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Dynamics of a lake-eutrophication model with nontransient/transient impulsive dredging and pulse inputting
Advances in Difference Equations volume 2021, Article number: 280 (2021)
Abstract
In this work, we present a lake-eutrophication model with nontransient/transient impulsive dredging and pulse inputting. We obtain globally asymptotically stable conditions for the phytoplankton-extinction periodic solution of system (2.1). Furthermore, we gain the permanent conditions for system (2.1). Finally, we employ computer simulations to illustrate the results. Our results indicate the effective controlling strategy for water resource management.
1 Introduction
Lakes are very important water resources; many lakes have water supply, shipping, flood control, irrigation, aquaculture, tourism, and other functions [1]. Lake eutrophication has become a worldwide environmental problem. According to statistics, the proportion of eutrophic water bodies in Asia, Europe, North America, and Africa reached 54%, 53%, 46%, and 28%, respectively [2]. Bennett et al. [3] investigated human impact on erodable phosphorus and eutrophication. The main characteristic of lake pollution is eutrophication of water body. Because of human interference of activities, eutrophication process is very rapid. Deposing the sediment is an important reservoir of nutrients in lakes. After the nutrient load of the lake is reduced or completely cut off, the nutrient salt in the sediment will gradually released to become the dominant factor of lake eutrophication endogenous [4]. So the preventing and controlling phytoplankton in eutrophication lake ecosystem have also become an important subject of water environmental protection. Partly and periodically dredging sediments can protect lake ecosystem and water resource. At present, physical, chemical, and biological methods are the common methods of controlling phytoplankton (cyanobacteria) in eutrophication lake ecosystem [5]. The physical methods are relatively safe ways to remove algae. Impulsive differential equations are found in almost every domain of applied science and have been studied in many investigations [6–13]. However, the authors did not applied impulsive differential equations to describe the physical methods for water resource management. In this paper, we present a lake-eutrophication model for water resource management, which considers effects of nontransient/transient impulsive dredging and pulse inputting.
2 The model
For the diagram in Fig. 1, in this paper, we consider a like-eutrophication model with nontransient/transient impulsive dredging and pulse inputting on nutrients
Here \(s(t)\) represents the concentrations of the nutrients at time t, \(x_{i}(t)\) (\(i=1,2\)) represent the concentrations of phytoplankton in lake at time t, \(\lambda _{1}>0\) represents the input concentration of the nutrients from ravine streams around the eutrophication-lake in the interval \((n\tau ,(n+l)\tau )]\), \(d_{1}>0\) represents washout and loss rate of the nutrient in eutrophication-lake in the interval \((n\tau ,(n+l)\tau )]\), \(\beta _{11}>0\) represents the maximum growth rate of phytoplankton \(x_{1}\) in eutrophication-lake in the interval \((n\tau ,(n+l)\tau )]\), \(0<\delta _{11}<1\) represents the yield of the nutrients for phytoplankton \(x_{1}\) in eutrophication-lake in the interval \((n\tau ,(n+l)\tau )]\), \(\beta _{12}>0\) represents the maximum growth rate of phytoplankton \(x_{2}\) in eutrophication-lake in the interval \((n\tau ,(n+l)\tau )]\), \(0<\delta _{12}<1\) represents the yield of the nutrients for phytoplankton \(x_{2}\) in eutrophication-lake in the interval \((n\tau ,(n+l)\tau )]\), \(d_{12}>0\) represents the death and loss rate of the phytoplankton in the eutrophication-lake in the interval \((n\tau ,(n+l)\tau )]\), \(0<\mu _{s}<1\) represents the impulsive dredging effect on the nutrients in the eutrophication-lake at moment \(t=(n+l)\tau \), \(0<\mu _{1}<1\) represents the impulsive dredging effect on phytoplankton \(x_{1}\) in the eutrophication-lake at moment \(t=(n+l)\tau \), \(0<\mu _{2}<1\) represents the impulsive dredging effect on phytoplankton \(x_{2}\) in the eutrophication-lake at moment \(t=(n+l)\tau \), \(\lambda _{2}>0\) represents the input concentration of the nutrients from ravine streams around the eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(d_{2}>0\) represents washout and loss rate of the nutrient in eutrophication-lake on interval \(((n+l)\tau ,(n+1)\tau )]\), \(E_{s}>0\) represents the nontransient impulsive dredging effect on the nutrients in the eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(\beta _{21}>0\) represents the maximum growth rate of phytoplankton \(x_{1}\) in eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(0<\delta _{21}<1\) represents the yield of the nutrients for phytoplankton \(x_{1}\) in eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(\beta _{22}>0\) represents the maximum growth rate of phytoplankton \(x_{2}\) in eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(0<\delta _{22}<1\) represents the yield of the nutrients for phytoplankton \(x_{2}\) in eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(d_{22}>0\) represents the death and loss rate of the phytoplankton in the eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(E_{1}>0\) represents the nontransient impulsive dredging effect on phytoplankton \(x_{1}\) in the eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(E_{2}>0\) represents the nontransient impulsive dredging effect on phytoplankton \(x_{2}\) in the eutrophication-lake in the interval \(((n+l)\tau ,(n+1)\tau )]\), \(\mu >0\) represents the pulse inputting amount of the nutrients with seasonally rainstorm washing from soil around the lake at moment \(t=(n+1)\tau \). The time interval \((n\tau ,(n+l)\tau ]\) represents the nondredging period, the time interval \(((n+l)\tau ,(n+1)\tau ]\) represents the dredging period, and \(0< l<1\) represents the interval length of the nondredging.
3 Some lemmas
The solution \(X(t)=(s(t), x_{1}(t), x_{2}(t))^{T}\) of system (2.1) is a nonsmooth function X: \(R_{+}\rightarrow R_{+}^{3}\). It is continuous on \((n\tau ,(n+l)\tau ]\) and \(((n+l)\tau ,(n+1)\tau ]\), \(n\in Z_{+}\), and the limits \(X(n\tau ^{+})=\lim_{t\rightarrow n\tau ^{+} }X(t)\) and \(X((n+l)\tau ^{+})=\lim_{t\rightarrow (n+l)\tau ^{+} }X(t)\) exist. Obviously, the global existence and uniqueness of solutions of system (2.1) are guaranteed by the smoothness properties of f defined by right-side of system (2.1) [6].
Lemma 3.1
For solution \((s(t),x_{1}(t),x_{2}(t))\) of system (2.1), there exists a constant \(M>0\) such that \(s(t)\leq M\), \(x_{1}(t)\leq M\), and \(x_{2}(t)\leq M\) for all t large enough.
Proof
Defining \(V(t)=\delta s(t)+x_{1}(t)+x_{2}(t)\) and taking \(\delta =\max \{\delta _{11},\delta _{12},\delta _{21},\delta _{22}\}\) and \(d=\min \{d_{1},d_{11},d_{12}, d_{2},d_{21},d_{22}\}\), we have \(D^{+}V(t)+dV(t)\leq \delta \lambda _{1} \) for \(t\in ( n\tau ,(n+l)\tau ]\). We also have \(D^{+}V(t)+dV(t)\leq \delta \lambda _{2} \) for \(t\in ( (n+l)\tau ,(n+1)\tau ]\). Denoting \(\xi =\max \{\delta \lambda _{1},\delta \lambda _{2}\}\), we have the following inequality for \(t\neq n\tau \), \(t\neq (n+l)\tau \):
We haver \(V(n\tau ^{+})= V(n\tau )+\mu \) for \(t=n\tau \) and \(V((n+l)\tau ^{+})\leq V((n+l)\tau )\) for \(t=(n+l)\tau \). By the lemma of [6] we have
So \(V(t)\) is uniformly ultimately bounded. By the definition of \(V(t)\) we have that there exists a constant \(M>0\) such that \(s(t)\leq M\), \(x_{1}(t)\leq M\), and \(x_{2}(t)\leq M\) for t large enough.
If \(x_{i}(t)=0\) (\(i=1,2\)), then a subsystem of system (2.1) is
Between the impulsive points, system (3.1) has the analytic solution
Considering the second and fourth equations of system (3.1), the stroboscopic map of system (3.1) is presented by
The unique fixed point \(s^{\ast }\) of (3.3) is
□
Similarly to [12], we can easily obtain the following two lemmas.
Lemma 3.2
The fixed point \(s^{\ast }\) of (3.3) defined in (3.4) is globally asymptotically stable.
Lemma 3.3
The periodic solution of system (3.1) is globally asymptotically stable, where is defined as
where \(s^{\ast }\) is defined in (3.4), and \(s^{\ast \ast }\) is defined as
4 The dynamics
Theorem 4.1
If
and
then the phytoplankton-extinction periodic solution of system (2.1) is globally asymptotically stable, where \(s^{\ast }\) is defined in (3.4), and \(s^{\ast \ast }\) is defined in (3.6).
Proof
We first prove that the phytoplankton-extinction solution of (2.1) is locally stable. Defining , \(x_{1}(t)=x_{1}(t)\), and \(x_{2}(t)=x_{2}(t)\), we have the following linearly similar system for system (2.1), which is concerning one periodic solution of system (2.1):
and
It is easy to obtain the fundamental solution matrix on interval \((n\tau ,(n+l)\tau ]\)
There is no need to calculate the exact form of \(\ast _{1j}\) (\(j=1,2,3\)) as they are not required in the analysis that follows, and the fundamental solution matrix on the interval \(((n+l)\tau ,(n+1)\tau ]\) is
where . There is no need to calculate the exact form of \(\star _{2j}\) (\(j=1,2,3\)) as they are not required in the analysis that follows.
For \(t=(n+l)\tau \), the linearization of the fourth, fifth, and sixth equations of (2.1) is
For \(t=(n+1)\tau \), the linearization of the tenth, eleventh, and twelfth equations of (2.1) is
The stability of the periodic solution is determined by the eigenvalues of
The eigenvalues of (4.9) are represented as
and
From (4.1) and (4.2) we have \(| \lambda _{2}| <1\) and \(| \lambda _{3}| <1\). Then, according to the Floquet theory [6], we can obtain that the phytoplankton-extinction solution of system (2.1) is locally stable.
In the next step, we prove that the phytoplankton-extinction solution of system (2.1) is globally attractive. Choosing \(\varepsilon >0\) such that
and
we have the following two inequalities by the first and seventh equations of (2.1):
and
Therefore we find the comparatively impulsive differential equation
From Lemma 3.3. and the comparison theorem of impulsive equation [6] we have \(s(t)\leq s_{1}(t)\) and as \(t\rightarrow \infty \). Then there exists \(\varepsilon >0\) small enough such that
for all t large enough. For convenience, we assume that (4.13) holds for all \(t\geq 0\). From (2.1) and (4.13) we have
Therefore
and
Hence \(x_{i}(n\tau )\leq x_{i}(0^{+})\rho _{i}^{n}\) (\(i=1,2\)). So \(x_{i}(n\tau )\rightarrow 0\) (\(i=1,2\)) as \(n\rightarrow \infty \). Therefore \(x_{i}(t)\rightarrow 0\) (\(i=1,2\)) as \(t\rightarrow \infty \).
In the third step, we prove that as \(t\rightarrow \infty \). For \(\varepsilon _{1}>0\) small enough, there exists \(t_{0}>0\) such that \(0< x_{1}(t)<\varepsilon _{1} \) and \(0< x_{2}(t)<\varepsilon _{1} \) for all \(t\geq t_{0}\). Without loss of generality, we assume that \(0< x_{1}(t)<\varepsilon _{1} \) and \(0< x_{2}(t)<\varepsilon _{1} \) for all \(t\geq {0}\). Then we have
and
and \(z_{2}(t)\leq s(t)\leq z_{1}(t)\) and , as \(t\rightarrow \infty \), where \(z_{1}(t)\) and \(z_{2}(t)\) are the solutions of
and
respectively. Similarly to Lemma 3.3, the periodic solution of (4.17), which is globally asymptotically stable, is
where
and
Therefore, for any \(\varepsilon >0\), there exists \(t>t_{1}\) such that
Letting \(\varepsilon _{1}\rightarrow 0\), we have
for t large enough, which implies as \(t\rightarrow \infty \). This completes the proof. □
Theorem 4.2
If
and
then system (2.1) is permanent, where \(s^{\ast }\) is defined in (3.4) and \(s^{\ast \ast }\) is defined in (3.6).
Proof
By Lemma 3.1, \(s(t)\leq M\), \(x_{1}(t)\leq M\), and \(x_{2}(t)\leq M\) for t large enough. We may assum that \(s(t)\leq M\), \(x_{1}(t)\leq M\), and \(x_{2}(t)\leq M \) for \(t\geq 0\). Therefore we have
and \(s(t)\geq z_{3}(t)\) and as \(t\rightarrow \infty \), where \(z_{3}(t)\) is the globally asymptotically stable solution of the comparatively impulsive differential equation
with
where
and
Therefore, for any \(\varepsilon _{2}>0\),
for t large enough, which implies that
Thus we only need to find \(m_{1}>0\) such that \(x_{1}(t)\geq m_{1}\) and \(x_{2}(t)\geq m_{1}\) for t large enough.
By the conditions of this theorem we can select \(m_{3}>0\) and \(\varepsilon _{1}> 0\) small enough such that
and
We prove that \(x_{1}(t)< m_{3}\) and \(x_{2}(t)< m_{3}\) cannot hold for \(t\geq 0\). Otherwise,
By Lemma 3.3 we have \(s(t)\geq z(t)\) and \(z(t)\rightarrow \overline{z(t)}\), \(t\rightarrow \infty \), where \(z(t)\) is the globally asymptotically stable solution of
with
where
and
Therefore there \(T_{1}>0\) such that, for \(t\geq T_{1}\),
and
Let \(N_{1}\in N\) and \(N_{1}\tau > T_{1}\), Integrating (4.36) on \((n\tau ,(n+1)\tau )\), \(n\geq N_{1}\), we have
and
Then \(x_{1}((N_{1}+k)\tau )\geq (1-\mu _{1})^{k}x_{1}(N_{1}\tau ^{+})e^{k \sigma _{1}}\rightarrow \infty \) and \(x_{2}((N_{1}+k)\tau )\geq (1-\mu _{2})^{k}x_{2}(N_{1}\tau ^{+}) e^{k \sigma _{2}}\rightarrow \infty \) as \(k\rightarrow \infty \), which is a contradiction to the boundedness of \(x_{1}(t)\) and \(x_{1}(t)\). Hence there exists \(t_{1}>0\) such that \(x_{1}(t)\geq m_{1}\) and \(x_{2}(t)\geq m_{1}\). The proof is complete. □
5 Discussion
According to the fact of water management, we propose a periodic lake-eutrophication model with nontransient/transient impulsive dredging and pulse inputting on nutrients. We proved that the phytoplankton-extinction boundary periodic solution of system (2.1) is globally asymptotically stable and obtained the conditions for the permanence of system (2.1).
If we suppose that \(s(0)=0.3\), \(x_{1}(0)=0.3\), \(x_{2}(0)=0.3\), \(\lambda _{1}=0.5\), \(d_{1}=0.2\), \(\beta _{11}=0.5\), \(\delta _{11}=1\), \(\beta _{12}=0.5\), \(\delta _{12}=1\), \(d_{11}=0.4\), \(d_{12}=0.4\), \(\lambda _{2}=0.1\), \(d_{2}=0.2\), \(\beta _{21}=0.3\), \(\delta _{21}=1\), \(\beta _{22}=0.3\), \(\delta _{22}=1\), \(d_{21}=0.3\), \(d_{22}=0.3\), \(E_{s}=0.3\), \(E_{1}=0.2\), \(E_{2}=0.2\), \(\mu _{s}=0.28\), \(\mu _{1}=0.1\), \(\mu _{2}=0.1\), \(\mu =0.1\), \(l=0.8\), \(\tau =1\), then these parameter values satisfy Theorem 4.1. Then the phytoplankton-extinction periodic solution of system (2.1) is globally asymptotically stable (see Fig. 2). If we assume that \(s(0)=0.3\), \(x_{1}(0)=0.3\), \(x_{2}(0)=0.3\), \(\lambda _{1}=0.5\), \(d_{1}=0.2\), \(\beta _{11}=0.8\), \(\delta _{11}=1\), \(\beta _{12}=0.8\), \(\delta _{12}=1\), \(d_{11}=0.4\), \(d_{12}=0.4\), \(\lambda _{2}=0.2\), \(d_{2}=0.2\), \(\beta _{21}=0.5\), \(\delta _{21}=1\), \(\beta _{22}=0.5\), \(\delta _{22}=1\), \(d_{21}=0.3\), \(d_{22}=0.3\), \(E_{s}=0.3\), \(E_{1}=0.2\), \(E_{2}=0.2\), \(\mu _{s}=0.2\), \(\mu _{1}=0.1\), \(\mu _{2}=0.1\), \(\mu =0.1\), \(l=0.8\), \(\tau =1\), then these parameter values satisfy Theorem 4.2. Then system (2.1) is permanent (see Fig. 3). From Theorems 4.1 and 4.2, and Figs. 2 and 3 we can deduce that the parameter \(\lambda _{2}\) has a controlling threshold \(\lambda _{2}^{\ast }\). When \(\lambda _{2}<\lambda _{2}^{\ast }\), the phytoplankton-extinction periodic solution of system (2.1) is globally asymptotically stable. When \(\lambda _{2}>\lambda _{2}^{\ast }\), system (2.1) is permanent. That is to say, we should reduce the nutrients indraughting lake-ecosystem during nontransient impulsive dredging.
The parameter values \(s(0)=0.3\), \(x_{1}(0)=0.3\), \(x_{2}(0)=0.3\), \(\lambda _{1}=0.5\), \(d_{1}=0.2\), \(\beta _{11}=0.5\), \(\delta _{11}=1\), \(\beta _{12}=0.5\), \(\delta _{12}=1\), \(d_{11}=0.4\), \(d_{12}=0.4\), \(\lambda _{2}=0.1\), \(d_{2}=0.2\), \(\beta _{21}=0.3\), \(\delta _{21}=1\), \(\beta _{22}=0.3\), \(\delta _{22}=1\), \(d_{21}=0.3\), \(d_{22}=0.3\), \(E_{s}=0.3\), \(E_{1}=0.2\), \(E_{2}=0.2\), \(\mu _{s}=0.28\), \(\mu _{1}=0.1\), \(\mu _{2}=0.1\), \(\mu =0.1\), \(l=0.6\), \(\tau =1\) satisfy Theorem 4.1. Then the phytoplankton-extinction periodic solution of system (2.1) is globally asymptotically stable (see Fig. 4). From Theorems 4.1 and 4.2 and from the simulation experiments of Figs. 3 and 4 we can easily deduce that there exists a threshold \(l^{\ast }\). If \(l>l^{\ast }\), then system (2.1) is permanent. If \(l < l^{\ast }\), then the phytoplankton-extinction periodic solution of system (2.1) is globally asymptotically stable. That is to say, a too long nontransient impulsive period will confuse the lake-ecosystem. Then appropriate extending the nontransient impulsive period will be beneficial to water resource management. A similar discussion may do with thresholds of the parameters \(\lambda _{1}\), \(\mu _{s}\), \(\mu _{1}\), \(\mu _{2}\), and so on. Therefore the method of dredging sediment engineering should be combined with implementing ecological engineering to restore and rebuild healthy and stable aquatic ecosystem, which should be an effective way to control eutrophic lakes. Our results also provide reliable tactic basis for the practical water resource management.
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This paper is supported by National Natural Science Foundation of China (11761019, 11361014), the Science Technology Foundation of Guizhou Education Department (20175736-001, 2008038), the Science Technology Foundation of Guizhou (2010J2130), the Project of High Level Creative Talents in Guizhou Province (20164035), and the Joint Project of Department of Commerce and Guizhou University of Finance and Economics (2016SWBZD18).
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Jiao, J., Tang, W. Dynamics of a lake-eutrophication model with nontransient/transient impulsive dredging and pulse inputting. Adv Differ Equ 2021, 280 (2021). https://doi.org/10.1186/s13662-021-03434-5
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DOI: https://doi.org/10.1186/s13662-021-03434-5