Discrete Dynamics in Nature and Society

Volume 2018, Article ID 3467405, 18 pages

https://doi.org/10.1155/2018/3467405

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

^{1}Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China^{2}Key Laboratory of Biologic Resources Protection and Utilization, Hubei Minzu University, Enshi, Hubei 445000, China^{3}College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, China

Correspondence should be addressed to Jin Yang; moc.621@mohees

Received 17 April 2018; Accepted 3 October 2018; Published 18 October 2018

Academic Editor: Zhengqiu Zhang

Copyright © 2018 Jin Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 .*