Abstract
Based on the Lotka–Volterra system, a pest-natural enemy model with nonlinear feedback control as well as nonlinear action threshold is introduced. The model characterizes the implementation of comprehensive prevention and control measures when the pest density reaches the nonlinear action threshold level depending on the pest density and its change rate. The mortality rate of the pest is a saturation function that strictly depends on their density while the release of natural enemies is also a nonlinear pulse term depending on the density of real-time natural enemies. The exact impulsive and phase sets are given. The definition and properties of the Poincaré map corresponding to the pulse points on the phase set are provided. We investigate the existence and stability of boundary and interior order-1 periodic solution. The theoretical analysis developed in the present paper combined with nonlinear controlling measures as well as nonlinear action threshold methods and techniques laid the foundation for the establishment and analysis of other state-dependent feedback control models.
1. Introduction
Pest control [1–6] is not only an ancient problem but also a new challenge faced by the modern world. Various scientific and effective methods [7–13] are needed to comprehensively prevent and control pest outbreaks and reinfestation. The most common early method was chemical control [14, 15], that is, the method of controlling pest by spraying pesticides during pest outbreaks. The main advantages of chemical control are quick effect and convenient use. It can eradicate or maintain the number of pests at a lower level within a short period of time. Therefore, chemical control is still one of the important means to control pest population. Biological control [16–18] is another important control method, which has the advantages of strong effect and long duration, and is also an environmental friendly control method. Maiti et al. [19] used a valuable technique known as sterile insect release method (SIRM) to manage the pest population. The authors discussed the effect of uncertain ecological variations on sterile and fertile insects. Other main methods are physical control and agricultural control. For example, the agricultural control method is a method to reduce or control pests through measures such as crop rotation, intercropping, and reasonable adjustment of cultivation procedures.
Each pest control method has its advantages and disadvantages. Due to long-term and high-dose use, pests can easily develop resistance to specific pesticides, resulting in pest control failure and pest reemergance. However, other control strategies cannot effectively reduce the number of pests in a short time because of their slow effectiveness. Therefore, how to effectively and reasonably use multiple methods is the best choice for pest control. Based on this, the Food and Agriculture Organization of the United Nations (FAO) proposed the concept of integrated pest management (IPM) [1, 20, 21] and defined it as follows: “IPM is a pest control system that comprehensively considers the population dynamics of the pest and its related environment and uses all appropriate control techniques and methods that work as closely as possible to maintain levels at which pest populations do not cause economic harm.” Both experimentally [22, 23] and theoretically [24, 25], it has been proved that IPM is more practical than the classic approach. This is one of the most useful methods which minimizes damage to individuals and the environment in addressing pest control.
In this perspective, researchers have studied the mathematical problems based on impulsive differential equations in order investigate the dynamics of IPM and compass biped robotic systems. In numerous realistic problems, impulses often occur at state-dependent. Therefore, it is more feasible to apply the procedure of state-dependent feedback control to model real-world issues. Znegui et al. [26] used an impulsive hybrid nonlinear system to construct a passive biped robot model that demonstrates complicated behaviors. In [27], the authors constructed a Poincaré map which was further utilized to examine the existence and stability of order-1 periodic type solution of the problem under consideration. Many new systems on the design of specific analytical expression of the hybrid state-dependent Poincaré were studied in [28, 29]. The authors in [26–29] portrayed an expression of the controlled Poincaré map to discuss the stabilization of passive dynamic walking of the compass-gait biped robot. The compass-gait biped robot is a two-DoF legged mechanical system which is identified by its passive dynamic walking. The one-DoF mechanical systems are also of great importance. Some articles related to one-DoF state-feedback control with respect to different perspectives can be found in [30, 31].
The impulsive differential equations are also used proficiently in epidemic dynamics [32] and population dynamics [33–35]. A basic assumption of the above series of studies is that regardless of how huge the number of pests or the growth rate is, as long as the number of pest populations touches economic threshold (ET) [33–35], the IPM strategy can be implemented. However, there are two basic situations of actual pest growth that require high attention: first, the number of pests is comparatively large, and the rate of change is small; second, the population is small, but the rate of change is high. A fundamental problem illustrated by these two situations is that when the pest population is large (such as exceeding ET), the growth rate is small or even negative at this time. In this case, even if the IPM strategy is not implemented, the number of pests may not exceed economic injury level (EIL) [36]. Another situation is that the number of pests is not large, and the rate at which the pest population is growing is very large. In this case, if the control strategy is not implemented in time, it may lead to a large outbreak of pests. Next, in order to establish appropriate and effective integrated controlling strategies, the IPM process needs precise inspection of the pest quantity. The mortality rate should be fluctuated according to the saturating function which relies upon the density of pest, and the releasing quantity of natural enemies should be a function of their density. Therefore, keeping in mind the above factors, a feasible new state-feedback control pest-natural enemy ecosystem with nonlinear controlling measures as well as nonlinear action threshold system is proposed. The corresponding analytical techniques and numerical methods are developed to examine the dynamical aspects of the system under consideration.
The main research contents are reflected in the following aspects. We construct a Lotka–Volterra prey-predator model involving both nonlinear feedback and action threshold depending on the density of pest and its change rate. In the model, we use the action threshold instead of the economic threshold to characterize the implementation of control measures, that is, when the number of pests reaches the action threshold depending on the density of pest and its change rate, a comprehensive pest control tactic is applied so that the number of pests does not exceed the nonlinear ratio-dependent AT. On the other hand, the use of nonlinear controlling factors in the feedback control makes the model closer to reality. Properties of the nonlinear ratio-dependent AT are given. Then, the classification is performed according to the positional relationship between the action threshold level and the stable equilibrium point of the corresponding ordinary differential system. By using the definition and properties of Lambert W function, the analytical expression of the Poincaré map is given. Furthermore, by using the analytical properties of Poincaré map, the existence, uniqueness, and stability of the pest-free and interior-order one periodic solution of the pest-natural enemy system are given, and corresponding sufficient conditions are obtained. The main results are confirmed by numerical simulations.
2. Model Construction and Main Properties of Action Threshold
2.1. Construction of Model
In view of the above objective factors, we propose the following nonlinear state-dependent feedback control model combined with nonlinear ratio-dependent AT:
It can be seen that without pulse control measures, the model is simply based on the classical Lotka–Volterra type problem which is extensively used to describe the relation between the populations of pest and natural enemy shown by and , respectively. Weighted parameters , , and are positive constants, which satisfy . The discontinuous mapping shown in the third and fourth equations in system (1) represents that the implementation of comprehensive control measures depends on the action level, that is, once the pest density reaches action threshold, the densities of pests as well as the natural enemies are immediately updated to and , respectively. represents the semisaturation constant, is defined as the maximum instantaneous killing rate after the use of pesticides, and is the maximum natural enemy when executing the control strategy. The amount is the natural enemy density adjustment parameter. The nonlinear term shows a function of which decreases monotonically, and the maximum amount of natural enemy release does not exceed. The symbols with , respectively, represent the initial populations of pests and natural enemies and satisfy . In model (1), there always exist a stable centre and a saddle point which is unstable.
The special cases of the above model for different parameters were considered in [37–39]. The biological significance and main properties of the corresponding ODE model can be seen in [37]. In [38], Tian et al. extended the classic pest-natural enemy model with linear state-dependent control measures to a model with nonlinear state-dependent impulsive control tactics. In [39], the authors for the first time introduced and provided the concept of action threshold depending on the density of pest and its rate of change. They used the definition and properties of the Lambert W function to construct the analytical expression of the Poincaré map. Furthermore, by using the analytical properties of Poincaré map, the existence, uniqueness, and stability of the natural enemy free periodic solution and internal periodic solution were discussed in detail. The results explain the significance of nonlinear ratio-dependent AT in integrated pest control and the important guiding role in IPM strategy.
2.2. Properties of Action Threshold
The quantities and are dependent weighted parameters. If , then the ratio-dependent AT converts into ET. Therefore, we can say that ET is a special case of ratio-dependent AT for . Combining the first equation of ODE model (1) with ratio-dependent AT, we get
If we put , then the ratio-dependent AT converts into . In this case, if , then is bounded and reaches its highest value . Further, with the utilization of the control actions on , we get another curve . For , the curve changes into showing a vertical straight line. Let ; then, for convenience, we denote the two curves and by and , respectively, as shown in Figure 1.
(a)
(b)
3. Impulsive and Phase Sets
This section is devoted to present the dynamical aspects of the system (1), and we can use the Poincaré map on the sequence of pulse points which will be formulated later. Let be the abscissa of the curves at .
Then, we take the following cases based on the equilibrium and curve .
The necessary and primary component is to examine the section that is not used during the pulse effect process, which means that the trajectory initiating from cannot touch the curve in the case of maximum impulsive set. In the following part of the paper, we address the definition of impulsive sets.
3.1. Impulsive Set
In Case , the solution is tangent to the curve at point . If we denote the impulsive set by , then it can be written as
Now based on the corresponding horizontal coordinate, we search the exact value of in the following lemma. The point is actually the maximum value of the impulsive set for Case .
Lemma 1. For Case , the maximum impulsive set is defined as with
Proof. Let be a trajectory tangent at , and it touches the curve at point . Then, and must satisfy the following equation:Solving this equation for , we getwhere . The above equation obviously gives two solutions when we solve it by using Lambert W function. The minimum solution can be written as follows:which is well defined because .
For Case , it is clear from Figure 1(b) that at point , is tangent to the curve where . Then, taking into account the locations of equilibrium and the curve , we can write the maximum impulsive set for Case asThe above information shows that for this case, the tangent point with varies due to small changes in and .
If the weighted parameter decreases, then the quantity approaches its maximum value .
3.2. Phase Set
To determine the exact phase set of system (1) under different conditions, we need to know whether the solution from initial point reaches the corresponding impulsive set and whether the pulse action occurs or not. To provide the exact domain of phase sets, we first discuss the interval which is free of impulsive effect.
Lemma 2. For Case , any solution starting from the phase set with initial point (where ) will not reach the impulsive set , whereprovided that .
Proof. Assume that the closed trajectory starts from and touches the curve at point . Then, and must satisfy the following relationship:Rearranging this equation for , we getwhere . The above equation can be easily solved utilizing the Lambert W function approach which clearly will result in two solutions of the problem. The maximum solution can be written asThe value of can be found in the similar way as above, i.e.,with .
As a result, any solution curve initiating from with will be free from the effect of impulsive set.
For the case when , the trajectory shown by becomes tangent at . So, and becomeThe impulsive function described by satisfies some properties which are very important.
To do this, we indicateand then we get and at .
. From Lemma 1, we can describe the impulsive set as . Further, we can take three subclasses as follows.(i). For this subcase, for all , which shows that . Then, the corresponding phase set to can be expressed as with(ii). For this subcase, for , which denotes that . Then, the corresponding phase set to is expressed as follows: with(iii).For the present subcase, the impulsive set becomes , wherewith withHence, the corresponding phase set to the impulsive set = is , wherewith with. For this case, we express the impulsive set as follows.
. In order to give the exact domain of phase sets for Case , based on Lemma 2, we describe the following sets:The following three subcases can be taken based on the definition of the phase set.(i). For this subcase, for all values of belongs to . This shows that . The corresponding phase set to can be expressed as with (ii). For this subcase, for , which denotes that . Hence, the phase set corresponding to is given as with (iii).If , then and . If , then and .
The impulsive set is now can be explained in the form , wherewithandwithHence, the phase set corresponding to the impulsive set = can be expressed as , wherewith with For Case , if , then the solution from the phase set does not reach the interval . It is also important to note that if and , then . For Case , it can be seen from the vector field of system (1) that if the closed orbit is tangent or does not touch the curve , then there must be a trajectory that is tangent to the curve at a point , and the trajectory intersects the curve at lower point . This proves that the impulsive set in this case is defined by , as shown in Figure 1(b).
If the closed trajectory is tangent to at point and intersects the curve at two points, then it can be seen that for any solution from the phase set, it is impossible to reach the interval . The above theory shows that nonlinear terms of the controlling measure combined with nonlinear action threshold make impulse system (1) quite complicated, and it is very difficult to analyze each situation in detail.
4. Poincaré Map
Poincaré map [40–42] plays a very helpful role in examining the qualitative behavior of a dynamical system, most prominently the asymptotic stability of periodic or almost periodic orbits. Based on the impulse and phase sets discussed above, the following related theorem for Poincaré map can be obtained.
Theorem 1. For the impulsive points of model (1), the Poincaré map for Cases has the following form.(A):(B):where
Proof. Suppose that a trajectory initiating from repeats (finite or infinite) times pulse action. Let the points of the impulse set be represented by , and after the pulse action, the corresponding points of phase set are represented by . If and are on the same trajectory above, then the coordinates of the two points satisfy the following trajectory equation:Solving the above equation for , we getwhereand thereforeFrom above equation, we can see that the Poincaré map given in (47) depends on both the Lambert W function and the sign of .
Case. . If , then for , the above expressions defined in (9) and (10) are well defined. Further, if we define , then it is easy to prove that achieved its minimum value at . Therefore, for all and . This denotes that the Poincaré map defined relative to Case is (7).
For Case , if , then . From this, we obtain the following:This solution further simplifies as , and from Lemma 2 we know thatHence, in the same way, the Poincaré map domain for all remaining cases provided in Section 3 and Table 1 can be found. This finalized the proof.
5. Characteristics of Poincaré Map
To discuss the existence as well as the stability for the order-1 periodic solution of problem (1), we first analyze the different characteristics of Poincaré map for the above existing cases. For this, we define an important point which will be used in the following discussion. If , then after one time pulse, the corresponding impulse point can be presented as .
Theorem 2. The Poincaré map for Cases and provided in Table 2 satisfies different properties as follows:(A) and .(i)It shows increasing behavior on and decreasing behavior on for .(ii)It is increasing on and decreasing on for .(iii)It is decreasing on and and increasing on and for , where .(B) and .(i)It shows increasing behavior over the closed interval and decreasing behavior on for .(ii)It is increasing on and decreasing on for .(iii)It is decreasing on and and increasing on and for , where , .
Proof. Assuming that , the solution initiating from intersects the curve at . If and lie in one trajectory, then is established by and can be expressed as . The corresponding vector field relative of the system given in (1) confirms that the domain of consideration of Poincaré map for Case is defined by . Furthermore, for this case, the corresponding impulsive function has increasing behavior over the closed interval . Therefore, based on the definition of , it is increasing on and decreasing on . The function is decreasing upon in Case , which shows that is decreasing over the interval and increasing over the closed interval . For Case , is decreasing over and increasing upon . Therefore, is decreasing on and and increasing on and .
By using the same methods as above, we can prove that the monotonicities of the Poincaré map for Cases in Theorem 2 are true.
Lemma 3. If and , then the inequalityis fulfilled for the corresponding Poincaré map shown by .
Proof. Let a solution originate from , and it touches the curve at point . We assume that ; then,From (55), we getIf , then we get the inequalityLet ; then, if and if . The inequality is satisfied for all . We also know that and . Hence, we deduce that for all .
In light of the above explained properties of Poincaré map, the existence of the fixed point of Poincaré map for is discussed in following section.
6. Characteristics of Boundary Periodic Solution
In Section 4, the formula for Poincaré map has been attained. We will use this formula to study the existence of fixed point, where the fixed point is indicated as , satisfying , such as
For , we get the following equation from above:
If , the fixed point shown by of the respective Poincaré map becomes
This shows that if , , then every point is the fixed point of . If , , then (a fix point) of the fulfils
In this case, holds . Thus, we deduced that is a unique fixed point for system (1).
In the following result, we present the conditions of global stability for boundary order-1 periodic solution. To demonstrate it, we first discuss an important lemma [43, 44].
Lemma 4. The -periodic solution of systemis orbitally asymptotically stable if the Floquet multiplier satisfies , wherewithand is continuously differentiable corresponding to both . and are evaluated at , and , and (, is the set of nonnegative integers) is the time of the -th jump.
Theorem 3. If and , then the fixed point of Poincaré map is stable in the phase set. If and , then is globally asymptotically stable. If and , then it is unstable.
Proof. If , , then in the phase set is a fixed point of the Poincaré map . This case confirms the stable solution of the problem but is not asymptotically stable. We first show that when if and only if , and then boundary order-1 periodic solution exists for system (1). For , system (1) is converted into the subsystem given below:The first equation of the subsystem (14), combining with the respective initial condition shown as , where , gives us the solutionTaking the equation and evaluating it for , we get . This shows that T-periodic boundary order-1 solution exists for system (1) asNext, we show that is asymptotically stable. For this, we apply Lemma 4 and present the following.
.From the above, we getBased on the above information, the Floquet multiplier denoted by is defined asIf and , then we get . This indicates that for the problem described in (1), the boundary order-1 periodic solution is orbitally stable asymptotically. If , the sequence of pulse points is increasing strictly and additionally will be free from more pulse action only after limited time pulse effects.
. The asymptotic stability of boundary order-1 periodic solution can also be discussed directly from Poincaré map portrayed in (47). Let ; then,Taking the derivative of (72), we getThe boundary order-1 periodic solution is stable . By utilizing the limit of , we getThis denotes that if , then for , and hence is asymptotically stable.
In the following, we show the global attractivity of the boundary order-1 periodic solution . Let and be the points of the same trajectory; then,Let ; then, from (75), it is clear that . If , then . This indicates that if , then is monotonically increasing.
If, then . Since , the inequality becomes . This shows that . Therefore, if, then the impulsive sequence is monotonically decreasing and . These kinds of information affirm that the boundary order-1 periodic solution is globally attractive. In the same way as above, we can prove that if , then . Therefore, the sequence will be free from impulsive effect after finite time pulse actions, as shown in Figure 2(b). Hence, from all the above outcomes, it can be concluded that if , then the boundary order-1 periodic solution, i.e., , is globally asymptotically stable.
The numerical calculation in Figure 2(a) shows that if , then the boundary order-1 periodic solution is stable while Figure 2(b) confirms that if , then it is unstable.
(a)
(b)
7. Existence of Order-1 Periodic Solution
In this section, we will discuss and analyze the order-1 periodic solution for system (1) when .
Theorem 4. For Case, the fixed point of Poincaré mapexists, and therefore an order-1 periodic solution exists for system (1).
Proof. For Case , the trajectory is tangent to the curve at point and intersects the curve at lower point . If , then the curve forms an order-1 periodic solution for system (1).
For Case , if or , then the solution originating from the point touches the curve at a point with . The pulse action is applied and the point maps to a point , and . For Case , is increasing on . Therefore, satisfies the inequalityThe point being the lowest impulsive point satisfiesInequalities (17) and (18) confirm that a fixed point of the Poincaré map exists, and therefore an order-1 periodic solution exists for system (1).
For Case , is decreasing on . If or , we getMoreover, the highest impulsive point is , and we getInequalities (19) and (20) confirm that there exists a fixed point for the Poincaré map, and therefore an order-1 periodic solution exists for system (1). This completes the proof.
Theorem 5. For Case, the fixed point of Poincaré mapexists, and therefore an order-1 periodic solution exists for system (1).
Proof. If , then the curve forms an order-1 periodic solution for the problem given in system (1). If , then the following two cases are taken into consideration.For Case (1), if , then we can writeAs is the lowest impulsive point, it satisfiesThus, inequalities (21) and (22) confirm that we can find a fixed point of Poincaré map .
If , then we can writeMoreover, if is the least impulsive point, then it leads to the following:Thus, the above two inequalities (83) and (84) confirm that there exists a fixed point of Poincare map .
For Case (2), if , then . On the other hand, if the highest impulsive point is , then . The above two inequalities affirm that there exists a fixed point of the Poincaré map .
If , then . Moreover, as is the least impulsive point, we get . It confirms that there exists a fixed point for the map shown by , and hence an order-1 periodic solution exists for system (1).
Theorem 6. For Case, if, then the fixed point of Poincaré mapexists, and therefore an order-1 periodic solution exists for system (1).
Proof. For Case , we know that there exists a curve , which is tangent to at point and intersects the curve at two points and . If , then the curve forms an order-1 periodic solution for the problem stated in (1).
Further, for Case , if , then the point demoted by lies above the point , and we getIn addition, the solution initiating from the point meets the curve at a point which lies below the point , i.e., . As is increasing on , we have , i.e., . All the above results affirm that the Poincaré map for Case satisfiesInequalities (25) and (26) confirm that a fixed point in will exist. Hence, an order-1 periodic solution exists for problem (1).
If , then after a one time impulsive effect, the solution will directly map to the interval . Thus, if , then according to inequality (1), any trajectory originating from with will intersect the curve and experience a limited time of pulse actions and at last enter into Int and will be free from more pulse action. If , then each solution curve of problem (1) will map to the Int after a one time impulsive effect. Hence, if , then a fixed point does not exist.
For Case , if , then . We also know that the function is decreasing on . So, the solution initiating from will map to the interval after a one time impulsive effect. Therefore, the trajectory originating from the point will satisfy . From the above inequalities, it follows that the fixed point exists in the interval .
Theorem 7. For Case, if, then the fixed point of Poincaré mapexists, and therefore an order-1 periodic solution exists for system (1).
Proof. If , then for system (1), the curve forms an order-1 periodic solution. If , then we consider the following two cases.For Case (1), if , then . Moreover, according to the exact domain of the Poincaré map , the impulsive point of lies below the point , i.e., for . Therefore, inequality is true, which shows that the fixed point exists in the interval .
If , then applying the same techniques as those given in Theorem 6, it can easily be shown there must exist a finite number of pulse effects for any solution of system (1). Furthermore, the solution enters into Int and becomes free from more pulse actions.
For Case (2), if , then holds true. We also know that the highest impulsive point is because . Therefore, we get , and hence the theorem is true.
If , then any trajectory of system (1) tends into Int only after finite pulse effects. This completes the proof.
8. Stability of Order-1 Periodic Solution
The monotonicities of Poincaré map and existence of its fixed point were discussed in previous sections. Now, based on these, we will discuss the stability of fixed point of Poincaré map for system (1).
Theorem 8. For Case , if the fixed point of Poincaré map is unique and one of the following two conditions is satisfied, then the corresponding fixed point denoted by is stable globally.(a)If.(b)Ifandfor.
Proof. From Theorem 4, we know that for Case , the fixed point of Poincaré map exists. Let the fixed point be unique; then, the global stability can be discussed as follows:(a)If , then for all . This means that as increases, increases monotonically and satisfies . If , then we take two cases. (1) If , then according to the relation , decreases monotonically, i.e., for all and we get . (2) If , then and . Therefore, the conclusion in is true.(b)If , then we take three intervals: (1) ; (2) ; (3) . For interval (1), since and Poincaré map is monotonically decreasing in this interval, it is easy to get . At the same time, by using the second condition , we get . This means that for all , . This shows that increases monotonically, and .For intervals (2) and (3), using the same method as those in (1), we can prove that there must exist such that , and hence the fixed point of Poincaré map is globally stable under conditions (2) and (3). This completes the proof.
Theorem 9. For Case , if the fixed point of Poincaré map is unique and one of the following two conditions is true, then is globally stable.(a)If .(b)If and for .
Proof. Theorem 4 shows that for Case , there exists a fixed point of the map . Assuming that the fixed point is unique, we have the following conclusions regarding its stability:(a)From Theorem 2, it is clear that the Poincaré map is monotonically increasing in the interval and monotonically decreasing in the interval . If , then the fixed point satisfies for any and increases with the increasing value of such that for all . decreases as increases, and . For all , there is ; therefore, or . In summary, the only fixed point is globally stable.(b)The Poincaré map is monotonically decreasing in the interval , and for , the condition is satisfied. So, it is easy to get . By induction, there is a relation for all . This shows that monotonically decreases with increasing value of , and . In addition, for all , there must exist such that , and hence .
Theorem 10. For Case , if the fixed point is unique and one of the following conditions is true, then it is globally stable.(a)If .(b)If , and for all .(c)If , , and , for when .(d)If , , and .
Proof. Theorem 5 shows that there exists a fixed point of Poincaré map for Case . Moreover, if is unique, then its global stability can be described as follows:(a)If , then we take three intervals: (1) ; (2) ; (3) . For all , we get . The Poincaré map is monotonically increasing in the interval , and . By induction, we get for all , which means that monotonically increases as increases, and , . For all , we get . From the monotonicity of , we have , which means that decreases with increasing value of and for all . For all , it is easy to get , and according to the previous conclusion, we get . Therefore, the result in Case is true.(b)If , then we take two cases: (1) ; (2) . For all and according to the monotonicity of the Poincaré map, satisfies . From this, it is easy to get . By induction, the inequality for all holds, which means that as increases, the mapping monotonically decreases, and for all . For all , there exists , such that . From this, we get for all . All the above conclusions indicate that Case is true.(c)We again take two conditions: (1) ; (2) . For all , the Poincaré map is monotonically decreasing, and the inequality is satisfied. We can easily get the relationship , and by induction, for all . This means that as increases, the mapping monotonically decreases, and . For all , there must exist , such that . Therefore, we get for all , which means that Case (c) is true.(d)If the conditions given in statement are satisfied, we consider two intervals: (1) ; (2) . If , then according to the monotonicity of the Poincaré map , monotonically increases as increases, and . If , then monotonically decreases as increases, and . For all , it is easy to know that there must exist a positive integer , such that , and at the same time, or . Hence, the Case (d) is true.
Theorem 11. For Case , if , then the fixed point of Poincaré map is globally asymptotically stable provided that for all .
Proof. From Theorem 6, we know that for Case , a fixed point of Poincaré map exists.
According to the inequality given in Lemma 3, for all . At the same time, the inequality is satisfied. So, the fixed point does not lie in the interval . This shows that the unique fixed point belongs to the interval .
If , then from the monotonicity of the mapping , we get . By applying the inequality for all , we get . By induction, there exists a relationship for all . This means that as increases, increases monotonically, and hence .
Theorem 12. For Case, if, then the fixed point of Poincaré mapis globally stable.
Proof. From Theorem 6, there exists a fixed point of Poincaré map for Case . Using the same method as in Theorem 11, there is no fixed point on the interval , and is located in the interval . Moreover, under the uniqueness of , the global stability can be described as follows.
For Case , the Poincaré map is monotonically decreasing in the interval and monotonically increasing in the interval . If , then according to the relationship , it is obvious that increases monotonically towards as increases, i.e., . For all , according to the relationship and properties of Poincaré map , we know that monotonically decreases with the increasing value of , and .
If , then there must exist some such that , and therefore . Hence, the result in Theorem 12 is correct.
Theorem 13. For Case , if , then the unique fixed point of Poincaré map exists. If one of the conditions (a) and (b) given below is true, then is globally stable.(a)If .(b)If , and for all .
Proof. For Case , if , then from Lemma 3 and Theorem 7, we know that Poincaré mapping has at least one fixed point belonging to the interval . Under the uniqueness of , the global stability can be demonstrated as follows:(a)If , then only exists in the interval . From Theorem 2, we can see that Poincaré map is monotonically increasing in the interval . For any , we get , which shows that for increases monotonically, and . For any , we get the relation . Therefore, from the monotonicity of , monotonically decreases with increasing value of , and we get . For all , it is obvious that there exists an integer , such that . Hence, for all , we get . All these results show that if , then the unique fixed point of the mapping is globally stable.(b)If , then combined with the inequality given in the statement, it is clear that there exists only one in the interval . The mapping monotonically decreases in the interval , i.e., for all , we have . In addition, by applying the condition , we get . Hence, we get for . This shows that monotonically increases with the increasing value of and for all .If and , then there must exist such that . By using the same way as above, we get for all .
Therefore, if and for all , then the fixed point is globally stable.
9. Conclusions
The IPM strategy is a dynamic management system. From a mathematical perspective, this is actually an optimal control problem under multiple objectives. The IPM approach’s purpose is to monitor the number of pest populations in real time and decide whether to implement a control strategy based on the size of the population. The state-dependent impulsive differential equation [20, 45–47] is needed to truly characterize the IPM strategy and the dynamic evolution of pest-natural system. Moreover, in recent years, researchers have proposed a variety of state-dependent pest-natural enemy feedback control systems.
The change rate of pest population plays an important role in state-dependent prey-predator ecological system. There are two fundamental circumstances in the previous studies which require high attention. First, the pest population is comparatively high and the change rate is little; second, the population of pest is small, but the change rate is high. A crucial issue illustrated by these two situations is that when the pest population is large, the growth rate is small or even negative at this time. In this case, even if the IPM strategy is not implemented, the number of pests may not exceed EIL. Another situation is that although the number of pests is not large, the growth rate of the pest population is very large. If the control strategy is not implemented in time, it may lead to a large outbreak of pests. Next, the IPM process needs precise checking of the pest populations, and consequently suitable integrated control strategies can be prepared. The pest killing rate should be a function of their density, whereas the releasing quantity of natural enemies should be a function of their density. Based on this, a feasible new nonlinear state-feedback system with nonlinear ratio-dependent AT is proposed.
The use of nonlinear pulse as state-dependent feedback control with nonlinear ratio-dependent AT is more reasonable and closer to reality in a biological sense, but the impulsive model becomes very difficult because of the existence of two population quantities in the control actions. By including the densities of pest and its natural enemy in controlling measures, we can develop the pest control model based on the practical importance according to the growth direction of agriculture and forestry. Corresponding analytical techniques and numerical methods were developed, the dynamic behavior of the system was examined, and the important role of the main conclusions in integrated pest control was given.
To avoid the complexity, in this paper, we proposed the simple Lotka–Volterra impulsive mathematical model. Our aim is to reveal how nonlinear pulse control with nonlinear ratio-dependent AT affects the whole dynamics and concentrate on the biological implications. The definition and properties of Poincaré map for phase-concentrated pulse points in various cases are discussed and studied. The existence, uniqueness, and global stability of boundary and interior periodic solutions of order 1 for model (1) are analyzed by using the definition of Poincaré map. In the present paper, some basic techniques were used for the qualitative analysis of nonlinear pulsed model with nonlinear ratio-depended AT, which can be widely used in the study of feedback control systems with critical conditions, such as the blood glucose-insulin regulation system.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Basem Al Alwan would like to thank the Deanship of Scientific Research at King Khalid University, Abha, K.S.A., for funding this work through a research group program under grant no. RGP.2/204/42.