**R E S E A R C H**

**Open Access**

## Dynamics of water eutrophication

## model with control

### Jianjun Jiao

1*_{, Pei Li}

1_{and Dahe Feng}

1
*_{Correspondence:}

jiaojianjun05@126.com

1_{School of Mathematics and}

Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China

**Abstract**

In this work, we consider a water eutrophication model with impulsive dredging. We prove that all solutions of the investigated system are uniformly bounded. There exists globally asymptotically stable periodic solution of microorganism-extinction when some condition is satisﬁed. The condition for permanence of the system is also obtained. It is concluded that the approach of impulsive dredging provides a reliable theoretical basis for the management of water eutrophication.

**Keywords:** Impulsive dredging; Water eutrophication; Extinction; Permanence

**1 Introduction**

Due to human activities, there are more and more nutrients in water, which causes exces-sive reproduction of algae and other aquatic organisms, decrease in water transparency, dissolved oxygen reduction, deterioration of water quality, and so on. This kind of pollu-tion phenomenon, which is caused by the change in the ecological balance of the whole water body, is called the eutrophication of water body [1].

Eutrophication occurs in ponds, reservoirs, lakes, and so on and it has attracted the attention of many experts and scholars in the early twentieth century. Especially in re-cent years, the study on the eutrophication of water body has become quite active like fuzzy mathematics, stochastic model, gray system, artiﬁcial intelligence, and other theo-retical methods combined with computer technology [2–4]. For instance, based on the analysis of the present situation and causes of pollution in Chaohu, Li et al. [5] proposed speciﬁc measures for eutrophication control of Chaohu. Yu et al. [6] used three methods— Carlson comprehensive index method, Grey clustering method, and principal compo-nent analysis—to analyze the main factors of eutrophication, and they provided theo-retical basis for controlling water pollution in Poyang Lake. In addition, many scholars [7–9] have studied the dynamics model of a single population in the polluted environ-ment.

Many scholars have done in-depth research on the problems of eutrophication. But most of them studied the reasons and the prevention and treatment of water eutrophication from the technical aspects. Few researchers studied the problem of eutrophication from the viewpoint of biological dynamics. In this paper, we discuss water eutrophication with impulsive dredging.

**2 Model formulation**

In this paper, we consider the following eutrophication model with control:

⎧

where*s*(*t*) denotes the concentration of nutrients in water.*x*(*t*) denotes the concentration
of microorganism in water.*λ*> 0 is the original concentration of nutrients*s*(*t*) in water,

*β*> 0 is called the transmission coeﬃcient.*β _{δ}* > 0 is called the transmission of

*x*(

*t*) absorbed nutrients.

*d*1> 0 is called the loss of

*s*(

*t*).

*d*2> 0 is called the death coeﬃcient of

*x*(

*t*).

*μ*> 0

the global existence and uniqueness of the solutions of (2.1) are guaranteed by the
smooth-ness of function*y*, which denotes the mapping deﬁned by the right-hand side of system
(2.1) (see [10]). Before discussing the main results, we will give some deﬁnitions,
no-tations, and lemmas. Since *s*(*t*) = 0 whenever*s*(*t*) =*λ*,*x*(*t*) = 0 whenever*x*(*t*) = 0, and

**Lemma 3** *There exists a constant M*> 0*such that S*(*t*)≤*M*,*x*(*t*)≤*M for each solution*

So*V*(*t*) is uniformly ultimately bounded. Hence, by the deﬁnition of*V*(*t*), there exists a
constant*M*> 0 such that*S*(*t*)≤*M*,*x*(*t*)≤*M*for*t*large enough. The proof is complete.

We can easily obtain the analytic solution of (3.1) between pulses as follows:

Considering the third equation of (3.1), we have

*s*(*n*+*l*)*τ*+=1 –*μ*1

*d*1

*λ*–*λ*–*d*1*s*

*nτ*+*e*–d1*lτ*_{.} _{(3.3)}

Considering the ﬁfth equation of (3.1), we also have

*s*(*n*+ 1)*τ*+= 1

*d*1

*λ*–*λ*–*d*1*s*

(*n*+*l*)*τ*+*e*–d1(1–l)*τ*_{+}_{μ}_{.} _{(3.4)}

Substituting (3.3) into (3.4), we have

*s*(*n*+ 1)*τ*+= *λ*

*d*1
–*λe*

–d1(1–l)*τ*

*d*1

+*λ*(1 –*μ*1)*e*
–d1(1–l)*τ*

*d*1

–*λ*(1 –*μ*1)*e*
–d1*τ*

*d*1

+ (1 –*μ*1)*s*

*nτ*+*e*–d1*τ*_{+}_{μ}_{.} _{(3.5)}

Equation (3.1) has one ﬁxed point as follows:

*s*∗=*λ*–*λμ*1*e*

–d1(1–l)*τ*_{–}_{λ}_{(1 –}_{μ}

1)*e*–d1*τ* +*μd*1

*d*1[1 – (1 –*μ*1)*e*–d1*τ*_{]} . (3.6)

Then the periodic solution of (3.1) is

˜

*s*(*t*) =

1

*d*1[*λ*– (*λ*–*d*1*s*

∗_{)}* _{e}*–d1(t–n

*τ*)

_{],}

*∈*

_{t}_{[}

_{n}_{τ}_{, (}

_{n}_{+}

_{l}_{)}

_{τ}_{),}

1

*d*1[*λ*– (*λ*–*d*1*s*

∗∗_{)}* _{e}*–d1(t–(n+l)

*τ*)

_{],}

*∈*

_{t}_{[(}

_{n}_{+}

_{l}_{)}

_{τ}_{, (}

_{n}_{+ 1)}

_{τ}_{).}(3.7)

Here*s*∗is determined as (3.6),*s*∗∗is deﬁned as

*s*∗∗=1 –*μ*1

*d*1

*λ*–*λ*–*d*1*s*∗

*e*–d1*lτ*_{+}_{d}_{1(1 –}_{μ}_{1)}* _{s}*∗

*–d1*

_{e}*lτ*

_{.}

_{(3.8)}

It is similar to reference [8], we can obtain the following lemma.

**Lemma 4** *System*(3.1)*has a positive periodic solution*˜*s*(*t*).*For every solution s*(*t*)*of system*

(3.1),*we have s*(*t*)→ ˜*s*(*t*)*as t*→ ∞.

**4 The dynamics**

From the above discussion, we know that system (2.1) has a microorganism-extinction
periodic solution (˜*s*(*t*), 0). So we can have the following theorem.

**Theorem 1** *If*

ln 1

1 –*μ*2
>*λτ*

*d*1

–*λ*–*d*1*s*
∗

*d*2_{1}

1 –*e*–d1*lτ*_{–}*λ*–*d*1*s*

∗∗

*d*2_{1}

1 –*e*–d1(1–l)*τ _{e}*–d1

*τ*

_{–}

_{d}2*τ* (4.1)

*Proof* First we prove the local stability. Deﬁning*S*(*t*) =*s*(*t*) –*s*˜(*t*),*X*(*t*) =*x*(*t*), we have the
following linearly similar system for (2.1) which concerns one periodic solution (˜*s*(*t*), 0):

It is easy to obtain the fundamental solution matrix

(*t*) =

There is no need to calculate the exact form of∗as it is not required in the analysis that follows. The linearization of the third and fourth equations of (2.1) is as follows:

The linearization of the ﬁfth and sixth equations of (2.1) is as follows:

The stability of the periodic solution (˜*s*(*t*), 0) is determined by the eigenvalues of

*M*=

The following work is to prove the global attraction. Choose*ε*> 0 such that

From the ﬁrst equation of (2.1), we notice that *ds(t) _{dt}* ≤

*λ*–

*d*1

*s*(

*t*). So we consider the fol-lowing impulsive diﬀerential equations:

⎧ ⎪ ⎨ ⎪ ⎩

*dy(t)*

*dt* =*λ*–*d*1*y*(*t*), *t*= *nτ*,*t*= (*n*+*l*)*τ*,

*y*(*t*) =*μ*, *t*=*nτ*,

*y*(*t*) = –*μ*2*y*(*t*), *t*= (*n*+*l*)*τ*.

(4.2)

From Lemma2and the comparison theorem of impulsive diﬀerential equations (see
The-orem 3.1.1 in [11]), we have*s*(*t*)≤*y*(*t*), and,*y*(*t*)→ ˜*s*(*t*), as*t*→ ∞, that is,

*s*(*t*)≤*y*(*t*)≤ ˜*s*(*t*) +*ε* (4.3)

for all*t*large enough. For convenience, we may assume (4.3) holds for all*t*> 0. From (2.1)
and (4.3), we get

⎧ ⎪ ⎨ ⎪ ⎩

*dx(t)*
*dt* ≤

*β*

*δ*(*s*˜(*t*) +*ε*)*x*(*t*) –*d*2*x*(*t*), *t*=*nτ*,*t*= (*n*+*l*)*τ*,

*x*(*t*) = 0, *t*=*nτ*,

*x*(*t*) = –*μ*2*x*(*t*), *t*= (*n*+*l*)*τ*.

(4.4)

So

*x*(*n*+ 1)*τ*≤*xnτ*+(1 –*μ*2)exp

(n+1)*τ*

*nτ*

*β*
*δ*

˜

*s*(*s*) –*ε*–*d*2

*ds*

.

Hence

*x*(*n*+*l*)*τ*≤*x*0+*ρn*

and*x*((*n*+*l*)*τ*)→0, as*t*→ ∞, therefore*x*(*t*)→0, as*t*→ ∞.

In what follows, we prove that*s*(*t*)→ ˜*s*(*t*), as*t*→ ∞, let 0 <*ε*≤*d*1, there must exist

*t*0> 0 such that*t*≥*t*0, 0 <*x*(*t*) <*ε*. Without loss of generality, we assume that 0 <*x*(*t*) <*ε*
for all*t*> 0, then, for system (2.1), we have

*λ*– (*d*1+*ε*)*s*(*t*)≤

*ds*(*t*)

*dt* ≤*λ*–*d*1*s*(*t*). (4.5)

Then we have*z*1(*t*)≤*s*(*t*)≤*z*2(*t*) and*z*1(*t*)→ ˜*s*(*t*),*z*2(*t*)→ ˜*s*(*t*) as*t*→ ∞. While*z*1(*t*) and

*z*2(*t*) are the solutions of

⎧ ⎪ ⎨ ⎪ ⎩

*dz*1(t)

*dt* =*λ*– (*d*1+*ε*)*z*1(*t*), *t*=*nτ*,*t*= (*n*+*l*)*τ*,

*z*1(*t*) =*μ*, *t*=*nτ*,

*z*1(*t*) = –*μ*1*z*1(*t*), *t*= (*n*+*l*)*τ*

(4.6)

and

⎧ ⎪ ⎨ ⎪ ⎩

*dz*2(t)

*dt* =*λ*–*d*1*z*2(*t*), *t*= *nτ*,*t*= (*n*+*l*)*τ*,

*z*2(*t*) =*μ*, *t*=*nτ*,

*z*2(*t*) = –*μ*1*z*2(*t*), *t*= (*n*+*l*)*τ*,

respectively

*z*1(*t*) =

1

*d*1+*ε*[*λ*– (*λ*– (*d*1+*ε*)*z*
∗

1)*e*–(d1+*ε*)(t–n*τ*)], *t*∈[*nτ*, (*n*+*l*)*τ*),
1

*d*1+*ε*[*λ*– (*λ*– (*d*1+*ε*)*z*
∗∗

1 )*e*–(d1+*ε*)(t–(n+l)*τ*)], *t*∈[(*n*+*l*)*τ*, (*n*+ 1)*τ*).

(4.8)

Here*z*∗_{1}and*z*∗∗_{1} are deﬁned as

*z*_{1}∗=*λ*–*λμ*1*e*

–(d1+*ε*)(1–l)*τ*_{–}_{λ}_{(1 –}_{μ}_{1)}* _{e}*–(d1+

*ε*)

*τ*

_{+ (}

_{d}1+*ε*)*μ*
(*d*1+*ε*)[1 – (1 –*μ*1)*e*–(d1+*ε*)*τ*]

and

*z*_{1}∗∗= *λ*–*μ*1
(*d*1+*ε*)

*λ*–*λ*– (*d*1+*ε*)*z*∗1

*e*–(d1+*ε*)l*τ*
.

Therefore, for any*ε*1> 0, there exists*t*1,*t*>*t*1, such that

*z*1(*t*) –*ε*1<*s*(*t*) <˜*s*(*t*) +*ε*1.

Let*ε*→0, so we have

˜

*s*(*t*) –*ε*1<*s*(*t*) <˜*s*(*t*) +*ε*1

for*t*large enough, which implies*s*(*t*)→ ˜*s*(*t*) as*t*→ ∞. This completes the proof.

The following work is to investigate the permanence of system (2.1). Before starting this work, we should give the following deﬁnition.

**Deﬁnition 1** System (2.1) is said to be permanent if there are constants*m*,*M*> 0
(inde-pendent of the initial value) and a ﬁnite time*T*0such that for all solutions (*s*(*t*),*x*(*t*)) with
all initial values*s*(0+) > 0,*x*(0+) > 0,*m*≤*s*(*t*)≤*M*,*m*≤*x*(*t*)≤*M*hold for all*t*≥*T*0. Here

*T*0may depend on the initial values (*s*(0+),*x*(0+)).

**Theorem 2** *Let*(*s*(*t*),*x*(*t*))*be any solution of system*(2.1).*If*

ln 1

1 –*μ*2
<*λτ*

*d*1

–*λ*–*d*1*s*
∗

*d*2
1

1 –*e*–d1*lτ*

–*λ*–*d*1*s*
∗∗

*d*2
1

1 –*e*–d1(1–l)*τ*

*e*–d1*τ*
–*d*2*τ*

*holds*,*then system*(2.1)*is permanent*.

*Proof* Let (*s*(*t*),*x*(*t*)) be a solution of (2.1) with*s*(0) > 0,*x*(0) > 0. By Lemma3, we have
proved there exists a constant of*M*> 0 such that*s*(*t*)≤*M*,*x*(*t*)≤*M*for*t*large enough.
We may assume*s*(*t*)≤*M*,*x*(*t*)≤*M*for*t*≥0. Thus we only need to ﬁnd*m*1> 0 and*ε*3
such that*x*(*t*)≥*m*1for*t*large enough. Otherwise, we can select*m*3> 0 small enough and
prove*x*(*t*) <*m*3does not hold for*t*≥0. By the condition of Theorem2, let

*σ*=

(n+1)*τ*

*nτ*

*β*
*δ*

˜

*z*(*t*) –*ε*1

–*d*2

We will prove that*x*(*t*) <*m*3cannot hold for*t*≥0. Otherwise,

⎧ ⎪ ⎨ ⎪ ⎩

*ds(t)*

*dt* ≥*λ*– (*d*1+*βm*3)*s*(*t*), *t*=*nτ*,*t*= (*n*+*l*)*τ*,

*s*(*t*) =*μ*, *t*=*nτ*,

*s*(*t*) = –*μ*1*s*(*t*), *t*= (*n*+*l*)*τ*.

(4.9)

By Lemma3, we have*s*(*t*)≥*z*(*t*) and*z*(*t*)→ ˜*z*(*t*),*t*→ ∞, where*z*(*t*) is the solution of

⎧ ⎪ ⎨ ⎪ ⎩

*dz(t)*

*dt* =*λ*– (*d*1+*βm*3)*z*(*t*), *t*=*nτ*,*t*= (*n*+*l*)*τ*,

*z*(*t*) =*μ*, *t*=*nτ*,

*s*(*t*) = –*μ*1*z*(*t*), *t*= (*n*+*l*)*τ*

(4.10)

and

˜

*z*(*t*) =

1

(d1+*βm*3)[*λ*– (*λ*– (*d*1+*βm*3)*z*

∗_{)}* _{e}*–(d1+

*βm*3)(t–n

*τ*)

_{],}

*∈*

_{t}_{[}

_{n}_{τ}_{, (}

_{n}_{+}

_{l}_{)}

_{τ}_{),}

1

(d1+*βm*3)[*λ*– (*λ*– (*d*1+*βm*3)*z*

∗∗_{)}* _{e}*–(d1+

*βm*3)(t–(n+l)

*τ*)

_{],}

*∈*

_{t}_{[(}

_{n}_{+}

_{l}_{)}

_{τ}_{, (}

_{n}_{+ 1)}

_{τ}_{).}

(4.11)

Here*z*∗and*z*∗∗are deﬁned as

*z*∗=*λ*(1 –*μ*1*e*

–(d1+*βm*3)(1–l)*τ*_{) –}_{λ}_{(1 –}_{μ}_{1)}* _{e}*–(d1+

*βm*3)

*τ*

_{) + (}

_{d}1+*βm*3)*μ*
(*d*1+*βm*3)[1 – (1 –*μ*1)*e*–(d1+*βm*3)*τ*]

and

*z*∗∗= 1 –*μ*1
(*d*1+*βm*3)

*λ*–*λ*– (*d*1+*βm*3)*z*∗

*e*–(d1+*βm*3)l*τ*_{.}

Therefore, there exists*T*1> 0 such that

*s*(*t*)≥*z*(*t*)≥ ˜*z*(*t*) –*ε*1,

and

⎧ ⎪ ⎨ ⎪ ⎩

*dx(t)*
*dt* ≥(

*β*

*δ*(*z*˜(*t*) –*ε*1) –*d*2)*x*(*t*), *t*=*nτ*,*t*= (*n*+*l*)*τ*,

*x*(*t*) = 0, *t*=*nτ*,

*x*(*t*) = –*μ*2*x*(*t*), *t*= (*n*+*l*)*τ*

(4.12)

for*t*≥*T*1. Let*N*1∈*N*and*N*1*τ*>*T*. Integrating (4.12) on (*nτ*, (*n*+ 1)*τ*] (*n*≥*N*1), we have

*x*(*n*+ 1)*τ*≥(1 –*μ*2)*xnτ*+exp

(n+1)*τ*

*nτ*

*β*
*δ*

˜

*z*(*t*) –*ε*1

–*d*2

*dt*

= (1 –*μ*2)*xnτ*+*eσ*.

Thus, we can also obtain that *ds(t) _{dt}* ≥

*λ*– (

*d*1+

*βM*)

*s*(

*t*), then the following comparatively impulsive diﬀerential equation is

⎧ ⎪ ⎨ ⎪ ⎩

*dz*3(t)

*dt* =*λ*– (*d*1+*βM*)*z*3(*t*), *t*=*nτ*,*t*= (*n*+*l*)*τ*,

*z*3(*t*) =*μ*, *t*=*nτ*,

*z*3(*t*) = –*μ*2*z*3(*t*), *t*= (*n*+*l*)*τ*.

(4.13)

Similar to (3.7), we have

˜

*z*3(*t*) =

1

(d1+*βM)*[*λ*– (*λ*– (*d*1+*βM*)*z*

∗

3)*e*–(d1+*βM)(t–nτ*)], *t*∈[*nτ*, (*n*+*l*)*τ*),
1

(d1+*βM)*[*λ*– (*λ*– (*d*1+*βM*)*z*

∗∗

3 )*e*–(d1+*βM)(t–(n+l)τ*)], *t*∈[(*n*+*l*)*τ*, (*n*+ 1)*τ*).
(4.14)

Here*z*∗_{3}and*z*∗∗_{3} are deﬁned as

*z*_{3}∗=*λ*(1 –*μ*1*e*

–(d1+*βm*3)(1–l)*τ*_{) –}_{λ}_{(1 –}_{μ}_{1)}* _{e}*–(d1+

*βm*3)

*τ*

_{) + (}

_{d}1+*βm*3)*μ*
(*d*1+*βm*3)[1 – (1 –*μ*1)*e*–(d1+*βm*3)*τ*]

and

*z*_{3}∗∗= 1 –*μ*1
(*d*1+*βm*3)

*λ*–*λ*– (*d*1+*βm*3)*z*∗

*e*–(d1+*βm*3)l*τ*_{.}

For any*ε*4small enough, we obtain

*z*3(*t*) >*z*˜3(*t*) –*ε*4.

From the comparison theorem of impulsive diﬀerential equations, we have

*s*(*t*) >*z*3(*t*) >˜*z*3(*t*) –*ε*4>

*z*∗_{3}(*t*) +*z*∗∗_{3} (*t*)–*ε*4=*m*4,

i.e.,*s*(*t*) >*m*4. This completes the proof.

*Remark* Let

*F*(*τ*) =ln 1

1 –*μ*2
–*λτ*

*d*1

–*λ*–*d*1*s*
∗

*d*2
1

1 –*e*–d1*lτ*_{–}*λ*–*d*1*s*

∗∗

*d*2
1

1 –*e*–d1(1–l)*τ _{e}*–d1

*τ*

_{–}

_{d}2*τ*.

We easily know*F*(0) =_{ln}_{(1–}1_{μ}

2)> 0,*F*(*τ*)→–∞as*τ*→ ∞, and*F*

_{(}_{τ}_{) > 0, so}_{F}_{(}_{τ}_{) = 0 has a}
unique positive root, denoted by*τ*max. From Theorems1and2, we know*τ*maxis a

thresh-old. If*τ*<*τ*max, the microorganism-extinction periodic solution (*s*(*t*), 0) of (2.1) is globally

asymptotically stable. If*τ* <*τ*max, system (2.1) is permanent.
**5 Discussion**

*μ*∗_{2}. If*μ*2>*μ*∗2, the microorganism-extinction periodic solution (*s*(*t*), 0) of (2.1) is globally
asymptotically stable. If*μ*2<*μ*∗2, system (2.1) is permanent. Our results indicate that the
impulsive dredging and the period of impulsive dredging play an important role in water
eutrophication management.

**Acknowledgements**

The authors are grateful to the anonymous referees for their excellent suggestions, which greatly improved the presentation of the paper.

**Funding**

The research was supported by the National Natural Science Foundation of China (11761019, 11361014), the Project of High Level Creative Talents in Guizhou Province (No. 20164035), and the Joint Project of Department of Commerce and Guizhou University of Finance and Economics (No. 2016SWBZD18).

**Competing interests**

The authors declare that they have no competing interests.

**Authors’ contributions**

JJ corrected the manuscript, PL carried out the main part of this article, HF gave some suggestions for this article. All authors have read and approved the ﬁnal manuscript.

**Publisher’s Note**

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 26 January 2018 Accepted: 13 August 2018
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