Analysis of a Periodic Impulsive Predator-Prey System with Disease in the Prey

We investigate a periodic predator-prey system subject to impulsive perturbations, in which a disease can be transmitted among the prey species only, in this paper. With the help of the theory of impulsive differential equations and Lyapunov functional method, sufficient conditions for the permanence, global attractivity, and partial extinction of system are established, respectively. It is shown that impulsive perturbations contribute to the above dynamics of the system. Numerical simulations are presented to substantiate the analytical results.


Introduction
As a relatively new branch of study in theoretical biology, ecoepidemiology can be viewed as the coupling of ecology and epidemiology.Ecoepidemiological model is more appropriate than the ecological model (or epidemiological model) when species spreads the disease and is predated by other species.Following Anderson and May [1] who were the first to propose an ecoepidemiological model, a number of sophisticated predator-prey models with disease in prey population only are extensively investigated in the ecoepidemiological literature (see [2][3][4][5]).
Notice that periodic phenomenon often occurs in many realistic ecoepidemiological models.The effect of a periodically varying environment is important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment.Nicholson [6] has suggested that any periodic change of climate tends to impose its period upon oscillations of internal origin or to cause such oscillations to have a harmonic relation to periodic climatic changes.Thus, it is reasonable to assume that the coefficients in the systems are periodic functions.On the other hand, in reality, many evolution processes are characterized by the fact that they experience changes of state suddenly.These processes are subject to short-term perturbations whose duration is negligible in comparison with the duration of the process.Consequently, the abrupt changes can be well approximated as impulses.A natural description of the motion of impulsive processes can be expressed by impulsive differential equations.Some impulsive equations have been introduced in ecoepidemiological models in relation to chemotherapeutic [7] and vaccination [8,9] and population disease control [10,11].
Considering the above facts, in this paper, we will consider a periodic predator-prey model subject to impulsive perturbations, in which a disease can be transmitted among the prey species only.Our motive comes from a delayed nonautonomous predator-prey system with disease in the prey in [5], and we consider the effect of impulsive perturbations on a corresponding undelayed periodic version in this paper.Here, we will establish the sufficient conditions for the permanence and partial extinction of the system by using the theory of impulsive differential equations and inequality analytical technique.By Lyapunov functional method, we will also establish sufficient conditions for the global attractivity of the system.

Assumptions and Formulation of Mathematical Model
The periodic predator-prey model with disease in the prey which is studied in this paper is the system of impulsive differential equations below.To formulate the mathematical model, we need to make the following assumptions which are the same as those in [5].
(A1) All newborns are susceptible in the model, in which only susceptible prey is capable of reproducing with logistic law while the infected species does not recover or become immune.The disease only spreads among the prey species and it is not genetically inherited.
(A2) The mortality terms for susceptible and infected prey are density dependent, and both contribute to population growth toward the environmental carrying capacity.
(A3) The predator species hunts on susceptible and infected prey with possibly different predation rates.For example, in some situations, the infected individuals may be caught easily, but the predators eat more fewer infected ones in other situations.

Notations and Preliminary Lemmas
Before establishing our main results, we summarize several useful lemmas for the later sections.
Let J ⊂ R = (−∞, +∞).Denote by (J, R) the set of functions V : J → R which are continuous for  ∈ J,  ̸ =   , are continuous from the left for  ∈ J, and have discontinuities of the first kind at the points   ∈ J. Denote by  1 (J, R) the set of functions V : J → R with a derivative V/ ∈ (J, R).

Main Results
In this section, we will establish sufficient conditions for the permanence, global attractivity of positive solutions, and partial extinction of system (1), respectively.We first give the result on permanence.29), (33), and (38), respectively.

(I) Consider the uniformly ultimately upper boundary (or UUUB) of 𝑆(𝑡), 𝐼(𝑡), 𝑃(𝑡).
First of all, we discuss the UUUB of ().It follows from the first equation of (1) and impulsive condition that Now, we consider two cases to obtain the UUUB of ().
For (11), applying Lemma 1, we have When  ∈ (, ( + 1)],  ∈ N ∪ {0}, we set It follows from (13) that If If This fact implies that lim  → +∞ () = 0.That is, there are a sufficient small constant  1 > 0 and  1 > 0 such that It follows from (2) in Lemma 4 that the comparison system of ( 12) has a unique positive globally asymptotically stable periodic solution denoted by  1 ().Let  1 () be the solution of (19) with  1 (0) = (0) > 0. The asymptotic property of  1 () shows that there exist a sufficient small constant  2 > 0 and  2 > 0 such that Applying the comparison theorem of impulsive differential equations, one has Equations ( 18) and (21) show that there must be Next, we discuss the UUUB of ().From the second equation of (1) and impulsive condition, for  ≥  1 , one has Using (2) in Lemma 3, we can see that the comparison system of ( 23) is permanent, which implies that there are  2 > 0 and  2 ≥  1 such that Using the comparison theorem of impulsive differential equations, we get Finally, we verify the UUUB of ().For  ≥  2 , from the third equation of ( 1) and impulsive condition, we obtain Similar to the proof of the UUUB of (), it follows from (10) that there exist  3 > 0 and  3 > 0 such that (II) Consider the uniformly ultimately lower boundary (or UULB) of (), (), ().
(I) and (II) yield and hence system (1) is permanent.The proof is completed.

Global Attractivity.
In this subsection, we will discuss the global attractivity of system (1) based on Theorem 6.  (