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Almost Periodic Solution for an Epidemic Prey-Predator System with Impulsive Effects and Multiple Delays
A nonautonomous epidemic prey-predator system with impulsive effects and multiple delays is considered; further, there is an epidemic disease in the predator. By the mean-value theorem of multiple variables, integral inequalities, differential inequalities, and other mathematical analysis skills, sufficient conditions which guarantee the permanence of the system are obtained. Furthermore, by constructing a series of Lyapunov functionals it is proved that there exists a unique uniformly asymptotically stable almost periodic solution of the system.
It is reported that there were more than twenty kinds of newly emerging viruses in the recent 30 years, such as Ebola, Hantavirus, AIDS, SARS, H5N1, and H7N9. Authoritative experts express the fact that the fundamental cause of these epidemic viruses is the loss of ecological balance and environmental degradation, and the basic rule between human and the nature is destroyed, which lead to the revenge of the nature and the invasion of the virus. Therefore, human, environment, and the disease are mutually affective and mutually restrictive.
Ecological epidemiology is a new subject to study the distribution and propagation rules between humans and other species; it is the combination of ecology and epidemiology, starting from the changes of ecological environment. And one of the most important aims of the ecological epidemiology is to study the propagation trend of the disease in the ecological systems so that the government can give corresponding controlling strategies in first time. Thus, it is very meaningful and valuable to study the dynamics of an epidemic ecological system. And in the recent years, more and more ecologists, epidemiologists, and mathematicians have devoted themselves to the study of the epidemic ecological models (see [1–20]), and most of the models are ordinary differential equations (see [1–15]) or functional differential equations (see [16–20]), respectively. For example, Xiao and Bosch  derived the following ecoepidemic model with disease in the predator in 2003: in which mathematical analyses of the model equations with regard to invariance of nonnegativity, boundedness of solutions, nature of equilibria, permanence, and global stability are analyzed.
The environment varies due to the factors such as seasonal effects of weather, food supplies, mating habits, and harvesting. So it is more reasonable to assume the periodicity or almost periodicity of parameters in the systems. And considering this factor, Tian et al.  considered the following nonautonomous epidemic prey-predator system in 2009:
On the other hand, it is known that the ecoepidemic system will often be perturbed by the factors of human exploitation activities, such as planting and harvesting. Among these human disturbances, periodic or almost periodic and instantaneous perturbations are the most common, and these perturbations can be regarded as the impulsive effects when modeling the system. If the factors of time delays are also considered, then the corresponding models should be described as impulsive functional differential equations. However, to the best our knowledge, references on the periodic ecoepidemic model with impulsive perturbations seem relatively fewer. Since the rapid development in the theory of the impulsive equations and the functional differential equations (see [21–26]), many excellent results have been derived in ecological models, epidemic models, and even neural network models (see [27–37] and so on).
Enlightened by the above work, in this paper we will propose an epidemic ecological prey-predator model with impulsive perturbations and three kinds of time delays. First, we generalized the term into in the present model, which means the negative feedback intermediate prey crowding. Second, we consider there is a time of digestion of the susceptible predator. Third, we consider there is a latency period before an infected predator could infect a susceptible predator. The final model we will study in this paper is as follows: where denotes the density of the prey, and denote the density of the susceptible and the infected predators, means the intrinsic rate of natural increase, and the minus before means that the susceptible predator is dependent on the prey. denotes the coefficient of the density dependence of the prey, denotes the competitive coefficients between the predators, means the preying capacity of the susceptible and the infected predators, means the relative preying capacity of the susceptible predator, and means the touching rate between the susceptible predators.
denotes the impulsive effects. When , the effects represent planting, when , the effects denote harvesting.
Throughout the present paper, we define for any bounded function defined on .
Further, we assume that(C1), , , , and are all bounded and positive almost periodic functions;(C2) is almost periodic functions and there exist positive constants and such that
The rest of this paper is organized as follows. In Section 2, we will give several useful lemmas for the proof of our main results. In Section 3, we will state and prove our main results such as the permanence of the system and the existence and the uniqueness of almost periodic solution which is uniformly asymptotically stable by constructing a series of Lyapunov functional. In the last section, we will give some discussions and give a brief summary for the paper.
, and we denote the set of all sequences that are unbounded and increasing. Let , , , . Also, we denote as the space of all functions having points of discontinuity at of the first kind, being left continuous at these points.
For , is the space of all piecewise continuous functions from to with points of discontinuity of the first kind , at which it is left continuous.
Let , denote , , , and is the solution of system (3) satisfying the following initial conditions:
Lemma 1 (see ). (1) Assume that, for , the following holds:with initial conditions , where , are positive constants; then there exists a positive constant such that (2) Assume that, for , the following holds: with initial conditions , where and are positive constants; then there exists a positive constant such that
Also, as a direct application of the conclusion in the proof of Corollary 4.1 in , we can obtain the following lemma.
Lemma 2. For , .(1)If , then for any , there exists a constant , such that That is, (2)If , then for any , there exists a constant , such that That is,
If we consider the following almost periodic system with delay: and the associate product system of system (14) is in the form of then, by the conclusions of , one has the following lemma.
Lemma 3 (see ). For , suppose that there exists a Lyapunov function defined on satisfying the following three conditions:(1), where is continuous increasing function and ;(2), where is a positive constant;(3), where is a positive constant. Further, assume that (14) has a solution such that for , . Then system (14) has a unique almost periodic solution which is uniformly asymptotically stable.
Lemma 4 (see ). For any , one has the following inequality:
Now, we will study the nonimpulsive system of system (3): with the initial condition where , , . Further expressions of functions , , , , , are given as follows:
Then we have the following lemma.
Proof. LetThen it follows from the first equation of system (17) that From the second equation of system (17), we have Integrating the above inequality on the interval yields Similarly, from the last equation of system (17), we have Integrating the above inequality on the interval yields This completes the proof of this lemma.
Lemma 6. For system (3) and system (17), one has following results:(1)If is a solution of system (17), then is a solution of impulsive system (3);(2)If is a solution of system (3), then is a solution of the nonimpulsive system (17).
Proof. (1) If is a solution of system (17), then, for any , , is a solution of system (17), which yields By previous definitions of the function , , , and , it follows from the simplification of (29) that On the other hand, when , we have Similarly, we can verify that Thus, from (30)–(32) we know that is the solution of impulsive system (3).
(2) Since , , and are continuous on each interval , then we only need to check the continuity of them at the impulsive point In fact, That is, Similarly, we can check that Therefore, , , and are continuous on .
On the other hand, similar to the above proof process of step , we can verify that satisfies system (17).
Then, is a solution of nonimpulsive system (17).
This completes the proof of Lemma 6.
3. Main Results
Theorem 7. Assume that (C1)-(C2) hold, and suppose further that (C3);(C4);(C5);(C6);then any positive solution of system (17) satisfies where
Proof. From the first equation of system (17) we have It follows from the first conclusion of Lemma 1 that Thus, there exists a , such that when .
At the moment, from the second equation of system (17), when , we have Since condition (C3) holds, then it follows from the second conclusion of Lemma 2 that Therefore, there exists a , such that when , and when from the last equation of system (17) we have Then it follows from the second conclusion of Lemma 2 again that Therefore, there exists a , such that when .
At the moment, considering the inequality estimations on the opposite direction from all the equations of system (17), we can obtain Since condition (C4) holds, then by the second conclusion of Lemma 1 we have Then there exists a , such that when . When it follows from the second equation of system (17) that Since condition (C5) holds, and by the second conclusion of Lemma 2 again, we have Then there exists a , such that when . When , it follows from the last equation of system (17) that Since condition (C6) holds and it follows from Lemma 2 again, we have that Thus, combining (39), (41), and (43) with (45), (47), and (49), we can see that the proof of this theorem is completed.
Theorem 8. Assume that (C1)–(C6) hold; then any positive solution of system (3) satisfies
Proof. Since is a solution of system (3), then by the second conclusion of Lemma 6is a solution of system (17).
Then it follows from Theorem 7 that which implies that This completes the proof of this theorem.
Remark 9. Suppose that (C1)–(C6) hold; then system (3) is permanent.
In the following, we will discuss the uniformly asymptotic stability of a unique almost periodic solution of system (3) by Lemma 4. And for the sake of convenience, we give some notations before the theorem as follows.
Theorem 10. Assume that (C1)–(C6) hold, and further assume that(C7)there exists a set of positive parameters , such that the following linear matrix inequality holds: Then system (3) admits a unique almost periodic solution which is uniformly asymptotically stable, where
Proof. At first, we will prove that system (17) has a unique uniformly asymptotically stable almost periodic solution. In order to achieve this aim, we take a transformation: Then system (17) is transformed into the following system: Suppose that and are any two solutions of system (59); then the product system of (59) reads Denote
For any , we can choose such that Consider a Lyapunov functional defined on , where According to the definitions of and , there is some positive constant large enough such that By the structure of , it is easy to see Let , and then Moreover, by the integrative inequality and the absolute-value inequality properties we havewhere