Abstract
A pest management model with stage structure and impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semicontinuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincarè criterion. Finally, we analyze numerically the theoretical results obtained.
1. Introduction
Banana leaves diseases are divided into epiphyte and virus. Banana bunchy top disease (i.e., Prawn banana, Green banana, Banana) is one of virus diseases, caused by Banana bunchy top virus. Banana farmers call it an incurable disease. Banana aphids are the major propagation medium of banana virus diseases. The development of banana aphids includes three stages: egg, nymph, and adult (winged form). Eggs do not carry and spread virus. Nymphs transmit virus to healthy plants only through short-distance crawling since genitalia and wings of nymphs are not fully developed yet, and therefore infected nymphs have slight infective power. After 4 instars, nymphs grow into adults which have fully developed genitalia and wings and can oviposit and transmit virus to healthy plants through migrating after piercing and sucking the virus of diseased plants, so infected adults have strong infective power. To avoid the outbreak of banana aphids, we will use ovicides to kill eggs or use insecticides to kill nymphs and adults.
In pest management, we spray pesticides only when pest density increases to a certain level called ET (economic threshold, i.e., pest population density at which control measures should be adopted to prevent an increasing pest population from reaching the economic injury level). ET is the index of pest density. Crop output will not decrease much when pest density is lower than ET; thus, we need not adopt any control measure. Once pest density rises to ET, some measures must be carried out to prevent EIL (economic injure tolerate level) from happening. To control pests, such a measure for spraying pesticides is always adopted when pest density arrives at a given ET.
Considering that immature pests cause a minor damage to crops, in this paper, we will spray insecticides when the density of immature pests increases to ET, which is a more effective preventive measure than we do when the density of mature pests increases to ET. Usually, insecticides have specificity; in other words, insecticides (such as 2000 to 2500 times dilution of acetamiprid 3% EC, 15000 times dilution of imidacloprid 70% WG, 1000 times dilution of omethoate 40% EC, and 2500 to 3000 times dilution of sumicidin 20% EC) can only kill nymphs and adults but cannot kill eggs. Therefore, a pest management model with stage structure and impulsive state feedback control is constructed as follows: where denote the proportions of immature pests (nymphs) and mature pests (adults) at time , respectively, denotes the transformation rate from mature to immature pests, where denotes the birth rate of mature pests, denotes the transformation rate from eggs to immature pests, denotes the transformation rate from immature to mature pests, denote the death rate of immature and mature pests, respectively, are positive constants, are the ratios of killing immature and mature pests by spraying pesticides, respectively, and denotes ET.
At present, for stage structure pest management model with impulsive effect, extinction and permanence have been proved by using Floquet theorem and comparison theorem [1–4]. For impulsive state feedback control systems, the sufficient condition for the existence and the orbitally asymptotically stability of the order-1 periodic solutions have been obtained by differential equation geometry theory, the method of successor function, and analog of Poincarè criterion [5–14]. However, for the pest management model with stage structure and impulsive state feedback control, almost no one investigates. In this paper, we try to obtain a new judgement method for the stability of the order-1 periodic solution by referencing the stability analysis of limit cycles for continuous systems. This is a superior method, by which the more perfect and simple conclusions than the others are obtained.
In the next section, we give some preliminaries. In Section 3, we get the sufficient condition for the existence of the order-1 periodic solution of system (1) by differential equation geometry theory and successor function. In Section 4, referencing the stability analysis of the limit cycles for continuous dynamic systems, we prove the order-1 periodic solution of system (1) is orbitally asymptotically stable under some conditions. In Section 5, we analyze numerically the theoretical results obtained.
2. Preliminaries
Definition 1 (see [15]). Suppose impulsive state differential equationwhose solution mapping composes the system called semicontinuous dynamic system, denoted by . Set initial point of mapping ; is a continuous mapping, , and is called impulse mapping, where and are straight lines or curves on the plane , denotes impulse set, and denotes phase set.
In system (1), impulse set , impulse mapping : , phase set . Therefore, system (1) composes a semicontinuous dynamic system .
Definition 2. Let be the semicontinuous dynamical system mapping described by system (2) at , and is a mapping in itself. If there are a point in phase set and a such that (pulse mapping is ), then is said to be the order-1 periodic solution.
Definition 3 (see [15]). Suppose is the phase set of system (1), is the impulse set of system (1), and both and are straight lines (see Figure 1). The intersection point of and -axis is , the distance between point () and point is noted by , denotes the intersection point of trajectory passing through point and , phase point of is (), and the distance between and is noted by . One defines subsequent point of as , and the successor function of is .
Remark 4. If , the trajectory passing through point is the order-1 periodic solution of the system.
Lemma 5 (see [15]). Successor function is continuous.
According to Lemma 5, we can get the following lemma.
Lemma 6 (see [15]). Assume continuous dynamical system ; if there exist two points in the phase set such that successor function , , we can find a point between and in the phase set satisfying . So there must exist an order-1 periodic solution passing through point .
3. Existence of the Order-1 Periodic Solution
For system (1), if , , that is, without impulse effects, there is a unique singular point if and only if
Taking the transform system (1) turns into
There are four forms for Jordan standard of the two-dimensional matrix: where are two real roots of characteristic equation where , and obviously, , .
By theory of stability, singular point has the following two cases [16]:(i)if , is an asymptotically stable node; see Figures 2(a) and 2(b);(ii)if , is a saddle point; see Figures 3(a) and 3(b).
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Theorem 7. If , that is, , there exists a point satisfying ; that is to say, there exists an order-1 periodic solution of system (1).
Proof. If , that is, , for system (1), the impulse set is straight line , the phase set is straight line , and intersect -axis at , respectively (see Figure 4). Denote isoclinic lines , and boundary between the two isoclinic lines as , where intersects at and intersects at , respectively.
Firstly, we analyze the existence of order-1 periodic solution of system (1) in the domain .
Suppose is the phase point of ; then , where is the coordinate of , respectively. Choose between and ; the trajectory passing through intersects the phase set at after impulse effect, and then is the subsequent point of . Since distinct trajectories do not intersect, must be below ; we have , where is the coordinate of , respectively. Therefore, .
Suppose the trajectory passing through intersects the phase set at after impulse effect; then is the subsequent point of , and there are two cases.
Case 1. If is above , then , where is the coordinate of , respectively; we have . By Lemma 6, there exists a point such that . Therefore, there exists an order-1 periodic solution of system (1) passing through . The proof is completed.
Case 2. If is below , then ; we have (see Figure 5). In addition, the trajectory passing through intersects the phase set at after impulse effect; then is the subsequent point of , and must be below and above because distinct trajectories do not intersect. Therefore, we have , where is the coordinate of , respectively. By Lemma 6, there exists a point such that . The proof is completed.
Secondly, suppose intersects at , respectively, and intersects at . Draw a straight line which is perpendicular to and -axis; the foot points are , respectively (see Figure 6). Let us analyze the existence of order-1 periodic solution of system (1) in the trapezoid .
Suppose are the phase points of , respectively. On the one hand, the trajectory passing through intersects the phase set at after impulse effect; then is the subsequent point of , and must be below because distinct trajectories do not intersect. We have , where is the coordinate of , respectively.
On the other hand, choose a point between and . The trajectory passing through intersects the phase set at after impulse effect, and then is the subsequent point of , and must be above because distinct trajectories do not intersect. We have , where is the coordinate of , respectively.
By Lemma 6, there exists a point such that . Therefore, there exists an order-1 periodic solution of system (1) passing through . The proof is completed.
4. Stability of the Order-1 Periodic Solution
Definition 8 (see [17]). On the positive half-trajectory of semicontinuous dynamic system denoted by , , choose any time series such that . If is the limit point of point range , One calls the limit point of point range The set made up of all limit points of point range , is called limit set.
Definition 9. Assume is the order-1 periodic solution of semicontinuous dynamic system. If there exists a neighborhood sufficiently small such that limit set of trajectory starting from any point is always , the order-1 periodic solution is stable. Otherwise, the order-1 periodic solution is unstable.
In system (1), is any point of the phase set (see Figure 7); assume the single-closed curve consisting of curve and line segment is an order-1 periodic solution of system (1), denoted by . Get point near ; there exists a point range: where are the subsequent points of , respectively.
Establish coordinates at phase set and near , the coordinate of is . Let denote the coordinates of pointsrespectively.
Proposition 10. For any point near , when , the point range that is, and then the order-1 periodic solution is stable (unidirectional).
Proposition 11 (königs). Assume is a continuous transform from line segment to itself; is a fixed point under the transform. If the part near origin of curve on the plane lies in the interior of the domainthe fixed point is stable (unstable).
Proof. We prove firstly that the fixed point is stable. Choose sufficiently small such that for any point in noncentral neighborhood of the fixed point , .
Let and we haveFor any point range where , we get sequence From Figure 7, we have .
Let , ; it is easy to deduce that and hence when . Upon that, the fixed point is stable.
In the same way, we prove the fixed point is unstable. The proof is completed.
Corollary 12. Assume the derivative of function at exists; then is stable when .
From Figure 8, assume the closed orbit consisting of the curve and line segment is the order-1 periodic solution of system (20), denoted by , where , , is the phase set, and is impulse set. Draw normal line passing through and establish coordinate system on point . Choose any point in small enough neighborhood of . The trajectory starting from intersects vertically -axis at and intersects impulse set at . denotes the phase point of , the trajectory passing through point intersects vertically -axis at as increases.
Assume rectangular coordinate of is ; then for , there is the relation between its rectangular coordinates and curvilinear coordinates : wherewhere denote the values of at the point , respectively; we have
From (20), it is easy that we have and hence
Since there is a zero solution for (24), when there exist continuous partial derivatives for functions , there exists the continuous partial derivative of with respect to also; (24) is written as In order to calculate we first get where denote partial derivatives of when , respectively. Since when , it is easy to know . By (24) and (27), we have where denotes the curvature of orthogonal trajectory at for system (1). Therefore, the approximate equation of (25) is whose solution is
Theorem 13. Assume is the length of curve which is a section of the order-1 periodic solution of system (1). The order-1 periodic solution is stable when
Proof. Let us investigate trajectory (see Figure 8). In the coordinate system , the ordinate of is denoted by and the ordinate of is denoted by . From (30), we have when , where is the length of curve . By Propositions 10 and 11, the order-1 periodic solution is stable.
Corollary 14 (see Diliberto [18]). If the integral along the order-1 periodic solution satisfies , the order-1 periodic solution is stable.
Let ; the left of (31) can be rewritten as that is, Consider the integral along the periodic solution of continuous system and we suppose the integral along the order-1 periodic solution of semicontinuous system has the same result.
Denote .
Lemma 15. If function is continuous and differentiable, the integral along the order-1 periodic solution of system (1) satisfies where period of the order-1 periodic solution is .
Proof. Let be an order-1 periodic solution of system (1) (see Figure 9); denotes the curve of system (1) from to , when , and when . denotes line segment .
Take the transform ; system (1) can be written as where the trajectory of system (37) is similar to system (1) except for time variable.
Let denote the curve of system (37) from to (see Figure 10); when and when . denotes line segment ; the parameter equation of is where when and when .
Obviously, system (37) system (1); that is, , when , and thus we havethat is, the integral along the order-1 periodic solution of system (1) satisfies The proof is completed.
According to (34), we have the following theorem.
Theorem 16. If the integral along the order-1 periodic solution of system (1) satisfies is stable.
Theorem 17. The order-1 periodic solution of system (1) is stable.
Proof. Sinceby Theorem 16, the order-1 periodic solution of system (1) is stable. The proof is completed.
5. Numerical Analysis and Discussion
Without impulse effects, there is an equilibrium point for system (1). If , is an asymptotically stable node; if , is a saddle point. According to Theorem 7, if , that is, , there exists an order-1 periodic solution of system (1).
To verify the theoretical results obtained in this paper, we choose as the parameter and analyze numerically the following cases.
Case 1. Let ; we have (see Figure 11). According to the above discussion, is an asymptotically stable node when . It implies that tend to be extinct as increases without any control measures.
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Case 2. Choose as the control parameter; let ; we have , the impulse set , and the phase set . According to Theorem 7, if , that is, , there exists an order-1 periodic solution (see Figure 12). We can observe that there exists an order-1 periodic solution of system (1) which lies between the phase set and the impulse set (i.e., between and ).
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Changing , the rest of the parameters are the same as Figure 12, and we obtain Figure 13. From the phase portrait of Figure 13, we can observe that there exists an order-1 periodic solution of system (1) which lies between the phase set and the impulse set (i.e., between and ).
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Figures 12 and 13 give the time series and phase portraits when and , respectively, and show different positions of the periodic solution under different parameter values and different initial values. Furthermore, the phase portrait of Figure 12 indicates that the mature pests always keep increasing, but Figure 13 indicates that the mature pests firstly decrease and then begin to increase. Therefore, the control parameter or can result in different change in density of mature pests and different efficiencies of killing mature pests by spraying pesticides which will give a conclusion theoretically to the researchers in killing mature pests. Researchers should give suitable control parameter and appropriate initial values in order to obtain a steady and optimal control. In fact, immature pests (nymphs) are more easily to be killed by pesticides than mature pests (adults); thus, Figure 13 is more feasible than Figure 12.
Time series portraits of Figures 12 and 13 show that the order-1 periodic solution of system (1) is stable, and it is consistent with Theorem 17. The numerical analysis illustrates that we can achieve the aim of controlling immature and mature pests by impulsively spraying pesticides when immature pests density increases to .
According to the obtained conclusions, we can predict the cycle time without repeated measurements, which can save a lot of labor and material resources. Obviously, the model with impulsive state feedback control is closer to the reality than the periodic impulsive model where there is no density dependence.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to sincerely thank the reviewers for their careful reading of the original paper and many valuable comments and suggestions that greatly improved the presentation of this paper. This work is supported by the National Natural Science Foundation of China (11161052, 61364020, 11361068), the Natural Science Foundation of Guangxi Province (2011jjA10044), the Scientific Research Foundation of Guangxi Education Office (201012MS183), and the Sustentation Fund of the Elitists for Guangxi Universities (GJRC0831).