Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control

In this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré map are investigated in detail which play important roles in the proofs of the existence of limit cycles, and it is concluded that the definition domain of the Poincaré map has a complicated shape with discontinuity points under certain conditions. Subsequently, the existence of the boundary order−1 limit cycle is discussed and it is shown that this limit cycle is unstable. Furthermore, the conditions for the existence and stability of an order−1 limit cycle are provided, and the existence of order−k(k ≥ 2) limit cycle is also studied. Moreover, numerical simulations are carried out to substantiate our results. Finally, biological implications related to the mathematical results which are beneficial for successful pest control are addressed in the Conclusions section.


Introduction
Since Lotka-Volterra systems have made great efforts on the mathematical models of predator-prey interactions, many studies were carried out to develop predator-prey systems with the aim of solving problems originating from real world phenomena.In particular, one of the most important and famous biological models named the Holling-Tanner model or known as the model of R. M. May has become a hot topic and has been studied by many scholars [1][2][3][4][5][6][7]; it can be described by the following differential equations: where (), () are the densities of the prey and predator populations at time , respectively. represents the intrinsic growth rate of the prey,  represents the carrying capacity of the prey,  is the maximal predator per capita consumption rate, that is, the maximum number of preys that can be eaten by a predator in each time unit,  is the number of preys necessary to achieve one-half of the maximum rate ,  represents the intrinsic growth rate of the predator, and ℎ is a measure of the food quality that the prey provides for conversion into predator births.Note that the dynamics of system (1) have been investigated by many scholars [1][2][3][4][5][6][7].
Naturally, the () of system (1) usually denotes the pests population; integrated pest management (IPM) is needed to be implemented in order to control the pest population within a safe range [8][9][10], where IPM includes biological control, chemical control, or their combinations.Furthermore, biological control often consists of releasing enemies, harvesting, and catching, etc., whilst chemical control involves spraying pesticides [11,12].
To investigate the global dynamics of the predator-prey systems concerning IPM and to further explore how IPM affects the corresponding successful control strategies, the predator-prey systems with impulsive control strategy are commonly proposed to model the IPM with releasing natural enemies and spraying pesticide at different fixed periods [11][12][13].In these studies, the permanence, the stability of the pest-free periodic solution, and the conditions for the coexistence of pest and natural enemies are addressed.Although the applications of IPM at fixed times can achieve the purpose of pest control, many negative effects have been detected.For example, the overuse of pesticides has resulted in the enhancement of drug resistance, environmental pollution, and cost increases, etc.
In practice, state-dependent feedback control is proved to be more reasonable to depict problems originating from real world phenomena than fixed time pulses [14], and it is often described by using impulsive dynamical systems, which have received a lot of attention [14][15][16][17][18][19][20][21], and revealing that the control tactics should only be applied once the states of the model reach a prescribed given threshold.However, none of the authors expanded the system (1) to include the effects of state-dependent feedback control with IPM owing to the complexity of the system (1).Therefore, the main subjects are to investigate the model (1) with effects of IPM by considering the impulsive strategy; these modifications derive the following model: where  > 0 represents the fatality rate for the prey due to chemical control,  > 0 and  > 0 denotes the release number of ().Denote (0 + ) and (0 + ) as the initial densities of () and ().In this paper we assume that (0 + ) is always less than  for biological implications.When the number of preys reaches  at time , then control strategies are initiated and the number of () and () becomes (1 − ) and (1 + )() + , respectively.Note that a more general case of system (2) has been studied by Nie and coauthors without concerning the dynamics of system (1) [22].In this paper, we will present novel analytical methods to study system (2) based on the dynamics of system (1); we will not only provide exact domains of the phase sets and impulsive sets when system (1) exhibits different dynamical behaviour, but also discuss the main properties of the Poincaré map, in addition to the existence of an order−3 limit cycle, which are different from reference [22].The paper is arranged as follows: we introduce many important definitions and lemmas of the planar impulsive dynamical systems in Section 2. In Section 3, we first construct the Poincaré map and then the complex properties of the Poincaré map are discussed.Further the conditions which guarantee the existence of the boundary order−1 limit cycle are obtained, and then it is concluded that this limit cycle is unstable.Subsequently, the existence and stability of an order−1 limit cycle will be addressed, and the existence of order−( ≥ 2) limit cycles is also studied.In Section 4, the complex domains of impulsive set and phase set are provided for system (2) and many interesting results are indicated.Moreover, numerical studies are employed not only to verify the results but also to reveal the complexity of system (2).Finally, some biological implications of the results are discussed and some conclusions are presented.

Preliminaries and Main Properties of System (1)
The generalized planar impulsive semidynamical systems with control are usually described by (, ) ∉ I, where (, ) ∈  2 ; we denote  + = ( + ) and  + = ( + ) for simplicity, and , , ,  are continuous functions from  2 into ; I ⊂  2 represents the impulsive set.For each point (, ) ∈ I, the map  :  2 →  2 is defined via and  + is called an impulsive point of .
Let P = (I) be the phase set (that is, for any  ∈ I, () =  + ∈ P), and P ∩ I = Ø.In the following some definitions related to impulsive semidynamical systems will be listed briefly, which are used in this work.

Lemma 4. (i)
If () ≥ 0, then  * is globally asymptotically stable in the interior of the first quadrant.
(iii) Under certain conditions (for example, 0 <  * <  1 and  1 > 0, for details see [3,7]), then  * is locally stable and system (1) has two limit cycles with the outermost being stable and the innermost being unstable.
In the light of the above Definitions and Lemmas, we next focus on the constuctions of the Poincaré map and the global dynamical behaviours of system (2).

Poincaré Map and Order−𝑘 Limit Cycle
In order to study the dynamics of system (2), the Poincaré map which is determined by the impulsive points in the phase set needs to be constructed first.
To define the impulsive semidynamical system for model (2), the exact domains of impulsive sets and phase sets should be addressed.To this end, based on the positions between the threshold  and the equilibrium  * , we consider the following two cases: For case ( 1 ),  3 and  4 are both located to the left of the equilibrium  * .According to the vector fields of the model (2), any solution initiating from the line  3 will reach the line  4 in a finite time.Then the impulsive set I for system (2) can be determined as follows: which is a closed subset of  2 .Moreover, define the continuous function  as follows: So the phase set P can be defined, where with  0 = [, (1 + )  + ].Therefore, based on the above analysis model ( 2) defines an impulsive semidynamical system (, Π; I, ).
For case ( 2 ),  4 is located to the right of the equilibrium  * , while the locations of  3 could lie on the left (or right) of the equilibrium  * .According to Lemma 4, system (2) could possess a global stable equilibrium  * , or unique stable limit cycle Ω, or two limit cycles with the outermost Ω 2 being stable and the innermost Ω 1 being unstable under different set of conditions.Thus, any solution initiating from ((1 − ),  + 0 ) with  + 0 > 0 will experience infinitely many pulses or will be free from impulsive effects, depending on the initial conditions.For example, for case (i) of Lemma 4, there exists a curve Λ 2 which is tangential to the line  4 at a point (,   ), and the curve Λ 2 must intersect the line  2 at a point  1 (  1 ,   1 ) such that Λ 2 is tangential to the line  2 at this point.If (1 −  0 ) <   1 , then any solution initiating from ((1 − ),  + 0 ) with  + 0 ∈  0 experiences infinitely many pulses.If (1 −  0 ) ≥   1 , then the curve Λ 2 must intersect the line  3 at two points, denoted by  1 = ((1 −  0 ),    ) and  2 = ((1 −  0 ),    ).Moreover, any solutions initiating from ( + 0 ,  + 0 ) with    <  + 0 <    will be free from impulsive effects.Therefore, the exact domains of impulsive sets and phase sets of system (2) could vary which will lead to complex dynamical behavior for system (2) under case ( 2 ); those will be discussed after investigations for case ( 1 ).
(VI) Denote the closure of Ω by For case ( 1 ), the set Ω is an invariant set of system (1).In fact, denote If the vector field of model ( 1) is flowing into the boundary Ω, then Ω is an invariant set provided where ⋅ denotes the scalar product of two vectors, and the inequality ( 34) is equivalent to calculate  → +∞, as can be seen from Figure 1.This completes the proof.
Since the properties of the Poincaré map  have been investigated, the fixed point of  can be discussed which corresponds to the existence of an order-1 limit cycle for system (2).In the following subsection, the boundary order-1 limit cycle will be addressed in detail first.

Existence and Stability of the Boundary Order-1 Limit
Cycle.It follows from system (2) that a boundary order-1 limit cycle exists when () = 0 if and only if  = 0. To show this, consider the following subsystem: Solving the first equation with initial condition (0 + ) = (1 − ) one yields and letting  be the time at which () meets the line Solving the above equation with respect to , we have Therefore, there is a boundary order-1 limit cycle for system (2) with period , which is denoted by (  (),   ()) and can be described as follows: In the following, we show that the boundary order-1 limit cycle of system (2) is always unstable.Theorem 6.The boundary order-1 limit cycle (  (), 0) of system ( 2 Then and ( The Floquet multiplier  2 can be calculated as It is obvious that holds true.Therefore, it follows from | 2 | > 1 that the boundary order-1 limit cycle (  (), 0) of system (2) is unstable.This completes the proof.
If we fix the parameter values as shown in Figure 2, then it can be seen that the boundary order-1 limit cycle of system ( 2) is unstable and an interior order-1 limit cycle is generated.In addition, all solutions tend to the interior order-1 limit cycle, as shown in Figure 2.

Existence and Stability of Limit
Cycles for  > 0. In this subsection, the existence of the order- limit cycle of system (2) under case ( 1 ) will be investigated, which is equivalent to the discussion of fixed point of the Poincaré map.For simplicity, the generalized result for the stability of the order-1 limit cycle ((), ()) will be provided first.To this end, we have the following generalized result.
For case ( 1 ), as mentioned before, any solution initiating from ((1−),  + 0 ) with  + 0 ∈  0 experiences infinitely many pulses; we write the corresponding impulsive point series as  +  ( = 1, 2, . ..) with  +  =   ( + 0 ).In the following, what we want to show is the existence of the fixed point of the Poincaré map , which corresponds to the existence of the order-( ≥ 1) limit cycle for system (2).Theorem 8.If (  ) <   , then the Poincaré map  has a unique fixed point   with  *  <   ≤   which is globally asymptotically stable, and it implies system (2) exists with an order-1 limit cycle.
Proof.From the property (V) of the Poincaré map, as shown in Theorem 5, if (  ) <   , then there is a unique fixed point   with  *  <   ≤   for the Poincaré map.Further, the results of Theorem 7 indicate that this unique order-1 limit cycle is orbitally asymptotically stable.To show the global stability of the order-1 limit cycle, we only need to show that this unique order-1 limit cycle is globally attractive.
For any solution starting from ((1 − ),  + 0 ), if  + 0 >   , then there are two possible cases for   ( + 0 ): (a)   ( + 0 ) >   for all ; (b)   ( + 0 ) >   does not hold true for all .For the former, it follows from ( + 0 ) <  + 0 that   ( + 0 ) is monotonically decreasing as  increases, and lim →+∞   ( + 0 ) =   .For the latter, assume that there exists a smallest positive integer  1 such that   1 ( + 0 ) <   .From the analysis of case (a), it suggests that   2 ( + 0 ) is monotonically increasing as  2 increases ( 2 is a positive integer and  2 >  1 ), and lim  2 →+∞   2 ( + 0 ) =   .Therefore, the unique order-1 limit cycle is globally attracting and consequently is globally asymptotically stable.This completes the proof.Remark 9.In particular, if (  ) =   , then it follows from Theorem 8 that there is a unique globally asymptotically stable fixed point   for the Poincaré map, which refers to a unique globally asymptotically stable order-1 limit cycle for system (2).Theorem 10.If (  ) >   and  2 (  ) ≥   , then there is a stable fixed point or a period two-point cycle for the Poincaré map, and it indicates that system (2) does not permit an order-( ≥ 3) limit cycle except for the stable order-1 limit cycle or order-2 limit cycle.
Proof.For any  + 0 ∈ [0,   ], it follows from the properties of the Poincaré map that there does not exist a fixed point for  on [0,   ], and  is monotonically increasing on [0,   ].Thus, there will be an integer  such that
From Theorem 10, the sufficient conditions for the existence and stability of an order-1 or order-2 limit cycle of system (2) are provided when (  ) >   .In the following, what we want to do is to show the global stability of the order-1 limit cycle for system (2).Theorem 11.If (  ) >   , then the order-1 limit cycle of system ( 2) is globally stable if and only if  2 () >  for all  ∈ [  ,   ).
For case ( 3 ), assume that the solution starting from ((1 − ), ) experiences  impulsive effects; then for   () there may be two cases; that is,   () >   holds true for all  or there will be a positive integer  such that   () ∈ (0,   ) (or   () ∈ [  ,   )).For the former,   () is monotonically decreasing due to () <  with lim →+∞   () =   .For the latter, the results are also true by using the same methods as cases ( 1 ) and ( 2 ).
Remark 13.Note that the conditions for the existence of an order-3 limit cycle have been provided in Theorem 12, and it suggests that system (2) exists with order-( ≥ 3) limit cycles, which implies chaos [31,32].
Based on case ( 1 ), we have not only studied the existence and stability of the order−1 limit cycle, but also provided the conditions for the existence of the order-( ≥ 2) limit cycle.In the following, numerical simulations will be carried out to verify the theoretical results.Fixed parameters are shown in Figure 3; it is shown that system (2) exists with an order-1 limit cycle (Figure 3(a)), an order-2 limit cycle (Figure 3(b)), an order-3 limit cycle (Figure 3(c)), and chaos (Figure 3(d)) when  decreases, which confirms our main results.Note that Theorem 7 provides a condition for the orbitally asymptotically stable of an order-1 limit cycle; an example is given to show the possibility that parameters satisfy Theorem 7. In the case of an order-1 limit cycle (Figure 3 (57) It follows from Theorem 7 that the order-1 limit cycle ((), ()) is orbitally asymptotically stable (Figure 3(a)).Now we choose control parameters as bifurcation parameters to show the complex dynamics of system (2), In Figure 4, the simulation reveals complexities of system (2).In Figure 4(a), as  increases, period-doubling bifurcations lead system (2) to chaos.When  continues to increase, period-halving bifurcations lead system (2) to an order−4 limit cycle.Furthermore, system (2) exhibits sharp changes from order-( + 1) limit cycles to order- ( = 1, 2, 3) limit cycles via period-decreasing bifurcations with chaotic bands.Moreover, as  increases system (2) also exhibits perioddecreasing bifurcations with sharp changes (Figure 4(b)).In addition, the bifurcation diagrams of the outbreak period of the pest population with respect to  and  are also indicated in Figures 4(c) and 4(d).All these confirm that the theorems of case ( 1 ) are correct and further indicate that small changes in  and  would result in the pests and enemies oscillate periodically.

Complexity of Poincaré Map for Case (𝐴 2 )
4.1.Complex Domains of Poincaré Map.For case ( 2 ),  4 is located to the right of the equilibrium  * , while the locations of  3 could vary.According to Lemma 4, the equilibrium  * of system (2) could be stable or unstable under certain conditions.Without loss of generality, unless otherwise specified, we assume that  * is a focus for system (2) in the following, and let Is and Ps denote the impulsive set and phase set.
For case (i) of Lemma 4,  * is a globally stable focus.To derive domains for Is and Ps, we only need to consider the case  ≥  * .For this case, as shown in Figure 5(a), there is a solution denoted by Λ 2 which is tangential to the line  4 at the point , and then Λ 2 intersects the line  2 at a point  1 (  1 ,   1 ) such that Λ 2 is tangential to  2 at this point.If (1− ) <   1 , then the curve Λ 1 starting from the point ((1 − ),   ) which is tangential to  2 will meet  4 at point  1 (,   1 ).So we derive Is Figure 5: The definition domains of phase set and impulsive set.(a) Case (i) of Lemma 4, that is, there exists a globally stable focus of system (2).(b) Case (ii) of Lemma 4, that is, there exists a unique limit cycle Ω of system (2).(c) Case (iii) of Lemma 4, that is, there exists two limit cycles Ω 1 and Ω 2 of system (2).The blue lines represent the phase sets and the red line denotes the impulsive sets under different cases.
3 must intersect Λ 2 at two points  1 ((1 − ),    ) and  2 ((1 − ),    ).Thus, Is is I and we obtain the Ps For case (ii) of Lemma 4,  * is an unstable focus and the unique stable limit cycle is denoted by Ω, which intersects  2 at two points  11 ( 11 ,  11 ) and  12 ( 12 ,  12 ) with  11 <  12 ; see Figure 5(b).In order to provide the domains for the Poincaré map, based on the positions of (1 − ), ,  11 , and  12 , we need to consider three possible cases: For case ( 1 ), from the analysis for case (i) of Lemma 4, we obtain Is I 1 and Ps P 1 .Similarly, for case ( 2 ), if (1 − ) <   1 , then we obtain Is I 1 and Ps P 1 (Figure 5(b)).If (1 − ) ≥   1 , then Is is I and Ps is P 2 .For case ( 3 ), note that  3 and  4 are located between  11 and  12 , any trajectory starting from P (not include  * ) may meet impulsive set or have multiple intersection points with  3 before meeting impulsive set, thereafter undergoes impulsive effects.Thus, we obtain Is I and Ps P/{ * }, and P/{ * } indicates  * is excluded from P.
For case (iii) of Lemma 4 (for details see Figure 5(c)), it is revealed that  * is locally stable and there are two limit cycles in system (1)   In conclusion, we obtain all domains of Is and Ps of model ( 2) and list them in Table 1 for case ( 2 ).For convenience, the same notations are used to denote cases with the same domains of Is and Ps.In particular, let () be case when Is and Ps are I and P; let ( 1 ) be case when Is and Ps are I 1 and P 1 ; let ( 2 ) be case when Is and Ps are I and P 2 ; let ( 3 ) be case when Is and Ps are I and P 3 ; let ( 4 ) be case when Is and Ps are I and P/{ * }.Since the Poincaré map is well defined for cases () and (  ) ( = 1, 2, 3, 4), and the properties of the Poincaré map as shown in Theorem 5 are all satisfied, the existence and stability of order- limit cycles can be investigated similarly by using the same methods as case ( 1 ).
Based on the above analyses, the complex domains of the Poincaré map under case ( 2 ) have been discussed; then the complex dynamics of system (2) can be investigated by using the same methods as those shown in Section 3. In order to avoid redundancy, we just show the results which are different from case ( 1 ) in the following.Theorem 14.For case ( 2 ), if (   ) ≤    , then there exists a stable order-1 limit cycle for system (2).If (   ) ≥    , then there also exists a unique order-1 limit cycle for system (2).If (   ) >    and  2 (   ) ≥    , then system (2) does not have an order-( ≥ 3) limit cycle other than the stable order-1 limit cycle or order-2 limit cycle. ,   ) and 0 <  1  <   ) will be free from impulsive effects after one single impulsive effect.Consequently, the properties of the Poincaré map are satisfied on the interval  2 /[0,   ) (or  2 /( 1  ,   )).So system (2) has a stable order-1 limit cycle.If (   ) >    and  2 (   ) ≥    , then, by using the same methods as in the proof of Theorem 10, system (2) does not have an order-( ≥ 3) limit cycle other than the stable order-1 limit cycle or order-2 limit cycle.This completes the proof.

Proof. If 𝜑(𝑦
Under the conditions of Theorem 14, there does not exist an order-( ≥ 3) limit cycle other than the stable order-1 limit cycle or order-2 limit cycle for system (2).By using the same methods, it is claimed that system (2) does not have an order-( ≥ 3) limit cycle for two other cases: case ( From the proof of Theorem 14, we conclude that the multistable behaviors corresponding to coexistence of the limit cycle and order− limit cycles could happen under the conditions of Remark 15.For example, an order-1 limit cycle, the limit cycle Ω 1 , and the limit cycle Ω 2 can coexist, as shown in Figure 6.It implies that the pests and enemies may oscillate periodically along the limit cycles with different amplitudes; in other words, the final quantities at the equilibrium level for pests and enemies mainly rely on initial densities [33,34].For case ( 4 ), the Is and Ps are both located in the interior of the stable limit cycle Ω; any solution initiating from Ps will meet Is and then experience impulsive effects.Furthermore, the Poincaré map has a fixed point, which corresponds to the limit cycle of system (2) (see Figure 7(a)).Compared to the results obtained from case ( 1 ), the difference is that the Poincaré map in this case has complex properties.More precisely, the definition domain of the Poincaré map for system (2) has a complicated shape with discontinuity points.For example, if we fix all the parameters as those shown in Figure 7(a), then there is an order−1 limit cycle for system (2) with three discontinuity points, and then denoted by  1 ,  2 , and  3 .Moreover, it is clear that the number of discontinuity points of Poincaré map depends on the number of intersection points of spiral orbits initiating from the phase set with the impulsive set before it reaches the impulsive set.Finally, it is found that the number of discontinuity points of Poincaré map first increases and reaches its maximum value around  * and then decreases as the key parameter  decreases, as shown in Figure 8.For case (), similar results can be obtained and we do not address them in detail (Figure 7(b)); in this case this order−1 limit cycle lies between Ω 1 and Ω 2 .

Conclusions
Recently, mathematical models with feedback control formulated by impulsive dynamical systems [23,24] have been applied in many fields since the problems originating from these areas can be modelled very well when human actions are taken for real word applications [14][15][16][17][18][19][20][21], rather than fixed pulsed models.Nevertheless, a very important and well-known biological model named the Holling-Tanner model, also known as the model of R. M. May, has never been studied when incorporating state-dependent feedback control because the system has very complex dynamics.It is known that mathematical analysis of the Holling-Tanner model with impulsive control not only possesses significant theoretical implications, but also indicates biological meanings.Therefore, the paper proposes a novel Holling-Tanner model with impulsive feedback control.The objective is to provide a comprehensive analysis for global dynamics of system (2) and show how state-dependent feedback control strategy affects the dynamics.
The Poincaré map is first constructed in the phase sets considering phase portraits of model (1).Based on the positions between the threshold  and the equilibrium  * , we consider two cases: ( 1 ) <  * and ( 2 ) ≥  * .For case ( 1 ), the main properties of the Poincaré map which play important roles in the proof of the existence of limit cycles are firstly investigated.Then the existence and stability of the boundary order−1 limit cycle are studied, and it is shown that this limit cycle is always unstable.It means that the pest itself can not be maintained below the threshold when chemical control only is applied, and biological control such as predators should be introduced for successful pest control.Subsequently, the conditions for the existence and stability of the order−1 limit cycle are provided by employing the fixed point theorems and analogue of Poincaré criterion, revealing that the pests and enemies will oscillate periodically below .Moreover, we have also studied the existence of order −( ≥ 2) limit cycle, and the conditions for the existence of an order−3 limit cycle are provided, which confirms that there exists limit cycles of any period for system (2) [31,32].After that, numerical investigations with respect to control parameters are carried out to substantiate our results, and these results further show that there exists rich and complex dynamics for system (2) which are consistent with the theoretical results.
For case ( 2 ), the complex domains of the impulsive set and phase set are given for system (2) and the results are listed in Table 1.In order to avoid redundancy, the results that are different from case ( 1 ) are shown.In particular, under certain conditions there does not exist an order−( ≥ 3) limit cycle other than the order−1 limit cycle or order−2 limit cycle for system (2), and multistable behaviors are observed which indicate that the final states of the pest and natural enemy populations mainly depend on their initial densities as well as on their ratios.Once the impulsive set and phase set are both located in the interior of the stable limit cycle Ω, it is found that the definition domain of the Poincaré map has a complicated shape with discontinuity points, and it is concluded that the number of discontinuity points first increases and reaches its maximum value around  * and then decreases as the control parameter  decreases.
Compared to the previous studies with impulsive feedback control [17,18,20], we list the differences: (1) in references [17,18,20], the impulsive and phase sets are often defined for some special cases, while in this paper all possible cases are discussed and listed in Table 1; (2) in previous studies, the boundary order−1 limit cycle could be stable for some parameter sets.However, in this paper, the boundary order−1 limit cycle is unstable over the whole parameter space due to the complexity of system (1), suggesting that chemical control on its own cannot reach the target of successful pest control; (3) Theorem 14 provided conditions   for the existence of the multistable behaviors; that is to say, the order−1 limit cycle, the limit cycle Ω 1 , and the limit cycle Ω 2 can coexist (Figure 6); this is much more complex compared to the results as shown in previous studies [17,18,20]; (4) the existence of order−3 limit cycle has been proved rather than order-1 or order-2 limit cycle; (5) the discontinuity points of the Poincaré map have also been addressed.In conclusion, all the results and methods presented in this paper enrich and improve the previous studies related to state-dependent feedback control.
There are studies about the development and extinction of predators affected by resource limitations [13,14].How could we incorporate the effect of resource limitation into the prey-predator system (2) and assess how resource limitation affects the dynamics of the system (2) and further affects the outcome of pest control?These topics will be studied in the near future.Moreover, an ideal control strategy is that integrated pest management (IPM) strategies should be implemented once the density of the pest population grows and exceeds the economic threshold (ET), while control tactics are suspended once the density of the pest population falls below the ET again.To address these, complex nonsmooth model or Filippov system needs to be developed and investigated within an IPM context [35,36], which we plan to do in the near future.

Table 1 :
Complex domains of impulsive set and phase set of system (2) for case ( 2 ).