Dynamical Analysis of Biological Systems 2021View this Special Issue
A State-Dependent Impulsive Nonlinear System with Ratio-Dependent Action Threshold for Investigating the Pest-Natural Enemy Model
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.
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.  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.  used an impulsive hybrid nonlinear system to construct a passive biped robot model that demonstrates complicated behaviors. In , 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  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) . 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 . In , 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 , 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.
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).
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 get