Research Article | Open Access
The Impulsive Model with Pest Density and Its Change Rate Dependent Feedback Control
The idea of action threshold depends on the pest density and its change rate is more general and furthermore can produce new modelling techniques related to integrated pest management (IPM) as compared with those that appeared in earlier studies, which definitely bring challenges to analytical analysis and generate new ideas to the state control measures. Keeping this in mind, using the strategies of IPM, we develop a prey-predator system with action threshold depending on the pest density and its change rate, and study its dynamical behavior. We develop new criteria guaranteeing the existence, uniqueness, local and global stability of order-1 periodic solutions. Applying the properties of Lambert function, the Poincaré map is portrayed for the exact phase set, which is helpful to provide the sufficient conditions for the existence and stability of the interior order-1 periodic solutions and boundary order-1 periodic solution, also confirmed by numerical simulations. It is studied in detail that how and under what conditions the fixed point of Poincaré map and its stability are affected by the newly introduced action threshold. The analytical methods developed in this paper will be very beneficial to study other generalized models with state-dependent feedback control.
The risk of pests to agricultural productions may be an enormous issue over the world, which makes pest control being a motivating topic and attracts great attention to the development of effective pest management strategies. Pests will cause vital crop yield declines, even colossal failure. Additionally, they will downsize the standard of farm items. Therefore, countries around the world have established special organizations to review the management procedure of agricultural pests [1–7].
Integrated pest management (IPM) is a useful methodology in prevailing pests that have been demonstrated to be more practical than the classic strategies both experimentally [8–10] and theoretically [11, 12]. It is a procedure that is used to solve pest problems while minimizing threats to individuals and the environment. IPM can be utilized to deal with all sorts of pests anywhere in rural, urban, and natural areas or wild land. IPM is an ecosystem-based approach that concentrates on long-term prevention of pests or their damage through a combination of strategies, such as biological control, adjustment of social practices, living space control, and utilization of safe assortments. The objective of IPM is not to eliminate pests, rather to manage the amount of the pests below an associated economic threshold (ET) and ensure ecosystem up to maximum level.
Recently, many researchers have proposed impulsive differential equations to examine the dynamics of pest control models [13–18]. Impulsive equations have been brought into population dynamics in relation to impulsive vaccination, chemotherapeutic handling of disease, population ecology, and impulsive birth. Especially, some impulsive differential equations have been presented effectively in population dynamics (agriculture or fishing) and epidemic dynamics. Numerous recent articles have mathematically exhibited a variety of IPM tactics using impulsive differential equations, for example, stage structure in the predator species and periodically changing environmental conditions . Relative models also have been studied in .
Most of the researchers considered systems with impulses at fixed moments [21–29]. The shortcomings of this kind of systems are that they did not pay enough attention to the management cost and the growth rules of the pest. Impulsive differential equations with impulses happening at fixed time emerge in the modelling of real-world phenomena in which the state of the inspected procedure fluctuates at fixed moments of time. In literature , the author extended a model with linear impulsive control tactics to a model with nonlinear impulsive control measures, which revealed further precise conditions for pest control. Wang et al.  discussed the threshold condition which guarantee the existence and stability criteria for the pest-free periodic solution. In addition, the complex dynamics for system is discussed when the forward and backward bifurcations could happen once the pest-free periodic solution becomes unstable.
State-dependent feedback control approach is generally expressed by an impulsive semi dynamical system, and they can be portrayed in comprehensive terms in real biological problems. For example, control tactics (i.e., pesticide application, harvesting, treatment, etc.) are applied only when a particular species size ranges an earlier known threshold density. Specifically, in [32–35] an excellent example in the series of models encouraged by IPM has been framed and examined. In [36–38], IPM has been exhibited by experiments, and it is demonstrated that IPM is more effective than classical methods.
In all the previous literatures, researchers projected models either with a single economic threshold or multiple thresholds [39–45]. There are few drawbacks to this sort of thresholds. However, there are two reasonable circumstances: one is that the number of the pest population is comparatively large, but its change rate is quite small; the other is that the number of population is small, but its change rate is significantly high. The latter case is more obvious at the initial stage of the occurrence of the pest. To overcome these drawbacks, we planned to take the model with action threshold depending on the pest density and its change rate (so-called ratio-dependent AT), and investigate its global dynamics.
The paper is ordered as follows: In Section 2, the commonly used generalized prey-predator model is proposed and the new ratio-dependent nonlinear action threshold is introduced. In Section 3, the exact impulsive and phase sets are determined for all existing cases. In view of the impulsive and phase sets, the Poincaré map is constructed in Section 4. In Section 5.1, some important relations and lemmas are provided that are very important for the next sections. The boundary order-1 periodic solution is given in Section 5.2. In Section 6, the global properties of system constructed in Section 2 are discussed, including the existence, local and global stability of order-1 periodic solution, and the effect of weighted parameters on the fixed point of Poincaré map. In the same section, the effect of weighted parameters on the different cases is also discussed. To sum up the whole work, a detailed conclusion is given in Section 7.
2. Construction of Model and Main Properties
In view of the reasons specified above, we consider the commonly used prey-predator system with pest density and its change rate dependent feedback control, i.e., the action threshold depends not only on pest density but also on its change rate, which can be modeled by
where , and are all positive constants with . and respectively, represent the quantities of prey and predator. denotes the intrinsic growth rate of the pest population and demonstrates the carrying capacity. The pest population dies at the rate of and is predated by the predator population at a rate . The quantity , which is actually a saturating function of the present quantity of pest, is the expand rate of predator response. The prey population breakdowns the predator response at a rate , and represents the decay rate of the predator in the absence of prey. The quantities and are known as the controlling quantities; whenever the pest population touches the action threshold, the management activities are adapted and the quantities of prey and predator are adjusted according to the controlling actions and respectively. Therefore denotes the instant killing rate and represents the releasing constant.
If the value of carrying capacity , then for model (1) is reduced to the following form
The quantities and are dependent weighted parameters. It is interesting to note that if the second weighted parameter disappears, the ratio-dependent will transform into ET [41–45]. Therefore, the ET is an exceptional case of ratio-dependent for . From ratio-dependent and the first equation of model (2), it follows that with
If the weighted parameter vanishes, the ratio-dependent AT transformed into . It is obvious that if again pest population tends to infinity, the predator population is bounded and approaches its maximum value . By applying the controlling quantities on , we get another curve . For , the curve transforms into the vertical straight line .
For convenience, we denote and by and respectively, as shown in Figure 1. is the initial value which curve attains at . At this point, the vertical coordinate with takes the value . If approaches one, then approaches and hence in this case, the vertical coordinate with attains the value . It is an essential assumption that the initial value must satisfy .
Our main objective is to discuss the global dynamics of model (2). We will see how the global dynamics are affected if the threshold is not a straight line but complex curve. For the first two equations (i.e., the ODE system without control measures), there always exist trivial equilibrium and two interior equilibria
provided that and . If , then the two roots will coincide with each other. It is also clear that is the centre and is a saddle point.
where AT signifies the economic threshold level, i.e., action threshold transforms into ET.
The special case of model (2) for and is
3. Impulsive and Phase Sets
In this section, we will find out the exact impulsive and phase sets for the existing cases. The foremost and necessary part is to search out the segment that is free from impulsive effect, i.e., the solution starting from cannot reach to curve for maximum impulsive set. Based on the positions of equilibria , and curve , we take the following three cases:
3.1. Impulsive Set
In Case , trajectory is tangent to curve at point with . If we represent point by , then the domain of the impulsive set becomes as:
It is obvious from the domain of impulsive set that in this case, no solution originating from the phase set will reach the interval . In the following lemma, we find out the exact value of which depends on the corresponding horizontal coordinate.
Lemma 1. For Case , the impulsive set is defined as . The maximum vertical coordinate of is , where provided that .
Proof. Let a solution is tangent to curve at point , and it touches curve at point . Then these points must satisfy the following equationSolving this equation for , we getwhere . We can solve the above equation with the help of Lambert W function. Obviously, the above equation will give us two solutions, but only the minimum value lies on curve as well as . If we denote it by , we obtainwhich is well defined due to .
From Figure 1(b), it can be seen that for Case , is tangent to curve at point , where . Then based on the positions of equilibria , and curve , we discuss the maximum impulsive set for this case as follows:The tangent point of the closed trajectory with curve varies with the changing estimations of weighted parameters and . From the domain of impulsive set , it is obvious that if then the interval cannot be used for any solution originating from the respective phase set.
Now we discuss the impulsive set for the Case . This case is more crucial than the previous cases. In this case, homoclinic trajectory exists. This homoclinic trajectory touches curve at points and , and its lower right branch touches curve at point (as shown in Figure 1(c)). This is actually the maximum impulsive point for Case . Before finding out the exact value of vertical coordinate , we first provide some necessary quantities which are not only helpful for finding the maximum vertical coordinate of the impulsive set , but also assume a significant role in finding the fixed point of the Poincaré map . These quantities are listed as follows:Replacing by in equations (15) and (16) and denoting the resultant equations by and respectively, then = . If we denote by , then we find the exact value of which depends on the respective horizontal coordinate.
Lemma 2. For Case , the impulsive set is defined as . The maximum vertical coordinate for this is , where provided that .
Proof. In this case, the lower right branch of the homoclinic trajectory touches curve at point . Combining point with must satisfy the following relation:which can be simplified asSolving the above equation for , we getFollowing the same way as in Lemma 1, applying the properties of Lambert W function, we get two solutions. From Figure 1(c) it is clear that only the minimum value lies both on curve and . If we denote it by , then we getwhich is well defined due to . If we represent the impulsive set by , then it can be expressed as
3.2. Phase Set
In this subsection, we aim to discuss the phase sets for all the existing cases expressed above. The most essential and tough task in the process of discussing phase sets is to find out the segment, which is free from the impulsive effect. To find the exact domain of phase sets, we provide the following intervals:
For Case , trajectory is tangent to curve at point . Thus, the corresponding phase set to the impulsive set can be expressed as:
For Case , the closed trajectory is tangent to curve at point , where . We indicate the intersection point of the closed trajectory with line (denoted by ) as . If we denote by and by , then the phase set corresponds to the impulsive set can be expressed as follows:
From the phase set , it is clear that the solution initiating from the interval will be free from impulsive effect. In the following lemma, based on the respective horizontal coordinates, the exact values of and are given.
Lemma 3. For Case , the impulsive set is defined as . In this case, any solution initiating from will be free from impulse effect, whereprovided that .
Proof. Suppose that the closed trajectory originates from , and tangent to curve at point . Then, these points must satisfy the relation:Rearranging this equation for , we getwhere . The above equation can be solved with the help of Lambert W function. If we denote the maximum solution by , we getThe value of , denoted by can also be found by using the same method as above, i.e.,with .
If weighted parameter , i.e., the threshold level only depends on the pest density then the closed trajectory becomes tangent at . In this case, and become aswith .
For Case , let us denote the intersection of the homoclinic trajectory with line (denoted by ) as . Trajectory touches curve at upper point and lower point , denoted by and respectively. In the following lemma, we find the exact values of and .
Lemma 4. For Case , the impulsive set is defined as . In this case any solution initiating from will be free from impulse effect, whereprovided that .
Proof. Suppose that the homoclinic trajectory touches curve at upper point . Combining point with must satisfy the following relation:Arranging this equation for , we getwhere . The above equation can be solved with the help of Lambert W function. If we denote the maximum solution by , we getThe value of , denoted by can also be found by following the same way as above, i.e.,with .
If we represent the phase set for Case by , then it can be expressed aswithand
If lies on the left and does not touch or then the impulsive and phase sets will be transformed into and respectively.
3.3. The Impulsive and Phase Sets for Model (8)
In the first case, trajectory is tangent to curve at point with (as shown in Figure 2(a)). If we represent point by , then the impulsive set can be expressed as
To discuss the exact domains of the phase set for both cases, we define the intervals , , and . Then, the phase set for Case becomes as:
In the following lines the analytical value of is given. The proof of the lemma is the same as previous section, so we omit it.
Lemma 5. For Case , the impulsive set is defined as . The maximum vertical coordinate for this is , where provided that .
For Case , we denote the intersection point of the closed trajectory with line (denoted by ) as . In this case, the closed trajectory is tangent to curve at point , where . If we denote by and by , then based on the positions of curves and , we discuss the impulsive and phase sets as follows:
From the phase set, it is clear that the solution initiating from the interval will be free from impulsive effect.
Lemma 6. For Case , the impulsive set is defined as . In this case any solution initiating from will be free from impulse effect, whereprovided that
The proof of the Lemma 6 can also be shown as previous section, so we also omit it. If the weighted parameter , i.e., the threshold level only depends on the pest density then the closed trajectory becomes tangent to curve at . In this case, and become as:
In the upcoming discussions, for convenience, we use rather than , , or . For Case , is equivalent to . For Case , and to avoid the complexity, we will focus only on . Similarly, for Case it will be more convenient to denote and by , and by , and by .
4. Formation of Poincaré Map
Theorem 1. The Poincaré map for the impulsive points of model (2) can be defined as follows:where
Proof. Assume that a trajectory originate from and repeats the pulse action times, which can be finite or infinite. Let and be two points of the same trajectory. Then for these points, the following relation can easily be obtained:Applying the properties of Lambert W function and solving the above equation for , we getwhereFrom (58), we get Case: If , then for , equations (58) and (60) are well defined. Actually, if we define thenand it can easily be shown that achieves its minimum value at point . Therefore, for all and . This shows that the Poincaré map is defined by (53) for Case .
Now in order to demonstrate the exact domains of the Poincaré map for Cases and , the most important part is to find the section of the phase set that is free from impulsive effect, i.e., the solution originating from cannot reach to point . Case: From Lemma 3, it is obvious that if the starting point lies inside of the closed trajectory , then trajectory cannot reach to curve . This indicates that points and cannot lie in the same trajectory, as shown in Figure 1(b). From Lemma 3, it also follows that in this case, we have and we need .If , then . From this, we getThe solution gives , and from Lemma 3 we know thatThe Case can be attained directly from the domains of the Poincaré map and applying the proof of Lemma 4. For this case , regardless of or (as shown in Figure 3), the Poincaré map is given by the case (55). This completes the proof.
Difference equation (56) which explains the Poincaré map reveals the relations between the impulsive points and , so the existence and stability of fixed point of equation (56) indicate the existence and stability of order-1 periodic solution of system (2). Therefore, we conclude that the properties of the Poincaré map play an essential role in exploring the impulsive semi-dynamical system.
Corollary 1. The Poincaré map for model (8) can be defined as:
5. Characterization of Periodic Solution for
In this section, we will focus on the boundary order-1 periodic solution for system (2). To prove this, we first provide some significant relations and lemmas in the following subsection.
5.1. Some Important Relations and Notations
In view of the domains of the Poincaré map characterized in Section 4 or the signs of and , we modify the Cases – as:
This shows that the sign of is crucial for coming analysis. So, while choosing the parameters, we should be very careful. If we change the value of weighted parameters, the sign of not necessarily remains the same.
The fixed point of the Poincaré map can be found directly from the analytical formula of the Poincaré map derived in Section 4. To do this, let
By applying the properties of Lambert W function, we get
This demonstrates that there exists a unique fixed point
Whenever weighted parameter turned out to be zero, the closed trajectory becomes tangent at with extreme vertical coordinate. In this case, if the fixed point exists then it must belong to the basic phase set . We will demonstrate that under what condition the fixed point of the Poincaré map belongs to the maximum phase set . We have the following two positions for , i.e., (i) and (ii) . If , then it can easily be shown that and furthermore, the inequality
holds true. This shows that if , then . If , then the fixed point exists provided that . This also ensures that is positive and greater than . We take
which is equivalent to
The following inequality can easily be obtained after simple rearrangement
Solving inequality (74) with respect to gives either or . The first inequality is impossible due to . This shows that , and hence when .
If weighted parameter , the tangency point of the closed trajectory moves to some other point having vertical coordinate less than , and in this case, the basic impulsive set can be written as or , where . This shows that or when