Dynamics of water eutrophication model with control

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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

1School 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.

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2 Model formulation

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

wheres(t) denotes the concentration of nutrients in water.x(t) denotes the concentration of microorganism in water.λ> 0 is the original concentration of nutrientss(t) in water,

β> 0 is called the transmission coefficient.βδ > 0 is called the transmission ofx(t) absorbed nutrients.d1> 0 is called the loss ofs(t).d2> 0 is called the death coefficient ofx(t).μ> 0

the global existence and uniqueness of the solutions of (2.1) are guaranteed by the smooth-ness of functiony, 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 whenevers(t) =λ,x(t) = 0 wheneverx(t) = 0, and

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

SoV(t) is uniformly ultimately bounded. Hence, by the definition ofV(t), there exists a constantM> 0 such thatS(t)≤M,x(t)≤Mfortlarge enough. The proof is complete.

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

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Considering the third equation of (3.1), we have

s(n+l)τ+=1 –μ1

d1

λλd1s

+e–d1. (3.3)

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

s(n+ 1)τ+= 1

d1

λλd1s

(n+l)τ+e–d1(1–l)τ+μ. (3.4)

Substituting (3.3) into (3.4), we have

s(n+ 1)τ+= λ

d1 –λe

–d1(1–l)τ

d1

+λ(1 –μ1)e –d1(1–l)τ

d1

λ(1 –μ1)e –d1τ

d1

+ (1 –μ1)s

+e–d1τ+μ. (3.5)

Equation (3.1) has one fixed point as follows:

s∗=λλμ1e

–d1(1–l)τλ(1 –μ

1)e–d1τ +μd1

d1[1 – (1 –μ1)e–d1τ] . (3.6)

Then the periodic solution of (3.1) is

˜

s(t) =

1

d1[λ– (λd1s

)e–d1(t–nτ)], t[nτ, (n+l)τ),

1

d1[λ– (λd1s

∗∗)e–d1(t–(n+l)τ)], t[(n+l)τ, (n+ 1)τ). (3.7)

Heres∗is determined as (3.6),s∗∗is defined as

s∗∗=1 –μ1

d1

λλd1s

e–d1+d1(1 –μ1)se–d1. (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 –μ2 >λτ

d1

λd1s

d21

1 –e–d1λd1s

∗∗

d21

1 –e–d1(1–l)τe–d1τd

2τ (4.1)

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Proof First we prove the local stability. DefiningS(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. Chooseε> 0 such that

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From the first equation of (2.1), we notice that ds(t)dtλd1s(t). So we consider the fol-lowing impulsive differential equations:

⎧ ⎪ ⎨ ⎪ ⎩

dy(t)

dt =λd1y(t), t= ,t= (n+l)τ,

y(t) =μ, t=,

y(t) = –μ2y(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 haves(t)≤y(t), and,y(t)→ ˜s(t), ast→ ∞, that is,

s(t)≤y(t)≤ ˜s(t) +ε (4.3)

for alltlarge enough. For convenience, we may assume (4.3) holds for allt> 0. From (2.1) and (4.3), we get

⎧ ⎪ ⎨ ⎪ ⎩

dx(t) dt

β

δ(s˜(t) +ε)x(t) –d2x(t), t=,t= (n+l)τ,

x(t) = 0, t=,

x(t) = –μ2x(t), t= (n+l)τ.

(4.4)

So

x(n+ 1)τxnτ+(1 –μ2)exp

(n+1)τ

β δ

˜

s(s) –εd2

ds

.

Hence

x(n+l)τx0+ρn

andx((n+l)τ)→0, ast→ ∞, thereforex(t)→0, ast→ ∞.

In what follows, we prove thats(t)→ ˜s(t), ast→ ∞, let 0 <εd1, there must exist

t0> 0 such thattt0, 0 <x(t) <ε. Without loss of generality, we assume that 0 <x(t) <ε for allt> 0, then, for system (2.1), we have

λ– (d1+ε)s(t)≤

ds(t)

dtλd1s(t). (4.5)

Then we havez1(t)≤s(t)≤z2(t) andz1(t)→ ˜s(t),z2(t)→ ˜s(t) ast→ ∞. Whilez1(t) and

z2(t) are the solutions of

⎧ ⎪ ⎨ ⎪ ⎩

dz1(t)

dt =λ– (d1+ε)z1(t), t=,t= (n+l)τ,

z1(t) =μ, t=,

z1(t) = –μ1z1(t), t= (n+l)τ

(4.6)

and

⎧ ⎪ ⎨ ⎪ ⎩

dz2(t)

dt =λd1z2(t), t= ,t= (n+l)τ,

z2(t) =μ, t=,

z2(t) = –μ1z2(t), t= (n+l)τ,

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respectively

z1(t) =

1

d1+ε[λ– (λ– (d1+ε)z

1)e–(d1+ε)(t–nτ)], t∈[, (n+l)τ), 1

d1+ε[λ– (λ– (d1+ε)z ∗∗

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

(4.8)

Herez1andz∗∗1 are defined as

z1∗=λλμ1e

–(d1+ε)(1–l)τλ(1 –μ1)e–(d1+ε)τ+ (d

1+ε)μ (d1+ε)[1 – (1 –μ1)e–(d1+ε)τ]

and

z1∗∗= λμ1 (d1+ε)

λλ– (d1+ε)z∗1

e–(d1+ε)lτ .

Therefore, for anyε1> 0, there existst1,t>t1, such that

z1(t) –ε1<s(t) <˜s(t) +ε1.

Letε→0, so we have

˜

s(t) –ε1<s(t) <˜s(t) +ε1

fortlarge enough, which impliess(t)→ ˜s(t) ast→ ∞. 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 constantsm,M> 0 (inde-pendent of the initial value) and a finite timeT0such that for all solutions (s(t),x(t)) with all initial valuess(0+) > 0,x(0+) > 0,ms(t)≤M,mx(t)≤Mhold for alltT0. Here

T0may 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 –μ2 <λτ

d1

λd1s

d2 1

1 –e–d1

λd1s ∗∗

d2 1

1 –e–d1(1–l)τ

e–d1τd2τ

holds,then system(2.1)is permanent.

Proof Let (s(t),x(t)) be a solution of (2.1) withs(0) > 0,x(0) > 0. By Lemma3, we have proved there exists a constant ofM> 0 such thats(t)≤M,x(t)≤Mfortlarge enough. We may assumes(t)≤M,x(t)≤Mfort≥0. Thus we only need to findm1> 0 andε3 such thatx(t)≥m1fortlarge enough. Otherwise, we can selectm3> 0 small enough and provex(t) <m3does not hold fort≥0. By the condition of Theorem2, let

σ=

(n+1)τ

β δ

˜

z(t) –ε1

d2

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We will prove thatx(t) <m3cannot hold fort≥0. Otherwise,

⎧ ⎪ ⎨ ⎪ ⎩

ds(t)

dtλ– (d1+βm3)s(t), t=,t= (n+l)τ,

s(t) =μ, t=,

s(t) = –μ1s(t), t= (n+l)τ.

(4.9)

By Lemma3, we haves(t)≥z(t) andz(t)→ ˜z(t),t→ ∞, wherez(t) is the solution of

⎧ ⎪ ⎨ ⎪ ⎩

dz(t)

dt =λ– (d1+βm3)z(t), t=,t= (n+l)τ,

z(t) =μ, t=,

s(t) = –μ1z(t), t= (n+l)τ

(4.10)

and

˜

z(t) =

1

(d1+βm3)[λ– (λ– (d1+βm3)z

)e–(d1+βm3)(t–nτ)], t[nτ, (n+l)τ),

1

(d1+βm3)[λ– (λ– (d1+βm3)z

∗∗)e–(d1+βm3)(t–(n+l)τ)], t[(n+l)τ, (n+ 1)τ).

(4.11)

Herez∗andz∗∗are defined as

z∗=λ(1 –μ1e

–(d1+βm3)(1–l)τ) –λ(1 –μ1)e–(d1+βm3)τ) + (d

1+βm3)μ (d1+βm3)[1 – (1 –μ1)e–(d1+βm3)τ]

and

z∗∗= 1 –μ1 (d1+βm3)

λλ– (d1+βm3)z

e–(d1+βm3)lτ.

Therefore, there existsT1> 0 such that

s(t)≥z(t)≥ ˜z(t) –ε1,

and

⎧ ⎪ ⎨ ⎪ ⎩

dx(t) dt ≥(

β

δ(z˜(t) –ε1) –d2)x(t), t=,t= (n+l)τ,

x(t) = 0, t=,

x(t) = –μ2x(t), t= (n+l)τ

(4.12)

fortT1. LetN1∈NandN1τ>T. Integrating (4.12) on (, (n+ 1)τ] (nN1), we have

x(n+ 1)τ≥(1 –μ2)xnτ+exp

(n+1)τ

β δ

˜

z(t) –ε1

d2

dt

= (1 –μ2)xnτ+.

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Thus, we can also obtain that ds(t)dtλ– (d1+βM)s(t), then the following comparatively impulsive differential equation is

⎧ ⎪ ⎨ ⎪ ⎩

dz3(t)

dt =λ– (d1+βM)z3(t), t=,t= (n+l)τ,

z3(t) =μ, t=,

z3(t) = –μ2z3(t), t= (n+l)τ.

(4.13)

Similar to (3.7), we have

˜

z3(t) =

1

(d1+βM)[λ– (λ– (d1+βM)z

3)e–(d1+βM)(t–nτ)], t∈[, (n+l)τ), 1

(d1+βM)[λ– (λ– (d1+βM)z

∗∗

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

Herez3andz∗∗3 are defined as

z3∗=λ(1 –μ1e

–(d1+βm3)(1–l)τ) –λ(1 –μ1)e–(d1+βm3)τ) + (d

1+βm3)μ (d1+βm3)[1 – (1 –μ1)e–(d1+βm3)τ]

and

z3∗∗= 1 –μ1 (d1+βm3)

λλ– (d1+βm3)z

e–(d1+βm3)lτ.

For anyε4small enough, we obtain

z3(t) >z˜3(t) –ε4.

From the comparison theorem of impulsive differential equations, we have

s(t) >z3(t) >˜z3(t) –ε4>

z3(t) +z∗∗3 (t)–ε4=m4,

i.e.,s(t) >m4. This completes the proof.

Remark Let

F(τ) =ln 1

1 –μ2 –λτ

d1

λd1s

d2 1

1 –e–d1λd1s

∗∗

d2 1

1 –e–d1(1–l)τe–d1τ d

2τ.

We easily knowF(0) =ln(1–1μ

2)> 0,F(τ)→–∞asτ→ ∞, andF

(τ) > 0, soF(τ) = 0 has a unique positive root, denoted byτmax. From Theorems1and2, we knowτmaxis a

thresh-old. Ifτ<τmax, the microorganism-extinction periodic solution (s(t), 0) of (2.1) is globally

asymptotically stable. Ifτ <τmax, system (2.1) is permanent. 5 Discussion

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μ2. Ifμ2>μ∗2, the microorganism-extinction periodic solution (s(t), 0) of (2.1) is globally asymptotically stable. Ifμ2<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 References

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