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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 962639, 21 pages
http://dx.doi.org/10.1155/2010/962639
Research Article

Modelling and Analysis of a Pest-Control Pollution Model with Integrated Control Tactics

Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei 445000, China

Received 24 August 2010; Accepted 11 October 2010

Academic Editor: Zhen Jin

Copyright © 2010 Yiping Chen 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

A hybrid impulsive pest control model with stage structure for pest and Holling II functional response is proposed and investigated, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Sufficient conditions for global attractiveness of the pest-extinction periodic solution and permanence of the system are obtained, which show that there exists a globally asymptotically stable pest-extinction periodic solution when the number of natural enemies released is more than some critical value, whereas the system can be permanent when the number of natural enemies released is less than another critical value. Furthermore, numerical simulations are carried out to illustrate our theoretical results and facilitate their interpretation.

1. Introduction

Since the beginning of recorded history, outbreaks of pests have plagued humanity, coming in direct competition with people for life-sustaining food. Reportedly, an estimated 67,000 different pest species attack agricultural crops, and about 35% of the yearly agricultural crop production is lost to pests worldwide [1, 2]. That problem is one of how to control or suppress damaging populations of pests over widespread areas. As we know, the most effective strategy for controlling pests may be to combine methods in an approach known as integrated pest management (IPM) that emphasizes preventing pest damage. In IPM, information about pests and available pest-control methods (including biological, cultural, and chemical) is used to manage pest damage by the most economical means and with the least possible hazard to people, property, and environment [35].

Biological control of pests in agriculture is a method of controlling pests (including insects, mites, weeds, and plant diseases) that relies on predation, parasitism, herbivory, or other natural mechanisms. It can be an important component of integrated pest management (IPM) programs. It is defined as the reduction of pest populations by natural enemies and typically involves an active human role such as augmentation which involves the supplemental release of natural enemies. Biological control is not a "quick fix" for most pest problems. Natural enemies usually take longer to suppress a pest population than other forms of pest-control, and farmers often regard this as a disadvantage. Cultural controls are manipulations of the agroecosystem that make the cropping system less friendly to the establishment and proliferation of pest populations. Although they are designed to have positive effects on farm ecology and pest management, negative impacts may also result, due to variations in weather or changes in crop management [6]. Another important method for pest-control is chemical control. Chemical control is the approach of controlling pests through the spraying pesticide which is liable to reduce the pest populations considerably and which is indispensable when there are not enough natural enemies to decrease pest populations. In most cropping systems, insecticides are still the principal means of controlling pests once the economic threshold has been reached. They can be relatively cheap and are easy to apply, fast acting, and in most instances reliable in controling the pests [7]. Despite the advantages of conventional insecticides, the problems associated with their use have been well documented. These include the resurgence of pest populations after decimation of the natural enemies, development of insecticide-resistant populations, and negative impacts on nontarget organisms within and outside the crop system [8]. When considering these actions, in the process of effective control of pest, excessive use of a single control strategy is undesirable. Wherever possible, different pest-control techniques should work together rather than against each other. Even so, in many cases, the most effective release rate or spraying rate has not been identified as it will vary depending on crop type and target host density. Therefore, human beings have been forced to face the new challenge in the integrated pest management (IPM) program. One of the most important questions in IPM is how many natural enemies should be released and what fraction of the pest population should be killed to avoid economic damage and reduce the pesticide applications when the pest population reaches or exceeds the economic threshold level.

According to the idea of IPM, many mathematical models have been constructed and studied for understanding the range of possible ecological interactions between pest, natural enemy, and pesticides in the last decades. In order to consider the consequences of especially spraying pesticide and introducing additional predators into a natural pest-predator system, impulsive differential equations have been employed to describe such a system by many researchers [913], and the references cited therein. Impulsive differential equations are found in almost every domain of applied sciences [14, 15] and have been studied in many investigations [1618]. They generally describe phenomena which are subject to steep or instantaneous changes. In IPM, impulsive reduction of the pest population is possible by trapping the pests and/or by poisoning them with chemicals. An impulsive increase of the natural enemy density can be achieved by releasing the natural enemy based on laboratory breeding into the field [5, 11]. Unfortunately, most of the pest-control models in the literature, which were modeled by impulsive differential equations, assume that at every impulsive spraying period, the pest population (including the natural enemies) may be killed immediately, and the instant killing rate of pesticide is a proportional constant. However, the actual situation is not always the same. Generally, pesticide appears in environment first, then it is absorbed by organism, and the individuals are affected, that is, the toxicity of pesticide does not act on the organism at once; in other words, it will last for some time before toxins are capable of decreasing the average growth rate of the species [19]. This fact urges us to consider the effect of pollution time delay on the extinction and permanence of population in a polluted environment. With this in mind, it is necessary to introduce the pollution model to model the process of pest-control problems and study its dynamics, and this is different from the previous pest-control model which assumed that pests were reduced proportionally by spraying pesticides.

As we know, since Hallam and his coworkers proposed a toxicant-population model in the early 1980s [2023], mathematical models of single or multiple populations with toxicant effect have been constructed and studied extensively [19, 2429]. However, the majority of these studies have been focused on the effects of toxicant emitted into the environment from industrial and household resources on biological species, and only a few attempts have been made to combine pollution model to study pest-control problems with pesticide (toxin) input. Recently, by using pollution model and impulsive delay differential equation, Liu et al. [19] constructed, and investigated, a pest-control model with age structure for pest by introducing a constant periodic pesticide input and releasing natural enemies at different fixed moment. It is assumed in their model that each individual has the same dose response parameter to the organismal toxicant concentration regardless of the difference in many aspects between the immature and mature pest populations. However, in the natural world, there are many species whose individual members have a life story that takes them through two stages, immature and mature. Those species hatch from egg. Moreover, the immature and mature species express great differences in many aspects. One of the facts is that only the mature individuals are affected by the toxin (pesticide) and the immature individuals are not. For example, locust and salt-cedar leaf beetle, and so forth, are such species whose immature individuals (eggs) are protected by their eggshell and hardly injured by pesticides.

Based on all the above points, in this paper, we propose and investigate a pest-control model with a constant periodic pesticide input and natural enemies release at different fixed moment, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Moreover, we assume that the pest individuals have two life stages: immature (egg) and mature with a constant maturation time delay, pesticide (toxin) has no effect on the immature individuals, and the capacity of the environment is so large that the change of toxin in the environment that comes from uptake and egestion by the organisms can be ignored. On the other hand, it is well known that functional response is a basic modeling unit in community ecology [30]. So, we further assume that natural enemy (predator) only feeds on mature pest (prey), and the functional response of natural enemy (predator) to mature pest (prey) species takes the Holling type II form. Meanwhile, because we may artificially pick on the appropriate releasing time when there is the lowest chance of adversely affecting natural enemies; thus, we further assume that pesticide input has little influence on the natural enemies, that is, the effect of pesticide input on natural enemies can be ignored. We are interested in a theoretical study about the effects of our control tactics on dynamical behavior of populations and attempt to obtain a theoretical threshold value which determines extinction of pest species and permanence of the system.

The organization of this paper is as follows. In Section 2, we set up our model and introduce some notations, definitions, and lemmas. In Section 3, sufficient conditions for extinction of the pest species and permanence of the system are given, respectively. The numerical simulations are carried out to study the effects of the impulsive varying parameters on the system as well as to illustrate our theoretical results in Section 4. Finally, a brief discussion is given to conclude this work.

2. Model and Preliminaries

According to the above analysis and assumption, we construct a pest-control pollution model with stage structure for the pest and Holling II functional response concerning integrated control tactics. The model takes the following form: The initial conditions are where ,  , and   represent the density of the immature pest (egg), mature pest and natural enemy at time , respectively, represents the concentration of pesticide in the environment at time ; represents the concentration of pesticide in the organism for the mature pest at time , is the growth rate of the immature pest; and show the death rate of the immature pest and natural enemy, respectively, represents a constant time to maturity, represents the intraspecific competition coefficient of mature species, expression is Holling II functional response function, , , represents the rate of conversion of consumed mature pest to natural enemy, represents the decreasing rate of the intrinsic growth rate associated with the uptake of pesticide in the organism for the mature pest, represents the organism's net uptake pesticide from the environment, and represent the egestion and depuration rates of pesticide in the organism for the mature pest, respectively, represents the loss of pesticide in the environment due to natural degradation, , , ; , is the period of impulsive effect, , is the releasing amount of the natural enemy at time, and  is the amount of pesticide input at time .

Obviously, the first equation of system (2.1) can be written as which means that the property of can be investigated by . Moreover, the condition (2.4) presents the total surviving immature population from the observed birth on . On the other hand, because the immature pest (egg) does little harm to the crops and it cannot breed, we just need to consider the control of the mature pest. Meanwhile, note that the variable does not appear in the second, third, fourth, and fifth equations of system (2.1); hence, we only need to consider the subsystem of (2.1) as follows: The initial conditions for system (2.5) are Furthermore, since and are the concentration of toxicant, to ensure and , we assume that condition , holds in this paper. Meanwhile, considering the biological meaning, we assume that .

In Sections 3 and 4, we mainly consider the global stability of pest-extinction solution and the uniform permanence of system (2.1); before introducing our main results, we give some preliminaries needed in next sections.

Let and . Denote as the map defined by the right hand of system (2.1). The solution of (2.1), denoted by , is continuous on and . and exist. Obviously, the global existence and uniqueness of solutions of (2.1) is guaranteed by the smoothness properties of (see [15]). Furthermore, the following lemma is easily obtained.

Lemma 2.1. If is a solution of system (2.1) with , then for all .

Consider the following system

Lemma 2.2 (see [19]). System (2.7) has a unique positive periodic solution given by which is globally asymptotically stable.

Consider the following system

Lemma 2.3 (see [19]). System (2.9) has a unique positive periodic solution given by which is globally asymptotically stable.

Now we consider some basic properties of the following subsystem of system (2.5)

Lemma 2.4 (see [19]). System (2.11) has a unique positive -periodic solution given by for , which is globally asymptotically stable.

Lemma 2.5 (see [19]). Considering the following equation where , ,, and are all positive constants, for , one has(1)if , then ;(2)if , then.

Definition 2.6. System (2.5) is said to be permanent if there are constants (independent of initial value) and a finite time such that for every positive solution with initial conditions, (2.6) satisfies , , , for all . Hence, may depend on the initial conditions (2.6).

3. Extinction and Permanence

Firstly, we show that all solutions of system (2.1) are uniformly ultimately bounded.

Theorem 3.1. There exists a constant such that , , , , for each solution of system (2.1) with large enough .

Obviously, when the pest individuals are entirely absent from the model, that is, for , satisfies the system (2.7). Accordingly, by Lemmas 2.2 and 2.4, we can get that system (2.1) exists for an immature and mature pest-extinction periodic solution , whose global attractiveness is equivalent to global attractiveness of the mature pest-extinction periodic solution of system (2.5). In the following, we give the sufficient conditions for global attractiveness of solution .

If we denote , , , , , , then we have the following.

Theorem 3.2. The mature pest-extinction periodic solution of system (2.5) is globally attractive provided that

holds true.

Remark 3.3. The pest-extinction periodic solution of system (2.1) is also globally attractive if the condition (3.1) holds.

Now, we give the sufficient conditions for permanence of system (2.5). If we denote , , and then we have the following.

Theorem 3.4. System (2.5) is permanent provided that

holds true.

Remark 3.5. System (2.1) is also permanent if the condition (3.2) holds.

For convenience, the proofs of Theorems 3.1, 3.2, and 3.4 are given in Appendices AC, respectively.

The above results show that many factors including maturation time delay, functional response of the predator, the organism's net uptake pesticide from the environment, the egestion and depuration rates of pesticide in an organism, the loss of pesticide in the environment due to natural degradation, the natural enemy releasing amount, the pesticide spraying amount, and the releasing and spraying period, can induce variation in the characteristics of populations. Meanwhile, the results imply that the modelling methods described can help in the design of appropriate control strategies and assist management decision-making. In fact, the conditions (3.1) and (3.2) imply that there exist two theoretical criteria values in system (2.1), which can be, respectively, denoted as follows: Moreover, if , the pest-extinction periodic solution is globally asymptotically stable; if , the insect pests and the natural enemies can coexist, that is, system (2.1) that we consider permanent. It is well known that, in a definitive ecological environment, the appropriate artificial release of natural enemies and spraying of pesticides play an important role in the success of pest-control. Due to the antagonism between chemical and biological methods, we should reduce the pesticide application to avoid antagonism and especially negative impacts on nontarget organisms. Theorems 3.2 and 3.4 indicate that we can choose the appropriate impulsive parameters to reduce pests to tolerable levels with little economical cost and minimal effect on the environment. Therefore, our impulsive strategy is more effective than the classical one if the chemical control is adopted rationally. To confirm our mathematical findings and facilitate their interpretation, we proceed to investigate further by using numerical simulations in the following section.

4. Numerical Simulations

In this section, numerical simulations are carried out to investigate effects of impulsive varying parameters on dynamical behaviors of system (2.1) as well as to illustrate our theoretical results. Owing to the lack of biologically realistic parametric values, the solution of the system with initial conditions in the first octant is obtained numerically for biologically feasible ranges of parametric values dominated by Theorems 3.2 and 3.4. For convenience, we assume that some parametric values of system (2.1) are kept as

Firstly, we give numerical results of the system, in which there are no impulsive perturbations (including natural enemy releasing and pesticide spraying), in other words, that is the unforced system of (2.1). The model takes the following form: where the value of parameters for model (4.2) can be seen in (4.1). We can easily plot the time series of every population and phase portrait of the system and find that the solution of (4.2) with initial values , ,  and would tend to a positive equilibrium solution (see Figures 1(a) and 1(b) in details). From the following discussion, we can observe that the solution of the unforced system would become unstable via impulsive perturbation. Further, it indicates that the system is impulsively controllable.

fig1
Figure 1: Dynamical behavior of system (4.2) with , , , , , , , , initial values , , . (a) Time series of system (4.2). (b) Phase portrait of system (4.2).

From theoretical criteria values formula (3.3) and the above parameter hypothesis (4.1), by a straightforward calculation, we can obtain that two theoretical criteria values of system (2.1) are and  , respectively.

Let , that is, the condition (3.1) holds true; we know that the pest-extinction periodic solution is globally asymptotically stable from Theorem 3.2. that is, if we let , a typical pest-eradication periodic solution of system (2.1) with initial values , , , , and is shown in Figure 2(a), where we observe how the predator (natural enemy) and the concentration and of pesticide in the environment and the concentration of pesticide in the organism periodically oscillate; in contrast, both the immature pest and mature pest rapidly decrease to zero. If we continue to increase and let and , from Figures 2(b) and 2(c), the same phenomenon as above can be observed, respectively. This illustrates that the pest-extinction periodic solution of system (2.1) is globally asymptotically stable.

fig2
Figure 2: Dynamical behavior (extinction) of system (2.1) with , , , , , , , , , , , , , , , , initial values , , , , (corresponding theoretical criteria value: . (a) Time-series of system (2.1) with . (b) Time-series of system (2.1) with  . (c) Time-series of system (2.1) with .

Let , that is, the condition (3.2) holds true; we know that system (2.1) that we consider is permanent from Theorem 3.4. That is, if we let , a positive periodic solution of system (2.1) with initial values , , , , is shown in Figure 3, where we observe that each population of system (2.1) can coexist on a stable limit cycle. If we continue to decrease , and let and , from Figures 4 and 5, the same phenomenon as above can be observed, respectively. This illustrates that system (2.1) is permanent.

fig3
Figure 3: Dynamical behavior (permanence) of system (2.1) with , , , , , , , , , , , , , , and , and ; initial value , , , , and (corresponding theoretical criteria value: ). (a) Time-series of system (2.1) with . (b) Solution , , with will finally tend to a -periodic solution.
fig4
Figure 4: Dynamical behavior (permanence) of system (2.1) with , , , , , , , , , , , , , , , and , initial values , , , , and (corresponding theoretical criteria value:  . (a) Time-series of system (2.1) with . (b) Solution ,, with will finally tend to a -periodic solution.
fig5
Figure 5: Dynamical behavior (permanence) of system (2.1) with , , , , , , , , , , , , , , , , initial values , , , , (corresponding theoretical criteria value: . (a) Time series of system (2.1) with . (b) Solution ,, with will finally tend to a -periodic solution.

We must emphasize here that condition (3.1) and condition (3.2) are the only sufficient conditions which, respectively, assure global attractiveness of the pest-extinction periodic solution of system (2.1) and permanence of the populations. Accordingly, and are only two theoretical criteria values, not the threshold. Concerning the mathematical formula of theoretical threshold, we leave this for future work. We only give here an approximate threshold which can be obtained by numerical simulations. Indeed, by plotting the bifurcation diagram, we may observe that the theoretical threshold of parameter is approximately equal to 0.29 (see Figure 6 in details). That is to say, when , the pest-extinction periodic solution of system (2.1) is globally asymptotically stable; reversely, when , system (2.1) that we consider is permanent.

fig6
Figure 6: Bifurcation diagrams of system (2.1), showing the effect of with , , , , , , , , , , , , , , , and , initial values , , , , and . (a) is plotted for over . (b) is plotted for  over .

According to the bifurcation theory, the properties of a dynamic system depend on certain parameter, and dynamic system with different parameters may have different dynamic behaviors. The above numerical results that we have investigated depend on parameter , that is, is control parameter. In fact, from condition (3.1) and condition (3.2), the control parameter may also choose the other parameter as , , , or , and then the same argument as above can be continued. We only give here two numerical examples. Figure 7(a) is plotted by changing the parameter of Figure 3 to . Figure 7(b) is plotted by changing the parameter of Figure 3 to . As against Figure 3, Figure 7 implies that long maturation time delay and short impulsive period may induce variation in the characteristics of populations and cause pests eradication.

fig7
Figure 7: Dynamical behavior (extinction) of system (2.1) with , , , , , , , , , , , ,  and , initial values , , , ,   and . (a) Time series of system (2.1) with ,, . (b) Time series of system (2.1) with , , and .

5. Conclusion

In this paper, in order to investigate the consequences of periodically spraying pesticides and releasing natural enemies at different fixed moment in pest-natural enemy system, a hybrid impulsive pest-control model with stage structure for pest and Holling II functional response is proposed, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Sufficient conditions for global attractiveness of the pest-extinction periodic solution and permanence of the system have been obtained, which shows that there exists a globally asymptotically stable pest-eradication periodic solution when the number of natural enemies released is more than some critical value (see Figure 2), whereas the system can be permanent when the number of natural enemies released is less than another critical value (see Figures 3, 4, and 5). Meanwhile, numerical simulation results for biologically feasible ranges of parametric values can confirm our mathematical findings and facilitate their interpretation. We also note that the conditions for the extinction or permanence in system (2.1) are quite different from the corresponding system (4.2) without impulse. For example, the system (4.2) has a positive equilibrium which is orbitally asymptotically stable (see Figure 1); however, this properties are changed via additional impulsive perturbation (see Figures 27). Furthermore, by plotting the bifurcation diagram (see Figure 6), we obtained the theoretical threshold of control parameter , which is crucial for extinction or permanence of the population if the other parameters of system (2.1) are fixed. Finally, the numerical results, which show that long maturation time delay and short impulsive period may cause pests eradicat, have been given (see Figure 7). Obviously, these results indicate that the models proposed in this paper can help us understand pest-natural enemy interactions, to design appropriate control strategies and to make management decisions in insect pest-control. We would like to mention here that an interesting but challenging problem associated with the studies of system (2.1) should be how to optimize the number of periodically releasing natural enemy and the dosage of spraying pesticides to reduce pests to tolerable levels with little economical cost and minimal effect on the environment. We leave this for future work.

Appendices

A. Proof of Theorem 3.1

Define ,  . When and , we have where . In addition, , .

By a straightforward calculation, when , we have and when , we have Accordingly, we have

So is uniformly ultimately bounded. By the definition of , we have , , , , for large enough . The proof is completed.

B. Proof of Theorem 3.2

Suppose that is any solution of system (2.5) with initial conditions (2.6). From system (2.5), we have

By Lemma (2.2), we know that is the unique positive periodic solution of impulsive differential equation as follows:

By comparison theorem of impulsive equation [15], for any small enough , there exists an integer such that Accordingly, we obtain On the other hand, from Lemma 2.4, we can easily obtain that for any small enough , there exists an integer such that Leting , from the first equation of system (2.5), (B.4), (B.5), and Theorem 3.1, we have

   where is obtained from Proof of Theorem 3.1. In the following we consider the comparison equation Because the condition (3.1) is equivalent to the condition , therefore, we can choose and small enough such that For any solution of (B.7), by Lemma 2.5 and (B.8), we can get . Thus by the comparison theorem in delay differential equation and Lemma 2.1, we obtain that , and .

Further, for any small enough  and large enough , we have . Without loss of generality, we may assume for . And then from system (2.5), we obtain By Lemma 2.3 and the comparison theorem in impulsive differential equation [15], for any is small enough, when large enough , we have where for is the unique positive periodic solution of impulsive differential equation as follows

Combining (B.3) with (B.10), when is large enough, we obtain which implies since are all sufficiently small positive constants. Moreover, by Lemma 2.4, when , we have . Thus the proof is completed.

C. Proof of Theorem 3.4

Suppose that is any solution of system (2.5) with initial conditions (2.6). By Theorem 3.1, we have proved that there exists a constant such that , , , for large enough . From Proof of Theorem 3.2, we know that , for large enough (see (B.4), and (B.5)). By Lemma 2.4, we easily obtain that for large enough . Thus, from Definition 2.6, we only need to find a constant such that for large enough. We will do it in the following two steps.

(1) we prove that there exists a constant such that cannot hold for all . Otherwise, there is a constant such that for all . Thus, from system (2.5), when , we have

By Lemma (2.4) and comparison theorem of impulsive equation [15], for any small enough, there exists a such that For the above , by Lemma 2.4, there exists a such that Because the first equation of (2.5) can be rewritten as

Now, we define

By calculating the derivative of along system (2.5), we have Let , then for , combining (C.2), (C.3), and (C.6), we have Since condition (3.2) holds, we can choose and to be small enough such that Leting , we show that for . Otherwise, there is a nonnegative constant such that for , , and . Further, from the first equation of (2.5), we obtain that

This is a contradiction. So, we obtain that for . Combining (C.7) and (C.8), we have

It implies that as . Meanwhile, by the definition of , we easily obtained that . This is contradiction. Hence, for any constant ,  cannot hold for all .

(2) If holds true for all large enough , then our aim is obtained. Otherwise, is oscillatory about . Thus there exist two positive constant such that and for . Let . In the following, we firstly show that for and then address that for large enough.

From system (2.5), we know that is continuous and bounded. So, there exists a constant ( and independent of the choice of ) such that for all . Moreover, when is large enough, by Theorem 3.1 and the first equation of (2.5), we have Accordingly, if , our aim is obtained; if , from (C.11), we have

It is obvious that for ; if , from (C.11), we can obtain that for . The same argument can be continued, so we can obtain that for . Since two positive constants, , , are arbitrarily chosen, we only assure to be large enough, and then we get that for large enough.

According to the above analysis, we can find a constant such that for large enough . Thus the proof is completed.

Acknowledgments

This work is supported by Science and Research Project Foundation of Educational Department of Hubei Province (D20101902), the Key Project of Chinese Ministry of Education (210134), Youth-group Innovation Project for Colleges and Universities in Hubei (T200804), and the NFS of Hubei Province (2008CDB068). The authors would like to thank Editor Professor Z. Jin and the referees for helpful remarks that improved the paper.

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