Research Article | Open Access
Dynamic Analysis of an Impulsive Predator-Prey Model with Disease in Prey and Ivlev-Type Functional Response
A predator-prey model with disease in prey, Ivlev-type functional response, and impulsive effects is proposed. By using Floquet theory and small amplitude perturbation skill, sufficient conditions of the existence and global stability of susceptible pest-eradication periodic solution are obtained. By impulsive comparison theorem, conditions ensuring the permanence of the system are established. Examples and simulation are given to show the complex dynamics for the key parameters.
It is well known that many evolution processes are characterized by the fact that at certain moments their stage changes abruptly. For example, for integrated pest management (IPM) strategy on ecosystem, the predators are released periodically every time , and periodic catching or spraying pesticides are also applied. Hence, the predator and prey experience a change of state abruptly. It is natural to assume that these processes act in the form of impulse. The effects of impulsion on the dynamics of predator-prey system have been investigated extensively, see [1–15].
On the other hand, in population dynamics, a functional response of the predator to the prey density refers to the change in the density of prey attached per unit time per predator as the prey density changes. Usually, functional response plays key role in the dynamics of predator-prey system [16–18]. Recently, many different functional responses are studied such as Holling-type [8, 11], Beddington-type [7, 12], and Watt-type [13, 14]. Ivlev-type functional response is the most common type of functional response among arthropod predators, and much progress has been seen in the study of predator-prey model with Ivlev-type functional response [19–21].
Considering the influence of periodic pesticide spraying on all species and periodically releasing predator at fixed different time, authors  proposed and studied the following predator-prey system with impulsive perturbation and Ivlev-type functional response: They obtained the local stability of prey-free periodic solution and permanence of the system.
However, for biological control, in addition to the approach to release natural enemies, another approach is to use microbial control with pathogens since diseases can be important natural controls of some pests. For example, insects can be infected by disease-causing organism such as bacteria and viruses. Under appropriate condition, these naturally occurring organisms may multiply to cause disease outbreak that can decimate an insect population. There is a large amount of literatures on applications of entomopathogens to suppress pests [22–26]. Then how does the disease in prey affect the dynamics of above system? Further, whether we can derive the global stability of the susceptible pest-eradication periodic solution?
Motivated by above discussion, in this paper, we are concerned with the following predator-prey model with complex influence of disease in prey, Ivlev-type functional response, and impulsive perturbation as follows: where and represent densities of susceptible prey (pest) population and infective prey (pest) population, respectively; is the density of predator (natural enemy), and , , exist.
For (1.2), we give the following biological assumptions.(i)The growth rate for prey is , is intrinsic growth rate, and is the carrying capacity.(ii)There are diseases among prey population, and the prey population is divided into susceptible class and infective class. The incidence rate is classic bilinear as , and is the contact number per unit time for every infective prey with susceptible prey such that .(iii)The predator only catches susceptible prey, and the predation functional response is Ivlev type. Parameters , are positive constants, and is conversion rate from prey to predator.(iv)Parameters , are death rates for infective prey and predator, respectively.(v) represents the fraction of prey and predator which due to the pesticide at , are positive constants representing the release amount of infective prey and predator periodically at time respectively, where (positive integer set), and is the period of impulsive effect.
By using Floquet theory, comparison method of impulsive differential equation, and numerical analysis skill, we aim to study the dynamics of (1.2) with disease in prey, Ivlev-type functional response, and impulsive effects.
The rest of the paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, by using the Floquet theory and small amplitude perturbation skill and comparison theorem of impulsive differential equation, the existence of susceptible pest-eradication periodic solution and permanence of system (1.2) are studied. In Section 4, some examples and numerical analysis are given to show the rich dynamics of (1.2). Finally, biological implications and a brief discussion are given in Section 5 to conclude this paper.
Let and . Denote by the map defined by right-hand sides of the first three equations of (1.2). Let , and then is said to belong to class if(i) is continuous in , and , , and exist, and(ii) is locally Lipschitzian in .
Definition 2.1. Let , then for and , the upper right derivative of with respect to system (1.2) is defined as
Definition 2.2. System (1.2) is said to be permanent if there exist positive constants and with such that each positive solution satisfying , ,?and for all is sufficiently large.
The solution of system (1.2) is continuously differentiable on and , . Obviously, the global existence and uniqueness of solutions to system (1.2) are guaranteed by the smoothness properties of function . For more details see .
Lemma 2.3. Suppose that is a solution of system (1.2) with , . Then , , for all . Furthermore, ,? if , , .
Lemma 2.4 (see ). Let . Assume that where is continuous in , and , , and exist; and are nondecreasing. Let be the maximal solution of the scalar impulsive differential equation existing on . Then implies that for , where is any solution of system (1.2). Assume that all the inequalities “” in system (2.2) are replaced by “” in the preceding equations, and let be the minimal solution of (2.3) existing on . Then implies that for .
Lemma 2.5. There exists a positive constant such that , , for each solution of system (1.2) with positive initial values, where is sufficiently large.
Proof. Define , then it is clear that . For the continuity points of (1.2), that is, and , then , by simple computation, we have
where and .
When , by system (1.2) directly, we have Similarly, when , we have
According to Lemma 2.4, we can obtain that Thus, is uniformly ultimately bounded from above. By the definition of , we follow the conclusion immediately. This completes the proof.
3. Extinction and Permanence
Lemma 3.1. System (3.1) has a unique positive periodic solution with initial values
Next, we investigate the stability of the susceptible pest-eradication periodic solution of system (1.2).
Theorem 3.2. Let be any solution of (1.2). Then is globally asymptotically stable provided that
Proof. First, by using Floquet theory, we show the local stability of periodic solution of (1.2). Considering the behavior of small amplitude perturbation, let
where , and are all small perturbations. By using Taylor expansion and after neglecting higher-order terms, (1.2) can be linearized, and the linearized equations read as
Suppose that is the fundamental solution matrix of system (3.6), and then satisfies
is the identical matrix. Then the resetting impulsive conditions of (1.2) become
By Floquet theory, the local stability of is determined by the eigenvalues of
The exact expression of and are omitted since they are not required in the analysis that follows. The eigenvalues of are
Obviously, . It follows that if and only if
By computation, we have
Therefore, we can derive that if and only if holds true. Thus, by Floquet theory, under condition , the positive periodic solution is locally asymptotically stable.
Next, we prove the globally attractive property.
Choose such that Besides, we have From Lemma 3.1 and comparison theorem of impulsive equation, for sufficiently large, we have Similarly, for sufficiently large, we have For simplification, we suppose that (3.16) and (3.17) hold for all . Hence, it follows that Integrating (3.18) on leads to Thus . In virtue of the assumption , we can follow that as . Noting that for , hence as .
Next, we prove as . For an arbitrary positive constant small enough such that , since for large enough, without loss of generality, we assume that hold for all . Then By Lemma 3.1 and comparison theorem again, for all , there exists , for all , and we obtain where is defined in Lemma 3.1, and is the solution of the following system: that is, Let , it follows that holds for sufficiently large. Note that is a constant small enough, then letting , and we have as . By the same method, we can similarly derive that as . This completes the proof.
Finally, we study the permanence of system (1.2).
Theorem 3.3. System (1.2) is permanent if
Proof. By Lemma 2.5, without loss of generality, we suppose that , and for all , where is a constant satisfying .
On the other hand, from (3.16) and (3.17), we can obtain for large enough. Therefore, and are ultimately positively bounded from below. Hence, we only need to prove that there exists a constant such that for sufficiently large. By the proof of Theorem 3.2, assumption is equivalent to By the density of real number, we can select positive constant and small enough with such that where We claim that cannot hold for all . Otherwise, we can follow from (1.2) that By comparison theorem of impulsive differential equation, then there exists a such that where are defined in (3.29) and (3.30), respectively; that is, they are the solutions of systems (3.31) and (3.32), respectively. Then, for , we have Let and . Integrating (3.34) on , we have Thus, as , which is a contradiction to the boundedness of . Hence, there exists a such that . Then there are two cases.
Case??1 ( for all ). Then let , our aim is obtained. Otherwise, we consider Case 2.
Case??2. We consider those solutions which leave the region and reenter it again. Let , then there are two possible cases for .
Subcase??2.1 (, ). Then for , and . Select such that where .
Let , then we claim that there must exist a such that . Otherwise, considering system (3.31), we can derive that and .
From (3.36) and (3.38), we have That is, Similarly, we can derive that which implies that (3.34) holds for . In view of the discussion after (3.34), we have . Integrating the following system (3.42) on , we have It follows from (3.37) and (3.43) that which is a contradiction to the above assumption, then our claim is true.
Let , then for , , . Therefore, For , the same argument can be continued since .
Subcase??2.2 (). Then for and .
Suppose , , . Then either for all , or there exists a such that .
If for all , similar to the former discussion, there must be a such that . Let , and then for . Therefore, . That is, holds for . For , due to , the same argument can be continued.
If there exists a such that , the same argument in Subcase 2.1 can also be continued, and one can follow the conclusion easily. We omit it here.
Incorporating all the cases above, we deduce that system (1.2) is permanent. The proof is complete.
Remark 3.4. For system (1.2), if there is no disease in prey, then Theorems 3.2 and 3.3 reduce to the corresponding results of , while Theorems 3.2 and 3.3 imply that disease in prey affects dynamics of (1.2), which will be shown by simulation in Section 4. It is interesting and valuable for biological control. Specially, the globally asymptotical stability of susceptible pest-eradication periodic solution is studied here, but authors  only give locally asymptotical stability of the prey-free periodic solution. Therefore, we improve and generalize the main results of .
4. Examples and Simulation
Theorems 3.2 and 3.3 show that dynamics of system(1.2) is affected by complicated factors such as contact number per unit time for every infective prey , impulsive period , releasing amount of infective prey , and releasing amount of predator . In this section, by numerical analysis, we show the effects of parameters , and on dynamics of (1.2), respectively. For example, take , , , , , , , , , , and . If , then Theorem 3.2 implies that system (1.2) has a susceptible pest-eradication periodic solution, which is globally asymptotically stable. If , then Theorem 3.3 implies that (1.2) is permanent. By simulation, the results can be seen directly, see Figures 1 and 2, respectively. Actually, by simulation, there exists a critical value , and if , then the system is permanent; otherwise, it has a susceptible pest-eradication periodic solution, see Figures 3, 4, and 5, respectively.