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We propose an algae-fish semicontinuous system for the Zeya Reservoir to study the control of algae, including biological and chemical controls. The bifurcation and periodic solutions of the system were studied using a Poincaré map and a geometric method. The existence of order-1 periodic solution of the system is discussed. Based on previous analysis, we investigated the change in the location of the order-1 periodic solution with variable parameters and we described the transcritical bifurcation of the system. Finally, we provided a series of numerical results to illustrate the feasibility of the theoretical results. These results may help to facilitate a better understanding of algal control in the Zeya Reservoir.

The economic development of human society means that the waters of lakes, marshes, and reservoirs are experiencing increasingly serious eutrophication, which can cause sustained algal growth. With a high level of eutrophication, algae with rapid growth characteristics may form algal blooms, which can lead to ecological failure and even cause harm to humans. For example, algal blooms due to eutrophication appear frequently in the Zeya Reservoir in Wenzhou, which is located in a subtropical region, and this may cause deterioration in the water quality that could deprive millions of people of drinking water.

Therefore, it is necessary to control algal growth. Indeed, many researchers have studied these ecological systems, including the use of biological and chemical controls, and these systems have been described using impulsive differential equations. The theory of impulsive differential equations has experienced a period of intensive development [

In many practical cases, however, such as algal blooms and pest control, the impulses often depend on the state rather than fixed time periods. Thus, semicontinuous dynamic systems have been introduced for these purposes. In this study, the so-called semicontinuous dynamic system is defined using a set of impulsive state-dependent differential equations [

In this paper, we consider a semicontinuous ecological system. The main difference between our results and those described in [

This paper is organized as follows. Section

We consider an autonomous system with an impulse effect as

Let

Let

orbitally stable if for all

orbitally semistable if for all

orbitally attractive if for all

orbitally asymptotically stable if it is orbitally stable and orbitally attractive.

In this discussion,

The phase plane is divided into two parts by the trajectory of the differential equations that constitute the order-1 cycle. The section containing the impulse line and the trajectory is known as the inside of the order-1 cycle.

We assume that

The successor function

The

Let

In system (

In system (

For all

Set

If

Similar to (i), we have

(ii)

(iii)

If

If

Similar to (iii), we have

(iv)

Therefore, the trajectory with the initial point

According to (iv), the trajectory with the initial point

Similar to Lemma

In system (

First, we consider the case of system (

A vector graph of system (

In this subsection, we will derive some basic properties for the following subsystem of system (

Setting

Thus, the following theorem is obtained.

There exists a semitrivial order-1 periodic solution (

It is known that

Furthermore,

Therefore, it is possible to obtain the Floquet multiplier

Thus,

If

There exists a positive order-1 periodic solution in system (

Because

The proof is completed.

(a) is the proof on Theorem

In this case, we suppose that

There exists a positive order-1 periodic solution for system (

The method for this proof is similar to the method for Theorem

In summary, system (

For any

We suppose that the period of the order-1 periodic solution is

In this subsection, we will discuss the bifurcation near the semitrivial periodic solution. The following Poincaré map

Using Lemma

A transcritical bifurcation occurs when

The values of

Let

Using (

Furthermore,

The next step is to check whether the following conditions are satisfied.

It is easy to see that

Using (

Because (

Finally, inequality (

These conditions satisfy the conditions of Lemma

In this subsection, we will discuss the movement of the order-1 periodic solution with variable parameters. The following theorem is required.

The rotation direction of the pulse line is clockwise if

Let

The existence of an order-1 periodic solution was proved in the previous analysis, so we assume that there exists an order-1 periodic solution when

In system (

The order-1 periodic solution breaks when

Next, the orbital stability can be established based on the following proof (see Figure

The order-1 periodic solution

If

While

In addition,

Obviously,

It represents the proof on Theorem

Similar to the method used for the proof of Theorem

In system (

The following numerical results are provided to illustrate the feasibility of the theoretical results. In this section, the parameters are fixed as follows:

Based on the previous analysis, there exists a semitrivial solution when

Trajectories with the initial point (0.02, 0.01) in system (

According to Theorems

(a) an order-1 periodic solution of system (

Figure

Existence of an order-1 periodic solution for system (

From Theorems

Transition of the order-1 periodic solution of system (

To study the dynamics of system (

Bifurcation diagrams for system (

According to Theorem

In this paper, we developed an algae-fish semicontinuous model, which we studied analytically and numerically. Theoretical mathematical studies have investigated the existence and stability of a semi-trival periodic solution and an order-1 periodic solution of system (

In the semicontinuous system, the movement of the order-1 periodic solution was first studied theoretically, which will be useful for studying the control of algae. In system (

In addition, our results are useful for others systems. For example, some applications refer to the mathematical model proposed in the literature [

This work was supported by the National Key Basic Research Program of China (973 Program, Grant no. 2012CB426510), by the National Natural Science Foundation of China (Grant nos. 31170338 and 31370381), and by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001).