Dynamics of a lake-eutrophication model with nontransient/transient impulsive dredging and pulse inputting

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.


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][7][8][9][10][11][12][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.

Lemma 3.2
The fixed point s * of (3.3) defined in (3.4) is globally asymptotically stable.

5)
where s * is defined in (3.4), and s * * is defined as The dynamics and then the phytoplankton-extinction periodic solution ( s(t), 0, 0) of system (2.1) is globally asymptotically stable, where s * is defined in (3.4), and s * * is defined in (3.6).
Proof We first prove that the phytoplankton-extinction solution ( s(t), 0, 0) of (2.1) is locally stable. Defining s 1 (t) = s(t)s(t), 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 ( s(t), 0, 0) of system (2.1): and ⎛ ⎜ ⎝ It is easy to obtain the fundamental solution matrix on interval (nτ , (n + l)τ ] There is no need to calculate the exact form of * 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)τ , (n+1)τ ] is There is no need to calculate the exact form of 2j (j = 1, 2, 3) as they are not required in the analysis that follows.
For t = (n + l)τ , the linearization of the fourth, fifth, and sixth equations of (2.1) is For t = (n + 1)τ , the linearization of the tenth, eleventh, and twelfth equations of (2.1) is The stability of the periodic solution ( s(t), 0, 0) is determined by the eigenvalues of The eigenvalues of (4.9) are represented as From (4.1) and (4.2) we have |λ 2 | < 1 and |λ 3 | < 1. Then, according to the Floquet theory [6], we can obtain that the phytoplankton-extinction solution ( s(t), 0, 0) of system (2.1) is locally stable.

.22)
and and s(t) ≥ z 3 (t) and z 3 (t) → z 3 (t) as t → ∞, where z 3 (t) is the globally asymptotically stable solution of the comparatively impulsive differential equation )M](t-nτ ) ], t ∈ (nτ , (n + l)τ ], Therefore, for any ε 2 > 0, for t large enough, which implies that Thus we only need to find m 1 > 0 such that x 1 (t) ≥ m 1 and x 2 (t) ≥ m 1 for t large enough. By the conditions of this theorem we can select m 3 > 0 and ε 1 > 0 small enough such that We prove that x 1 (t) < m 3 and x 2 (t) < m 3 cannot hold for t ≥ 0. Otherwise,

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 we can deduce that the parameter λ 2 has a controlling threshold λ * 2 . When λ 2 < λ * 2 , the phytoplankton-extinction periodic solution ( s(t), 0, 0) of system (2.1) is globally asymptotically stable. When λ 2 > λ * 2 , system (2.1) is permanent. That is to say, we should re-  (t), 0, 0) 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 * . If l > l * , then system (2.1) is permanent. If l < l * , then the phytoplankton-extinction periodic solution ( s(t), 0, 0) 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 λ 1 , μ s , μ 1 , μ 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.