Dynamic Complexity of an Ivlev-Type Prey-Predator System with Impulsive State Feedback Control

Abstract

The dynamic complexities of an Ivlev-type prey-predator system with impulsive state feedback control are studied analytically and numerically. Using the analogue of the Poincaré criterion, sufficient conditions for the existence and the stability of semitrivial periodic solutions can be obtained. Furthermore, the bifurcation diagrams and phase diagrams are investigated by means of numerical simulations, which illustrate the feasibility of the main results presented here.

1. Introduction

The theoretical investigation of predator-prey systems in mathematical ecology has a long history, beginning with the pioneering work of Lotka and Volterra. During this time, the theory and application of differential equations with impulsive perturbations were significantly advanced by the efforts of Lakshmikantham et al. [1]. In fact, many systems in physics, chemistry, and biology can be modeled by impulsive differential equations which can represent the abrupt jumps that occur during their evolutionary processes [2].

Many factors in the environment must be considered in predator-prey systems [3]. Impulsive perturbations are an important element because some factors, such as fires, floods, and similar disturbances, are not well suited to be considered in a continuous manner. In general, impulsive perturbations can be classified into two cases [4]. The first is perturbations caused by nature, and the second is perturbations that arise as a result of human efforts to control prey density, for instance, controlling pest outbreaks. There are many strategies to control agricultural pests, including chemical and biological controls. Chemical control methods, such as crop dusting, are useful because they quickly kill a significant portion of a pest population and sometimes provide the only feasible method for preventing economic loss. However, pesticide pollution is a major hazard to human health and the populations of natural enemies. Another important control method is biological control. Biological control is the purposeful introduction and establishment of one or more natural enemies of a pest [5, 6]. The key to successful biological pest control is to identify the pest and its natural enemy and to release the natural enemies for pest control. Proportional harvesting, for example, of fish, is also considered in this category. Consequently, it is natural to assume that these perturbations are instantaneous, that is, in the form of an impulse.

Generally speaking, there are three possible cases of impulsive perturbation: systems with impulses at fixed times, systems with impulses at variable times, and autonomous impulsive systems. In recent years, most investigations of impulsive differential equations have concentrated on systems with impulses at fixed times [7–14], while the other two kinds of impulsive differential equations have been relatively less studied. As a matter of fact, in many practical cases, impulses often occur at state-dependent times rather than at fixed times. For example, it may be desirable to control a population size by catching, crop-dusting, or releasing the predator when prey numbers reach a threshold value.

As is well known, significant developments have recently been achieved in the bifurcation theory of continuous dynamic systems [15–20]. The study of impulsive systems mainly involves the properties of their solutions, such as existence, uniqueness, stability, boundedness, and periodicity. This paper also considers bifurcation behaviors. Recently, Lakmeche and Arino [21] transformed the problem of a periodic solution into a fixed-point problem, discussed the bifurcation of periodic solutions from trivial solutions, and obtained the existence conditions for the positive period-1 solution. Tang and Chen [22] developed a complete expression for a period-1 solution and investigated the bifurcation of periodic solutions numerically using a discrete dynamic system determined by a stroboscopic map. Many papers have been devoted to the analysis of mathematical models with state-dependent impulsive effects [23]. For instance, Tang, Jiang, Zeng, Qian, Nie, and others [24–29] have studied the dynamic behaviors of predator-prey systems with impulsive state feedback control and have determined the existence and stability of positive periodic solutions using the Poincaré map and the properties of the Lambert 𝑊 function.

Recently, the continuous model with Ivlev-type has been extensively studied [30–36]. The Ivlev-type functional response describes a cyrtoid or Holling II prey-dependent functional response because the feeding rate declines with increasing resource abundance until it reaches a constant rate [34]. Although a direct link between the predator and prey cannot be established unless quantitative methods are used, the precious works clearly show that the amount of two species is often related, and a change in one species can cause a change in another, especially predator. Thus, we apply Ivlev-type functional response to describe their relationship with sufficient accuracy in this paper. Using the method of impulsive perturbations, a predator-prey model with Ivlev-type and state impulsive perturbations will be considered, as follows:𝑥̇=𝑟𝑥1−𝑥𝑦̇=((1Δ−Δ𝑘𝑦e𝑥x=p=−𝑞((𝑦−−1𝑝+𝑎−𝑥𝑥𝜏,)e,)x−p𝑥(𝑚−=)𝑎𝑦ℎ𝑥,),)𝑦,𝑥≠ℎ,(1.1) where 𝑥(𝑡) and 𝑦(𝑡) are functions of time representing the population densities of the prey and predator, respectively. 𝑎 is the efficiency with which predators extract preys from their environment, which sometimes is called the apparency of the preys, 𝑘 is the carrying capacity of prey 𝑥, 𝑚 is the death rate of predator 𝑦, 𝑝∈(0,1) is the average lost rate of prey 𝑥 during this time the amount of prey 𝑥 reaches to critical threshold ℎ>0, 𝑞(𝑞>0) describes a released parameter for juvenile predator 𝑦, 𝜏(𝜏>0) represents a released parameter for adult predator 𝑦, Δ𝑥(𝑡)=𝑥(𝑡⁺)−𝑥(𝑡), and Δ𝑦(𝑡)=𝑦(𝑡⁺)−𝑦(𝑡). When the amount of prey 𝑥 reaches to critical threshold ℎ, a control strategy is used; then the numbers of prey and predator become (1−𝑝)ℎ and (1+𝑞)𝑦(𝑡_𝑖(ℎ))+𝜏, respectively.

The rest of this paper is organized as follows. Section 2 presents certain preliminaries, important definitions, and lemmas that are frequently used in the following discussions. In Section 3, the existence and stability of a positive periodic solution of system (1.1) are stated and proved. Section 4 presents a numerical analysis to illustrate the theoretical results. Finally, conclusions and remarks are presented in Section 5.

2. Preliminaries

The dynamic behavior of system (1.1) without impulsive effects can be interpreted as follows. It has one saddle at (0,0), and calculations reveal that (0,𝑘) is also a saddle, while (−ln(1−𝑚)/𝑎,−𝑟ln(1−𝑚)(𝑎𝑘+ln(1−𝑚))/𝑎²𝑘𝑚) is a stable positive focus when 𝑎𝑘+ln(1−𝑚)>0 and 𝑎𝑘+2ln(1−𝑚)<0 hold.

Throughout this paper, it is assumed that ℎ<−ln(1−𝑚)/𝑎, 𝑎𝑘+ln(1−𝑚)>0 and 𝑎𝑘+2ln(1−𝑚)<0 always hold. Only solutions with nonnegative components, continuously differentiable in the region 𝐷={(𝑥,𝑦)∶𝑥≥0,𝑦≥0} based on the biological background of system (1.1), will be considered.

Let 𝑅=(−∞,∞) and let 𝑧(𝑡)=(𝑥(𝑡),𝑦(𝑡)) be any solution of system (1.1). The positive orbit through point 𝑧₀∈𝑅²⁺={(𝑥,𝑦)∶𝑥≥0,𝑦≥0} for 𝑡≥𝑡₀≥0 is defined as 𝑂⁺𝑧₀,𝑡₀=𝑧∈𝑅²⁺∶𝑧=𝑧(𝑡),𝑡≥𝑡₀,𝑧𝑡₀=𝑧₀.(2.1)

Definition 2.1. A trajectory 𝑂⁺(𝑧₀,𝑡₀) of system (1.1) is said to be order-𝑘 periodic if there exists a positive integer 𝑘≥1 such that 𝑘 is the smallest integer for which 𝑥₀=𝑥_𝑘.

The next step is to construct the Poincaré map. To discuss the dynamics of system (1.1), consider its vector field. As shown in Figure 1, denote 𝑆₀={(𝑥,𝑦)∣𝑥=(1−𝑝)ℎ,𝑦≥0} and 𝑆₁={(𝑥,𝑦)∣𝑥=ℎ,𝑦≥0}. It is clear that the line 𝑥=(1−𝑝)ℎ and the line 𝑥=ℎ intersect the isoclinal line 𝑟𝑥(1−𝑥/𝑘)−(1−exp(−𝑎𝑥))𝑦=0, or in other words, 𝑑𝑥/𝑑𝑡=0, at point 𝐴((1−𝑝)ℎ,𝑟ℎ(1−𝑝)(𝑘−(1−𝑝)ℎ)/𝑘(1−exp(−𝑎(1−𝑝)ℎ))), and that 𝐵(ℎ,𝑟ℎ(𝑘−ℎ)/𝑘(1−exp(−𝑎ℎ))) intersects the line 𝑦=0 at point 𝐶((1−𝑝)ℎ,0), 𝐷(ℎ,0). Denote Ω={(𝑥,𝑦)∣0<𝑦<𝑟𝑥(𝑘−𝑥)/𝑘(1−exp(−𝑎𝑥)),(1−𝑝)ℎ<𝑥<ℎ}, and Ω₁=Ω∪𝐶𝐷. It is obvious that 𝑑𝑥/𝑑𝑡=0,𝑑𝑦/𝑑𝑡<0 are satisfied at point (𝑥,𝑦)∈𝐴^∩𝐵, where 𝐴^∩𝐵 is represented as 𝑦=𝑟𝑥(𝑘−𝑥)/𝑘(1−exp(−𝑎𝑥)) and (1−𝑝)ℎ<𝑥<ℎ. Any orbit passing through segment 𝐴^∩𝐵 and into the interior of Ω will exit Ω by passing through segment 𝐵𝐷.

Figure 1 

Poincaré map of system (1.1).

Assume that point 𝑆_𝑛((1−𝑝)ℎ,𝑦_𝑛) is on section 𝑆₀. Then the trajectory 𝑂⁺(𝑆_𝑛,𝑡_𝑛) of system (1.1) intersects section 𝑆₁ at point 𝑆_𝑛+1(ℎ,𝑦_𝑛+1), where 𝑦_𝑛+1 is determined by 𝑦_𝑛. Then the point 𝑆_𝑛+1(ℎ,𝑦_𝑛+1) jumps to point 𝑆^+𝑛+1((1−𝑝)ℎ,(1+𝑞)𝑦_𝑛+𝜏) on 𝑆₀ due to the impulsive effects, and section 𝑆₀ is a Poincaré section. The following Poincaré map 𝑓 can thus be obtained: 𝑦^+𝑛=(1+𝑞)𝑔𝑦^+𝑛−1+𝜏.(2.2)

Now choose section 𝑆₁ as another Poincaré section. Another Poincare map 𝑓₁ can be obtained for 𝑆₁: 𝑦_𝑛+1=𝑔(1+𝑞)𝑦_𝑛+𝜏≡𝐹𝑞,𝜏,𝑦_𝑛.(2.3)

In this discussion, 𝑦_𝑘+1 is determined by 𝑦_𝑛 and parameters 𝑞 and 𝜏.

Next, an autonomous system with impulsive effects will be considered: 𝑑𝑥𝑑Δ𝑡𝑥==𝑃𝜉((𝑥𝑥,,𝑦𝑦)),,Δ𝑑𝑑𝑦𝑦𝑡==𝜂𝑄((𝑥𝑥,,𝑦𝑦)),,𝜑𝜑((𝑥𝑥,,𝑦𝑦))=≠00,,(2.4) where 𝑃(𝑥,𝑦) and 𝑄(𝑥,𝑦) are continuous differential functions and 𝜑(𝑥,𝑦) is a sufficiently smooth function with grade 𝜑(𝑥,𝑦)≠0. Let (𝜉(𝑡),𝜂(𝑡)) be a positive 𝑇-periodic solution of system (2.4). The following technical lemma will now be introduced.

Lemma 2.2 (see [37]). If the Floquet multiplier 𝜇 satisfies the condition |𝜇|<1, where 𝜇=^𝑛_𝑘=1Δ_𝑘exp^𝑇0𝜕𝑃𝜕𝑥(𝜉(𝑡),𝜂(𝑡))+𝜕𝜕𝑄𝑦(𝜉(𝑡),𝜂(𝑡))𝑑𝑡,(2.5) with Δ_𝑘=𝑃⁺((𝜕𝛽/𝜕𝑦)(𝜕𝜙/𝜕𝑥)−(𝜕𝛽/𝜕𝑥)(𝜕𝜙/𝜕𝑦)+𝜕𝜙/𝜕𝑥)+𝑄𝑃(𝜕𝜙/𝜕𝑥)+𝑄(𝜕𝜙/𝜕𝑦)₊((𝜕𝛼/𝜕𝑥)(𝜕𝜙/𝜕𝑦)−(𝜕𝛼/𝜕𝑦)(𝜕𝜙/𝜕𝑥)+𝜕𝜙/𝜕𝑦)𝑃(𝜕𝜙/𝜕𝑥)+𝑄(𝜕𝜙/𝜕𝑦),(2.6) and 𝑃, 𝑄, 𝜕𝛼/𝜕𝑥, 𝜕𝛼/𝜕𝑦, 𝜕𝛽/𝜕𝑥, 𝜕𝛽/𝜕𝑦, 𝜕𝜙/𝜕𝑥, 𝜕𝜙/𝜕𝑦 are calculated at point (𝜉(𝑡^+𝑘),𝜂(𝑡^+𝑘)), 𝑃₊=𝑃(𝜉(𝑡^+𝑘),𝜂(𝑡^+𝑘)), 𝑄₊=𝑄(𝜉(𝑡^+𝑘),𝜂(𝑡^+𝑘)) and 𝑡_𝑘(𝑘∈𝑁) is the time of the 𝑘-th jump, then (𝜉(𝑡),𝜂(𝑡)) is orbitally asymptotically stable.

Lemma 2.3 (see [38]). Let 𝐹∶𝑅×𝑅→𝑅 be a one-parameter family of 𝐶² maps satisfying(i)𝐹(0,𝜇)=0, (ii)(𝜕𝐹/𝜕𝑥)(0,0)=1, (iii)(𝜕²𝐹/𝜕𝑥𝜕𝜇)(0,0)>0, (iv)(𝜕²𝐹/𝜕𝑥²)(0,0)<0.
Then 𝐹 has two branches of fixed points for 𝜇 near zero. The first branch is 𝑥₁(𝜇)=0 for all 𝜇. The second bifurcating branch 𝑥₂(𝜇) changes its value from negative to positive as 𝜇 increases through 𝜇=0 with 𝑥₂(0)=0. The fixed points of the first branch are stable if 𝜇<0 and unstable if 𝜇>0, while those of the bifurcating branch having the opposite stability.

3. Dynamic Properties

3.1. Case 𝜏=0

It should be stressed that the semitrivial periodic solution with 𝑦=0 of system (1.1) exists if and only if 𝜏=0. Therefore, the discussions start with 𝜏=0.

When 𝜏=0, system (1.1) can be stated in the following form: 𝑥̇=𝑟𝑥1−𝑥𝑦̇=((1Δ−Δ𝑘𝑥e𝑦x==p−−((𝑞−𝑝1𝑦𝑥𝑎.−,𝑥)e)x𝑥−p=(𝑚−ℎ)𝑎𝑦,𝑥,))𝑦,𝑥≠ℎ,(3.1) Let 𝑦(𝑡)=0 for 𝑡∈[0,∞); then from system (3.1), 𝑥̇=𝑟𝑥1−𝑥Δ𝑥=−𝑝𝑥𝑘,𝑥,=𝑥ℎ≠.ℎ,(3.2) Setting 𝑥₀=𝑥(0)=(1−𝑝)ℎ leads to the solution of system (3.2), 𝑥(𝑡)=𝑘(1−𝑝)ℎexp(𝑟(𝑡−𝑛𝑇))/[𝑘−(1−𝑝)ℎ+(1−𝑝)ℎexp(𝑟(𝑡−𝑛𝑇))]. Let 𝑇=ln[𝑘−(1−𝑝)ℎ/(𝑘−ℎ)(1−𝑝)]^1/𝑟; then 𝑥(𝑇)=ℎ and 𝑥(𝑇⁺)=(1−𝑝)ℎ. Hence, system (3.1) has the following semitrivial periodic solution: 𝑥(𝑡)=𝑘(1−𝑝)ℎexp(𝑟(𝑡−𝑛𝑇))𝑘−(1−𝑝)ℎ𝑦+(𝑡)(1=−0𝑝,)ℎexp(𝑟(𝑡−𝑛𝑇)),(3.3) where 𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁, and which is denoted by (𝜉(𝑡),0).

Now the stability of this semitrivial periodic solution will be discussed.

Theorem 3.1. The semitrivial periodic solution (3.3) is said to be orbitally asymptotically stable if 0<𝑞<𝑘−(1−𝑝)ℎ(𝑘−ℎ)(1−𝑝)^{(𝑑−1)/𝑟}exp^𝑇0exp(−𝑎𝜉(𝑡))𝑑𝑡−1.(3.4)

Proof. In fact, 𝑃(𝑥,𝑦)=𝑟𝑥1−𝑥(𝜉(𝑇𝛼)(,𝑥𝜂,(𝑦𝑇))=𝑄)=−(𝑥𝑝(,ℎ𝑥𝑦,,)0=),(𝛽(1(𝑥−,𝑘𝜉𝑦e)x𝑇−=p₊((𝑞1−𝑦,𝑎−𝑥e)x)p−𝜙((𝑚−𝑥),𝑎𝑦𝑦𝑥,)))=𝑦,𝑥−ℎ,,𝜂𝑇⁺=((1−𝑝)ℎ,0).(3.5) According to Lemma 2.2, a straightforward calculation yields 𝜕𝑃𝜕𝛼𝜕𝑥=𝑟−2𝑘𝑟𝑥−𝑎𝑦exp(−𝑎𝑥),𝜕𝜕𝑄𝑦=1−exp(−𝑎𝑥)−𝑚,𝜕𝑥=Δ−𝑝,𝜕𝜕𝛼𝑦=0,𝜕𝜕𝛽𝑥=0,𝜕𝜕𝛽𝑦=𝑞,𝜕𝜕𝜙𝑥=1,𝜕𝜕𝑦𝜙=0,₁=𝑃⁺((𝜕𝛽/𝜕𝑦)(𝜕𝜙/𝜕𝑥)−(𝜕𝛽/𝜕𝑥)(𝜕𝜙/𝜕𝑦)+𝜕𝜙/𝜕𝑥)+𝑄𝑃(𝜕𝜙/𝜕𝑥)+𝑄(𝜕𝜙/𝜕𝑦)₊((𝜕𝛼/𝜕𝑥)(𝜕𝜙/𝜕𝑦)−(𝜕𝛼/𝜕𝑦)(𝜕𝜙/𝜕𝑥)+𝜕𝜙/𝜕𝑦)=𝑃𝑃(𝜕𝜙/𝜕𝑥)+𝑄(𝜕𝜙/𝜕𝑦)₊𝜉𝑇⁺,𝜂𝑇⁺(1+𝑞)𝑃(𝜉(𝑇),𝜂(𝑇))=(1−𝑝)(1+𝑞)𝑘−𝑘(1−−ℎ𝑝)ℎ.(3.6) Furthermore, exp^𝑇0=exp𝜕𝜕_𝑇0𝑃𝑥(𝜉(𝑡),𝜂(𝑡))+𝜕𝜕𝑄𝑦(𝜉(𝑡),𝜂(𝑡))𝑑𝑡𝑟+1−𝑑−2𝑟=𝑘−(1−𝑝)ℎ𝑘𝜉(𝑡)−exp(−𝑎𝜉(𝑡))𝑑𝑡(𝑘−ℎ)(1−𝑝)^{(𝑟+1−𝑚)/𝑟}𝑘−𝑘(1−−ℎ𝑝)ℎ⁻²exp^𝑇0−exp(−𝑎𝜉(𝑡))𝑑𝑡.(3.7) Hence, the Floquet multiplier 𝜇 can be obtained by direct calculation as follows: 𝜇=^𝑛_𝑘=1Δ_𝑘exp^𝑇0=(1+𝑞)𝑘−(1−𝜕𝜕𝑃𝑝𝑥)(ℎ𝜉(𝑡),𝜂(𝑡))+𝜕𝜕𝑄𝑦(𝜉(𝑡),𝜂(𝑡))𝑑𝑡(𝑘−ℎ)(1−𝑝)^{(1−𝑚)/𝑟}exp−^𝑇0exp(−𝑎𝜉(𝑡))𝑑𝑡.(3.8)
Therefore, |𝜇|<1 holds if and only if (3.4) holds. This completes the proof.

Remark 3.2. Set 𝑞^∗=((𝑘−(1−𝑝)ℎ)/((𝑘−ℎ)(1−𝑝)))^{(𝑚−1)/𝑟}exp(∫^𝑇0exp(−𝑎𝜉(𝑡))𝑑𝑡)−1; a bifurcation may occur at 𝑞=𝑞^∗ for |𝜇|=1, and a positive periodic solution may appear when 𝑞>𝑞^∗. Hence, the problem of bifurcations will now be discussed.
First, in the case 𝜏=0, consider the Poincaré map (2.2). Set 𝑢=𝑦^+𝑛 and 𝑢≥0 small enough. The map then takes the following form: 𝑢⟼(1+𝑞)𝑔(𝑢)≡𝐺(𝑢,𝑞),(3.9) where the function 𝐺(𝑢,𝑞) is continuously differentiable with respect to both 𝑢 and 𝑞, 𝑔(0)=0; then lim_𝑢→0⁺𝑔(𝑢)=𝑔(0)=0.
Second, by examining the bifurcation of map (3.9), it is possible to obtain the following theorem.

Theorem 3.3. A transcritical bifurcation occurs when 𝑞=𝑞^∗. Therefore, a stable positive fixed point appears when parameter 𝑞 changes through 𝑞^∗ from left to right. Correspondingly, system (3.1) has a stable positive periodic solution if 𝑞∈(𝑞^∗,𝑞^∗+𝛿) with 𝛿>0.

Proof. The values of 𝑔^(𝑢) and 𝑔^(𝑢) must be calculated at 𝑢=0, where 0≤𝑢≤𝑟ℎ(1−𝑝)(𝑘−(1−𝑝)ℎ)/𝑘(1−exp(−𝑎(1−𝑝)ℎ))≡𝑢₀. From system (1.1), 𝑑𝑦𝑑𝑥=𝑄𝑃((𝑥𝑥,,𝑦𝑦)),(3.10) where 𝑃(𝑥,𝑦)=𝑟𝑥1−𝑥𝑄(𝑥,𝑦)=((1−exp𝑘(−−𝑎𝑥(1))−−e𝑚xp)𝑦(.−𝑎𝑥))𝑦,(3.11) Let (𝑥,𝑦(𝑥;𝑥₀,𝑦₀)) be an orbit of system (3.10), and set 𝑥₀=(1−𝑝)ℎ,𝑦₀=𝑢,0≤𝑢≤𝑢₀; then 𝑦(𝑥;(1−𝑝)ℎ,𝑢)≡𝑦(𝑥,𝑢),(1−𝑝)ℎ≤𝑥≤ℎ,0≤𝑢≤𝑢₀.(3.12) Using (3.12), 𝜕𝑦(𝑥,𝑢)𝜕𝑢=exp^{𝑥(1−𝑝)ℎ}𝜕𝜕𝜕𝑦𝑄𝑃((𝑠𝑠,,𝑦𝑦((𝑠𝑠,,𝑢𝑢))))𝑑𝑠₂,𝑦(𝑥,𝑢)𝜕𝑢²=𝜕𝑦(𝑥,𝑢)𝜕𝑢^{𝑥(1−𝑝)ℎ}𝜕²𝜕𝑦²𝑄𝑃((𝑠𝑠,,𝑦𝑦((𝑠𝑠,,𝑢𝑢))))𝜕𝑦𝜕(𝑠𝑢,𝑢)𝑑𝑠.(3.13) Clearly, it can be deduced that 𝜕𝑦(𝑥,𝑢)/𝜕𝑢>0 and 𝑔^(0)=𝜕𝑦(ℎ,0)𝜕𝑢=exp^{ℎ(1−𝑝)ℎ}𝜕=exp_{ℎ(1−𝑝)ℎ}𝜕𝑦𝑄𝑃((𝑠𝑠,,𝑦𝑦((𝑠𝑠,,00))))𝑑𝑠1−𝑚−exp(−𝑎𝑠)=𝑘−(1−𝑝)ℎ𝑟𝑠(1−𝑠/𝑘)𝑑𝑠(𝑘−ℎ)(1−𝑝)^{(1−𝑟)/𝑚}exp^{ℎ(1−𝑝)ℎ}−exp(−𝑎𝑠)=𝑘−(1−𝑝)ℎ𝑟𝑠(1−𝑠/𝑘)𝑑𝑠(𝑘−ℎ)(1−𝑝)^{(1−𝑟)/𝑚}exp^𝑇0−exp(−𝑎𝜉(𝑡))𝑑𝑡.(3.14) Furthermore, 𝑔^(0)=𝑔^(0)^{ℎ(1−𝑝)ℎ}𝑙(𝑠)𝜕𝑦(𝑠,0)𝜕𝑢𝑑𝑠,(3.15) where 𝑙(𝑠)=𝜕²=𝜕2𝑦(1²−𝑄𝑃ex((p𝑠𝑠,,(𝑦𝑦−((𝑎𝑠𝑠𝑠,,)00−))))𝑚)(1−exp(−𝑎𝑠))(𝑟𝑠(1−𝑠/𝑘))²,𝑠∈[(1−𝑝)ℎ,ℎ].(3.16) Using the previous assumption, ℎ<−ln(1−𝑚)𝑎.(3.17) It can be determined that 𝑙(𝑠)<0,𝑠∈[(1−𝑝)ℎ,ℎ).(3.18) Therefore, 𝑔^(0)<0.(3.19) The next step is to check whether the following conditions are satisfied.(a) It is easy to see that 𝐺(0,𝑞)=0,𝑞∈(0,∞).(3.20)(b) Using (3.14), 𝜕𝐺(0,𝑞)𝜕𝑢=(1+𝑞)𝑔^=(1+𝑞)(𝑘0)−(1−𝑝)ℎ(𝑘−ℎ)(1−𝑝)^{(1−𝑟)/𝑚}exp^𝑇0−exp(−𝑎𝜉(𝑡))𝑑𝑡,(3.21) which yields 𝜕𝐺0,𝑞^∗𝜕𝑢=1.(3.22) This means that (0,𝑞^∗) is a fixed point with eigenvalue 1 of map (3.9).(c) Because (3.14) holds, 𝜕²𝐺0,𝑞^∗𝜕𝑢𝜕𝑞=𝑔^(0)>0.(3.23)(d) Finally, (3.19) implies that 𝜕²𝐺0,𝑞^∗𝜕𝑢²=1+𝑞^∗𝑔^(0)<0.(3.24)These conditions satisfy the conditions of Lemma 2.3. This completes the proof.

3.2. Case 𝜏>0

In this subsection, the existence of a positive periodic solution with 𝜏>0 will be discussed using the Poincaré map (2.3). Sufficient conditions will be given for the existence and stability of positive periodic solutions. The following theorem will now be proved.

Theorem 3.4. For any 𝑞>0 and 𝜏>0, system (1.1) has a positive order-1 periodic solution.

Proof. Let point 𝑀₁((1−𝑝)ℎ,0) be on section 𝑆₀. Then the trajectory 𝑂⁺(𝑀₁,𝑡₀) of system (1.1) starting from the initial point 𝑀₁ intersects section 𝑆₁ at point 𝑁₁(ℎ,0). In state 𝑁₁, the trajectory 𝑂⁺(𝑀₁,𝑡₀) is subjected to impulsive effects, jumps to point 𝑀₂((1−𝑝)ℎ,𝜏) on section 𝑆₀, and then returns to 𝑁₂(ℎ,𝛼₁) on section 𝑆₁. Because 𝜏>0, point 𝑀₂ is above point 𝑀₁. Furthermore, point 𝑁₂ is above point 𝑁₁, and 𝛼₁>0. From (2.3), 𝛼₁=𝐹(𝑞,𝜏,0)=𝑔(𝜏)>0, and 0−𝐹(𝑞,𝜏,0)=0−𝛼₁<0.(3.25) In addition, assuming that the initial point of the trajectory 𝑂⁺(𝐴,𝑡₀) is point 𝐴 where 𝑑𝑦/𝑑𝑡<0 and 𝑑𝑥/𝑑𝑡=0, obviously, 𝑂⁺(𝐴,𝑡₀) is tangent to the line 𝑆₀, intersects 𝑆₁ at point 𝐻(ℎ,𝑣₁), and then jumps to point 𝐻⁺((1−𝑝)ℎ,(1+𝑞)𝑣₁+𝜏) on 𝑆₀, and returns to point 𝐻^(ℎ,𝑣₂) on 𝑆₁. Assume further that there exists a positive 𝑞 such that (1+𝑞)𝑣₁+𝜏=𝑟ℎ(1−𝑝)(𝑘−(1−𝑝)ℎ)/𝑘(1−exp(−𝑎(1−𝑝)ℎ)). Then point 𝐻⁺ coincides with point 𝐴 for 𝑞=𝑞, and point 𝐻⁺ is above point 𝐴 for 𝑞>𝑞, but below point 𝐴 for 𝑞<𝑞. However, for any 𝑞>0, the point 𝐻′ is not above the point 𝐻 in view of the geometrical structure of the phase space of system (1.1).
In conclusion, the following results can be obtained from the previous discussion:(i)if 𝑣₁=𝑣₂(𝑞=𝑞), then system (1.1) has a positive order-1 periodic solution;(ii)if 𝑣₁>𝑣₂(𝑞≠𝑞), then 𝑣₁−𝐹𝑞,𝜏,𝑣₁=𝑣₁−𝑣₂>0.(3.26)
From (3.25) and (3.26), it follows that the Poincaré map (2.3) has a fixed point; that is, system (1.1) has a positive order-1 periodic solution. This completes the proof.

According to the following discussion, a positive periodic solution exists when 𝜏=0,𝑞≥𝑞^∗ or 𝜏>0,𝑞>0. Next, the stability of a positive order-1 periodic solution of system (1.1) will be proved. This will be accomplished by means of the following theorem.

Theorem 3.5. For any 𝜏=0,𝑞≥𝑞^∗ or 𝜏>0,𝑞>0, let (𝜉(𝑡),𝜂(𝑡)) be a positive order-1 𝑇-periodic solution of system (1.1) which starts from point (ℎ,𝜔). If the condition ||𝜇||=(1+𝑞)Γexp^𝑇0Ψ(𝑡)𝑑𝑡<1(3.27) holds, where Γ=𝑟ℎ(1−𝑝)(𝑘−(1−𝑝)ℎ)−𝑘(1−exp(−𝑎ℎ(1−𝑝)))[(1+𝑞)𝜔+𝜏]Ψ(𝑡)=𝑟ℎ𝜕𝑃(𝑘−ℎ)−𝑘𝜔(1−exp(−𝑎ℎ)),𝜕𝑥(𝜉(𝑡),𝜂(𝑡))+𝜕𝜕𝑄𝑦(𝜉(𝑡),𝜂(𝑡)),(3.28) then (𝜉(𝑡),𝜂(𝑡)) is a positive order-1 periodic solution of system (1.1) which is orbitally asymptotically stable and has the asymptotic phase property.

Proof . Based on the conclusion of Theorem 3.4, it is necessary only to verify the stability of the positive order-1 periodic solutions (𝜉(𝑡),𝜂(𝑡)) of system (1.1). In what follows, it is assumed that a periodic solution with period 𝑇 passes through points 𝐾⁺((1−𝑝)ℎ,(1+𝑞)𝜔+𝜏) and 𝐾(ℎ,𝜔), in which 𝜔≤𝑣₁ holds because of the properties of the vector field of system (1.1) as outlined in the following discussion. Because the mathematical form and the period 𝑇 of the solution are not known, the stability of this positive periodic solution will be discussed using Lemma 2.2. The difference between this case and that of Theorem 3.1 lies in the fact that (𝜉(𝑇),𝜂(𝑇))=(ℎ,𝜔),𝜉𝑇⁺,𝜂𝑇⁺=((1−𝑝)ℎ,(1+𝑞)𝜔+𝜏),(3.29) while the others are just the same. Then Δ₁=𝑃⁺((𝜕𝛽/𝜕𝑦)(𝜕𝜙/𝜕𝑥)−(𝜕𝛽/𝜕𝑥)(𝜕𝜙/𝜕𝑦)+𝜕𝜙/𝜕𝑥)+𝑄𝑃(𝜕𝜙/𝜕𝑥)+𝑄(𝜕𝜙/𝜕𝑦)₊((𝜕𝛼/𝜕𝑥)(𝜕𝜙/𝜕𝑦)−(𝜕𝛼/𝜕𝑦)(𝜕𝜙/𝜕𝑥)+𝜕𝜙/𝜕𝑦)=𝑃𝑃(𝜕𝜙/𝜕𝑥)+𝑄(𝜕𝜙/𝜕𝑦)₊𝜉𝑇⁺,𝜂𝑇⁺(1+𝑞)𝑃(𝜉(𝑇),𝜂(𝑇))=(1+𝑞)Γ,(3.30) where Γ=𝑟ℎ(1−𝑝)(𝑘−(1−𝑝)ℎ)−𝑘(1−exp(−𝑎ℎ(1−𝑝)))[(1+𝑞)𝜔+𝜏]𝑟ℎ(𝑘−ℎ)−𝑘𝜔(1−exp(−𝑎ℎ)).(3.31) Let Ψ(𝑡)=(𝜕𝑃/𝜕𝑥)(𝜉(𝑡),𝜂(𝑡))+(𝜕𝑄/𝜕𝑦)(𝜉(𝑡),𝜂(𝑡)); then ||𝜇||=Δ₁exp^𝑇0=(1+𝑞)Γexp𝜕𝜕𝑃𝑥_𝑇0(𝜉(𝑡),𝜂(𝑡))+𝜕𝜕𝑄𝑦(𝜉(𝑡),𝜂(𝑡))𝑑𝑡Ψ(𝑡)𝑑𝑡.(3.32) If |𝜇|<1, that is: ||||(1+𝑞)Γexp^𝑇0Ψ(𝑡)𝑑𝑡||||<1,(3.33) then the periodic solution is stable. This completes the proof.

Remark 3.6. From the previously mentioned, it is known that if there exists a 𝑞^>𝑞 such that |𝑢|=1, a flip bifurcation occurs at 𝑞=𝑞^. If a flip bifurcation occurs, there exists a stable positive order-2 periodic solution of system (1.1) for 𝑞^>𝑞, which may also lose its stability as 𝑞 increases.

4. Numerical Analysis

As is well known, system (1.1) cannot be solved explicitly, so it must be studied by numerical integration and the long-term dynamic behavior of the solution by numerical simulation.

To study the dynamic complexity of an Ivlev-type system with state-dependent impulsive perturbation on the predator, a semitrivial periodic solution of system (1.1) with initial conditions is first obtained numerically for a biologically feasible range of parameter values. The bifurcation diagram provides a summary of the essential dynamic behavior of system (1.1).

Next, two control parameters, 𝑞 and 𝜏, are chosen. Other parameters are set to 𝑟=0.95,𝑘=20,𝑎=2.8,𝑚=0.45 and provide some representative values to help with the analysis.

Note that the corresponding focus (−ln(1−𝑚)/𝑎,−𝑟ln(1−𝑚)(𝑎𝑘+ln(1−𝑚))/𝑎²𝑘𝑚)=(0.2135,0.4459), so ℎ≤0.2135. System (1.1) has a semitrivial periodic solution when 𝜏=0. Taking 𝑝=0.8 and ℎ=0.2, from Theorem 3.1, 𝜇≈0.7(1+𝑞). Note that 𝜇>1 is always true for any 𝑞>3/7 and that the periodic semitrivial solution is unstable (Figure 1(A)).

Let 𝑝=0.8 and ℎ=0.15; then 𝑞^∗≈0.56 can be obtained from Remark 3.2. Setting 𝑞=0.5, the solution of system (1.1) tends to a stable semitrivial periodic solution as 𝑡 increases (Figure 2(b)).

(a)

(b)

Figure 2 

Trajectories with initial point (0.02, 0.001) of system (1.1) with 𝑝=0.8, 𝑞=0.5, 𝜏=0: (a) ℎ=0.2, (b) ℎ=0.15.

When 𝜏>0, there is no semitrivial solution of system (1.1). Figures 3(a) and 3(b) show typical bifurcation diagrams for population 𝑦 in system (1.1) as 𝑝 increases from 0 to 35 and 𝜏 increases from 0 to 0.16 with initial 𝑋(0) = (0.02, 0.01). As 𝑝 and 𝜏 increase, the bifurcation diagrams clearly show that system (1.1) has rich dynamics, including period-doubling bifurcations, periodic windows, chaotic bands, period-halving bifurcations, and crises.

(a)

(b)

(c)

Figure 3 

Bifurcation diagram of system (1.1) with initial conditions 𝑋(0) = (0.02, 0.01), ℎ = 0.21, 𝑝 = 0.8: (a) 𝜏=0.065, (b) 𝑞 = 18, (c) 𝜏=0.

In Figure 3(a), there is no fold bifurcation. The positive order-1 periodic solution is stable for 𝑞∈(0, 3.92). At 𝑞≈ 3.92, a positive order-2 periodic solution bifurcates from the positive order-1 period solution by means of a flip bifurcation. Furthermore, order-4 and order-8 periodic solutions arise through flip bifurcation. The period-doubling bifurcation leads to chaos. Finally, a cascade of period-halving bifurcations leads to stable order-4 periodic solutions for 𝑞> 29.68. Now let 𝑞=18, and consider 𝜏 as a control parameter. Figure 3(b) shows a plot of the solution as a function of the bifurcation parameter 𝜏. In this case, there is a route from chaos to a stable periodic solution via a period-halving bifurcation in which complex dynamic behaviors exist, such as periodic windows, chaotic bands, and chaotic crises (Figure 3(b)).

In Figure 3(c), 𝑞 is considered as a parameter, and the bifurcation diagram of the periodic solution of system (1.1) with 𝜏=0 is shown. It is obvious that the semitrivial periodic solution is stable for 𝑞∈ (0, 0.74) and unstable for 𝑞∈ (0.74, +∞). A transcritical bifurcation leads to a positive order-1 periodic solution from the semitrivial periodic solution at 𝑞≈ 0.74. This positive order-1 periodic solution is stable for 𝑞∈ (0.74, 4.05) and unstable for 𝑞∈ (4.05, +∞). In addition, a positive period-2 solution bifurcates from the positive order-1 periodic solution by means of a flip bifurcation at 𝑞≈ 4.05. Due to the period-doubling bifurcation, chaos arises, in which periodic windows, chaotic bands, and crises also exist (Figure 3(c)).

From Theorem 3.5, Remark 3.6, and analysis of the bifurcations described previously, it is known that system (1.1) has a positive order-1 periodic solution, which is shown in Figure 4(a). A flip bifurcation occurs at 𝑞=4.05 according to the numerical simulations. Figure 4 also shows the period-𝑖 (𝑖=2,4,8) solutions for different value of 𝑞. Figure 5 presents the phase diagram and time series of population 𝑦 for a chaotic solution.

(a)

(b)

(c)

(d)

Figure 4 

Periodic solutions of system (1.1): ℎ = 0.21, 𝑝 = 0.8, 𝜏=0.065; (a) 𝑞=3, (b) 𝑞=6, (c) 𝑞=10, (d) 𝑞=11.5.

(a)

(b)

Figure 5 

(a) Phase diagram; (b) time series of 𝑦 for system (1.1) with ℎ = 0.21, 𝑝 = 0.8, 𝜏=0.065, 𝑞=14.

Based on the previous analysis, it can be seen that the impulsive state feedback control can enhance the predator 𝑦 biomass level with the increasing of 𝑞, in which result is agreed with some results in reality. Further, it is also interesting to point out that the two different parameters of the impulsive state feedback control can come into rich and complex dynamical behaviors, but these dynamical behaviors are different. Moreover, the use of mathematical model with impulsive state feedback control is considered to investigate some biological problems, and the numerical simulation provides an approximation of the real biological system behaviors; hence, these results can promote the study of ecological dynamics.

5. Conclusions

In this paper, a predator-prey model with Ivlev-type function and impulsive state feedback control has been built and studied analytically and numerically. Mathematical theoretical arguments have investigated the existence and stability of semitrivial periodic solutions of system (1.1) and have proved that the positive periodic solution comes into being from the semitrivial periodic solution through a transcritical bifurcation according to bifurcation theory. Numerical simulations illustrate the theory and show the complex dynamics of the impulsive system. All these results are expected to be useful in the study of the dynamic complexity of ecosystems.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions on this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 31170338 and no. 30970305) and also by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001).