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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 304568, 10 pages
http://dx.doi.org/10.1155/2014/304568
Research Article

The Dynamics of a Nonautonomous Predator-Prey Model with Infertility Control in the Prey

1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
2Department of Mathematics, Yuncheng University, Yuncheng 044000, China

Received 16 December 2013; Accepted 20 January 2014; Published 10 March 2014

Academic Editor: Weiming Wang

Copyright © 2014 Xiaomei Feng 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 nonautonomous predator-prey model with infertility control in the prey is formulated and investigated. Threshold conditions for the permanence and extinction of fertility prey and infertility prey are established. Some new threshold values of integral form are obtained. For the periodic cases, these threshold conditions act as sharp threshold values for the permanence and extinction of fertility prey and infertility prey. There are also mounting concerns that the quantity of biological sterile drug is obtained in the process of the prevention and control of pest in the grasslands and farmland. Finally, two examples are given to illustrate the main results of this paper. The numerical simulations shown that, when the pest population is permanet, different dynamic behaviors may be found in this model, such as the global attractivity and the chaotic attractor.

1. Introduction

Small mammals living in the grasslands, such as the plateau pika, not only burrow, but also accumulate the soil outside the hole, which makes the grass cease growing. More seriously, after a rainstorm, the soil would be washed away which increased soil erosion. So a greater range of damages resulted. And the lack of protective vegetation exacerbated the desertification and degradation of pastures. Besides, pirates of pikas also eat grass, which reduced the carrying capacity. When the number of these small mammals increased sharply, it would cause a lot of trouble and loss to economy, ecology, and people’s lives on the grassland. So at this moment, they are referred to as harmful animals.

As the change of the natural environment by the human production activities, agricultural, and the rapid development of cities provide plenty of food resources and good habitat for rodent, rat increases seriously, the management of pest also will be more difficult. Mouse control strategy from the traditional damage caves and machinery catch to fumigation, acute rodenticide, anticoagulant therapy, and the application of many chemical methods has made important progress. At present, the chemical prevention and control play an important role in the mouse control technology. However, chemical control is effective for short and harmful rat will soon come again and reproduction rapidly leads to the quick rebound in this species. In the fields, the application of acute rodenticide reaches 80; the population in the two years can be restored to the original level. In addition, there still exist many problems such as environmental pollution, secondary poisoning, and fungicide resistance in chemical control, which makes chemical Rodenticide restricted to the application of the rodent sustainable control. And integrating multidisciplinary approach and means, the sterility control technology based on ecological security has gradually become the development direction of rodent control. Infertility control technology has both directly and indirectly reduced the rodent population density, and will not lead to sharp fluctuations in ecological system, so it has a very good advantage in the environmental safety and cost-effectiveness.

Now, there are very serious rat in many areas of China, such as Xinjiang, Inner Mongolia, Gansu, Shanxi, and The Tibetan Plateau. In the Inner Mongolia grasslands, it is predicted that pest harm area is about 100 million mu and the serious disaster area is about 50 million mu [1]. Prairie mousehole per hectare is 300 at least and even 900 at most. In 2006, eleven silver foxes were introduced for the first time at Alxa League in Inner Mongolia grassland in order to control the prairie mice. Those foxes can catch large amounts of prairie gerbil, Meriones unguiculatus, and jerboa [2]. In addition, in 2011, Beowulf Biological antisterility rodenticide was used. The purpose is to test the effect of preventing grassland rat and whether or not achieves these requirements such as restraining the birth rates of harmful rat population, reducing the pest population density, slowing population growth benefiting Environmental Health and Safety [3].

At present, the research about infertility control is at most laboratory studies [49] and theoretical analysis even less. Based on the above understanding of the facts and mathematical biology background, the study about a class of predator-prey model with infertility control in the prey (harmful rat) is very meaningful. Moreover, the result indicated that species and quantity are different by vegetation and physiognomy, and change of density is more distinct along with changing season. Therefore, it is a very basilic problem to research this kind of nonautonomous population dynamic systems.

It is interesting to note that rodents living in the North generally have seasonal breeding, such as plateau pika nearby Qinghai lake breed from April to August, Brandt’s voles breed from March to September, and Mongolian gerbil in Inner Mongolia breed from April to August. Obviously, this kind of periodic phenomenon, extensively exists in the real world. Therefore, the dynamical behavior of the -periodic system is also worthy of being discussed.

Now, we only consider infertility control in the prey (harmful rat) population. It is composed of two population classes: one is the class of fertility prey, denoted by , and the other is the class of infertility prey, denoted by . Therefore, at any time , the total density of prey population is . Fertility rodents will become infertile after eating the sterilant. Therefore, is assumed to the rate at which infertility prey contacts occur. In this paper, we study the following nonautonomous predator-prey model with infertility control in the prey: with initial conditions where Here, is the fertility prey population density, is the sterility prey population density, is the predator population density, , are the intrinsic growth rate and density-dependent coefficient of the prey, respectively, , are the intrinsic growth rate and density-dependent coefficient of the predator, respectively, is the capturing rate of the predator, and is the rate of conversion of nutrients into the reproduction of the predator.

2. Preliminaries

For a continuous bounded function defined on , we denote If is -periodic, then the average value of on a time interval can be defined as

For system (1), we introduce the following assumptions.(H1)Functions , , , , , and are all negative, continuous, and bounded on and are continuous and bounded functions.(H2)There exist positive constants such that

In particular, when model (1) degenerates into -periodic system, that is, , , , , , , , and are continuous periodic functions with period , then assumption (H2) is equivalent to the following forms:, , , and .

In the following, we state several lemmas which will be useful in the proof of main results in the paper.

Firstly, we consider the following nonautonomous logistic equation: where functions and are bounded continuous defined on and for all . We have the following result.

Lemma 1 (see [10]). Suppose that there are constants such that Then,(a) there exist positive constants and such that for any positive solution of (7) (b) each fixed positive solution of (7) is globally uniformly attractive;(c) if , then for any positive solution of (7) (d) if (7) is -periodic, then condition (8) reduces to and ; thus (7) has a uniformly attractive positive -periodic solution.

Further, we consider the following nonautonomous equation: where and are defined as in (7) and is continuous and bounded function defined on .

Let be the solution of (11) with initial condition and let be some fixed positive solution of (7). We have the following result.

Lemma 2 (see [11]). Suppose that all conditions of Lemma 1 hold. Then for any constants and there exist constant and such that for any and , when for all , one has

Next, we consider the following nonautonomous linear equation: where and are bounded continuous defined on and for all . We have the following result.

Lemma 3 (see [12]). Suppose that there are constants such that Then, (a) there exist positive constants and such that for any positive solution of (13) (b) each fixed positive solution of (13) is globally uniformly attractive;(c) if , then for any positive solution of (13) (d) if (13) is -periodic, then the condition (14) reduces to and ; thus (13) has a uniformly attractive positive -periodic solution.

Further we investigate the following nonautonomous linear equation: where and are defined as in (13) and is continuous and bounded function defined on .

Let be the solution of (17) with initial condition and let be some fixed positive solution of (13). We have the following result.

Lemma 4 (see [13]). Suppose that there exists a constant such that Then for any constants and there exist constants and such that for any and , when for all , one has

In (17), if function , then we can obtain that . We have the following Corollary 5 of Lemma 4.

Corollary 5. Suppose that for all and there exists a constant such that Then for any constants and there exist constants and such that for any and , when for all , one has

3. Main Results

It is obvious that the solution of model (1) with initial condition (2) is positive; that is, , , for all in the maximum interval of existence of the solution. On the ultimate boundedness of solutions of system (1), we get the following theorem.

Theorem 6. Suppose that (H1) and (H2) hold. Then system (1) is ultimately bounded in the sense that there is a positive constant M such that for any positive solution of system (1).

The ecological implication of Theorem 6 is that the fertility prey is ultimately bounded. The sterility prey , when assumptions (H1) and (H2) hold, if is not ultimately bounded, then will expand unlimitedly. But the conversion of the fertile prey lies on the sterile prey by sterile drugs. So, the prerequisite for the unlimited increase of the sterility prey is that the fertility prey must be expanding unlimitedly. In short, the number of harmful rat will not go on rising forever.

Proof. Let be any positive solution of system (1). From the first equation of system (1) we have From (H2), it is easy to verify that the comparison equation satisfies all conditions of Lemma 1. So, the comparison theorem and Lemma 1 imply that we obtain there is a constant such that for any positive solution of system (1), there is a such that we have for all . Further, from the second equation of system (1) we have for all . From Lemma 3 it can be obtained that under assumption (H2) any positive solution of the following nonautonomous linear equation: is ultimately bounded. Hence, using the comparison theorem, we further can obtain that there is a constant such that for any positive solution of system (1), there is a such that for all . Lastly, from the third equation of equation of system (1) we have for all . Consider the following nonautonomous equation: the comparison theorem and Lemma 1 imply that there is a constant such that for any positive solution of system (2), there is a such that for all .
Now, let ; then from the above proofs, we have Therefore, solution is ultimately bounded. This completes the proof.

Remark 7. Applying the comparison theorem and combining conclusion (c) of Lemmas 1 and 3, we can obtain that if assumptions (H1) and (H2) hold in system (1), , , and , then constants given above can be chosen by

Next, we discuss the permanence and extinction of fertility prey and infertility prey .

Let be some fixed positive solution of the following nonautonomous logistic equation: Particularly, if , using conclusion (c) of Lemma 1, we can obtain

Theorem 8. Suppose that (H1) and (H2) hold and there exists a constant such that Then, for any positive solution of system (1).

Theorem 8 shows that if we guarantee that assumptions (H1),  (H2) and (35) hold, then the prey species must be permanent. In the ecological system, each component part, including the animal, plant and microorganism, plays its own role, and they are indispensable and irreplaceable. Every creature may deviate from its original trajectory, which lead to the outbreak of this population and the negative effect on human beings, such as harmful rat. Even if it happens, this species should not be extinct through the human activity. What we should do is to control the rat population to such a degree that will not be harmful to human beings. Therefore, the permanence of harmful rat given by Theorem 8 is very necessary.

Proof. Let be any positive solution of system (1). From condition (17) there are positive constants and such that for all According to Theorem 6, there exists a constant such that for all . Consider (11), that is, from Lemma 2, for and given in above there exist constants and such that for any and , when for all , we have where is the solution of (11) with initial condition .
Choose constant as follows: Consider the following nonautonomous linear equation: From Corollary 5, for and given in above there exist constants and such that for any and , when for all , we have
Let , we will discuss the following three cases.
Case  1. There exists a constant such that for all .
Case  2. There exists a constant such that for all .
Case  3. There exists a time sequence satisfying , and such that
If Case  1 appears, we have for all . Considering the auxiliary system Let be the solution of the above equation satisfying initial condition , by the comparison theorem, we have for all . Since for all , Hence, for all and . By (41), we have for all . Then, we obtain for all . So, Hence, for all . In (38), choosing and , by (38), we can get Then, For any , we have Integrating the above inequality from to , we can obtain From this and (35), it follows which leads to a contradiction.
If Case  2 appears, then obviously is permanent.
If Case  3 appears, for any we have and for all . If , choosing constant integrating the first equation of model (1) in interval , we get
If , because for all , we have for all and . Hence, we have for all . Then, we obtain for all . So, Hence, for any . In (38), choosing , and , by (38), we can get Then, For any , when , we can obtain from the above discussion on the case , In particular, we have . When , then we choose an integer such that ; integrating the first equation of system (1) from to we can obtain where . Choose then from above discussion we finally obtain
In addition, we have for all . Then, we finally obtain
Considering the second equation of system (1), according to Theorem 6, we have for all .
Considering the auxiliary equation According to Lemma 3, there exists a constant such that for any positive solution of (61). By the comparison theorem and (60), we have Let ; from (59) and (62) we obtain This completes the proof.

Theorem 9. Suppose that (H1) and (H2) hold and there exists a constant such that Then, for any positive solution of system (1).

The biological meaning of Theorem 9 is that if (H1), (H2), and (64) hold, the prey species will be extinct. Form the viewpoint of the Nature Conservancy and Human Health, the best way for our human beings is to keep the existence of the species, and, meanwhile, guarantee such existence do no harm to us. Thus, the condition of making harmful rat extinct for management expert is very important. Therefore, it is a critical threshold value.

Proof. By (64), we have for any , there are positive constants and and such that for all . From the third equation of system (1) we have applying the comparison theorem and conclusion (b) of Lemma 1, there exists a constant such that for all . For any , we have For any , if for all , integrating (68) from to , we obtain From (66), it follows that as which leads to a contradiction. Hence, there exists a such that . Let we prove If (71) is not true, then there exists a such that . From , there exists a such that and for all . Let be an integer such that , integrating (68) from to , which leads to a contradiction. Hence, (71) holds. From the arbitrariness of , we finally obtain as . Considering the second equation for all . Using Corollary 5, we can easily obtain as . This completes the proof.

Further, from conclusion (c) of Lemma 1, as consequence of Theorems 8 and 9, we also have the following corollaries.

Corollary 10. Suppose that (H1) and (H2) hold, and there exists a constant such that Then, for any positive solution of system (1).

Corollary 11. Suppose that (H1) and (H2) hold, and there exists a constant such that Then, for any positive solution of system (1).

As consequences of Theorems 8 and 9, we have the following corollaries. Firstly, from Lemmas 1 we obtain that if and hold, then (31) have the globally uniformly attractive nonnegative -periodic solutions .

Corollary 12. Suppose that system (1) is -periodic and and hold. Then the fertility prey and infertility prey in model (1) are permanent if and only if

Corollary 13. Suppose that system (1) is -periodic and and hold. Then the fertility prey and infertility prey in model (1) are extinct if and only if

Remark 14. In the process of the prevention and control of rat in the grasslands and farmland, we are concerned about how many biological sterile drug should be put in a period in order to make the population of the harmful rat reduce to a very low level. From Corollaries 12 and 13, we can easily obtain that is a critical value. If , we can control the population of rat at a very low level. The results are very meaningful and significant.

When system (1) is simplified into the corresponding autonomous system, that is,

Remark 15. For system (80), we know that is a critical value. Then, we can obtain that the quantity of biological sterile drug should be .

4. Example and Numerical Simulation

In this section, we give some examples and numerical simulations to the above theoretical analysis.

Example 1. Take , , , , , , and in system (1). We easily verify that assumptions , hold. From Lemma 1, some fixed positive solution of system (32) satisfies . Moreover, condition (35) holds, therefore, by Theorem 8, system (1) with these parameters is permanent. The corresponding numerical simulations are given in Figure 1, and this figure illustrates that the solutions will tend towards periodic oscillation along with time passing. It means that there exists a periodic solution, and it is seemed that this periodic solution is globally attractive.

fig1
Figure 1: Permanence of system (1) with parameters in Example 1.

Example 2. Take , , , , , , and in system (1). By similar calculation, we can obtain that , and (35) hold. Therefore, by Theorem 8, system (1) with these parameters is permanent, as shown in numerical simulations of Figures 2(a)2(d), which not only illustrate the validity of the proposed results, but also display the interesting complex dynamic behaviors; that is, there is not periodic oscillation along with time passing as like Figure 1, and from (a)–(d) in Figure 2, it can be obviously seen that there is a strange chaotic attractor, which may contribute to a better understanding of the complex chaotic behaviors which can be a high risk of the uncertain number of the population due to the unpredictability.

fig2
Figure 2: Chaotic behavior of system (1) with parameters in Example 2.

5. Conclusion

Based on the mouse rampant phenomenon in some areas, a predator-prey model with infertility control in rat species is established in the situation where all coefficients depend on time. For the nonautonomous system threshold conditions for the permanence and the extinction of fertility prey and infertility prey are established. The condition for permanence has the form of a lim inf condition for some time-dependent sterility conversion rate while the condition for extinction assumes the form of a lim sup condition. Hence, in the general case the main results are not threshold criteria in a strict sense. However, in the periodic cases, the conditions merge into a sharp threshold criterion and sterile drug dosage can be obtained. Two numerical examples are carried out to support theoretical results, and the second simulation result suggests that there may be interesting dynamic behaviors in this model-a strange chaotic attractor. Furthermore, chaos may cause the number of pests approaching to the uncontrollable state due to the unpredictability. Thus, how to control chaos in the population model is very important, which needs further investigation.

Conflict of Interests

The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work, there is no professional or other personal interests of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this paper.

Acknowledgments

This work was supported by The National Natural Science Foundation of China [11271312,11371313,11241005], The Natural Science Foundation of Shanxi Province [2013011002-5], and the Research Project at Yuncheng University [XK2012001,XK2012007].

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