give the affirm answer to this issue, more
precisely, we will prove the following result.
Theorem 3.1 A4x∗, y∗, u∗
1, u∗
2is globally attrac-
tive.
To prove this result, we need the following Lem-
ma. The result of the Lemma seems simple, however,
sine our proof of Theorem 3.1 deeply depend on the
Lemma, for the sake of completeness, we also give a
detail proof of the Lemma.
Let us consider the following single species
feedback control ecosystem.
dx
dt =x(a−bx −cu),
du
dt =−eu +fx,
(17)
where a, b, c, d, e are all positive constants. The
system (17) admits a unique positive equilibrium
A(x∗, u∗), where
x∗=a
b+cf
e
, u∗=f
ex∗.
Concerned with the stability property of this equilib-
rium, we have the following result.
Lemma 3.1 A(x∗, u∗)is globally stable.
Proof. Obviously, A(x∗, u∗)satisfies the equation
a−bx∗−cu∗= 0,
−eu∗+fx∗= 0,(18)
Now let's construct a Lyapunov function
V=x−x∗−x∗ln x
x∗+c
2f(u−u∗)2,(19)
Calculating the derivative along the solution of
system (17), we have
dV
dt
= (x−x∗)(a−bx −cu)
+c
f(u−u∗)(−eu +fx)
= (x−x∗)(bx∗+cu∗−bx −cu)
+c
f(u−u∗)(−eu +fx +eu∗−fx∗)
= (x−x∗)b(x∗−x) + c(u∗−u)
+c
f(u−u∗)−e(u−u∗) + f(x−x∗)
=−b(x−x∗)2−ce
f(u−u∗)2.
(20)
Thus dV
dt <0strictly for all x > 0, u > 0except
the positive equilibrium P(x∗, u∗), where dV
dt = 0.
Thus, V(t)satisfies Lyapunov's asymptotic stability
theorem, and the positive equilibrium P(x∗, u∗)of
system (17) is globally stable. This ends the proof of
Lemma 3.1.
Proof of Theorem 3.1. Noting that in (1), the second
and forth equations are independent of the variable x
and u1, hence, we could consider the following sub-
system previously.
˙y=y(b2−a22y−α2u2),
˙u2=−η2u2+a2y,
(21)
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Form Lemma 3.1, the unique positive equilibrium
(y∗, u∗)of system (20) is globally stable, where
y∗=b2
a22 +α2a2
η2
,
u∗
2=a2y∗
η2
.
(22)
Hence,
lim
t→+∞
y(t) = y∗,
lim
t→+∞
u2(t) = u∗
2.(23)
For ε > 0enough small, (23) implies that there exists
a enough Tsuch that
y∗−ε < y(t)< y∗+ε. (24)
For t>T, from the first and third equations in (1)
and (23), we have
˙x=xb1−a11x+a12y
a13 +a14y−α1u1
≤xb1−a11x+a12(y∗+ε)
a13 +a14(y∗+ε)
−α1u1,
˙u1=−η1u1+a1x,
(25)
Now let us consider the system
˙w1=w1b1−a11w1+a12(y∗+ε)
a13 +a14(y∗+ε)
−α1v1,
˙v1=−η1v1+a1w1.
(26)
Form Lemma 3.1, the unique positive equilibri-
um (w∗
1(ε), v∗
1(ε)) of system (26) is globally stable,
where
w∗
1(ε) =
b1+a12(y∗+ε)
a13 +a14(y∗+ε)
a11 +α1a1
η1
,
v∗
1(ε) = a1w∗
1
η1
.
(27)
From (25)-(27), we have
lim sup
t→+∞
x(t)≤lim
t→+∞
w1(t) = w∗
1(ε),
lim sup
t→+∞
u1(t)≤lim
t→+∞
v1(t) = v∗
1(ε).(28)
For t > T , from the first and third equations in (1)
and (24), we also have
˙x=xb1−a11x+a12y
a13 +a14y−α1u1
≥xb1−a11x+a12(y∗−ε)
a13 +a14(y∗−ε)
−α1u1,
˙u1=−η1u1+a1x,
(29)
Now let us consider the system
˙w2=w2b1−a11w2+a12(y∗−ε)
a13 +a14(y∗−ε)
−α1v2,
˙v2=−η1v2+a1w2.
(30)
Form Lemma 3.1, the unique positive equilibri-
um (w∗
2(ε), v∗
2(ε)) of system (29) is globally stable,
where
w∗
2(ε) =
b1+a12(y∗−ε)
a13 +a14(y∗−ε)
a11 +α1a1
η1
,
v∗
2(ε) = a1w∗
2
η1
.
(31)
From (29)-(31), we have
lim inf
t→+∞
x(t)≥lim
t→+∞
w2(t) = w∗
2(ε),
lim inf
t→+∞
u1(t)≥lim
t→+∞
v2(t) = v∗
2(ε).(32)
From (28) and (32) we have
w∗
2(ε) = lim
t→+∞
w2(t)≤lim inf
t→+∞
x(t)
≤lim sup
t→+∞
x(t)≤lim
t→+∞
w1(t) = w∗
1(ε),
v∗
2(ε) = lim
t→+∞
v2(t)≤lim inf
t→+∞
u1(t)
≤lim sup
t→+∞
u1(t)≤lim
t→+∞
v1(t) = v∗
1(ε).
(33)
Noting that
wi(ε)→x∗, vi(ε)→u∗
1as ε →0, i = 1,2.
(34)
Since εis enough small positive constant, setting ε→
0in (33) leads to
lim
t→+∞
x(t) = x∗lim
t→+∞
u1(t) = u∗
1.(35)
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(23) and (35) show that A4x∗, y∗, u∗
1, u∗
2is globally
attractive. This ends the proof of Theorem 3.1.
4 Numeric simulations
In this section, we provide an example to illustrate the
theoretical result by numerical simulations.
Example 4.1.
˙x=x1−x+3y
2+2y−3u1,
˙y=y(3 −2y−2u2),
˙u1=−2u1+ 3x,
˙u2=−u2+ 2y.
(36)
Here, corresponding to system (1), we choose b1=
1, a11 = 1, a12 = 3, a13 =a14 = 2, α1= 3, b2=
3, a22 = 2, α2= 2, η1= 2, a1= 3, η2= 1, a2=
2.By simple computation, one could easily see that
the system (36) admits a unique positive equilibrium
A4(0.27,0.5,0.41,1), from Theorem 3.1, A4is glob-
ally attractive. Numeric simulations (Fig.1 and 2) also
support this assertion.
Figure 1: Phase portraits of the first componen-
txand third component u1in system (36) with
the initial condition (x(0), y(0), u1(0), u2(0)) =
(0.5,2,0.5,0.5),(1,2,1,1),(1.5,2,1.5,1.5) and
(2,2,2,2), respectively.
5 Conclusion
In [20], Han and Chen proposed a Lotka-Volterra
commensalism system with feedback controls (i.e.,
system (2), by constructing some suitable Lyapunov
function, they showed that system admits a unique
Figure 2: Phase portraits of the second and
third component yand forth componen-
tu2in system (36) with the initial condition
(x(0), y(0), u1(0), u2(0)) = (0.5,2,0.5,0.5),
(1,2,1,1),(1.5,2,1.5,1.5) and (2,2,2,2),
respectively.
positive equilibrium which is globally stable. Stim-
ulated by the works of Han and Chen[20], Wu, Li and
Zhou[7] and Li, Lin and Chen[17], we propose a com-
mensalism system with Holling II functional response
and feedback controls.
With the introduction of the nonlinear functional
response, the method used in [20] could not be ap-
plied to our case. However, we find that in system
(1), the second and forth equations are independent
of variable xand u1, this stimulate us to investigate
the dynamic behaviors of subsystem (1) firstly. By
applying the differential inequality theory and com-
parison theorem of differential equation, we finally
show that the unique positive equilibrium of system
(1) is globally attractive.
It is well known that the system with Allee effec-
t may have very complex dynamic behaviors, to this
day, still no scholar propose the commensalism model
with both Allee effect and feedback controls, whether
the idea of this paper could be applied to that case is
still unknown, we will try to do some work on this
direction.
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