About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 935923, 15 pages
http://dx.doi.org/10.1155/2014/935923
Research Article

Time Delayed Stage-Structured Predator-Prey Model with Birth Pulse and Pest Control Tactics

1Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province, Enshi, Hubei 445000, China
2Department of Mathematics, Hubei Minzu University, Enshi, Hubei 445000, China
3Department of Mathematics and Information Sciences, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China

Received 21 November 2013; Revised 16 January 2014; Accepted 18 January 2014; Published 7 April 2014

Academic Editor: Yanni Xiao

Copyright © 2014 Mei Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Normally, chemical pesticides kill not only pests but also their natural enemies. In order to better control the pests, two-time delayed stage-structured predator-prey models with birth pulse and pest control tactics are proposed and analyzed by using impulsive differential equations in present work. The stability threshold conditions for the mature prey-eradication periodic solutions of two models are derived, respectively. The effects of key parameters including killing efficiency rate, pulse period, the maximum birth effort per unit of time of natural enemy, and maturation time of prey on the threshold values are discussed in more detail. By comparing the two threshold values of mature prey-extinction, we provide the fact that the second control tactic is more effective than the first control method.

1. Introduction

The outbreak of pest often triggers serious ecological and economic problems. In recent years, the management of pest has increasingly become the focus of attention. How to effectively control pest is one primarily concern problem. In practice, lots of factors can affect the efficiency of pest control, for instance, the time of impulsive effects, the number of prey stocked or naturally released, and the proportion of killing or catching pests. Mathematical modelling is one of the main ways for estimating and predicting the range of ecological interactions between pest and predator. Lately, many papers have been devoted to propose and analyze the predator-prey systems [14].

In one aspect, many species have the life history that goes through two stages, immature and mature, which has significant morphological and behavioral differences between them. Therefore, it is necessary to account for these differences, and the dynamics of mathematical models with stage-structured prey-predator model has been widely studied [511] in recent years. In other aspects, in the natural world, the immature predator always undergo a certain time (which is called maturation time delay) to be mature, so time delays play an important role in biological meanings in age-structured models. Time delay and stage-structured systems are introduced into predator-prey models, which greatly enriches biologic background. Many age-structured models with time delay were extensively studied [1214]. A single species model with stage structure and time delay was invested by Aiello and Freedman [15].

As far as the population dynamics is concerned, most models often considered that the population reproduces throughout all year. However, many species give birth seasonally or in regular pules. In this regard, the continuous reproduction of mature species should be removed from the model and termed this growth form as birth pulse. For instance, Tang and Chen study an age-structured model with density-dependent birth pulse in [16] and Xiang et al. [17] have considered a delayed Lotka-Volterra model with birth pulse and impulsive effect at different moment on the prey. Theories of impulsive differential equations have been studied lately [1820]. Impulsive systems are found in many domains of applied sciences. The application of impulsive differential equations to population dynamics is an interesting topic since it is reasonable and corrects in modeling the evolution of populations. Virtually, the pulses occur at different moments more realistic, which is more meaningful than the pulses that occur at the same time. Recently, the qualitative analysis of impulsive differential equations at different moments has been widely investigated in many works [2123].

Although, many authors have devoted to study the effects of pesticide on pest and its natural enemies and lots of instructive control strategies also have been given. An optimal time of pesticide applications still seems to be a novel interesting area. Following the practical pest management, we firstly propose the predator-prey model with pulse at the same time. Further, we assumed that the birth pulse and pest control tactics occur at different time. Discussing and comparing the mature prey-extinction of the two models, we get some new effective pest management.

2. Predator-Prey Model with Impulsive Effects at the Same Time

The purpose of this paper is to address how the time of impulsive effects influences the pest control. On the basis of the above discussion and motivated by [8, 16], we first will extend the following impulsive model which introduces the impulsive control tactics at the same time: where and are densities of the immature prey and mature prey, respectively. denotes the density of predator. represents a constant time to maturity and are positive constants. is the birth rate of immature prey, , are the mortality rates of the immature prey and mature prey, and is the death coefficient of . is the intraspecific competition rate of the mature prey. is the transmission coefficient. The term represents the immatures who were born at time and survive at time (with the immature death rate ) and therefore represents the transformation of immatures to matures (). is the rate of conversing prey into predator. represent the killing (or poisoning) rate of the prey and the predator, respectively. is the stocking amount of the immature prey at . denotes the birth effort of predator population at , where means the maximum birth effort per unit of time and the predator size at which 50% saturation measures how soon saturation occurs.

Lemma 1 (see [24]). Consider the following equation: where for . We have(i)if , then ;(ii)if , then .

Lemma 2. Consider the following system: (i)if , the fixed point is globally asymptotically stable;(ii)if , the fixed point is globally asymptotically stable.

Proof. Integrating the first equation of (3) on , we have With the successive pulse, we can obtain the following stroboscopic map of system (3): Let . If , then the fixed point , since and unstable if , since If , (5) has a unique positive fixed point: By the same way, we can obtain the fact that the fixed point is globally asymptotically stable if . This completes the proof.

which implies that corresponding periodic solution of (3) on is Thus (1) has a mature prey-extinction periodic solution , where .

Theorem 3. The mature prey-extinction periodic solution is globally asymptotically stable if holds, where .

Proof. Let be any solution of system (1). Following from the third equation of system (1), we notice that . Consider the following impulsive differential system: According to Lemma 2, we obtain the fact that , for . By using the comparison differential theorem of impulsive equation, we have and as . Therefore, for an arbitrarily small positive constant and all large enough: holds true.
For simplicity, we assume that (12) holds for all . From (12), we have Consider the following comparison differential equation: From (10) and Lemma 1, we have . By the comparison differential theorem, we get . Noticing the positivity of , we know that . Without loss of generality, we assume that for any (sufficiently small) Following from (1) and (15), we can obtain that which yields and as , while are the solutions of respectively. Here, and  , for .
In view of the comparison differential theorem, for any , let ; then , for large enough, which means as .
Next, we will prove that as . From (1) and (15), we get For the left hand inequality, it follows from impulsive differential equation (5) that and as . For the right hand inequality, we consider the following impulsive differential equation: and .
Therefore, for any , . Let ; then we get for large enough, which means as . The proof is completed.

Corollary 4. (i) If holds, then the mature prey-extinction periodic solution is globally asymptotically stable.
(ii) If holds, then the mature prey-extinction periodic solution is globally asymptotically stable.
In biological terms, since or   , the mature prey will extinct and immature prey and predators will coexist.

3. Predator-Prey Model with Impulsive Effects at the Different Time

In particular, in order to avoid the adverse effects of pesticides on the newly released natural enemies, we consider the following method implemented in practice to avoid such antagonism. That is, we assume that the pulse occurs at and . The detailed changes are shown as follows: If the mature prey is absent, then system (22) reduces to We can easily obtain the analytic solution of system (23) at the interval :

Lemma 5. (i) If , the fixed point is globally asymptotically stable.
(ii) If , the fixed point is globally asymptotically stable.

Proof. Consider system (24), which yields the following stroboscopic map: Let . If , there exists a unique trivial fixed point . The trivial fixed point is locally stable if , since and unstable if , since If , (25) has a positive fixed point denoted by , where , when . The positive fixed point is locally stable since , if .
Further, we can show that is globally asymptotically stable, if the following statements are satisfied:(i)if , then ;(ii)if , then .
By calculation, we get +, which shows that , when ; we know This yields . So we have under the assumption ; otherwise if . Thus the statements (i) and (ii) are satisfied. This completes the proof.

Now we can deduce that the positive equilibrium of system (25) is globally asymptotically stable. So we have the corresponding positive periodic solution in the following, which is globally stable:

Theorem 6. Assume that holds, where . Then the mature prey-extinction periodic solution is globally asymptotically stable.

Proof. Following from the second equation of system (22), we notice that ; thus consider the following impulsive differential system: According to Lemma 5 and the comparison theorem on impulsive differential equations, we have and as . Then there exists an integer , , such that holds for all t large enough.
This is Since > , we have For simplicity, we assume that (34) holds for all . From (22) we have Consider the following comparison differential system: From (23) and Lemma 1, we have . By the comparison differential theorem, we have . Noticing the positivity of , we know that . Without loss of generality, we assume that there exists an integer , for any (sufficiently small), such that In view of (22) and (37), it is easy to obtain that we get , and as , where From the comparison differential theorem, for any , there exists an integer ; let ; then , for large enough, which means as , where Next, we will prove that as . From (22) and (37), we get For the left hand inequality, it follows from impulsive differential equation (31) that and as . For the right hand inequality, we consider the following impulsive differential equation: Therefore, for any , there exists an integer such that Let ; then we get for large enough, which means as . The proof is completed.

Corollary 7. (i) If holds, then the mature prey-extinction periodic solution is globally asymptotically stable.
(ii) If holds, then the mature prey-extinction periodic solution is globally asymptotically stable.
The biological significance of Corollary 7 is the same as Corollary 4, so we omit it.
In order to investigate the permanence of system (22), we should give the following Definition and Lemma.

Lemma 8. There exists a constant such that for each solution of system (22) with all large enough.

Proof. Define , note that , then and we have When , When , Let us denote .
It follows from the comparison theorem of impulsive differential equations (see lemma 2.2 [25], page 23) that So is ultimately bounded. Hence, by the definition of , there exists a constant , such that for all large enough. This completes the proof.

Remark 9. According to Lemma 8, it is clear that . For convenience, we note that .

Definition 10. The system (22) is said to be permanent if there are constants (independent of all initial values) and a finite time , such that for all solutions with all initial values , and holds for all .

In biological terms, the permanence of (22) implies that prey (both immature and mature) and predators will coexist, none of them facing extinction or growing indefinitely. Denote

Theorem 11. If holds, where , system (22) is permanent.

Proof. It is seen that the second equation of system (22) can be rewritten as Let us consider any positive solution of system (22). By (51), is defined as Taking the first derivative of with respect to , we have As seen in Lemma 8, (53) can be written as From the definition of , we have and then ; it is easy to know that there exists a sufficiently small such that We claim that, for any , it is impossible that for all . Otherwise, there is a such that for all . It follows from the third equation of (22) that for all . Consider the following comparison impulsive system of (57): and we have In view of the comparison differential theorem for equation, there exists a , such that for , which implies that , where Thus From (54), we get By (54) and (63), we have Let
In the following we show that for all . Otherwise, there is a such that for and , which implies that, it is a contradiction. Thus, for all . Hence, for , we get which means as . This is contrary to the boundedness of . The claim is proved.
By the claim, consider the following two possibilities.
Case  1. for all large enough.
Case  2. oscillates about for large enough. Set
We will show that for all large enough. The conclusion is evident in Case  1. For Case  2, suppose that there exists the positive constant satisfying and for all , where is enough large such that since is continuous and bounded and is not affected by impulses. Thus is uniformly continuous. Consequently, there exists a constant and is independent of the choice of  ) such that for . If , our aim is obtained. Then we consider the case . From the second equation of (22) and the above assumption, we have that for . Hence one obtains . It is clear that for . If , then we have for . The same arguments can be continued; we can obtain for . Since the kind of interval is chosen in arbitrary way, we have for large enough. In view of the above discussion, the selection of is independent of the positive solution of (22) which satisfies for all large enough.
Next, from the first, the fourth, and the seventh equation of system (22), we have Considering the comparison system, Therefore for any small enough such that for sufficiently large, In view of the comparison theorem of impulsive differential equation and the Theorem 6, we get From (34) and (68), set and .
Define , and thus we have . By Lemma 8 and the above discussion, the system is permanent. The proof is completed.

4. Numerical Analysis

In the previous sections, we introduced the analytical tools and used them for a qualitative analysis of the system obtaining some results about the dynamics of the system. In this section, we perform a numerical analysis of the model based on the previous results. What we are interested in is how the key factors affect the thresholds and . Let = . Taking the first derivatives with respect to and , respectively, one obtains Those inequalities demonstrate that threshold value is a monotonic increasing function with respect to and and is a monotonic decreasing function with . These results indicate that the smaller killing (or poisoning) rate or the shorter pulse period , the smaller the threshold value which follows, so, the result shows that it is more effective for pest control. Similarly, when pests have very short maturity , the bigger the threshold value which follows, and hence make it more difficult for pest control. Therefore, we must carefully select the impulsive catching (or poisoning) rate or the time of pesticide applications or reduce the dosages of the pesticide.

So far, we have considered the global attractive mature prey-eradication solution of systems (1) and (22). We first focus on the system (1); when threshold value , the mature prey is eradicated totally and the immature prey and the predator population will tend to a stable level; see Figure 1, where , , , , and , while if we chose the parameters set as those in Figure 2, then the all populations oscillate periodically, which indicates that the system (1) is permanent. Furthermore, we can carry out two-parameter bifurcation analysis for the threshold value , as those shown in Figures 3(a)3(d). In each subplot, we let two key parameters vary simultaneously and see how those parameters affect the threshold value . All simulation results shown in Figure 3 indicate that the appears to be quite sensitive to small changes in parameters, , and . However, we note that the dynamical behaviour of system (1) is dominated by several types of periodic solutions and their coexistence, even with very high intrinsic growth rates. In Figure 4, we studied the influence of killing rate on the complexity of system (1); numerical results show that when killing rate varies from 0.45 to 0.95, the dynamical behaviour of system (1) is very complex. A typical chaos oscillation is captured when (see Figure 5). Figure 6 shows the bifurcation diagrams of system (1); the system analyzed here could take on many forms of complexity, including period doubling bifurcation, symmetry-breaking pitchfork bifurcation, and chaotic solutions. Meanwhile, system (1) also has a certain type of coexistence if the intrinsic growth rate of the pest population is large enough.

fig1
Figure 1: Dynamical behavior of system (1); the parameter values are as follows: , , , , , and .
fig2
Figure 2: Dynamical behavior of system (1); the parameter values are as follows: , , , , and .
fig3
Figure 3: The influence of some key parameters on the threshold level ; the other parameter values are as follows: , , , and (a) , ; (b) ,