Permanence and Global Attractivity of a Non-autonomous Single

Species System with Michaelis-Menten-Type Feedback Control

FENGDE CHEN, YANBO CHONG, SHANGMING CHEN

College of Mathematics and Statistics

Fuzhou University

No. 2, Wulongjiang Avenue, Minhou County, Fuzhou

CHINA

Abstract: During the past decade, many scholars have investigated the dynamic behaviors of the ecosystem with

Michaelis-Menten-type harvesting; however, most of them assume that the harvesting effort does not change with

time. Such an assumption has its drawbacks. Generally speaking, the rate of increase in harvesting effort 

changes with the density of the species. Inspired by this, we put forth a novel form of single-population feedback

control model, in which the feedback control variable is of the Michaelis-Menten-type. Sufficient conditions that

ensure the permanence and global attractivity of the system are obtained.

Key-Words: species, Michaelis-Menten type feedback control, permanence, global attractivity

Received: August 15, 2022. Revised: March 29, 2023. Accepted: April 21, 2023. Published: May 22, 2023.

1 Introduction

Throughout this paper, for a continuous and

bounded function, we let fl=inft∈Rf(t)and fu=

supt∈Rf(t).

This paper aims to investigate the dynamic behav-

ior of the following model:

dt =xa(t)−b(t)x

−q(t)x(t)u(t)

k1(t)u(t) + k2(t)x(t),

dt =−e(t)u+f(t)x.

(1)

In system (1), we always assume:

(H1)a(t), b(t), q(t), k1(t), k2(t), e(t),and f(t)

are all continuous and strictly positive functions that

satisfy

min{al, bl, ql, kl

1, kl

2, el, fl}>0,

max{au, bu, qu, ku

1, ku

2, eu, fu}<+∞.

During the past two decades, ecosystems with

feedback controls have become one of the main topics

in the study of population dynamics. One could refer

to [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11],

[12], [13], [14], [15], [16], [17], [18], [19], [20], [21],

[22], [23], [24], [25], [26], [27], [28], [29], [30],[31]

and the references cited therein for more information.

Gopalsamy and Weng[23] proposed the following

single-species feedback control ecosystem:

˙n=rn1−a1n(t) + a2n(t−τ)

−cu(t)],

˙u=−au(t) + bn(t).

(2)

They investigated the stability property of the positive

equilibrium of the system. For the first time, they in-

troduced the feedback control variable, which can be

implemented by means of biological control or some

harvesting procedure.

In [26], the authors studied the following single-

species feedback control ecosystem:

N=r(t)N(t)1−N2(t−τ1(t))

K2(t)

−c(t)u(t−τ2(t)),

˙u=−a(t)u(t) + b(t)N(t).

(3)

Under the assumption that the coefficients of the

system are all continuous positive periodic func-

tions, they showed that the system admits at least

one positive periodic solution. For the general non-

autonomous case, Chen et al. [24] showed that the

system is always permanent.

In [31], the authors proposed the following sin-

gle species model with feedback regulation and dis-

tributed time delay:

N=Na(t)−b(t)+∞

0H(s)N(t−s)ds

−c(t)u(t),

˙u=−d(t)u(t) + e(t)+∞

0H(s)N2(t−s)ds.

(4)

They obtained a set of sufficient conditions to en-

sure the existence of a positive periodic solution to

the above system. By constructing some suitable

Lyapunov functionals, Chen [32] obtained a set of

sufficient conditions that ensure the permanence and

global stability of the positive solution of the system.

In [22], the author argued that it is necessary to

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consider the stage structure of the species, and he pro-

posed the following non-autonomous single species

system with stage structure and feedback control:

dx1(t)

dt =b(t)x2(t)−d1(t)x1(t)

−b(t−τ)e−t

t−τd1(s)dsx2(t−τ),

dx2(t)

dt =b(t−τ)e−t

t−τd1(s)dsx2(t−τ)

−a(t)x2

2(t)−c(t)x2(t)u(t),

du(t)

dt =−f(t)u(t) + e(t)x2(t).

(5)

The author showed that system (5) admits at least one

positive T-periodic solution if the coefficients are all

continuous T-periodic functions. In [23], the authors

showed that for the general non-autonomous case, the

system (5) is permanent.

Recently, [29], took the stocking as a feedback

control variable, and she proposed a single species

stage-structured model with positive feedback con-

trol. The system takes the form:

dx1

dt =αx2−βx1−δ1x1,

dx2

dt =βx1−δ2x2−γx2

2+dx2u,

dt =g−eu −fx2.

(6)

The author showed that if the positive feedback con-

trol is large enough, the species will finally live in the

long run.

In [27], the authors proposed the following single-

species discrete model with feedback control:

N(n+ 1) = N(n)exp r(n)1−N(n−m)

−c(n)µ(n),

∆µ(n) = −a(n)µ(n) + b(n)N(n−m).

(7)

They investigated the positive periodic solution of the

system. In [28], the author gave sufficient conditions

to ensure the permanence of the system (7).

Recently, in [2], the authors argued that in some

cases, species may have the Allee effect, since en-

dangered species may have difficulties finding mates

when population density is low; hence, they proposed

the following single-species feedback control system:

dt=x(1 −x)(x−m)−axu,

dt=−bu +cx,

(8)

where a,b,care all positive constants and 0<m<

1,mrepresents the Allee constant. They showed

that the system may have saddle-node bifurcation and

Bogdanov-Takens bifurcation of codimension 2.

In [1], the authors proposed the following Logistic

model with additive Allee effect and feedback con-

trol:

dt=x(1 −x−m

x+a)−bxu,

dt=−u+cx.

(9)

The authors showed that the system may have

saddle-node bifurcation and transcritical bifurcation.

The dynamical behaviors of the system are richer

and more complex than those in the traditional

logistic model with feedback control. Both the Allee

effect and feedback control can increase the species’

extinction property.

It brings to our attention that in systems (2)-(9),

the feedback control variable represents the harvest-

ing effect of human beings or biological control, such

an idea comes from the pioneering work of [23]. The

idea of [23], comes from linear harvesting. However,

such a kind of harvesting is not a sound one. As we

know, nonlinear harvesting is more realistic from

biological and economic points of view. In [33], the

author first proposed a harvesting term h=qEx

cE +lx ,

which is named Michaelis-Menten type functional

form of catch rate. Recently, many scholars have

investigated the dynamic behaviors of the ecosys-

tem with Michaelis–Menten-type harvesting. For

example, in [34], the authors proposed a two-species

amensalism model with Michaelis-Menten-type

harvesting and a cover for the first species:

dt =a1x(t)−b1x2(t)−c1(1 −k)x(t)y(t)

−qE(1 −k)x(t)

m1E+m2(1 −k)x(t),

dt =a2y(t)−b2y2(t).

(10)

Chen[35] studied the following Lotka-Volterra

commensal symbiosis model of two populations

with Michaelis-Menten-type harvesting for the first

species:

dt =r1x1−x

+αy

K1

−qEx

m1E+m2x,

dt =r2y1−y

K2.

(11)

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They showed that there are two saddle-node bifurca-

tions and two transcritical bifurcations under suitable

conditions. One could refer to [34], [35], [36], [37],

[38], [39], for more works in this direction.

It brings to our attention that in systems (10)-(11),

the authors all assume that the harvesting effect Eis a

constant, such an assumption has its drawbacks. Gen-

erally speaking, the rate of increased harvesting effort

changes with the density of the species. Now, if we

take the harvesting effort as the feedback control vari-

able, the Logistic equation with nonlinear feedback

control could be proposed:

dt =x(a−bx)−qux

k1u+k2x,

dt =−eu +fx.

(12)

It is well known that a non-autonomous system is

more suitable since circumstances change with time.

This stimulated us to propose the system (1).

This paper’s goal is to investigate the dynamic be-

haviors of the system (1). In the next section, we will

investigate the persistent property of the system, and

then, in Section 3, by constructing some suitable Lya-

punov functions, we will investigate the global attrac-

tivity of the system. Some numeric simulations are

carried out to show the feasibility of the main results

in Section 4. We end this paper with a brief discus-

sion.

2 Permanence

To investigate the persistent and extinct properties of

the system, we need the following lemmas, which are

Lemma 2.2 and 2.3 of [30], respectively.

Lemma 2.1. If a > 0, b > 0and ˙x≥b−ax, when

t≥0and x(0) >0, we have

lim inf

t→+∞x(t)≥b

If a > 0, b > 0and ˙x≤b−ax, when t≥0and

x(0) >0, we have

lim sup

t→+∞

x(t)≤b

Lemma 2.2. If a > 0, b > 0and ˙x≥x(b−ax),

when t≥0and x(0) >0, we have

lim inf

t→+∞x(t)≥b

If a > 0, b > 0and ˙x≤x(b−ax),when t≥0and

x(0) >0, we have

lim sup

t→+∞

x(t)≤b

Concerned with the persistent property of the sys-

tem (4), we have the following result.

Theorem 2.1. Assumes that

al>qu

(13)

holds, then system (1) is permanent.

Proof. From the first equation of the system (1), one

has

dt =xa(t)−b(t)x

−q(t)x(t)u(t)

k1(t)u(t) + k2(t)x(t)

≤xau−blx.

(14)

Applying Lemma 2.2 to inequality (14) leads to

lim sup

t→+∞

x(t)≤au

def

=M1.(15)

For ε > 0enough small, from (15) there exits a T1>

0such that

x(t)<au

bl+εfor all t ≥T1.(16)

From the second equation of (1) and (16), for t > T1,

one has

dt =−e(t)u+f(t)x.

≤ −elu+fuau

bl+ε.

(17)

Applying Lemma 2.1 to inequality (17) leads to

lim sup

t→+∞

u(t)≤

fuau

bl+ε

el.(18)

Since εis an arbitrary small positive constant, setting

ε→0in (18) leads to

lim sup

t→+∞

u(t)≤fuau

blel

def

=M2.(19)

From the first equation of the system (1), we also have

dt =xa(t)−b(t)x

−q(t)x(t)u(t)

k1(t)u(t) + k2(t)x(t)

≥xa(t)−b(t)x−q(t)x(t)u(t)

k1(t)u(t)

≥xal−qu

−bux.

(20)

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Applying Lemma 2.2 to inequality (20) leads to

lim inf

t→+∞x(t)≥al−qu

def

=m1.(21)

For ε1>0enough small, without loss of generality,

we may assume that ε1<1

2m1, from (21) there exits

aT3> T2such that

x(t)> m1−ε1for all t ≥T3.(22)

From the second equation of (1) and (22), for t > T3,

one has

dt =−e(t)u+f(t)x.

≥ −euu+flm1−ε1.

(23)

Applying Lemma 2.1 to inequality (23) leads to

lim inf

t→+∞u(t)≥flm1−ε1

eu.(24)

Since ε1is an arbitrary small positive constant, setting

ε1→0in (24) leads to

lim inf

t→+∞u(t)≥flm1

def

=m2.(25)

(15), (19), (21) and (25) show that under the assump-

tion (13) holds, the system is permanent.

This ends the proof of Theorem 2.1.

3 Global attractivity

The following lemma is from [40], and will be em-

ployed in establishing the global asymptotic stability

of (1).

Lemma 3.1. Let hbe a real number and fbe a non-

negative function defined on [h; +∞)such that fis

integrable on [h; +∞)and is uniformly continuous

on [h; +∞), then lim

t→+∞f(t) = 0.

Theorem 3.1 Let (x∗(t), u∗(t)) be a bounded posi-

tive solution of system (1). If

bl> fu+quku

2M2

(kl

1m2+kl

2m1)2(26)

and

el>quku

2M1

(kl

1m2+kl

2m1)2(27)

hold, where m1, m2, M1, M2are defined by (15),

(19), (21) and (25) respectively. Then (x∗(t), u∗(t))

is globally asymptotically stable.

Proof. Conditions (26) and (27) imply that for a

sufficiently small positive constant ε > 0(with-

out loss of generality, we may assume that ε <

2min{m2, m1}), the following inequality holds:

bl>quku

2(M2+ε)

kl

1(m2−ε) + kl

2(m1−ε)2(28)

and

el>quku

2(M1+ε)

kl

1(m2−ε) + kl

2(m1−ε)2.(29)

Let (x(t), u(t))Tbe any solution of (1) with a positive

initial value. It then follows from condition (26) and

Theorem 2.1 that for above ε > 0, there exists a T >

0such that for all t≥T,

m1−ε < x(t), x∗(t)< M1+ε,

m2−ε < x(t), x∗(t)< M2+ε. (30)

Consider a Lyapunov function defined by

V(t) = |ln{x(t)} − ln{x∗(t)}|

+|u(t)−u∗(t)|, t ≥0.(31)

Now we are calculating and estimating the upper right

derivative of V(t)along the solutions of system (9),

for t > T , it follows that:

D+V(t)

=sgn(x(t)−x∗(t))a(t)−b(t)x(t)

−q(t)u(t)

k1(t)u(t) + k2(t)x(t)

−a(t) + b(t)x∗(t)

+q(t)u∗(t)

k1(t)u∗(t) + k2(t)x∗(t)

+sgn(u(t)−u∗(t))e(t)u∗(t)−f(t)x∗(t)

−e(t)u(t) + f(t)x(t).

(32)

Noting that for t > T , by applying (30), one has

q(t)u∗(t)

k1(t)u∗(t) + k2(t)x∗(t)

−q(t)u(t)

k1(t)u(t) + k2(t)x(t)

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=q(t)u∗(t)∆2−q(t)u(t)∆1

∆1∆2

=q(t)k2(t)(u∗(t)x(t)−u(t)x∗(t))

∆1∆2

=q(t)k2(t)(u∗(t)x(t)−u(t)x(t))

∆1∆2

+q(t)k2(t)(u(t)x(t)−u(t)x∗(t))

∆1∆2

≤quku

2(M1+ε)|u∗(t)−u(t)|

∆2

+quku

2(M2+ε)|x(t)−x∗(t)|

∆2

(33)

where

∆1=k1(t)u∗(t) + k2(t)x∗(t),

∆2=k1(t)u(t) + k2(t)x(t),

∆3=kl

1(m2−ε) + kl

2(m1−ε)2.

(34)

Hence,

D+V(t)

≤ −bl−fu−quku

2(M2+ε)

∆2

3|x(t)−x∗(t)|

−el−quku

2(M1+ε)

∆2

3|u∗(t)−u(t)|

≤ −µ(|x(t)−x∗(t)|+|u(t)−u∗(t)|),

(35)

where

µ=min bl−fu−quku

2(M2+ε)

∆3

el−quku

2(M1+ε)

∆3.

(36)

Hence, for t≥T, one has

D+V(t)≤ −µ|x(t)−x∗(t)|+|u(t)−u∗(t)|.

(37)

Integrating on both sides of (37) from Tto tproduces

V(t) +µt

T|x(s)−x∗(s)|+|u(s)−u∗(s)|ds

≤V(T)<+∞, t ≥T.

Then

t

T|x(s)−x∗(s)|+|u(s)−u∗(s)|ds

≤µ−1V(T)<+∞, t ≥T,

and hence,

|x(t)−x∗(t)|+|u(t)−u∗(t)| ∈ L1([T, +∞)).

The boundedness of x∗(t)and u∗(t)and the ultimate

boundedness of x(t)and u(t)imply that x(t), x∗(t),

u(t),and u∗(t)all have bounded derivatives for t≥

T(from the equations satisfied by them). Then it fol-

lows that |x(t)−x∗(t)|+|u(t)−u∗(t)|is uniformly

continuous on [T, +∞). By Lemma 3.1, we have

lim

t→+∞|x(t)−x∗(t)|+|u(t)−u∗(t)|= 0.

The proof is completed.

4 Numeric simulations

Now let’s consider the following example:

Example 4.1

dt =x5−x

−(1 + 1

2sin(t))x(t)u(t)

(2 + sin(t))u(t) + x(t),

dt =−e(t)u+f(t)x,

(38)

where corresponding to system (1), we take a(t) =

5, b(t) = 1, q(t) = 1 + 1

2sin(t), k1(t) = 2 +

sin(t), k2(t) = 1, e(t) = 6+cos(t), f(t) = 2−sin(t).

Then, by simple computation, one has

M1= 1, M2=3

5, m1=7

2, m2=1

2.(39)

Thus

bl= 5 >3 + 9

160 =fu+quku

2M2

(kl

1m2+kl

2m1)2(40)

and

el= 5 >3

32 =quku

2M1

(kl

1m2+kl

2m1)2(41)

hold. Hence, it follows from Theorems 2.1 and

3.1 that the system is permanent and the positive

solutions of the system are globally attractive. Fig.

1 shows the globally asymptotically stable of the

species xand Fig. 2 shows the the globally asymp-

totically stable of the feedback control variable u.

5 Discussion

Over the last two decades, two topics in popula-

tion dynamics have received a lot of attention: the

feedback control ecosystem and the system with

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Figure 1: Dynamic behaviors of the first compo-

nent xin system (38) with the initial condition

(x(0), y(0)) = (1.2,0.4),(1,0.2),(0.8,0.8),and

(0.6,0.6), respectively.

Figure 2: Dynamic behaviors of the second com-

ponent uin system (38) with the initial condition

(x(0), y(0)) = (1.2,0.4),(1,0.2),(0.8,0.8),and

(0.6,0.6), respectively.

Michealis-Menten-type harvesting. However, there

are some drawbacks in these two areas. Firstly, the

study of the feedback control ecosystem comes from

the pioneering work of [23], whose study was based

on linear harvesting. Since then, most of the studies

on the feedback control ecosystems are based on the

linear harvesting of the system. Seldom did scholars

consider the other types of harvesting. Secondly, re-

cently, many scholars argued that Michaelis-Menten

type harvesting is more suitable, and they investi-

gated the dynamic behaviors of the ecosystem with

nonlinear harvesting. However, all of their works as-

sume that the harvesting coefficient is constant, which

is unrealistic since, generally speaking, the harvest-

ing effect will change according to the density of the

species. To overcome those two drawbacks, we pro-

posed the system (1). Sufficient conditions that en-

sure the permanence and global attractivity of the sys-

tem are obtained.

We mention here that for species with generation

overlap, it is appropriate to model them with discrete

systems. We will try to propose a discrete single-

species model with nonlinear feedback control, and

investigate the dynamic behavior of the system.

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