Research Article | Open Access
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.
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  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 [2–5]).
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  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  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 , 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.
2. 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 .(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.
Considering the above basic assumptions, we can now construct the following dynamic system: Here, , , . is intrinsic birth rate function of the susceptible prey species , is death rate function of the predator species , , , and represent the self-inhibition rate functions of the susceptible prey species , the infected prey species , and the predator , respectively. is the infection rate. , and , denote the capturing rates and conversion rates, respectively. The parameters , , , , , , , , and are positive continuous -periodic functions. The fixed impulsive points satisfy and for . For two real number sequences and , there is an integer such that , , for . The jump conditions reflect the possibility of impulsive effects on the prey and predator. For biological reality, it is natural to assume , and . An obvious example of ecological situation giving rise to system (1) concerns the impulsive harvesting and stocking of the above species; that is, when , , , the impulsive perturbations stand for stocking while , mean harvesting.
The organization of this paper is as follows. In the next section, we present some notations and preliminary lemmas. In Section 4, we establish sufficient conditions for the permanence, global attractivity of positive solutions, and partial extinction for the above system, respectively. Several concrete examples and numerical simulations are also presented to substantiate the analytical results in the last section.
3. Notations and Preliminary Lemmas
Before establishing our main results, we summarize several useful lemmas for the later sections.
Let . Denote by the set of functions which are continuous for , , are continuous from the left for , and have discontinuities of the first kind at the points . Denote by the set of functions with a derivative .
Lemma 2 (see [13, 14]). Consider the following single-species impulsive model: where , the constant , is a continuous -periodic function satisfying , and there is an integer such that , . System (5) has a unique positive -periodic solution which is globally asymptotically stable.
Lemma 3 (see ). Consider the following impulsive system: where , , the constant , and , are continuous -periodic functions. Assume that there is an integer such that , . Then(1)species is permanent;(2)any two positive solutions and of (6) satisfy .
Lemma 4 (see [13, 16]). Consider the following single-species system with impulsive perturbations: where , for all , , are continuous -periodic functions and is a real number sequence with . Meanwhile, there exists an integer such that and for . Then(1)any solution of (7) satisfies if ;(2)(7) has a unique positive -periodic solution which is globally asymptotically stable if .
Lemma 5 (see ). Suppose that is a continuous -periodic function, where and . Then the following inequality holds, where , and .
4. 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.
Proof. For (1), it is easy to verify , , for all if , , . To finish the proof of Theorem 6, we consider the following two steps (i.e., (I) and (II)).(I) Consider the uniformly ultimately upper boundary (or UUUB) of , , .
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 .
Case 1. Consider .
For (11), applying Lemma 1, we have When , , we set It follows from (13) that If , then If , then This fact implies that . That is, there are a sufficient small constant and such that Case 2. Consider .
It follows from (2) in Lemma 4 that the comparison system of (12) has a unique positive globally asymptotically stable -periodic solution denoted by . Let be the solution of (19) with . The asymptotic property of shows that there exist a sufficient small constant and such that Applying the comparison theorem of impulsive differential equations, one has Equations (18) and (21) show that there must be and , , such that
Next, we discuss the UUUB of . From the second equation of (1) and impulsive condition, for , one has Using (2) in Lemma 3, we can see that the comparison system of (23) is permanent, which implies that there are and such that Using the comparison theorem of impulsive differential equations, we get
Finally, we verify the UUUB of . For , 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 and such that (II) Consider the uniformly ultimately lower boundary (or UULB) of , , .
Firstly, we prove the UULB of . From (I), for , one has By (2) in Lemma 4 and the assumption in (9), we know easily that the comparison system of (30) has a unique positive globally asymptotically stable -periodic solution denoted by . Let be the solution of (31) satisfying . The asymptotic property of implies that there exist a sufficient small constant and such that where . It follows from the comparison theorem of impulsive differential equations that
Secondly, we prove the UULB of . Recall the above results, and it follows from (1) that for , where the constant satisfies From Lemma 2, we know that the comparison system of (34) has a unique positive globally asymptotically stable -periodic solution denoted by . The asymptotic property of implies that there are and a sufficient small constant such that where . Let be the solution of (36) with . Using the comparison theorem of impulsive differential equations, we have Finally, we prove the UULB of . For , (1), (33), and (38) reduce to Notice that the assumption in (10) is similar to the proof of the UULB of ; we obtain that there are and such that
(I) and (II) yield and hence system (1) is permanent. The proof is completed.
4.2. Global Attractivity
Proof. Suppose that and are any two positive solutions of system (1). It follows from Theorem 6 that there exists a large enough such that, for , Consider the following Lyapunov function: where , , and are positive constants. Calculating and estimating the upper right derivative along the solutions of system (1), for and , we obtain that Applying (43) and the differential mean value theorem, for any closed interval contained in , and , we thus have Hence, for , and , from (45)-(46), we get where For , , one has where is between and , . It follows from (46) that which, together with (49), leads to where . Combining (47) and (51), we have From Lemma 1, for , (52) yields Since is an impulsive point, we have from (43) and (44) that, which indicates that is bounded. As a consequence, we get and hence or equivalently which, together with (46), yields that is, This completes the proof.
4.3. Partial Extinction
This section concerns the partial extinction of system (1). We will see that species , tend to extinction while species stabilizes at a positive solution of an impulsive system.
Proof. We first focus on . We get from system (1) that It follows from (60) and (1) in Lemma 4 that any positive solution of the comparison system of (63) satisfies . By the comparison theorem of impulsive differential equations we can obtain that . This fact implies that there exist a sufficient small constant and such that Next, from system (1), we obtain that, for , Similar to (23)–(26), we get that there exist a constant and such that From system (1), (65), and (67), we have Similar to , when the assumption (61) holds, a simple proof can verify . This fact implies that there are a sufficient small constant and such that From system (1), (65), and (69), we obtain that for Similar to (23)–(26), we can obtain that there exist a constant and such that which, together with (67), leads to Based on the above discussion, we will investigate the global attractivity of . Let Then, Clearly, is continuous. Recalling (74) and using the formula of solutions of first-order linear differential equation, we get, for , Furthermore, applying Lemma 1, one obtains where Notice that , , and thus a calculation shows that Hence, Combining (76) and (79), we obtain that , which implies . This completes the proof.
5. Examples and Numerical Simulations
In this paper, we investigate the dynamic behaviors of a periodic predator-prey system subject to impulsive perturbations, in which disease only spreads among the prey species. A good understanding of the permanence, global attractivity, and partial extinction of the above system is obtained. Our three main results (i.e., Theorems 6, 7, and 8) show that the impulsive perturbations play important roles in shaping the dynamics. To substantiate the theoretical results, we consider the following system: Obviously, the period . Let , ; then, . When , , , from Figures 1(a)–1(c), that species of system (80) is permanent while , tend to extinction.
Provided that all the other coefficients remain unchanged, we choose , , and , a calculation showing that the assumptions of Theorem 6 hold. So species , , are permanent (see Figures 2(a)–2(c)), and moreover, we can prove that the assumptions of Theorem 7 are satisfied, so system (80) is globally attractive. From Figures 3(a)–3(c), it is true that the positive solution with tends to the other positive solution with . To verify partial extinction we choose , , and ; system (62) with has a positive solution (see Figure 4(b)). We can see that the assumptions of Theorem 8 hold, so species stabilizes at a certain solution of a impulsive system (see Figures 4(b) and 4(c)) while species , tend towards extinction (see Figures 4(a) and 4(d)). The above facts demonstrate that the use of impulsive control strategy can change the dynamic behaviors of the system, and hence a suitable harvesting or stocking policy is important.