Abstract
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 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 limit cycle are provided, and the existence of order 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.
1. 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 [1–7]; 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 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 [1–7].
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 [8–10], 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 [11–13]. 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 [14–21], 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 represents the fatality rate for the prey due to chemical control, and denotes the release number of . Denote and as the initial densities of and . In this paper we assume that 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 and , 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 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 limit cycle are obtained, and then it is concluded that this limit cycle is unstable. Subsequently, the existence and stability of an order limit cycle will be addressed, and the existence of order 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.
2. Preliminaries and Main Properties of System (1)
The generalized planar impulsive semidynamical systems with control are usually described bywhere ; we denote and for simplicity, and are continuous functions from into ; represents the impulsive set. For each point , the map is defined viaand is called an impulsive point of .
Let be the phase set (that is, for any ), and . In the following some definitions related to impulsive semidynamical systems will be listed briefly, which are used in this work.
Assume or to be impulsive dynamical system [23, 24], denotes a metric space, and denotes a set of positive reals. For arbitrary , assume defined as which satisfies for arbitrary , and for arbitrary and . Let and be the attainable set of at . Besides, denote . The following Definitions and Lemmas are also important for discussing the dynamics of system (2) [25–30].
Definition 1. An impulsive semidynamical system consists of a continuous semidynamical system together with a nonempty closed subset (or impulsive set) of and a continuous function such that the following property holds: no point is a limit point of ; is a closed subset of .
We denote the points of discontinuity of by , and we define a function from into the extended positive reals as follows: let ; if we set ; otherwise and we set , where for but .
Definition 2. A trajectory in is said to be periodic of period and order if there exist nonnegative integers and such that is the smallest integer for which and .
Lemma 3 (see [23, 24]). The -periodic solution of systemis orbitally asymptotically stable and enjoys the property of asymptotic phase if the Floquet multiplier satisfies the condition , wherewithand is in . We denote by . , , , , , , and can be calculated at ; let and . (, is positive integers) is -th jump time.
Noting that many studies about system (1) can be found [1–7], based on these studies, we let and be two isoclines of system (1):There are two equilibria in system (1), that is, and the unique interior equilibrium with , and with
Define the functionand then denote the two positive roots of the function as and (assume that ) if they exist. Based on the discussions in [1–7], we get the following result.
Lemma 4. (i) If , then is globally asymptotically stable in the interior of the first quadrant.
(ii) If and , then is unstable and system (1) exists with a stable limit cycle.
(iii) Under certain conditions (for example, and , 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).
3. Poincaré Map and Order 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.
Define two lines as follows:
Since , substituting into the line , one yields the intersection point of and , denoted as withThen denote the intersection point of and as withDenote the intersection point of and as with .
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 , and are both located to the left of the equilibrium . According to the vector fields of the model (2), any solution initiating from the line will reach the line in a finite time. Then the impulsive set for system (2) can be determined as follows:which is a closed subset of . Moreover, define the continuous function as follows:So the phase set can be defined, wherewith . Therefore, based on the above analysis model (2) defines an impulsive semidynamical system .
For simplicity, assume that . We use to define the Poincaré map because any trajectory of system (2) with initial condition will experience one time impulsive effect and then satisfy for all .
For case , is located to the right of the equilibrium , while the locations of 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 , or two limit cycles with the outermost being stable and the innermost being unstable under different set of conditions. Thus, any solution initiating from with 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 which is tangential to the line at a point , and the curve must intersect the line at a point such that is tangential to the line at this point. If , then any solution initiating from with experiences infinitely many pulses. If , then the curve must intersect the line at two points, denoted by and . Moreover, any solutions initiating from with 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 ; those will be discussed after investigations for case .
3.1. Poincaré Map for Case
To define the Poincaré map for system (2), denote two sections as follows:We choose section as a Poincaré section. Assume that the point lies in the section , and the trajectory initiating from point intersects the section at the point in a finite time, where is only determined by ; that is . Further, point experiences one time impulsive effect and then maps to the point which lies on the section , where . Therefore, the Poincaré map with respect to impulsive point series of system (2) can be defined as
To investigate the existence of periodic solutions for system (2), we define the Poincaré map in the phase set according to the phase portrait of model (1). Thus, we denoteleading to the following scalar differential equation in phase spaceFor model (21), we only focus on the region , whereand in this region the function is continuously differentiable. Denote , with ; that is to say . Then we getand it follows from model (21) thatThen the Poincaré map related to the phase portrait of model (1) takes the following form:
Since the formation of the Poincaré map has been investigated, it is possible to address the corresponding properties which are useful in the rest of the paper.
Theorem 5. For case , the Poincaré map of system (2) satisfies the following properties, as shown in Figure 1:
(I) The domain and range of are and , respectively. It is increasing on and decreasing on .
(II) For all , the inequality holds true.
(III) is continuously differentiable.
(IV) is concave on .
(V) has a unique positive fixed point .
(VI) has a horizontal asymptote as .
(a)
(b)
(c)
(d)
Proof. (I) According to the vector field of system (1), the domain of is defined on the interval . In phase set, for any with , we have due to the Cauchy-Lipschitz Theorem. In fact, according to the vector fields of system (2) we have and for , and and in the region below . Thereby, the solutions starting from and will first cross and then meet at points and with . If , then and , so the solutions starting from and will meet at at points and with . In short, we have . After one time impulsive effect, one yields . Similarly, for any with , the trajectories starting from points and first meet at points and with and then reach at points and with . After one time impulsive effect, one getsThus, the Poincaré map is increasing on and decreasing on , and the range of is .
(II) If , it follows from the vector fields of system (1) that we have and , and thus variables and both increase along the trajectory. Furthermore, the time for the occurrence of each impulsive effect is denoted by ; then holds true for any ; that is, . Therefore, provided .
(III) We need to consider two cases to prove that is continuous and differentiable, that is, and . For the former, if , then functions for anyIt follows from that we obtain thatis continuous on . Thus, is continuously differential when . For the latter, any solution from will turn around the point and cross the line when at and then meet . Define the map . Thereby, is the composite function of and when . Since is continuously differential on , is continuously differential because of the standard theory of Poincaré application (the Cauchy-Lipschitz Theorem with initial value); it is concluded that is also continuously differential on .
(IV) From (21), we haveIt is obvious that and when , implying that and hold true for all .
From the theorem of Cauchy and Lipschitz with parameters on the scalar differential equation one getsandBased on the above analysis, the inequality holds true. Therefore, is monotonically increasing and concave on (Figure 1).
(V) Since for all and is a decreasing function on , then the Poincaré map has at least one fixed point. If , then the Poincaré map has a fixed point with (Figure 1(a)). According to the concavity property of , there is not any other fixed point on . Since is decreasing on , no other fixed point exists for all . If , then it follows from the concavity of that there is no fixed point when , and because is decreasing on , there exists a unique fixed point for on (Figures 1(c) and 1(d)).
(VI) Denote the closure of byFor case , the set is an invariant set of system (1). In fact, denoteIf the vector field of model (1) is flowing into the boundary , then is an invariant set providedwhere denotes the scalar product of two vectors, and the inequality (34) is equivalent to calculatenote that leads to . Besides, if ; it means that for any solution initiating from with , its vertical coordinate obtains the minimum value at point . Thus, when . In addition, the solution starting from reaches the impulsive set in a finite time and then experiences impulsive effect; we have . It indicates that the horizontal asymptote exists for when , 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.
3.2. Existence and Stability of the Boundary Order-1 Limit Cycle
It follows from system (2) that a boundary order-1 limit cycle exists when if and only if . To show this, consider the following subsystem:Solving the first equation with initial condition one yieldsand letting be the time at which meets the line Solving the above equation with respect to , we haveTherefore, 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 of system (2) is unstable.
Proof. To show this, denoteThenandMoreover, denote ; thus,and letting one yieldsThe Floquet multiplier can be calculated asIt is obvious thatholds true. Therefore, it follows from that the boundary order-1 limit cycle 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.
(a)
(b)
(c)
3.3. Existence and Stability of Limit Cycles for
In this subsection, the existence of the order- limit cycle of system (2) under case 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- limit cycle will be provided first. To this end, we have the following generalized result.
Theorem 7. The order-1 limit cycle is orbitally asymptotically stable when and only whenwhere .
Proof. Without loss of generality, it is assumed that the order-1 limit cycle with period passes through the points and . Based on the proofs of Theorem 6, the Floquet multiplier can be calculated asTherefore, it follows from the inequality (48) that ; that is, the order-1 limit cycle is orbitally asymptotically stable.
For case , as mentioned before, any solution initiating from with experiences infinitely many pulses; we write the corresponding impulsive point series as with . 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- 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 , if , according to the properties (II) and (IV) of the Poincaré map, then holds true, which implies that is monotonically increasing as increases. Therefore, , as shown in Figure 1(a).
For any solution starting from , if , then there are two possible cases for : (a) for all ; (b) does not hold true for all . For the former, it follows from that is monotonically decreasing as increases, and . For the latter, assume that there exists a smallest positive integer such that . From the analysis of case (a), it suggests that is monotonically increasing as increases ( is a positive integer and ), and . 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 , 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- limit cycle except for the stable order-1 limit cycle or order-2 limit cycle.
Proof. For any , it follows from the properties of the Poincaré map that there does not exist a fixed point for on , and is monotonically increasing on . Thus, there will be an integer such that and . Since is monotonically increasing on , then it is obvious that . Therefore, we have . In addition, for any , after one time impulsive effect, because is monotonically decreasing on . Therefore, there will be an integer such that .
Since is monotonically decreasing on and is monotonically increasing on , one yieldsWithout loss of generality, for any , assume that , , and . For the relations among , , , , and , there may be the following four possible cases:
. In this case, it can be seen that and ; thus we have . By induction one obtains