The Permanence of a Nonautonomous Single-species Model with Stage-
Structure and Feedback Control
FENGDE CHEN, HAN LIN, QUN ZHU, QIANQIAN LI
College of Mathematics and Statistics
Fuzhou University
No. 2, wulongjiang Avenue, Minhou County, Fuzhou
CHINA
1 Introduction
During the last decade, many scholars investigated
the dynamic behaviors of the ecosystem, see [1]-[53]
and the references cited therein. Such topics as ex-
tinction, persistence, and stability are extensively in-
vestigated.
It is well known that many species take several
stages throughout their life, and to model such kind
of phenomenon, many scholars ([1]-[17],[30], [52]-
[53]) proposed the stage-structured population
system. Aiello Freedman [30] proposed the following
single-species stage-structured model
2
2
22
.()
dx e x t x
dt
−
= − −
(1.1)
The authors of [30] showed that the system (1.1) ad-
mits a unique positive equilibrium that is globally as-
ymptotically stable. Based on the work of [30], many
scholars proposed the delayed stage-structured
model, for example, Lin et al[10] studied the persis-
tent property of the following stage-structured preda-
tor-prey model
( )
11 1
1 1 2 11 1 1 2 1
( ) ( ) ( ) ,
d
x t r x t d x t re x t
−
= − − −
( )
11 1
2 1 2 1 12 2
( ) ( )
d
x t re x t d x t
−
= − −
21 2 2
2
21
( ) ( )
( ) ,
()
a y t x t
bx t x t k
−− +
( )
22 2
1 2 2 22 1 2 2 2
( ) ( ) ( ) ,
d
y t r y t d y t r e y t
−
= − − −
( )
22 2
2 2 2 2 21 2
( ) ( )
d
y t r e y t d y t
−
= − −
2
22
22
()
()
a y t
x t k
−+
(1.2)
Their study indicates that for a stage-structured
predator-prey community, both stage structure and
the death rate of the mature species are the important
factors that lead to the permanence or extinction of
the system. For more work on the stage-structured
model incorporating time delay, one could refer to
[1]-[12] and the references cited therein.
On the other hand, ecosystems in the real world are
continuously disturbed by unpredictable forces
which can result in changes in biological parameters
such as survival rates. Of practical interest in ecology
is the question of whether or not an ecosystem can
stand those unpredictable disturbances that persist for
a finite period of time. In the language of control
variables, we call the disturbance functions control
variables. Gopalsamy and Weng[26] proposed the
following single-species feedback control ecosystem
12
( ) ( )
1a n t a n t
n rn K
+−
=−
( )],cu t−
(1.3)
( ) ( ).u au t bn t= − +
They showed that the inequality a1 > a2 is enough
to ensure the existence of a unique globally
asymptotically stable positive equilibrium. Chen,
Yang, and Chen [29] studied the following single-
species feedback control ecosystem
( )
2
1
2
()
( ) ( ) 1 ()
N t t
N r t N t Kt
−
=−
Abstract: - A nonautonomous single-species model with stage structure and feedback control is revisited in this
paper. By applying the differential inequality theory, a set of delay-dependent conditions ensures the
permanence of the system is obtained; Next, by further developing the analytical technique of Chen et al, we
prove that the system is always permanent. Numeric simulation supports our findings. Also, the numeric
simulation shows that the feedback control variable harms the final density of the species, and this may increase
the chance of the extinction of the species. Our results supplement and complement some known results.
Key-Words: Systems Theory, Dynamical Systems, stage structure, feedback control, permanence.
Received: March 26, 2022. Revised: October 22, 2022. Accepted: November 16, 2022. Published: December 31, 2022.
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( )
2
( ) ( ) ,c t u t t
−−
(1.4)
( ) ( ) ( ) ( )u a t u t b t N t= − +
By establishing a new integral inequality, they can
show that the system (1.4) is always permanent.
That is to say, the feedback control variable does not
influence the persistent property of the system (1.4).
Though there are many works on feedback control
ecosystems ([26]-[49]), there is still little work on
stage-structured ecosystems with feedback controls.
Ding and Cheng[27] proposed the following single-
species stage-structured model with feedback control:
1
2 1 1
() ( ) ( )=−
dx t bx t d x t
dt
1
2( ),
d
be x t
−
−−
12
2
22
() ( ) ( )
−
= − −
d
dx t be x t cx t
dt
(1.5)
2( ) ( ),cx t u t−
2
() ( ) ( ).= − +
du t fu t ex t
dt
In [27], it shows that if then system (1.5)
admits a unique positive equilibrium that is globally
attractive. Recently, Yang[28] argued that the
nonautonomous case is more suitable since the
circumstance is varying with time. They proposed the
following non-autonomous feedback control
ecosystem
1
2 1 1
() ( ) ( ) ( ) ( )=−
dx t b t x t d t x t
dt
1()
2
( ) ( ),
t
td s ds
b t e x t
−
−
− − −
1()
2
2
() ( ) ( )
−
−
= − −
t
td s ds
dx t b t e x t
dt
2
22
( ) ( ) ( ) ( ) ( ),a t x t c t x t u t−−
2
() ( ) ( ) ( ) ( ).= − +
du t f t u t e t x t
dt
(1.6)
Under the assumption and e(t)
are all continuous positive T -periodic functions, by
using the coincidence degree theory, the author
showed that system (1.6) admits at least one positive
T -periodic solution.
Since the environment is varied with season, it is
natural to consider the general non-autonomous case
of the system (1.6), i.e., it is natural to consider the
system (1.6) under the following assumption:
and are all
continuous functions bounded above and below by
positive constants.
For general non-autonomous cases, the persistent
property is one of the most important topics in the
study of population dynamics, however, Yang[28]
did not investigate the persistent property of the
system (1.6). The aim of this paper is, by applying the
comparison theorem of the differential equation and
developing the analytical technique of Chen, Yang,
and Chen[29], to obtain two sets of sufficient
conditions that guarantee the permanence of the
system (1.6).
The rest of the paper is arranged as follows: We
will state several lemmas in the next section, and give
the first set of sufficient conditions in Section 3. Then
we will use the idea of Chen, Yang, and Chen[29] to
establish another set of sufficient conditions in
Section 4. An example together with its numeric
simulation is presented in Section 5 to show the
feasibility of the main results. We end this paper with
a brief discussion.
2 Lemmas
Now let us state several lemmas which will be useful
in proving the main results.
Lemma 2.1. [10]Consider the following equation:
2
12
( ) ( ) ( ) ( ),x t bx t a x t a x t
= − − −
( ) ( ) 0, 0,x t t t
= −
and assume that and is a
constant. Then
1
1
2
( ) , lim ( ) ;
→+
−
=
t
ba
i If b a then x t a
1
( ) , lim ( ) 0.
→+
=
t
ii If b a then x t
Lemma 2.2. [29] Assume that is a
bounded continuous function and Further
suppose that
(i)
() ( ) ( ),
dx t ax t b t
dt − +
Then for all
( ) ( )exp{ } ( )exp{ ( )} .
s
ts
x t x t s as b a t d
−
− − + −
Especially, if b(t) is bounded above with respect to M,
then
.l (imsup )
t
M
xt a
→+
(ii)
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() ( ) ( ),
dx t ax t b t
dt − +
Then for all
,
( ) ( )exp{ } ( )exp{ ( )} .
s
ts
x t x t s as b a t d
−
− − + −
Especially, if b(t) is bounded below with respect to m,
then
.l (iminf )
t
m
xt a
→+
3 Permanence of system (1.6) (I)
The aim of this section is, by developing the
analysis technique of Chen et al([1]-[4]), more
precisely, by applying the differential inequality
theory, to investigate the persistent property of the
system (1.6).
We adopt the following notations throughout this
paper:
[0, )
sup ( , )
+
=
M
t
gtg
[0, )
inf ( , ) +
=
L
t
g g t
(3.1)
where g(t) is a continuous function on [0,+
).
Lemma 3.1. The first equation of system (1.6) is
equivalent to
1()
12
( ) ( ) ( ) .
−
=
t
s
td u du
ts
x t b s e x s ds
Proof. From (3.2), one has
1()xt
1()
2
( ) ( )
−
=
t
td u du
b t e x t
1()
2
( ) ( )
t
td u du
b t e x t
−
−
− − −
()
1()
21
( ) ( ) ( )
t
s
tt
d u du
ts
t
b s e x s ds d u du
−
−
+−
1()
22
( ) ( ) ( ) ( )
−
−
= − − −
t
td u du
b t x t b t e x t
( )
1()
21
( ) ( ) ( )
t
s
td u du
tb s e x s ds d t
−
−
+−
2 1 1
( ) ( )=−b t x d t x
1()
2
( ) ( ).
t
td s ds
b t e x t
−
−
− − −
The above analysis shows that the conclusion of
Lemma 3.1 holds. This ends the proof of Lemma 3.1.
Theorem 3.1. In addition to , assume further
that
11
UL
dd
L L L U U U
b e f a c e b e
−−
(3.2)
holds, then system (1.6) is permanent.
The proof of Theorem 3.1 immediately follows
from the proof of Theorem 3.2 and 3.3.
Remark 3.1. If we assume that the coefficients of
the system (1.6) are all positive constants, then con-
dition (3.2) degenerates to
,fa ce
(3.3)
from the introduction section, we know that condition
(3.3) is enough to ensure that system (1.5) admits a
unique globally asymptotically stable positive equi-
librium.
Theorem 3.2. Let be any positive
solution of the system (1.6), then
sup ( ) , 1,2,lim ii
t
x t M i
→+
=
3
lim .sup ( )
t
y t M
→+
(3.4)
where
1
12
,
−
=L
d
U
M b e M
1
2 ,
−
=
L
d
U
L
be
Ma
2
3 . =
U
L
eM
Mf
Proof. Let be any positive solu-
tion of system (1.6), then from the second equation
of (1.6), we have
2
dx
dt
1()
2
( ) ( )
−
−
= − −
t
td s ds
b t e x t
2
22
( ) ( ) ( ) ( ) ( )a t x t c t x t y t−−
(3.5)
1()
2
( ) ( )
−
−
− −
t
td s ds
b t e x t
2
2
( ) ( )a t x t−
2
22
( ) ( ).
−
− −
L
U d L
b e x t a x t
By applying Lemma 2.1 to (3.5), it immediately
follows that
def
22
lim .sup ( )
L
Ud
L
t
be
x t M
a
−
→+
=
(3.6)
For any enough small positive constant , there
exists a such that
2 2 1 1
( ) .x t M for all t T
+
(3.7)
(3.7) together with the third equation of system (1.6)
leads to
2
( ) ( ) ( ) ( )
dy f t y t e t x t
dt = − +
( )
2
( ) .
LU
f y t e M
− + +
(3.8)
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Applying Lemma 2.2 to (3.8) leads to
( )
21
.l sup ( )im
U
L
t
eM
yt f
→+
+
(3.9)
Setting in (3.9) leads to
def
2
3
lim .sup ( )
U
L
t
eM
y t M
f
→+
=
(3.10)
From Lemma 3.1 we have
1()
12
( ) ( ) ( ) .
t
s
td u du
t
x t b s e x s ds
−
−
=
(3.11)
This, together with (3.7) leads to
( )
1
1 2 1
( ) .
L
d
U
x t b e M
−
+
(3.12)
Setting in (3.12) leads to
1
def
1 2 1
lim .sup ( ) L
d
U
t
x t b e M M
−
→+
=
(3.13)
(3.6), (3.10), and (3.13) show that the conclusion of
Theorem 3.1 holds. This ends the proof of the Theo-
rem
3.1.
Theorem 3.3. In addition to (H1), assume further that
(3.2) holds, then
inf ( ) , 1, 2,lim ii
tx t m i
→+ =
3
lim ,inf ( )
ty t m
→+
(3.14)
where
1
12
,
−
=U
d
L
m b e m
1
3
2 ,
−−
=
U
d
LU
L
b e c M
ma
(3.15)
12
3 .=
L
U
em
mf
Proof. Let be any positive so-
lution of system (1.6). One could easily check that
inequality (3.2) is equivalent to
1
3.
U
d
LU
b e c M
−
Hence, for enough small positive constant ,
the inequality
( )
1
32
U
d
LU
b e c M
−+
(3.16)
holds. It follows from (3.6) that for the above ,
there exists a
, for ,
32
()y t M
+
(3.17)
holds. Hence, for
, from the second equation
of (1.6), one has
2
dx
dt
1()
2
( ) ( )
−
−
= − −
t
td s ds
b t e x t
2
22
( ) ( ) ( ) ( ) ( )a t x t c t x t y t−−
1()
2
( ) ( )
−
−
− −
t
td s ds
b t e x t
(3.18)
( )
2
2 3 2 2
( ) ( ) ( ) ( )a t x t c t M x t
− − +
2
22
( ) ( )
−
− −
L
U d L
b e x t a x t
( )
3 2 2 ( ).
U
c M x t
−+
Applying Lemma 2.2 to (3.18) leads to
( )
32
2.l inf ( )im
L
U d U
L
t
b e c M
xt a
−
→+
−+
(3.19)
Setting in(3.19) leads to
def
3
22
lim .inf ( )
L
U d U
L
t
b e c M
x t m
a
−
→+
−
=
(3.20)
For any enough small positive constant (with-
out loss of generality, we may assume that
), it follows from (3.20) that there exist a
,such that
2 2 3 3
( ) . x t m for all t T
−
(3.21)
For
, it follows from the third equation of sys-
tem (1.6) that
2
( ) ( ) ( ) ( )= − +
dy f t y t e t x t
dt
( )
2
( ) .
− + −
UL
f y t e m
(3.22)
Applying Lemma 2.1 to (3.22) leads to
( )
23
.l inf ( )im
L
U
t
em
yt f
→+
−
(3.23)
Since is enough small positive constant, setting
in (3.23) leads to
def
2
3
lim .inf ( )
L
U
t
em
y t m
f
→+ =
(3.24)
It follows from Lemma 3.1 that
1()
12
( ) ( ) ( ) .
t
s
td u du
t
x t b s e x s ds
−
−
=
(3.25)
(3.25) combine with (3.21) leads to
( )
1
1 2 3 3
( ) .
−
−
U
d
L
x t b e m for all t T
Setting in the above inequality, we have
1
12
inf ( ) .lim U
d
L
tx t b e m
−
→+
(3.26)
(3.20), (3.24), and (3.26) show that the conclusion of
Theorem 3.3 holds. This ends the proof of Theorem
3.3.
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In the previous section, we showed that under the
assumption (3.2) holds. System (1.6) is permanent.
One interesting issue is to investigate the dynamic be-
haviors of the system (1.6) if inequality (3.2) does not
hold.
To give some hints on this direction, already,
Yang[28] had showed that system (1.6) admits at
least one positive
T
-periodic solution if the coeffi-
cients of the system are all positive T-periodic func-
tions. Since the periodic solution means the species
could be survived in a fluctuating form. This moti-
vated us to propose the conjecture:
Conjecture. System (1.6) is permanent if the condi-
tion
( )
2
H
holds.
The aim of this section is to give the affirmative
answer to this conjecture. More precisely, we will ob-
tain the following result.
Theorem 4.1. System (1.6) is permanent if condi-
tion
( )
2
H
holds.
Proof. We will prove Theorem 4.1 by developing the
idea of Chen et al[29].
Let
( ) ( ) ( )( )
tutxtx ,, 21
be any positive solution of
system (1.6). Theorem 3.2 had shown that
( )
( )
,suplim
,2,1,suplim
3
Mty
iMtx
t
ii
t
=
+→
+→
(4.1)
Hence, for any enough small positive constant
0
,
there exists a
0T
, such that
( )
( )
( )
.
,
,
33
222
111
MMty
MMtx
MMtx
def
def
def
=+
=+
=+
( )
2.4
for all
Tt
.
From the second equation of system (1.6), we have
( )
( )
( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( )
( )
,
2
322
2
2
2
2
2
2
2
21
tx
McMatx
tytxtctxta
tytxtctxta
txetb
dt
dx
UU
dssd
t
t
=
−−
−−
−−
−
−=
−−
( )
3.4
Obviously,
.0
( )
4.4
Integrate (4.3) from
1
to
t
lead to
( )
( )
.exp
1
12
2 tds
x
tx
( )
5.4
Thus,
( ) ( ) ( )
}.exp{ 1212
−− ttxx
( )
6.4
From the third equation of system (1.6), we have
( ) ( )
.
2txetyf
dt
dy UL +−
( )
7.4
Applying Lemma 2.2 to the above inequality, we
have
( )
( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
( ) ( )
( )
( ) ( )
,exp1
exp
exp
exp
exp
exp
exp
exp
exp
2
1112
11
112
1
1112
txs
e
sfsty
dtxe
sfsty
dtf
txe
tsty
dtfxe
sfsty
ty
U
L
t
st
U
L
L
t
st
U
t
st
LU
L
−−
+
−−
−−+
−−
−
−−+
−−−
−+
−−
−
−
−
( )
8.4
here we have to use the fact
( )
.10expexpmax 1
,
1
==−
− tf L
tst
Choose K enough large, such that
,0,
exp
2
ln
1
max
11
3
−
UL
U
Ldb
Mc
f
K
then
.exp
2
1
exp 13
ULLU dbKfMc −−
( )
9.4
For above K, there exists a
KTT +
1
, for
1
Tt
,
Take
Ks =
in (4.8) leads to
( )
( )
( ) ( )
( ) ( )
( )
,exp
exp1
exp
exp1
exp
23
2
3
2
1
tHxKfM
txK
e
KfM
txK
e
KfKty
ty
L
U
L
U
L
+−
−−
+
−
−−
+
−−
( )
10.4
where
( )
.0exp1 −−
=K
e
H
U
Substituting (4.9) and (4.10) to the second equa-
4 Permanence of system (1.6)(II)
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tion of the system (1.6), for
1
Tt
, one has
( )
( )
( )
( ) ( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
,exp
exp
23
2
22
223
2
22
2
2
2
2
2
1
1
1
txKfMc
txHcatxeb
txtHxKfMc
txatxeb
tytxtctxta
txetb
dt
dx
LU
UU
d
L
LU
U
d
L
dssd
U
U
t
t
−−
+−−=
+−−
−−
−−
−
−=
−
−
−
( )
11.4
Applying Lemma 2.1 to (4.11) leads to
( )
.inflim 22 m
Hca
tx
def
UU
t=
+
+→
( )
12.4
where
KfMceb LU
d
LU−−= −exp
3
1
. For any
enough small positive constant
0
1
, without loss
of generality, we may assume that
21 2
1m
, it fol-
lows from (4.12) that there exists a
12 TT
such that
( )
122
− mtx
for all
.
2
Tt
( )
13.4
From (4.13) and the third equation of system (1.6),
for
2
Tt
, we have
( ) ( )
.
12
−+− metyf
dt
dy LU
( )
14.4
Applying Lemma 2.2 to (4.14) leads to
( ) ( )
.inflim 12
U
U
tf
me
ty
−
+→
( )
15.4
Since
1
is an arbitrarily small positive constant, set-
ting
0
1→
in (4.15) leads to
( )
.inflim 2
U
L
tf
me
ty
+→
( )
16.4
Lemma 3.1 had shown that
( ) ( )
( )
( )
.
21
1dssxesbtx t
t
duud
t
s
−
−
=
( )
17.4
(4.17) together with (4.13) leads to
( ) ( )
121
1
− −mebtx U
d
L
for all
.
2
+Tt
( )
18.4
Setting
0
1→
leads to
( )
.inflim 21
1mebtx U
d
L
t
−
+→
( )
19.4
Theorem 3.2, (4.12), (4.16), and (4.19) show that sys-
tem (1.6) is permanent. This ends the proof of Theo-
rem 4.1.
5 Numeric simulations
The aim of this section is to give some numeric
simulations to show the feasibility of the main re-
sults.
Example 5.1.
( ) ( ) ( ) ( )
( ) ( )
,11sin
2
1
2
5
sin
2
1
2
5
2
1
12
1
−
−+−
−
+=
−txet
txtxt
dt
tdx
( ) ( ) ( )
( )( ) ( )
( ) ( ) ( )
,sin
2
1
1
1cos3
11sin
2
1
2
5
2
2
2
2
1
2
tutxt
txt
txet
dt
tdx
−−
−+−
−
−+= −
( ) ( )( ) ( )
( ) ( )
.
2
cos
1
sin4
2tx
t
tut
dt
tdu
−+
+−=
( )
1.5
Here, corresponding to the system (1.6), we take
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
.
2
sin
1,
2
cos
1
.cos3,sin4
.1,1,sin
2
1
2
5
1
t
tc
t
te
ttattf
ttdttb
−=−=
+=+=
==+=
It follows from Theorem 4.1 that system (5.1) is per-
manent, also, from the main result of Yang[27], sys-
tem (1.6) admits at least one positive
T
-periodic so-
lution. Fig.1-3 support those assertions.
Example 5.2.
( ) ( ) ( ) ( )
( ) ( )
,11sin
2
1
2
5
sin
2
1
2
5
2
1
12
1
−
−+−
−
+=
−txet
txtxt
dt
tdx
( ) ( ) ( )
( )( ) ( )
( ) ( ) ( )
,
cos3
11sin
2
1
2
5
2
2
2
2
1
2
tutxtc
txt
txet
dt
tdx
−
+−
−
−+= −
( )
2.5
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( ) ( )( ) ( )
( ) ( )
.
2
cos
1
sin4
2tx
t
tut
dt
tdu
−+
+−=
Figure 1: Dynamic behaviors of the first component
of the system (5.1).
Figure 2: Dynamic behaviors of the second compo-
nent of the system (5.1).
Figure 3: Dynamic behaviors of the third component
of the system (5.1).
Here, all the other coefficients are the same as that of
the system (5.1), only with
( )
tc
be determined late.
Now let’s choose
( ) ( ) ( )
tttc sin2,sin3 ++=
and
( )
,sin1 t+
respectively, Fig.4 shows that in this case,
for the same initial value, with the increasing of the
( )
tc
, the density of mature species
( )
tx2
is decreas-
ing.
Figure 4: Numeric simulation of the system (5.2).
Here we choose
( ) ( ) ( )
tttc sin2,sin3 ++=
and
( )
,sin1 t+
respectively.
6 Discussion
Yang[28] proposed system (1.6), under the as-
sumption all the coefficients are positive T -period
function, they showed that system (1.6) admits at
least one positive periodic solution.However, they
did not investigate the persistent property of the sys-
tem. In this paper, by using the differential inequality
theory,we first obtain a set of sufficient conditions
(Theorem3.1) which ensure the permanent of the sys-
tem. After that, by comparing the results of Yang [28]
and Chen et al[29], we propose a conjecture: the feed-
back control has no influence on the system. We give
a strict proof of this conjecture in section 4. Numeric
simulation (Fig. 1-3) also supports our findings.
Though feedback control has no influence to the
persistent property of the system, example 5.2 shows
that with the increasing of the coefficient
( )
tc
, the fi-
nal density of the species is decreasing, it is well
known that with the decreasing of the density, the less
chance for the species to meet suitable partner, and
this increasing the extinct chance of the species. It is
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264
Volume 21, 2022
in this sense that the feedback control variable has the
in stable effect.
We mention here that this is the first time that we
find the feedback control variables has no influence
to the persistent property of the stage structured eco-
logical modelling. However, whether this conclusion
still hold or not for the complicate system, for exam-
ple, stage structured predator prey system is still un-
known. We will try to investigate some more compli-
cated ecological modelling in the future.
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265
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Contribution of individual authors to the creation
of a scientific article (ghostwriting policy)
Han Lin wrote the draft. Qun Zhu and Qianqian Li
carried out the simulation. Fengde Chen proposed the
issue and revise the paper.
Sources of funding for research presented in a sci-
entific article or scientific article itself
his work is supported by the Natural Science Foun-
dation of Fujian Province(2020J01499).
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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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