Abstract
A general predator-prey model with disease in the prey and double impulsive control is proposed and investigated for the purpose of integrated pest management. By using the Floquet theory, the comparison theorem of impulsive differential equations, and the persistence theory of dynamical systems, we obtain that if threshold value , then the susceptible pest eradication periodic solution is globally asymptotically stable and if , then the model is permanent. The numerical examples not only illustrate the theoretical results, but also show that when the model is permanent, then it may possess a unique globally attractive -periodic solution.
1. Introduction
Integrated pest management (IPM) is a long term management tactic that uses a combination of chemical, biological strategies to reduce pests to tolerable level or below the threshold, with less cost to the farmers and minimal effect on the environment (see [1, 2]). Such techniques include mechanical methods (erecting pest barriers or using pest traps) and biological methods (breeding natural predators of the pest or using biological insecticides). Some successful biological control examples contain the use of the predatory arthropod Orius sauteri against the pest Thrips palmi Karny to protect eggplant crops in greenhouses (see [3]) and the use of the predatory mites Phytoseiulus persimilis and Neoseiulus californicus to regulate the red spider mite Tetranychus urticae Koch in field-grown strawberries (see [4]).
The discontinuity of human activities and the abrupt variation in the amount of the pest population, which occurs immediately after successful control measures (such as spraying pesticides, releasing natural enemies of the pest, and freeing infective pest individuals), may be described mathematically through making use of impulsive differential equations (see [5–16]).
Many scholars have been devoted to the analysis of impulsive differential equation models describing IPM strategies and some rich results have been obtained (see [6–15, 17]). They assumed that the disease incidence rate should be distinguished; as far as disease transmission is concerned, nonlinear, bilinear, and standard incidence rates have often been used in establishing ecoepidemic models, which depends on different infective disease and environment. Georgescu and Zhang (see [10]) investigated a predator-pest model with incidence rate given by , Pang and Chen (see [12]) discussed an model with bilinear incidence rate , Wang et al. (see [13]) analyzed an model with incidence rate given by , and so forth. Main results of these theses have focused on conditions of pest eradication and permanence of the system. According to the authors’ knowledge, at present stage, there are few studies of general incidence rate. So one of the goals of this paper is to generalize the incidence rate.
The functional response between pests and natural enemies plays an important role in assessing dynamical behavior of the system. People use natural enemy, as in some sense like a pesticide, to control pest via augmentation or releasing natural enemy once the quantity of pest has reached or exceeded the economic threshold (see [9, 10, 14, 15]). Shi et al. (see [14]) analyzed a predator-pest model with disease in the pest and functional response given by Holling-II type and the time-dependent impulsive strategy including release of infective pest individuals and those natural predators at different point in time; the threshold on pest eradication was obtained. However, little of paper has been devoted to analysis of models which combine release of infective pest individuals and those natural predators. The approach to biological control which we adopted is to release both infective pest individuals and natural predators periodically with constant amount at different point in time; what is more, the functional response is also a more general form. Motivated by the valuable contributions of Georgescu and Zhang [10], Wang et al. [13], and Shi et al. [14], in this paper general IPM model will be considered as follows: The model is based on the following assumptions:(H1)The pest is divided into the susceptible and the infective, and the infective cannot produce offsprings as a result of the disease, but the infective still consume crop. The incidence rate of the infective is given by function . The growth rate of the susceptible is assumed to function (H2)Parameter represents the amount of infective pest released periodically at time , where and Parameter represents the amount of natural enemy released periodically at time , (H3)The natural enemy only hunts the susceptible and the functional response is given by function is the conversion rate.(H4)Positive constants and are the death rates of the infective pest and the natural enemy, respectively.
In model (1), and denote the density of the susceptible pest and the infective pest (prey) population, respectively. is the density of natural enemy (predator) population. For model (1), in this paper we will investigate global stability of the susceptible pest eradication periodic solution and the permanence of model (1). In Section 2, the positivity and boundedness of solutions are presented. In Section 3, by using the Floquet theory for impulsive differential equations, the theorem on the global asymptotic stability of the susceptible pest eradication periodic solution is established. In Section 4, by using the persistence theory of dynamical systems, the theorem on the permanence of model (1) is established. In Section 5, we will give the numerical simulations to illustrate the main results obtained in this paper. Finally, in last section a brief discussion and some possible future researches are proposed.
2. Preliminaries
Denote and . For model (1), we introduce the following assumptions.(A1)Function is continuous on and is nonincreasing for and , respectively. Consider with .(A2)Function is continuously differentiable for and on , , and , for all (A3)Function is continuously differentiable for and on , , and , for all
The solution of model (1), denoted by , is piecewise continuous on and , , , and exist. The global existence and uniqueness of solution for model (1) with any initial value are guaranteed by the smoothness of the right-hand functions of model (1) (see [18]). Firstly, the following results are obtained easily.
Lemma 1. Assume that is the solution of model (1) with , and then for all . Furthermore, if , then for all .
Lemma 2. Let be a positive constant. Then system has a positive periodic solution where . Furthermore, any solution of system (2) with initial value can be expressed as and satisfies as
If susceptible pest is absent, then model (1) reduces to By Lemma 2, positive periodic solution of system (5) is where and Furthermore, any solution of system (5) with initial values and can be expressed as It is easy to get the following conclusion.
Lemma 3. System (5) has positive periodic solutions . For any solution of system (5) with initial value , one has as .
On the ultimate boundedness of solutions for model (1), we have the following conclusion.
Lemma 4. There exists a constant such that, for any solution of model (1) with initial value , one has , , and with large enough.
Proof. Define By calculating the derivative of with respect to model (1), when and ,where and When , and when By Lemma given in [18], it is obvious that From this, there exists a constant such that , and for large enough. This completes the proof.
In the following, we introduce some necessary definitions and lemma on the persistence of dynamical systems, which will be used for the discussion of permanence of model (1). For more details, see [19, 20].
Let be a metric space with metric and let be a continuous map. For any , we represent for any integer and is said to be compact in if, for any bounded set , set is precompact in is said to be point dissipative if there is a bounded set such that, for any ,For any , the positive semiorbit through is defined by , the negative semiorbit through is defined as a sequence satisfying for integers , the omega limit set of is defined by there is a sequence such that , and the alpha limit set of is defined by there is a sequence such that .
A nonempty set is said to be invariant if A nonempty invariant set of is called isolated in if it is the maximal invariant set in a neighborhood of itself. For a nonempty set of , set is called the stable set of
Let and be two isolated invariant sets and set is said to be chained to set , usually expressed as , if there exists a full orbit though some such that and A finite sequence of isolated invariant sets is called a chain if , and if the chain is called a cycle.
Let be a nonempty open set of We denote
Lemma 5 (see [19, 20]). Let be a continuous map. Assume that the following conditions hold:(C1)Map is compact and point dissipative and .(C2)There exists a finite sequence of compact and isolated invariant sets in such that(1) for any and ;(2);(3)no subset of forms a cycle in ;(4) for each
Then map is uniformly persistent with respect to ; that is, there exists a constant such that for all
3. Global Stability of Susceptible Pest Eradication Periodic Solution
From Lemma 3, we know that model (1) has a susceptible pest eradication periodic solution . On the global asymptotic stability of this periodic solution, we have the following theorem.
Theorem 6. Assume that Then periodic solution of model (1) is globally asymptotically stable.
Proof. To investigate the local stability of susceptible pest eradication periodic solution , let . We have The corresponding linearized system is Let be the fundamental matrix of system (15); then with , a identity matrix. By calculating, we get whereObviously, the eigenvalues of matrix are If (13) holds, then . By the Floquet theory (see [21]), is locally asymptotically stable.
From (13), we can choose a small enough constant such that From the second equation of model (1) and assumption , we obtain Consider the following impulsive differential equations: and, by the comparison theorem of impulsive differential equations (see [18]) and Lemma 3, it follows that and as . Therefore, for large enough. Similarly, for large enough.
From assumptions and , there exist and such that and From (23), (24), and the first equation of model (1), there exists a such that for all For any we can choose an integer such that , where . Integrating the above inequality from to , we obtain where , which implies as .
Next, we show that and as . Choose constant small enough such that . From , it follows that for large enough. From assumption and the second equation in model (1), Consequently, and , as , where is the solution of (22) and is the solution of the following system: Since , then Therefore, for any small enough, we have for large enough, which implies as .
Lastly, a similar argument as from (27) to (29), we also can obtain that as This completes the proof.
Corollary 7. When the right-hand functions in model (1) are then condition (13) is equivalent to the following form: where , and are positive constants.
Remark 8. In (31), if , then , which means that if only natural enemies are released periodically, then the release amount must be larger than to ensure the eradication of the pest. If , then the release amount must satisfy the inequality to ensure the eradication of the pest.
4. Permanence of the Model
Theorem 9. Assuming that then model (1) is permanent.
Proof. Since the impulsive effects in model (1) are periodic, model (1) can be regarded as periodic model with period . Therefore, we can use the persistence theory of dynamical systems to discuss the permanence of model (1). Define Thus From Lemma 1, we claim that and are positively invariant with respect to model (1). is a relatively closed set in .
Let be a Poincaré map associated with model (1); that is, where is the unique solution of model (1) with initial value . By Lemma 4, Poincaré map is compact and point dissipative on . Therefore, condition of Lemma 5 holds.
Let where , , and .
Firstly, we will testify Clearly, For any , by , the solution of model (1) with initial value satisfies , and for all . Therefore, for any integer , we obtain This implies Therefore, (38) holds.
Model (1) can be simplified as model (5) in By Lemma 3, model (1) has globally attractive periodic solution in This shows that map has a global attractor in It is clear that, in is isolated, invariant, and does not form a cycle. Therefore, conditions of in Lemma 5 hold.
Secondly, let By the continuity of solutions with respect to the initial value, for any , there is a ; when , we have Now, we claim that Suppose the conclusion is not true; then we havefor some For the sake of simplicity, one may assume that Also, from (40) we obtain Then, for any , let , where and is the greatest integer less than or equal to ; we can get Since and , (45) signifies that for all By condition (33), we can choose constant such that Further, from assumptions and and inequalities (46), we have where , For any , choose an integer such that , where Integrating (48) from to and noticing (47), then where Thus, , which is a contradiction with for all . Hence, claim (41) holds. This shows Therefore, condition of holds. Consequently, by Lemma 5, is uniformly persistent with respect to .
Lastly, since model (1) is periodic, we obtain that model (1) is uniformly persistent. From Lemma 4, model (1) also is permanent. This completed the proof.
Corollary 10. When functions , and are given in (30), then condition (33) is equivalent to the following condition:
Remark 11. In (50), if , then we have , which means that if only natural enemies are released periodically and the amount is less than , then the system is permanent and the pest will not be eradicated. If , then the release amount satisfies to ensure the system is permanent and the pest will not be eradicated.
Remark 12. Applying Theorem given in [16], it is clear that when condition (33) holds, model (1) at least has one positive -periodic solution.
Remark 13. Taking the functions in model (1) as the model has been discussed in [14]. Furthermore, noticing (30), we have reason to confirm that our study makes the model in [14] more general.
Corollary 14. Define constant If , then susceptible pest eradication periodic solution of model (1) is globally asymptotically stable, and if , then model (1) is permanent.
5. Numerical Example
In this section, we give some examples and numerical simulations to confirm the above theoretical analysis. Let us consider model (1) with functions , and given in (30) and parameters as follows.
Example 1. Take , , , , , , , , , , , and . By computing, we have Therefore, inequality (31) holds; from Theorem 6, periodic solution of model (1) is globally asymptotically stable, which is illustrated in Figure 1.
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Example 2. Take , , , , , , , , , , , and By computing, , in Lemma 4, and Therefore, inequality (50) holds; from Theorem 9, model (1) is permanent. Numerical simulation (see Figure 2) shows that there exists a unique positive -periodic solution of model (1) which is globally attractive.
From the above example, we can guess that only inequality (33) holds; then model (1) has a unique positive T-periodic solution which is globally attractive.
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6. Discussion
In this paper, a general ecoepidemic model with impulsive control strategy is proposed and its dynamical behavior is analyzed for the purpose of integrated pest management. Meanwhile, the model which the researchers obtained in [14] was generalized. By using Floquet theorem and theory of persistence of dynamical systems, we show that if condition (13) holds, the susceptible pest eradication periodic solution is globally asymptotically stable (see Figure 1), which means that the pest has been eradicated; when condition (33) holds, model (1) is permanent (see Figure 2), which means the pest and their natural enemy can coexist in the area. Furthermore, from Remark 12, model (1) has at least one positive -periodic solution.
Further study includes investigating whether or not a nontrivial periodic solution emerges when the threshold holds and analyzing dynamical behavior about model with delay.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China , the Doctoral Subject Science Foundation (Grant no. 20136501110001), the Xinjiang Agriculture University Foundation [XJAU201120, jqztp72013015], and the National College Students’ Innovation and Entrepreneurial Training Program .