# Dynamic Complexity in a Prey-Predator Model with State-Dependent Impulsive Control Strategy

Authors: Chuanjun Dai
Journal: Complexity
 (2020)

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## Abstract

In this paper, an ecological model described by a couple of state-dependent impulsive equations is studied analytically and numerically. The theoretical analysis suggests that there exists a semitrivial periodic solution under some conditions and it is globally orbitally asymptotically stable. Furthermore, using the successor function, we study the existence, uniqueness, and stability of order-1 periodic solution, and the boundedness of solution is also presented. The relationship between order-k successor function and order-k periodic solution is discussed as well, thereby giving the existence condition of an order-3 periodic solution. In addition, a series of numerical simulations are carried out, which not only support the theoretical results but also show the complex dynamics in the model further, for example, the coexistence of multiple periodic solutions, chaos, and period-doubling bifurcation.


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## Body

## 1. Introduction

Since mathematical models represented by differential equations were introduced into biological and ecological systems, it has been proved that they are very useful tools to deal with practical problems such as emerging disease [1] and population dynamics between plankton and nutrient [2–4], which also promote the development of the theory of differential equation further. Especially, in the process, there is a significant development in impulsive differential equations [5–9] because many reality systems exhibit the abrupt jumps phenomena in population sizes during the evolution processes. Additionally, a theoretical method is urgently needed to provide a guide for the management of some practical problems such as pest outbreaks, phytoplankton blooms, vaccination, and so on. Accordingly, impulsive differential equation theory may prove to be one of the most potential theories to strategize about the approach of management for these problems, as impulsive differential equation can model abrupt jump behaviors induced by management, for example, some management occurs at fixed time, which can be modeled using the models with impulses at fixed times [10–13].State-dependent impulsive differential equation, as a branch of impulsive differential equation, has the advantage to model the control behaviors depending on the population state. In recent years, the studies on state-dependent impulsive differential equation have been paid increasing attention [13–16]. Many studies devote to the properties of periodic solutions [17–20], including the existence, stability and periodicity, etc. It has been confirmed that the Poincaré map is a very useful tool to prove the existence of periodic solution in state-dependent impulsive differential equations [21–23]. In addition, Simeonov and Bainov [24] gave analogue of Poincaré criterion in ref. [25], which contributes to the stability of the periodic solution. Based on the Poincaré map, it can be proved theoretically that the positive periodic solution can bifurcate from the trivial periodic solution through a transcritical bifurcation [25]. Particularly, Chen [26] introduced the successor function a in semicontinuous dynamical system described by state-dependent impulsive differential equation to prove the existence of periodic solution. Using the successor function, Dai et al. [27] studied homoclinic bifurcation in a semicontinuous dynamical system [28]. The successor function has been extensively employed to investigate the existence of periodic solution and even stability and bifurcation in state-dependent impulsive differential equation [28–38]. These studies enrich the theory of state-dependent impulsive differential equation further.In this paper, we will present a prey-predator two-species model with state-dependent impulsive control strategy to study the dynamics induced by impulsive control. Firstly, a model describing population dynamics of prey and predator is given below.(1)dxdt=rx1−xK−bxya+x,dydt=αbxya+x−my,where x and y denote the densities of prey and predator, respectively; r is the intrinsic growth rate of the prey; K is the carrying capacity of the prey; b is the maximum growth rate; a represents the half-saturation concentration; α is the assimilation efficiency of predator; and m is the specific mortality rate of predator. Here, we employ a Holling II functional response to present the interaction between prey and predator because some results suggest that the population dynamic following Holling II may exist in reality [2, 3, 39, 40].For some species, when their densities are beyond a certain value, some seriously negative effects will emerge, for instance, algal bloom. For this reason, their densities should be kept below the critical value by harvest or/and released natural enemy. Obviously, the occurrence of control will rely on the density of the species. Here, under the assumption that the interaction between preyx and predator y follows model (1), we investigate the dynamics of impulsive control and control strategies for prey x using state-dependent impulsive differential equation, and the model can be described as(2)dxdt=rx1−xK−bxya+xdydt=αbxya+x−myx<h,Δx=−pxΔy=−qy+τx=h,where h>0 denotes the critical value below which the density of prey x should be kept. 0<p<1 is the harvest rate of prey x; τ≥0 is the releasing amounts of predator y; and 0<q<1 represents the harvest rate of predator y or the level of negative effect caused by the harvest of prey x on predator y.The rest of the paper is organized as follows. InSection 2, some notations, basic definitions, and lemmas are given. In Section 3, a semitrivial solution and its stability are studied; we also study the existence, uniqueness, and stability of an order-1 periodic solution; moreover, the boundedness of solution and some propositions of periodic solution are discussed as well. Whereafter, a series of numerical results are carried out to study the dynamics of model (2) further in Section 4. Finally, the paper ends with the conclusion in Section 5.


## 2. Preliminaries

In model (1), there exist three equilibria, E0=0,0, E1=K,0, and E∗=x∗,y∗, where x∗=ma/αb−m and y∗=rK−x∗a+x∗/bK. When the condition max0,αbK−a/K+a<m<Kαb/a+K holds, E∗ is a stable positive focus or node. We assume that the following condition holds throughout the paper: (H) 0<m<Kαb/a+K.The vertical isocline and the horizontal isocline can be defined as follows:(3)L1:y=rK−xa+xbK=fx,L2:x=maαb−m.Furthermore, letN=x,y|x=1−ph,y≥0 be the phase set, and M=x,y|x=h,y≥0 denotes the impulsive set. Under condition (H), we assume that L1 intersects with the phase set N at point OxO,yO and the impulsive set M at point HxH,yH in the first quadrant when h<x∗. We define Π=x,y|1−ph≤x≤h,0≤y≤fx and Ω=x,y|0≤x≤h,y≥0.Letzt=xt,yt be any solution of model (2); then, we define the positive trajectory initializing at point z0=zt0∈R+2=x,y|x≥0,y≥0 for t≥t0 as πt;z0,t0=zt|zt∈R+2,t≥t0≥0,zt0=z0.In this paper, some notions are always mentioned, including phased point, impulsive point, and successor point. From works in [20, 23, 26], we state the following definition.Definition 1.
Take the pointA0∈N; set πt;A0,t0 as the trajectory initializing at point A0 in model (2). If there exists tk>t0 such that the trajectory πt;A0,t0 intersects the impulsive set M at Ak=h,yk when t=tkk=1,2,3,…, then a point Ak+ exists in the phase set N, where point Ak+=1−ph,yk+=1−qyk+τ. The point Ak+ is called the phased point of point Ak, and the point Ak is called the impulsive point of point Ak+. The point Ak+ is called the order-k successor point of point A0, and the vector AkAk+⟶ is called an impulsive line.Definition 2.
Set pointE=N∩x,y|x≥0,y=0; then, we define a number axis L in the phase N, and set point E as the origin of the number axis L. ∀A0∈N, set the direction of vector EA0⟶ as the positive direction of number axis L, and set a0=EA0⟶ as the coordinate of point A0 in the number axis L. Then, a map Fk:R+⟶Rk=1,2,3,… can be defined as Fka0=ak−a0, where ak=EAk+⟶ and Ak+ is the order-k successor point of point A0. The map Fk is called the order-k successor function, that is, FkA0=yAk+−yA0.Definition 3.
Take a pointS in the phase set N; then, πt;S,t0 is a solution of model (2). If there exists a positive integer k≥1 such that k is the smallest integer for FkS=0, then the solution πt;S,t0 is called order-k periodic solution.Lemma 1 (see [26]).
The successor functionFkS is continuous, where the point S∈N.
Obviously, we can obtain the following lemma by Lemma1.Lemma 2.
If there exist two pointsS1=1−ph,yS1 and S2=1−ph,yS2 in model (2) and yS1−yOyS2−yO≥0 such that F1S1F1S2<0, then model (2) has an order-1 periodic solution.
For the pointO, we always assume the point Ok+=1−ph,yOk+ is the order-k successor point of the point O, and the point Ok=h,yOk is the impulsive point of point Ok+, where k=1,2,3,….


## 3. Main Results

When the values of parameters in model (1) are given, the dynamics of model (2) only depend on the parameters h, p, q, and τ. In this paper, we are interested in the effect of impulsive control on dynamics of model (2). Hence, we will investigate the dynamical behaviors of model (2) under the condition that the parameters of model (1) are fixed.For pointsO and H, it is obvious that(1)
yO>yH if 1−p/2h>K−a/2.(2)
yO<yH if 1−p/2h<K−a/2 and K>a.In this section, we just show the proof with respect to caseK−a/2<1−ph<1−p/2h in the figure of illustration. The proof corresponding to other cases is similar, which has been omitted.By model (1), it is obvious that h<x∗ implies dy/dt<0. Additionally, yt+<yt always holds when q∈0,1.In model (2) with τ=0, it is obvious that the following semitrivial periodic solution exists for t∈n−1T,nTn=1,2,3,… when the predator y=0,(4)ξt=1−phert−n−1TK1−phert−n−1T+K−1−ph,ηt=0,where the period T=1/rlnK−1−ph/1−pK−h. Set Γ=ξt,ηt. Then, we can get the following theorem.Theorem 1.
Under condition (H), if h<x∗ and τ=0, then semitrivial periodic solution Γ is globally orbitally asymptotically stable in model (2) for any q∈0,1 and p∈0,1.Proof.
Forτ=0, solutions of model (2) will enter the space Π eventually. Hence, we study the stability of the semitrivial periodic solution in this space.
For anyε>0, set Nε=1−ph,ε and Mε=h,ε. Then, we assume that the trajectory πt;Nε,t0 intersects with impulsive set M at point P=h,yP, and yP<yNε=ε under condition (H). Set δ=ε; then, for any point G=1−ph,yG where 0<yG<δ=yNε, we have yG1+<yP under condition (H) because h<x∗ and q∈0,1 (see Figure 1(a)), which suggests that the distance(5)d=dπt;G,t0,Γ=infP0∈Γπt;G,t0−P0<ε,holds for all t>t0, where Γ=ξt,ηt. Hence, according to the definition of orbital stability in ref. [40], the semitrivial periodic solution Γ is orbitally stable.
For anyG0∈N, take a sequence of number tkk=1,2,3,…, where 0<tk<tk+1 and limk⟶+∞tk=+∞, such that πtk;G0,t0=1−ph,yk∈N. Then, the sequence of number yk is a strictly decreasing positive sequence because h<x∗ and q∈0,1. Hence, there exists a y∗≥0, such that limk⟶+∞yk=y∗≥0, but yk≥y∗. Set Q∗=1−ph,y∗, and let point Q∗ be the order-1 successor point of point Q#=1−ph,y# if y∗>0, where y#>y∗. Then, for any point Q∈Q∗Q#¯ (i.e., yQ∗<yQ<yQ#), its order-1 successor point is below the point Q∗ (see Figure 1(b)), that is, yQ+≤y∗.
Becauselimk⟶+∞yk=y∗, there exists a positive integer Θ1 such that yQ∗<yk<yQ# for k>Θ1. According to above analysis, there exists a positive integer Θ2>Θ1 such that yk<y∗ when k>Θ2, which contradicts with yk≥y∗. Hence, limk⟶+∞yk=y∗=0, which means that the semitrivial periodic solution Γ is orbitally attractive. So, the semitrivial periodic solution Γ is orbitally asymptotically stable. Since the attraction domain is Ω, the semitrivial periodic solution Γ is globally orbitally asymptotically stable.
This completes the proof.
Suppose thatM is a positive constant, and then we have the following theorem for any M∈0,+∞.Figure 1
Illustration corresponding to the orbital (a) stability and (b) attractability of semitrivial periodic solution in model (2).
(a)(b)Theorem 2.
Under condition (H), if h<x∗ and τ∈0,M, then there exists a unique order-1 periodic solution in model (2) for any q∈0,1 and p∈0,1.Proof.
For anyτ>0, Figure 2(a) displays the relation between point O and its order-1 successor point O1+ under condition (H), where h<x∗. Obviously, OO1O¯ is an order-1 periodic solution if yO=yO1+. If yO<yO1+, we have F1O=yO1+−yO>0 and F1O1+=yO2+−yO1+<0, where O2+ is the successor point of point O1+. According to Lemmas 1 and 2, there exists an order-1 periodic solution in model (2).
IfyO>yO1+, we take a point R=1−ph,τ in phase set N. Then, let the point R1+ be the order-1 successor point of point R and point R1 be the impulsive point of point R1+, and yR1+=1−qyR1+τ>τ=yR (see Figure 2(a)), so F1R>0. By F1O=yO1+−yO<0, there exists an order-1 periodic solution in model (2) using Lemmas 1 and 2.
Hence, there exists an order-1 periodic solution for anyq∈0,1 and p∈0,1 when h<x∗ and τ>0.
Suppose there are two order-1 periodic solutions in model (2), which are Γ1 and Γ2, respectively. Let point A+=1−ph,yA+ and A=h,yA be the phase point and impulsive point of Γ1, respectively, and point B+=1−ph,yB+ and B=h,yB be the phase point and impulsive point of Γ2, respectively. Without loss of generality, set yA+>yB+. Then, the relation among A+, B+, and O is (i) B+∈OA+¯ (yO<yB+<yA+); (ii) A+∈B+O¯ (yB+<yA+<yO); and (iii) O∈B+A+¯ (yB+<yO<yA+).
Figure2(b) shows case (i); it is obvious that Γ1 and Γ2 cannot coexist by disjoint property of trajectory of model (1). Hence, the order-1 periodic solution is unique for this case.
For case (ii), becauseA+∈B+O¯, we only need to focus on the region Π. We rewrite the model (1) in the region Π as the following initial-value problem:(6)dydx=gx,yfx,y≜Ψx,y,yx0=y0,x,y∈Π,where fx,y=rx1−x/K−bxy/a+x,gx,y=αbxy/a+x−my.
Obviously, the functionΨx,y is continuously differentiable and monotone decreasing in y on the space Π. Then, the solution of model (6) is equivalent to the following integral equation:(7)yx,y0=y0+∫x0xΨs,ys,y0ds.
By equation (7), set(8)yix=yx,yi+=yi++∫1−phxΨs,yisds,where i=A,B; then, yA=yAh and yB=yBh. Because yA+>yB+, we have yAx>yBx on the space Π by the disjoint property of trajectory of model (1). Hence,(9)yA−yB=yAh−yBh=yA++∫1−phhΨx,yAxdx−yB+−∫1−phhΨx,yBxdx<yA+−yB+.
On the other hand, because bothΓ1 and Γ2 are order-1 periodic solutions, we have yA+−yB+=1−qyA−yB<yA−yB by model (2), which is a contradiction with inequality (9). Hence, the order-1 periodic solution is unique for this case.
For case (iii), the proof is similar with case (ii), so it is omitted.
Hence, there exists a unique order-1 periodic solution in model (2) for any q∈0,1 and p∈0,1 when h<x∗ and τ>0.
This completes the proof.
From Theorem2, we can find that the existence and stability of order-1 periodic solution depend on parameter τ. Suppose point 1−ph,yτ is the phase point of the order-1 periodic solution, then we can get the following proposition.Figure 2
Illustration with respect to (a) existence and (b) uniqueness of the order-1 periodic solution in model (2).
(a)(b)Proposition 1.
Under condition (H), when h<x∗ and τ>0, y=yτ is a monotonously increasing function with respect to τ.Proof.
Letθ be the angle between the impulsive line and the x-axis. Then, tanθ=−qy+τ/−ph, so dθ/dτ=d/dτtan−1−qy+τ/−ph<0. Hence, θτ is a monotonously decreasing function in τ. Given an order-1 periodic solution Γ∗, let points U=xU,yU and W=xW,yW be the phased point and impulsive point of Γ∗, respectively; then, τ1=yU−1−qyW guarantees the existence of the order-1 periodic solution Γ∗.
For anyε>0, when τ=τ1+ε, the order-1 periodic solution Γ∗ will disappear. Let point U+=xU+,yU+ be the order-1 successor point of point U; then, yU+>yU because θτ is a monotonously decreasing function. In addition, F1U>0 and F1U+<0. Therefore, there exists an order-1 periodic solution whose phase point belongs to UU+¯. By the uniqueness of order-1 periodic solution, y=yτ is a monotonously increasing function in τ.
This completes the proof.
By Theorem2, we know that the trajectory OO1O¯ is an order-1 periodic solution of model (2) when τ=yO−1−qyO1≜τc. Then, we can get the following theorem.Theorem 3.
Under condition (H), if h<x∗, then the order-1 periodic solution is globally orbitally asymptotically stable in model (2) for any q∈0,1 and p∈0,1 when τ∈0,τc.Proof.
Whenτ∈0,τc, the order-1 periodic solution of model (2) must belong to space Π under condition (H). Γ+ denotes an order-1 periodic solution of model (2), whose phased point and impulsive point are point C=xC,yC and point D=xD,yD, respectively. In addition, the solutions of model (2) will enter the space Π eventually when τ∈0,τc. Hence, we only need to study the stability of order-1 periodic solution on the space Π.
For anyε>0, we assume that the trajectory πt;Nε1,t0 meets impulsive set M at point P1=xP1,yP1, where Nε1=1−ph,ε+yC (see Figure 3(a)); then, yP1<yNε1=yC+ε. Take δ=ε; for any point G1=xG1,yG1, where xG1=1−ph and yC<yG1<yC+δ=yNε1, we assume that the trajectory πt;G1,t0 intersects with impulsive set at point G2=h,yG2; then, yG1>1−qyG2+τ; otherwise, another order-1 periodic solution exists above Γ+, which contradicts with the uniqueness of order-1 periodic solution. Hence, the distance(10)d=dπt;G1,t0,Γ+=infP0∈Γ+πt;G1,t0−P0<εholds for all t>t0 according to equation (9).
On the other hand, for any0<ε<yC, given δ=ε, when yC−δ<yG1<yC, we still have(11)d=dπt;G1,t0,Γ+=infP0∈Γ+πt;G1,t0−P0<ε,using the same method. Hence, according to definition of orbital stability in ref. [41], the order-1 periodic solution Γ+ is orbitally stable.
For anyG0∈N, take a numerical sequence tkk=1,2,3,…, where 0<tk<tk+1 and limk⟶+∞tk=+∞, such that πtk;G0,t0=1−ph,yk∈N. Due to disjoint of impulsive line, there are two cases on the numerical sequence yk when τ∈0,τc and h<x∗: (1) yk is a strictly decreasing positive sequence; (2) yk is a strictly increasing positive sequence.
For case (1) (see Figure3(b)), we have limk⟶+∞yk=y∗≥yC and yk≥y∗. Let Q∗=1−ph,y∗; there exists a point Q#=1−ph,y# when y∗>yC, where y#>y∗, such that point Q∗ is the order-1 successor point of point Q#. For any point Q∈Q∗Q#¯ (i.e., yQ∗<yQ<yQ#), its order-1 successor point is below the point Q∗. Let point Q+ be the order-1 successor point of Q, then yQ+<y∗.
Becauselimk⟶+∞yk=y∗ and yk≥y∗, there exists a positive integer Θ1 such that yQ∗<yk<yQ# for k>Θ1. According to above analysis, there exists a positive integer Θ2>Θ1 such that yk<y∗ when k>Θ2, which contradicts with yk≥y∗. So, limk⟶+∞yk=yC. Likewise, limk⟶+∞yk=yC also holds for case (2), which says the order-1 periodic solution Γ+ is orbitally attractive.
Hence, the order-1 periodic solutionΓ+ is orbitally asymptotically stable. Since the attraction domain is Ω, the order-1 periodic solution Γ+ is globally orbitally asymptotically stable.
This completes the proof.Figure 3
Illustration corresponding to the orbital (a) stability and (b) attractability of order-1 periodic solution in model (2).
(a)(b)Proposition 2.
Under condition (H), when h<x∗ and τ>0, if there exists an order-kk≥3 periodic solution in model (2), setting points 1−ph,yii=1,…,k as the phase points of the order-k periodic solution, then yO∈mini=1,…,kyi,maxi=1,…,kyi.Proof.
IfyO≥maxi=1,…,kyi, it is obvious that there is no order-kk≥2 according to the proofs in the previous theorems. Hence, we only need to consider yO≤mini=1,…,kyi. Let points h,yi+i=1,2,…,k be the impulsive point of the order-k periodic solution and yi+1=1−qyi++τ; then, y1=1−qyk++τ. Because yO≤mini=1,…,kyi, for any l,j∈1,2,…,k, if yl>yj, then yl+>yj+ by the disjoint of trajectory of model (1).
ForyO≤mini=1,…,kyi, the relationship among y1, y2, and y3 is one of the following cases because of the disjoint of the trajectory of model (1) and the disjoint of the impulsive line of model (2):①y1>y3>y2②y3>y1>y2③y2>y1>y3④y2>y3>y1
Fory1>y3>y2, we get y1+<y3+<y2+, so y2<y4<y3<y1 using yi+1=1−qyi++τ. Hence, we claim(12)y2<⋯<y2n<y2n−1<⋯<y1.
Whenn=1,2, it is obvious that inequality (12) holds. Now, suppose inequality (12) holds when n=σ. In the following, we will prove that inequality (12) also holds when n=σ+1. Because y2<⋯<y2σ<y2σ−1<⋯<y1, we have y2+>⋯>y2σ+>y2σ−1+>⋯>y1+. Consequently, y1>y3>⋯>y2σ+1>y2σ>⋯>y2, that is, inequality (12) holds when n=σ+1. Hence, inequality (12) holds when y1>y3>y2.
In this case, obviously, for an order-k periodic solution, we have 1−qyk++τ<y1 when y1>y3>y2, which contradicts to y1=1−qyk++τ. Hence, the order-k periodic solution does not exist. Using similar procedure, it is proved that the order-kk≥3 periodic solution does not exist for cases ②, ③, and ④. So, there is no order-kk≥3 periodic solution in model (2) when yO≤mini=1,…,kyi. Therefore, yO∈mini=1,…,kyi,maxi=1,…,kyi when an order-kk≥3 periodic solution exists in model (2).
This completes the proof.
By Proposition2, the following theorem will exist.Theorem 4.
Under condition (H), if h<x∗ and τ≥yO, then there exists a globally orbitally asymptotically stable and unique order-1 periodic solution or a unique order-1 periodic solution and an order-2 solution coexists in model (2) for any q∈0,1, and there is no order-kk≥3 periodic solution.Proof.
By Proposition2, it is obvious that there is no order-kk≥3 periodic solution in model (2) because 1−qy+τ>τ≥yO for any y>0. If order-2 periodic solution does not exist, then a globally orbitally asymptotically stable and unique order-1 periodic solution exists using Theorems 2 and 3. If an order-2 periodic solution exists, then the order-2 periodic solution coexists with a unique order-1 periodic solution by Theorem 2.
This completes the proof.Theorem 5.
Under condition (H), if h<x∗, for given τ>0, then all the non-negative solutions of model (2) which start in Ω are ultimately bounded, and the bound depends on parameter τ,q,p, and h.Proof.
By model (2) and the space Ω, obviously, 0<limt⟶+∞supxt≤h and 0≤limt⟶+∞supyt hold. In addition, set ΛN=y|yis the ordinate of phase point and ΛM=y|yis the ordinate of impulsive point. Then, yO1=maxΛM, so we can obtain yO1+=maxΛN by yO1+=1−qyO1+τ. Hence, limt⟶+∞supyt≤yO1+, which obviously depends on τ, q, p, and h.
This completes the proof.
From Theorem5, it is not difficult to find that there is a rough estimate of the bound, and we have the following remark.Remark 1.
Under condition (H), if h<x∗, then 0<limt⟶+∞supxt≤h and 0≤limt⟶+∞supyt<1−qyH+τ.Proposition 3.
Under condition (H), if there exists a point S such that FkS=0 in model (2) when h<x∗, then there exists an order-kf periodic solution, where the positive integer kf is a factor of the positive integer k, including 1 and k.Proof.
Suppose positive integerkf is not a factor of the positive integer k, then let kr be the remainder of k/kf and kq be the quotient of k/kf. Because there exists an order-kf periodic solution in model (2), without loss of generality, assume FkfS=0. Obviously, we have FkqkfS=0, but FkrS≠0. Hence, the equation Fkqkf+krS≠0, which is a contradiction with FkS=0 because k=kqkf+kr. So, the positive integer kf is a factor of the positive integer k. When the positive integer kf is a factor of the positive integer k, FkS=0 is obvious if there exists an order-kf periodic solution in model (2).
This completes the proof.Proposition 4.
Under condition (H), if there exists a point O0 below point O such that F3O0<0 holds when h<x∗, 0<τ<yO, p∈0,1, and q∈0,1, then model (2) has an order-3 periodic solution when F1O>0, and the order-3 periodic solution coexists with an order-1 periodic solution.Proof.
SettingE=N∩x,y|0<x≤h,y=0, we have F1E>0,F2E>0 and F3E>0. Hence, there exists a point O∗∈EO0¯ such that F3O∗=0 because F3O0<0, so an order-kk=1or3 periodic solution exists by Proposition 3. However, owing to F1O>0, it is obvious that there is no order-1 periodic solution in space Π by the proof of Theorem 2. Thus, the periodic solution is the order-3 one. Furthermore, because F1O>0, there exists an order-1 periodic solution according to the proof of Theorem 2, and its phased point is above point O. Therefore, the order-3 periodic solution coexists with an order-1 periodic solution.
This completes the proof.
Although it is not proved that pointO0 must exist in Proposition 4, it provides a method to search an order-3 periodic solution for numerical analysis. Furthermore, the existence of order-3 periodic solution means that order-kk>3 periodic solution and chaos exist in model (2) by Sarkovskii’s theorem [42] and Li and Yorke’s theorem [43].


## 4. Numerical Results

In the previous section, we have analyzed the properties of periodic solution in model (2), including the existence, uniqueness, stability, and boundedness. In this section, we will further show the complex dynamics induced by impulsive control using the numerical simulations. Since the focus of this paper is the role of impulsive control in the population dynamics, the parameters of model (1) are fixed: r=0.9,K=100,b=1.2,a=7,α=0.7,andm=0.7, and we can obtain the equilibrium E∗=35,20.475 with index +1. Obviously, condition (H) always holds under this parameter set.By Theorem1, there only exists a semitrivial periodic solution in model (2) when τ=0 and h<x∗. Given h=3,p=0.6,q=0.5, and τ=0, we get a semitrivial periodic solution as shown in Figure 4(a), and its period is about 1.309. In addition, we also obtain the theoretical value of the period by T=1/rlnK−1−ph/1−pK−h, which is the same as the numerical one. When h=3 and p=0.6, the point O is fixed at 1.2,6.0762. The numerical simulation shows that all the successor points of any trajectory except the semitrivial solution are below initial point in model (2) and these trajectories converge towards the semitrivial periodic solution (see Figure 4(b)).Figure 4
(a) The semitrivial periodic solution of model (2) (the red curve denotes the density of population x and the blue line denotes the density of population y). (b) The proof of trajectory whose successor points are below initial points.
(a)(b)Whenτ>0, however, the semitrivial periodic solution disappears. Then, an order-1 periodic solution emerges and it should be unique by Theorem 2. In phase set N, because the order-1 successor point of any point above point O is below point O1+, we compute the order-1 successor function F1y where y∈0,yO1+ for different values of τ (see Figure 5(a)), where h=32,p=0.1,andq=0.1. In the rest of this section, we will employ this control parameter set unless otherwise specified.Figure 5
(a) The order-1 successor functionF1y for y∈0,yS, where the solid square denotes the value of yS. (b) Order-1 periodic solutions corresponding to roots of F1y=0, where the dashed curves denote the unstable order-1 periodic solutions and the solid curves denote the stable ones.
(a)(b)From Figure5(a), it is not difficult to find that the equation F1y=0 has a unique root, which means that there exists a unique order-1 periodic solution. Hence, we carry out some numerical simulations with respect to these roots of F1y=0, and the results verify that these solutions are order-1 periodic solutions as shown in Figure 5(b). But we find that some of these order-1 periodic solutions are unstable, which indicates that there may exist order-k periodic solution for some values of τ.In order to illustrate the existence of order-k periodic solution, we further compute order-1 successor function, order-2 successor function, and order-3 successor function, and the results are very interesting and are shown in Figure 6(a). From Figure 6(a), it is easy to find that the equation F1y=0 has a unique root, the equation F2y=0 has three roots, and the equation F3y=0 has seven roots. Furthermore, we can find that the root of F1y=0 is the common root of three equations: F1y=0, F2y=0, and F3y=0, which means that there exists an order-1 periodic solution corresponding to this root. Except this root, the equation F2y=0 still has two roots which indicate an order-2 periodic solution exists in virtue of the uniqueness of order-1 periodic solution. For order-3 successor function F3y, the equation F3y=0 has six roots except for the root of F1y=0.Figure 6
(a) The order-1 successor functionF1y, the order-2 successor function F2y, and the order-3 successor function F3y, where the line L denotes line y=yO. (b) An order-1 periodic solution corresponding to the root of F1y=0, an order-2 periodic solution corresponding to roots of F2y=0, and two order-3 periodic solutions corresponding to roots of F3y=0, where the dashed curves denote the unstable periodic solutions and the solid curves denote the stable ones. (c) Time-series of population y corresponding to the periodic solutions in (b), where τ=8.
(a)(b)(c)Except for the root ofF1y=0, it is not difficult to find that F1y and F2y are positive or negative definite in a small neighborhood of each root of F3y=0, which means that there may exist two order-3 periodic solutions in model (2). Thus, we perform some numerical simulations to show the solutions corresponding to these roots. The results show that a stable order-3 periodic solution coexists with an unstable order-3 periodic solution, an unstable order-2 periodic solution, and an unstable order-1 periodic solution (see Figure 6(b)). Additionally, the numerical solutions for prey y are given to further exhibit these periodic solutions, as shown in Figure 6(c).The above results indicate that the order-1 periodic solutions are not always stable. Using order-1 and order-2 successor functions, we computey−value with respect to the phased point of order-1 periodic solution to simulate its variation as parameter τ increases (see Figure 7(a)). From Figure 7(a), the order-1 periodic solution is stable before parameter τ reaches line l1, and the order-1 periodic solution is still stable near l1 when parameter τ is beyond l1. This result is in accord with Theorem 3. Keeping parameter τ increasing, the order-1 periodic solution loses its stability until it is beyond l3. It is well worth noting that model (2) only has order-1 and order-2 periodic solution when parameter τ is between l2 and l3, which agrees with Theorem 4. Furthermore, Figure 7(a) demonstrates that the order-1 periodic solution is unique, which supports the result in Theorem 2. Additionally, Figure 7(a) also shows that y=yτ is a monotonously increasing function with respect to τ, which is consistent with Proposition 1.Figure 7
(a) The existence and stability of the order-1 periodic solution (the solid yellow curve represents stability of the order-1 periodic solution, the dashed red curve represents instability of the order-1 periodic solution, and the solid blue curve represents the stable order-2 periodic solution; the linel1 denotes τ=yS−1−qπt1;O,t0, where t1 is the moment when the trajectory πt1;O,t0 first met line x=h, the line l2 denotes τ=yO, and the line l3 denotes the critical value of parameter τ where the order-1 periodic solution obtain its stability again). (b) The proof of stability of the order-1 periodic solution (the solid squares represent initial points, the lines with circles denote the successor points, and the solid green circle denotes the phase point of order-1 periodic solution).
(a)(b)In addition, our numerical results also indicate that the phased point of order-1 periodic solution rises along phase set as parameterτ increases. In order to present the stability of order-1 periodic solution, we compute two groups of order-k successor points whose initial point is on the sides of the phased point of order-1 periodic solution, respectively. It is revealed that these points monotonously converge towards the phased point of order-1 periodic solution (see Figure 7(b)). Hence, this order-1 periodic solution is stable due to the nonexistence of other periodic solutions.In order to display complex dynamics, we chose parameterτ as the controlled parameter and ran numerous simulations using a wide range of parameter τ to show the bifurcation diagram (see Figure 8(a)). Since the stability of order-1 periodic solution has been discussed above (Figure 7(a)), we will present the dynamics when the order-1 periodic solution is unstable. From Figure 8(a), increased parameter τ can generate chaos by a cascade of period-doubling bifurcation. Keeping parameter τ increasing, an order-3 periodic solution bifurcates from chaos. Subsequently, chaos occurs again with the increase of parameter τ. Finally, chaos disappears again via a cascade of inverse period-doubling bifurcation at higher values in the simulated range of parameter τ. To confirm the occurrence of chaos further, we calculate the largest Lyapunov exponents in the same range of parameter τ as Figure 8(a), which agrees with the bifurcation diagram (see Figure 8(b)). In addition, dozens of time-series of population y for differently initial values are given in Figure 8(c) to further show chaos, which suggests that all states described by these time-series are in disorder.Figure 8
(a) The bifurcation diagram with respect to parameterτ for model (2). (b) The variation of the Lyapunov exponents with respect to the parameter τ, where the symbol “LLE” denotes the largest Lyapunov exponent. (c) Time-series of population y with different initial values for τ=6.
(a)(b)(c)


## 5. Conclusion

In this paper, we proposed an ecological model with state-dependent impulsive control strategy to investigate the effect of impulsive control on prey-predator dynamics. Theoretical results suggest that model (2) has a globally orbitally asymptotically stable semitrivial periodic solution when parameter τ=0 and h<x∗, where condition (H) holds to guarantee existence of positive equilibrium E∗. When τ>0, the semitrivial periodic solution disappears, but the theoretical analysis shows that a unique order-1 periodic solution emerges for any q∈0,1 and p∈0,1.Furthermore, we find that the order-1 periodic solution is globally orbitally asymptotically stable when the value ofτ is below τc. Additionally, when the value of τ is beyond yo, theoretical analysis reveals that a globally orbitally asymptotically stable order-1 periodic solution exists or coexistence of a unique order-1 periodic solution and order-2 periodic solution occurs.Additionally, the existence of order-k is theoretically analyzed based on equation Fky=0, which demonstrates that an order-kf periodic solution exists when Fky=0, and kf is a factor of the positive integer k, including 1 and k. According to this result, we discuss the existence of order-3 periodic solution. Although we cannot get the exact range of parameter τ with respect to the existence of order-3 periodic solution, our results imply that the order-3 periodic solution may exist for τ∈τc,yO, which even means that chaos may occur.In order to study dynamics induced by impulsive control further, a series of numerical simulations are carried out, which agreed with the theoretical results well. Numerical simulations show that order-1, order-2, and order-3 periodic solutions can coexist. Specially, we find the coexistence of double order-3 periodic solutions. Moreover, numerical bifurcation analysis shows that an unstable order-1 periodic solution and a stable order-2 periodic solution can coexist forτ>yo, but there exists only a stable order-1 periodic solution when τ is beyond some critical value. In addition, numerical simulations show that chaos occurs. Obviously, dynamics induced by impulsive control is much richer.The theoretical and numerical results predict that the populationx can be controlled using the state-dependent impulsive control strategies. Because h<x∗, the method by impulsive harvesting population x was a unique choice which can keep the density of population x below the critical value. Our results shows that the control period is much lower when the released amount τ is much smaller, which means that the control frequency is much higher. But from the viewpoint of management for some ecological problems (e.g., pest outbreak, phytoplankton blooms, etc.), it is always hoped that the control period is much longer. Thus, much more predators are needed to be released. However, our results also indicate that chaos may occur for some values of τ, which is unfavourable for prediction of control period. Hence, the released amount of predator should be chosen carefully. Obviously, our results are much more interesting, and we expect that these results are helpful to the management of ecological problems.



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*Source: 1614894-2020-07-06.xml*