Global Stability of Symbiotic Model of Commensalism and Parasitism
with Harvesting in Commensal Populations
FENGDE CHEN, QIMEI ZHOU, SIJIA LIN
College of Mathematics and Statistics
Fuzhou University
No. 2, wulongjiang Avenue, Minhou County, Fuzhou
CHINA
Abstract: - This article revisit the stability property of symbiotic model of commensalism and parasitism with
harvesting in the commensal population. The model was proposed by Nurmaini Puspitasari, Wuryansari Muhari-
ni Kusumawinahyu, Trisilowati (Dynamic analysis of the symbiotic model of commensalism and parasitism with
harvesting in commensal populations, Jurnal Teori dan Aplikasi Matematika, 2021, 5(1): 193-204). By establish-
ing three powerful Lemmas, sufficient conditions which ensure the global stability of the equilibria are obtained.
Key-Words: -Commensalism; Parasitism; Comparison theorem; Global attractivity
Received: August 17, 2021. Revised: May 9, 2022. Accepted: May 25, 2022. Published: June 17, 2022.
1 Introduction
The aim of this paper is to revisit the global stabili-
ty property of the following symbiotic model of com-
mensalism and parasitism with harvesting in the com-
mensal population:
dx
dt =r1x(1−x
k1
+ay
k1)
−qEx
m1E+m2x,
dy
dt =r2y(1−y
k2
−bz
k2),
dz
dt =r3z(1−z
k3
+cy
k3),
(1)
where x(t), y(t)and z(t)denote the commensal pop-
ulation, host population and parasite species, respec-
tively. All parameters used in this model are positive.
For the detail construction of model (1) and the in-
terpret of the biological meaning of the coefficients,
one could refer to Nurmaini Puspitasari, Wuryansari
Muharini Kusumawinahyu, Trisilowati[25]).
During the lase decade, many scholars investigat-
ed the dynamic behaviors of the mutualism model or
commensalism model ([1]-[30]), most of those work-
s are concerned with the two species case, recently,
Puspitasari, Kusumawinahyu and Trisilowati[25] be-
gan to study three species case. They proposed the
system (1). The system has eight equilibria, which
takes the form
T0(0,0,0), T1(0,0, k3), T2(0, k2,0),
T3(x∗
3,0,0), T4(0,k2−bk3
1 + bc ,k3+ck2
1 + bc ),
T5(x∗
5,0, k3), T6(x∗
6, k2,0), T7(x∗
7, y∗
7, z∗
7).
Concerned with the local stability property of those
equilibria, the authors gave a thoroughly study of the
locally stability property of the eight equilibria, and
finally, they declared `` Of the eight points, only two
points are asymptotically stable if they meet certain
conditions." Indeed, they showed that T4and T7is
locally asymptotically stable while the other six equi-
libria are all unstable.
Now, one natural problem is that the conclusion-
s of Puspitasari, Kusumawinahyu and Trisilowati[25]
are all locally ones, whether we could obtain some
sufficient conditions to ensure the globally stability
property of the equilibria T4and T7?
The aim of this paper is to give affirm answer to
above issue. For more works on the ecosystem with
Michaelis-Menten type harvesting, one could refer to
[31]-[39] and the references cited therein.
The rest of the paper is arranged as follows. In
next section, we will state the main results of this pa-
per. We state and prove four useful Lemmas. We then
prove the main results in Section 4. Numeric simula-
tions are presented in Section 5 to show the feasibility
of the main results. We end this paper by a briefly dis-
cussion.
2 Main Results
Following are the main results of this paper.
Theorem 2.1 Assume that
r1(1 + ay∗
k1)<qE
m1E+m2(k1+ay∗)(2)
and
1>bk3
k2
(3)
hold, then T4(0,k2−bk3
1 + bc ,k3+ck2
1 + bc )is globally at-
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tractive, where
y∗=k2−bk3
1 + bc .
Theorem 2.2 Assume that
r1(1 + ay∗
7
k1)>q
m1
(4)
and
1>bk3
k2
(5)
hold, then T7(x∗
7, y∗
7, z∗
7)is globally attractive, where
y∗
7=k2−bk3
1 + bc , z∗
7=k3+ck2
1 + bc .
3 Lemmas
To finish the proof of Theorem 2.1 and 2.2, we need
several powerful Lemmas.
As a direct corollary of Lemma 2.2 of Chen[40],
we have
Lemma 3.1. If a > 0, b > 0and ˙x≥x(b−ax),
when t≥0and x(0) >0, we have
lim inf
t→+∞
x(t)≥b
a.
If a > 0, b > 0and ˙x≤x(b−ax),when t≥0and
x(0) >0, we have
lim sup
t→+∞
x(t)≤b
a.
Consider the equation
dx
dt =x(a−bx)−cx
d+ex,(6)
where a, b, c, d, e are all positive constants.
Lemma 3.2. Assume that
a > c
d(7)
holds, then system (6) admits a unique positive equi-
librium x∗which is globally stable, where
x∗=
−A2+√A2
2−4A1A3
2A1
,(8)
and A1=be > 0,
A2=−ae +bd,
A3=c−ad < 0.
(9)
Proof. Since
F(x) = a−bx −c
d+ex
=−G(x)
ex +d,
(10)
where
G(x) = A1x2+A2x+A3.
Noting that G(x)is the quadratic function, and under
the assumption of Lemma 2.2, G(0) = A3<0.
Hence, from the properties of quadratic func-
tion, G(x)=0admits unique positive solution
x∗∈(0,+∞). From (10) one could see that
F(x) = 0 also admits unique positive solution
x∗∈(0,+∞),F(x)>0for x∈(0, x∗)and
F(x)<0for x∈(x∗,+∞).Hence, it immediately
follows from Theorem 2.1 in [32] that the unique
positive equilibrium x∗of system (6) is globally
stable.
The proof of Lemma 2.2 is finished.
Lemma 3.3. Assume that
c > a(d+ea
b)(11)
holds, then in system (6), species xwill finally be driv-
en to extinction, i.e.,
lim
t→+∞
x(t) = 0.(12)
Proof. From (11), for any enough small positive con-
stant ε > 0, the inequality
a < c
d+e(a
b+ε)(13)
holds. From (6) we have
dx
dt ≤x(a−bx).(14)
Applying Lemma 2.1 to (14) leads to
lim
t→+∞
x(t)≤a
b.(15)
For ε > 0enough small which satisfies (13), it fol-
lows from (15) that there exists an enough large T1>
0such that
x(t)<a
b+εfor all t≥T1.(16)
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For t≥T1, from (6) and (16), one has
dx
dt ≤x(a−bx)−cx
d+e(a
b+ε),(17)
and so,
x(t)≤x(T1)exp {(a−c
d+e(a
b+ε))(t−T1)}.
(18)
(18) together with (13) leads to
lim
t→+∞
x(t) = 0.(19)
This ends the proof of Lemma 2.3.
Now let's consider the system
dy
dt =r2y(1−y
k2
−bz
k2),
dz
dt =r3z(1−z
k3
+cy
k3).
(20)
Lemma 3.4. Assume that
1>bk3
k2
(21)
hold, then system (20) admits a unique positive equi-
librium (y∗
7, z∗
7), which is globally attractive, where
y∗
7=k2−bk3
1 + bc , z∗
7=k3+ck2
1 + bc .(22)
Proof. One could easily check that under the assump-
tion (21) holds, system (20) admits a unique positive
equilibrium (y∗
7, z∗
7). The positive equilibrium of (20)
satisfies the equation
1−y∗
7
k2
−bz∗
7
k2
= 0,
1−z∗
7
k3
+cy∗
7
k3
= 0.
(23)
Now let's consider the Lyapunov function
V(x, y) = l1(y−y∗
7−y∗
7ln y
y∗
7)+l2(z−z∗
7−z∗
7ln z
z∗
7).
(24)
By computation, from (23) we have
dV
dt
=l1r2(y−y∗
7)(1−y
k2
−bz
k2)
+l2r3(z−z∗
7)(1−z
k3
+cy
k3)
=l1r2(y−y∗
7)(y∗
7
k2
+bz∗
7
k2
−y
k2
−bz
k2)
+l2r3(z−z∗
7)(z∗
7
k3
−cy∗
7
k3
−z
k3
+cy
k3)
(25)
=−l1r2
k2
(y−y∗
7)2+l1r2(y−y∗
7)b
k2
(z∗
7−z)
−l2r3
k3
(z−z∗
7)2+l2r3
k3
(z−z∗
7)(y−y∗
7)
(26)
By choosing the positive constants as:l1= 1, l2=
r2bk3
k2r3c, the following is obtained:
dV
dt =−r2
k2
(y−y∗
7)2−r2b
k2c(z−z∗
7)2.(27)
Obviously, dV
dt <0strictly for all y, z > 0except the
positive equilibrium (y∗
7, z∗
7), where dV
dt = 0. Thus,
V(x, y)satisfies Lyapunov's asymptotic stability the-
orem, and the positive equilibrium (y∗
7, z∗
7)of system
(20) is globally stable. This ends the proof of Lemma
2.4.
4 Proof of the main results
Proof of Theorem 2.1. For ε > 0enough small, con-
dition (2) implies that
r1(1+a(y∗+ε)
k1)<qE
m1E+m2(k1+a(y∗+ε) + ε).
(28)
Noting that in system (1) the second and third equa-
tions are independent of x, hence, under the assump-
tion (3) hold, it follows from Lemma 3.4 that system
(20) admits a unique positive equilibrium (y∗
7, z∗
7),
which is globally attractive, i.e.,
lim
t→+∞
y(t) = y∗
7=k2−bk3
1 + bc =y∗,
lim
t→+∞
z(t) = z∗
7=k3+ck2
1 + bc =z∗.
(29)
For ε > 0which satisfies (28), there exists a T1>0
such that
y(t)< y∗+εfor all t ≥T1.(30)
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From the first equation of system (1), we have
dx
dt ≤r1x(1−x
k1
+ay
k1)
≤r1x(1−x
k1
+ay∗+ε
k1).
(31)
Applying Lemma 3.1 to above inequality leads to
lim sup
t→+∞
x(t)≤(1 + ay∗+ε
k1)k1=k1+a(y∗+ε).
(32)
It follows from (32) that there exists a T2> T1such
that
x(t)< k1+a(y∗+ε) + εfor all t ≥T2.(33)
For t≥T2, from (30), (33) and the first equation of
system (1), we have
dx
dt ≤r1x(1−x
k1
+ay∗+ε
k1)
−qEx
m1E+m2(k1+a(y∗+ε) + ε).
(34)
Now let's consider the equation
du
dt =r1u(1−u
k1
+ay∗+ε
k1)
−qEu
m1E+m2(k1+a(y∗+ε) + ε).
(35)
It follows from (28) and Lemma 3.3 that
lim
t→+∞
u(t) = 0.(36)
By the comparison theorem of differential equation,
(35) and (36), it immediately follows that
lim
t→+∞
x(t) = 0.(37)
(29) and (37) show that T4(0,k2−bk3
1 + bc ,k3+ck2
1 + bc )
is globally attractive. This ends the proof of Theorem
2.1.
Proof of Theorem 2.2. For ε > 0enough small, con-
dition (43) implies that
r1(1 + a(y∗
7−ε)
k1)>q
m1
.(38)
Noting that in system (1) the second and third equa-
tions are independent of x, hence, under the assump-
tion (5) hold, it follows from Lemma 3.4 that system
(20) admits a unique positive equilibrium (y∗
7, z∗
7),
which is globally attractive, i.e.,
lim
t→+∞
y(t) = y∗
7=k2−bk3
1 + bc ,
lim
t→+∞
z(t) = z∗
7=k3+ck2
1 + bc .
(39)
For ε > 0which satisfies (38), without loss of gen-
erality, we may assume that ε < 1
2y∗, there exists a
T1>0such that
y∗
7−ε < y(t)< y∗
7+εfor all t ≥T1.(40)
From the first equation of system (1) and (40), we
have
dx
dt =r1x(1−x
k1
+ay
k1)
−qEx
m1E+m2x
≤r1x(1−x
k1
+ay∗+ε
k1)
−qEx
m1E+m2x.
(41)
Now let's consider the equation
dw1
dt =r1w1(1−w1
k1
+ay∗+ε
k1)
−qEw1
m1E+m2w1
.
(42)
It follows from (43) that
r1(1 + a(y∗
7+ε)
k1)>q
m1
.(43)
Hence, from Lemma 2.2 system (42) admits a unique
positive equilibrium w1(ε)which is globally attrac-
tive, where
w1(ε) =
−B2+√B2
2−4B1B3
2B1
,(44)
and
B1=m2r1>0,
B2=r1(m1E−k1m2−am2(y∗
7+ε)),
B3=−E(k1m1r1−k1q+am1r1(y∗
7+ε))<0.
(45)
It follows from (41)-(45) that
lim sup
t→+∞
x(t)≤w1(ε).(46)
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From the first equation of system (1) and (40), we also
have
dx
dt =r1x(1−x
k1
+ay
k1)
−qEx
m1E+m2x
≥r1x(1−x
k1
+ay∗−ε
k1)
−qEx
m1E+m2x.
(47)
Now let's consider the equation
dw2
dt =r1w2(1−w2
k1
+ay∗+ε
k1)
−qEw2
m1E+m2w2
.
(48)
It follows from (38) and Lemma 3.2 that system (48)
admits a unique positive equilibrium w2(ε)which is
globally attractive, where
w2(ε) =
−C2+√C2
2−4C1C3
2C1
,(49)
and
C1=m2r1>0,
C2=r1(m1E−k1m2−am2(y∗
7−ε)),
C3=−E(k1m1r1−k1q+am1r1(y∗
7−ε))<0.
(50)
It follows from (47)-(50) that
lim inf
t→+∞
x(t)≥w2(ε).(51)
(46) and (51) show that
w2(ε)≤lim inf
t→+∞
x(t)≤lim sup
t→+∞
x(t)≤w1(ε).(52)
Noting that
wi(ε)→x∗
7as ε →0, i = 1,2.(53)
Since εis enough small positive constant, setting ε→
0in (52) leads to
lim
t→+∞
x(t) = x∗
7.(54)
(39) and (54) show that T7(x∗
7,k2−bk3
1 + bc ,k3+ck2
1 + bc )
is globally attractive. This ends the proof of Theorem
2.2.
5 Numeric simulations
Now let us consider the following two examples.
Example 5.1 Consider the following system
dx
dt =x(1−x
1+y
1)
−7x
2 + x,
dy
dt =y(1−y
2−z
2),
dz
dt =z(1−z
1+y
1).
(55)
Here, corresponding to system (1.1), we choose r1=
r2=r3=k1=k3=b=c=a=E=m2=
1, q = 7, m1= 2,then by simple computation, we
have
r1(1 + ay∗
k1)=3
2<2 = qE
m1E+m2(k1+ay∗)
(56)
and
1>1
2=bk3
k2
(57)
hold, then it follows from Theorem 2.1 that
T4(0,0.5,1.5)is globally attractive. Figure 1 shows
that the first component xin system (55) is approach
to zero as tapproach to infinite. Figure 2 shows that
the second and third components yand zapproach to
0.5and 1.5, respectively, as tapproach to infinite.
Figure 1: Dynamic behaviors of the first com-
ponent xin system (55) with the initial condi-
tion (x(0), y(0), z(0)) = (0.5,2,0.5),(1,2,1),
(1.5,2,1.5) and (2,2,2), respectively.
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Figure 2: Phase portrait of the second and third
component yand zin system (55) with the ini-
tial condition (x(0), y(0), z(0)) = (0.5,2,0.5),
(1,2,1),(1.5,2,1.5) and (2,2,2), respectively.
Example 5.2 Consider the following system
dx
dt =x(1−x
1+y
1)
−1x
2 + x,
dy
dt =y(1−y
2−z
2),
dz
dt =z(1−z
1+y
1).
(58)
Here, corresponding to system (1), we choose r1=
r2=r3=k1=k3=b=c=a=E=m2=
1, q = 1, m1= 2,then by simple computation, we
have
r1(1 + ay∗
k1)=3
2>1 = q
m1
(59)
and
1>1
2=bk3
k2
(60)
hold, then it follows from Theorem 2.2 that
T7(1.186,0.5,1.5)is globally attractive. Figure 3
shows that the first component xin system (58) is ap-
proach to 1.186 as tapproach to infinite. Figure 4
shows that the second and third components yand z
approach to 0.5and 1.5, respectively, as tapproach
to infinite.
6 Conclusion
Puspitasari, Kusumawinahyu and Trisilowati [25]
proposed the system (1.1). The system have eight e-
quilibria. By computation, they showed that T4and
T7is locally asymptotically stable while the other six
equilibria are all unstable. In this paper, by introduc-
ing three powerful Lemmas, we are able to obtain suf-
ficient conditions to ensure the globally attractive of
Figure 3: Dynamic behaviors of the first com-
ponent xin system (58) with the initial condi-
tion (x(0), y(0), z(0)) = (0.5,2,0.5),(1,2,1),
(1.5,2,1.5) and (2,2,2), respectively.
Figure 4: Phase portrait of the second and third
component yand zin system (58) with the ini-
tial condition (x(0), y(0), z(0)) = (0.5,2,0.5),
(1,2,1),(1.5,2,1.5) and (2,2,2), respectively.
these two equilibrium.
It is well known that a more plausible system
should consider the past state of the species, this will
lead to the system with delay, whether our method
could be applied to the delay system or not is still un-
known, we will leave this for future investigation.
We also notice that the nonautonomous system is
more appropriate ([39]-[43]), for such kind of model,
the existence of positive periodic solution or almost
periodic solution is main topic, we will try to do some
works on this direction.
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Volume 21, 2022
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