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In a polluted environment, considering the biological population infected with a kind of disease and hunted by human beings, we formulate a nonautonomous SIR population-epidemic model with time-varying impulsive release and general nonlinear incidence rate and investigate dynamical behaviors of the model. Under the reasonable assumptions, the sufficient conditions which guarantee the globally attractive of the disease-free periodic solution and the permanence of the infected fish are established, that is, the infected fish dies out if , whereas the disease persists if . To substantiate our theoretical results, extensive numerical simulations are performed for a hypothetical set of parameter values.

It is well known that Poyang Lake located in the middle and lower reaches of the Yangtze River is the current largest freshwater lake in China. Its wetland ecosystem has a significant impact on the change of China’s environment. The sufficient water resource and the superior natural environment nurture the abundant aquatic living resources of Poyang Lake. There are 136 kinds of fishes, 87 kinds of shells, 102 kinds of aquatic vascular plants and 266 kinds of identified plankton in Poyang Lake. The fishes in Poyang Lake take up 16.39% of the fresh water fish varieties in China, and 36.76% of the fish varieties of Yangtze River system. There are also first-level and second-level national protected precious rare aquatic animals such as white-flag dolphin, cowfish, chinese sturgeon, hilsa herring and so on in Poyang Lake, making it known as the treasury of fishery resources and the fish species genetic base with a significant position in the ecology system of the fish industry of Yangtze River reaches [

At present, the grand development of Poyang Lake ecological economy is under way in a large scale in province, which promotes the establishment of the ecological economy zone [

In recent years, many scholars have been conducted to investigate the effect of toxicant emitted into the environment from industrial, agricultural and household sources on biological species [

The organization of this paper is as follows. In the next section, we give some useful notations, definitions and preliminary lemmas which will be used to proof our main results. In Section 3, we mainly investigate a nonautonomous mathematical model with general nonlinear incidence rate and time-varying impulsive release, under some assumptions and the biological interpretation. In Section 4, we show that global attractivity of the disease-free periodic solution is determined by the threshold parameter

In this section, we introduce some notations, definitions and state some lemmas which will be useful in the subsequent sections. Let C denote the space of all bounded continuous functions. Given

If f is w-periodic, then the average value of f on a time interval

Before demonstrating the global attractivity of disease-free periodic solution of system (7), we need to intro- duce an important lemma.

Lemma 1. (see [

where

First of all, the total freshwater fish is divided into three groups: Susceptible fish (S), Infected fish (I) and Re- moved fish (R). Motivated by the above works and these literatures [

The model is derived with the following assumptions.

・

・

・

・

・ The coefficients

to be nonnegative, continuous and bounded w-periodic functions in the interval

・ There exists a positive integer q such that

・ The general nonlinear incidence rate

for all integer

In the following, we give some basic properties of the following subsystem of model (1), which are very im- portant for deriving our main results.

where

Lemma 2. System (2) has a unique positive w-periodic solution

for

Proof. Integrating and solving the first equation of system (2) between pulses for

where

and

It follows from above equation and using the third equation of system (2), we get

and

Obviously,

Set

f is the stroboscopic map. It is easy to see that system (6) has a unique positive equilibrium:

Since

By Lemma 1, it is easy to see that system (7) has a unique disease-free periodic solution

To discuss the attractivity of the disease-free periodic solution of system (7), we firstly give the following hypothesis:

(A) There exist positive, continuous, periodic functions

Theorem 1. If

Proof. Let

From the second equation of system (7), we obtain that

By the comparison theorem, we can get that there exists a constant

for all

It follows from (9) and the second equation of system (7) that, for

Then, we obtain that

By using the similar method, we can infer that for

Especially, when

Therefore, we have

From the (10) and (11), we get

Therefore, for above mentioned

for all

and

where

By using the similar method, we can see that

and

where

From (14) and (15), we can see that the disease-free periodic solution

In this section, we mainly obtain the sufficient conditions for the permanence of system (7). Therefore, we give the following hypotheses at first.

(B) There exist positive, continuous, periodic functions

According to Lemma 1, we can obtain that the system has a unique positive w-periodic solution

Theorem 2. If

Proof. Since

In order to illustrate the conclusion, we firstly obtain the disease is uniformly weakly persistent, that is, there

exists a positive constant

In view of the Hypothesis (A) and the first equation of system (7), we get

By comparison theorem, we have

Therefore, for above mentioned

for all

For above mentioned_{1} such that

Then we obtain that

By using the similar method, we can get that for

Furthermore, when

Therefore, for any positive integer

From above, we obtain that

Therefore, the claim is proved.

By the claim, we are left to consider the following two possibilities:

Case 1.

Case 2.

Define

large enough. The conclusion is evident in the first case. For the second case, let

and

for

the choice of

the case

(1) If

It follows from (19) and

Let

(2) If

On the other hand, similar to discussion in subcase (1), it is easy to know that we can choose a proper

Since

for

Then,

Since this kind of interval

Thus, we see that

According to our above discussion, the choice of

fore, system (7) is permanent.

In this paper, we have constructed an impulsive equation to model the process of periodic release of toxicant at time-varying and studied the effect of toxicant on the fish population. From a biological point of view, the most interesting results are the following. On the basis of Theorems 1 and 2, we can see that

In the following, we will give some numerical simulations to illustrate the usefulness of the results and study the impact of impulsive release strength on the basic reproductive number. Numerical values of parameters of system (1) are given in

Parameter | Value | Unit |
---|---|---|

100 | month^{−}^{1} | |

1 | month^{−}^{1} | |

month^{−}^{1} | ||

1 | month^{−}^{1} | |

month^{−}^{1} | ||

2 | month^{−}^{1} | |

1 | month^{−}^{1} | |

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

month^{−}^{1} | ||

12 | month |

We let

The research has been supported by the Natural Science Foundation of China (11261004, 11561004), the Natural Science Foundation of Jiangxi Province (20151BAB201016), and the Science and Technology Plan Pro- jects of Jiangxi Provincial Education Department (GJJ14673, GJJ150984, GJJ150995). The Supporting the Development for Local Colleges and Universities Foundation of China-Applied Mathematics Innovative Team Building.

Fumin Zhang,Shujing Gao,Yujiang Liu,Yan Zhang, (2016) Dynamics of a Nonautonomous SIR Model with Time-Varying Impulsive Release and General Nonlinear Incidence Rate in a Polluted Environment. Applied Mathematics,07,681-693. doi: 10.4236/am.2016.77062