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In this paper, a nonautonomous eco-epidemiological model with disease in the predator is formulated and analyzed, in which saturated predation rate is taken into consideration. Under quite weak assumptions, sufficient conditions for the permanence and extinction of the disease are obtained. Moreover, by constructing a Liapunov function, the global attractivity of the model is discussed. Finally, numerical simulations verified these results.

In the nature world, diseases for each species are inevitable. So it has practical ecological significance to consider the effects of disease in predator-prey model. Over the past decade, great attention has been paid to modelling and analyzing eco-epidemiological systems (see [

where

The models, which were proposed in the literatures [

Motivated by these factors, we modify a predator-prey model with disease in predator by introducing standard infection rate

where

The initial conditions are

It is obvious that the set

This paper is organized as follows. In the next section, some useful lemmas are proposed. In Section 3, we establish the sufficient conditions for the permanence and extinction of the disease. Also, by constructing a Liapunov function, we obtain the global attractivity of the model. Moreover, as applications of the main results, some corollaries are introduced. Particularly, the periodic model is discussed. In Section 4, our qualitative results for the periodic system are verified by numerical simulation. This paper is ended with a conclusion.

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

Definition 2.1. System (1.2) is said to be permanent if there exists a compact region

Definition 2.2. The disease is said to be extinct if the solution of system (1.2) with initial conditions (1.3) satisfy

Definition 2.3. The system (1.2) is said to be globally attractive if for any two solutions

To prove our main results, first, we give the results on the following nonautonomous Logistic differential equation:

where functions

Lemma 2.1 [

Then

(a) There exist

(b) Each fixed solution

(c) If

(d) When Equation (2.1) is W-periodic, then Equation (2.1) has a unique nonegative W-periodic solution which is globally uniformly attractive.

Second, we consider the following equation:

where functions

Lemma 2.2 If there exist positive constants

Then for any constants ε > 0 and

Lemma 2.2 can be easily proved and hence we omit it here.

Third, we give the following nonautonomous linear differential equation

where functions

Lemma 2.3 [

Then

(a) There exist

(b) Each fixed solution

(c) If

(d) When Equation (2.3) is W-periodic, then Equation (2.3) has a unique nonegative W-periodic solution which is globally uniformly attractive.

Finally, we investigate the following equation

where functions

Lemma 2.4 [

Then for any constants ε > 0 and

In this section, we will study the permanence and extinction of infected predator, and then, demonstrate the global attractivity of system (1.2).

First, as a preliminary, we make the following assumptions:

(B1) Functions

(B2) There exist positive constant

Next, we will discuss the ultimate boundness and the permanence of prey and predator of system (1.2).

Theorem 3.1 Suppose that assumptions (B1) and (B2) hold, if there exists a constant

hold, where

prey population

Proof. Let

Based on the assumption (B2), the conclusion (a) of Lemma 2.3 and the com- parison theorem, there exist constant

If

From the second and third equations of (1.2) and (3.2), we have obtain that for all

Based on the assumption (B2), we have

According to the conclusion (a) of Lemma 2.1 and the comparison theorem, there exist constant

If

Consequently, any solutions

Furthermore, from the first equation of system (1.2) and (3.4), we can obtain that for all

According to Lemma 2.3 (a) and the comparison theorem, there are constant

If

Moreover, it follows from the second and third equations of system (1.2) and (3.6) that for

Based on the assumption (B3), the comparison theorem and conclusion (a) of Lemma 2.1, there exist constant

If

Therefore, from (3.2), (3.4), (3.6) and (3.8), we can obtain that

and

This completes the proof of Theorem 3.1.

Remark 3.1. Suppose that assumptions (B1), (B2), (B3) hold, and

and

Let

Particularly, if

Let

If

Let

If

Let

If

Let

If

Then we can obtain the following results.

Theorem 3.2 Suppose that assumptions (B1), (B2), (B3) hold. If there exists a constant

then the infective predator of (1.2)

Proof. Let

According to (3.2), (3.4), (3.6) and (3.8), we can obtain that there exists a constant

Following, we will prove that there is a positive constant

Constructing an auxiliary system

In view of Lemma 2.2, for the given constants

where

Set

So there exists a constant

Hence, from the second equation of system (1.2), we obtain that for all

Let

Therefore, according to

Therefore

From the first equation of system (1.2), we can obtain

By comparison theorem, we have

From the second and third equation of (1.2), we can obtain for

By comparison theorem, there are constant

From the first equation of system (1.2), we can obtain for

By comparison theorem, we have that there is a constant

Hence, from the third equation of system (1.2) and (3.20) - (3.23), we get

Integrating the above equation from

Thus (3.15) implies that

Thus, for any

(i) There exists

(ii)

It is obvious that we only need to consider the case (ii).

In the following, we will prove

Let

If

If

For any

If

So we have that

In other words, the infective predator

Theorem 3.3 Suppose that assumptions (B1) - (B3) hold. If there exists a constant

Then the infective pedator of system (1.2)

Proof. From assumption (B2), we can choose constants

For any

for all

From the second equation of system (1.2), we have

for all

From the second and third equations of (1.2), we have obtain that for all

By the comparison theorem and Lemma 2.1 (b), there exists a constant

From the second and third equations of (1.2), we have obtain that for all

By the comparison theorem and Lemma 2.1 (b), there exists a constant

Moreover, from the first equation of system (1.2), we have

for all

Let

and

If

Then it follows that

Finally, we will prove

for all_{3} be a nonnegative integer such that

This leads to a contradiction. Therefore, inequality (3.31) holds. Furthermore, since

In particularly, when system (1.2) degenerates into

(A1) Functions

In view of Theorems 3.2 and 3.3, we can get the following corollaries.

Corollary 3.1 Suppose that assumptions (A1) - (A3) hold, and

then the infective predator of system (1.2)

Corollary 3.2 Suppose that assumptions (A1) - (A3) hold, and

then the infective predator of system (1.2)

In the following, we will discuss the global attractivity of system (1.2).

Theorem 3.4 Suppose that assumptions ((B1) - (B3) hold. If there exist con-

stants

and

Proof. Denote

Let

for all

Define a Liapunov function

Calculating the Dini upper right derivative of

According to the condition

for all

then we have,

From (3.34) and system (3.33), we can obtain that

Therefore, in view of (3.36), we obtain

In other words, the system (1.2) is globally attractive. This completes the proof.

Numerical verification of the results is necessary for completeness of the analytical study. In this section, we present some numerical simulations to verify our analytical findings of system (1.2) by means of the software Matlab.

In system (1.2), let

Increasing the infective rate

Moreover, in system (1.2), let

three-dimensional space, which implies that there exists a periodic solution of system (1.2), and it is globally attractive (see

Finally, we will perform some numerical simulations to show the importance of contact rate σ. For system (1.2), in which all the coefficients are time-dependent, we then also discuss the effect of the mean value of contact rate σ on the dyna- mics of the system. Let us fix

The research has been supported by The National Natural Science Foundation of China (11561004), The 12th Five-Year Education Scientific Planning Project of Jiangxi Province (15ZD3LYB031).

Luo, Y.Q., Gao, S.J. and Liu, Y.J. (2017) Analysis of a Nonautonomous Eco-Epidemiological Model with Saturated Predation Rate. Applied Mathematics, 8, 252-273. https://doi.org/10.4236/am.2017.82021