# Dynamic Complexity of a Phytoplankton-Fish Model with the Impulsive Feedback Control by means of Poincaré Map

Authors: Dezhao Li; Yu Liu; Huidong Cheng
Journal: Complexity
 (2020)

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## Abstract

The phytoplankton-fish model for catching fish with impulsive feedback control is established in this paper. Firstly, the Poincaré map for the phytoplankton-fish model is defined, and the properties of monotonicity, continuity, differentiability, and fixed point of Poincaré map are analyzed. In particular, the continuous and discontinuous properties of Poincaré map under different conditions are discussed. Secondly, we conduct the analysis of the necessary and sufficient conditions for the existence, uniqueness, and global stability of the order-1 periodic solution of the phytoplankton-fish model and obtain the sufficient conditions for the existence of the order-kk≥2 periodic solution of the system. Numerical simulation shows the correctness of our results which show that phytoplankton and fish with the impulsive feedback control can live stably under certain conditions, and the results have certain reference value for the dynamic change of phytoplankton in aquatic ecosystems.


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## Body

## 1. Introduction

Fisheries can provide people with quality food resources for the survival and development of human beings. Therefore, the healthy development of fishery resources is the focus of attention. If we cannot regulate the capture of fish regularly, it will lead to the depletion of fishery resources. People would be faced with increasing shortages of fish resources. Therefore, it is imperative to formulate effective fishing strategies, maintain the ecological balance of fishery resources, and protect the ecological environment [1–4].Due to the interaction of energy conversion and the nutrient cycle between plankton and herbivores (such as fish), they play an significant role in the most terrestrial and aquatic ecological system. In [5], a phytoplankton and zooplankton model is established by the interaction of nutrients, and the dynamics properties such as limit cycle are investigated in this paper. In [6], a commercially valuable model of phytoplankton and zooplankton predation is proposed, which analyzes the stability of the equilibrium point, explores methods to maintain the ecological balance of the population at different harvest levels, and discusses the impact of selective harvesting on fisheries.Recently, threshold state pulsed dynamic systems have been widely used [7–14]. The geometric theory of impulse dynamical system has been well-developed [15–23]. Many pulse equations have been studied which simulate the ecological processes of populations [24–32]. However, with the further development of the state feedback control model, we need more new methods to find out the complete dynamic properties and control strategies of dynamic systems and to discuss its biological significance. The primary purpose of this paper is to provide a comprehensive qualitative analysis of the global dynamics through analyzing a phytoplankton-fish model with impulse feedback control using the Poincaré map.The central arrangement of this paper is as follows. In Section2, we establish a phytoplankton-fish model for catching fish based on impulse feedback control. In Section 3, the Poincaré map for the phytoplankton-fish model is defined and the properties of monotonicity, continuity, and differentiability of the Poincaré map are analyzed. In Section 4, we discuss the existence, uniqueness, and global stability of the order-1 periodic solution of the model and obtain the conditions for the existence of order-kk≥2 periodic solution of the model. In Section 5, we perform numerical simulations.


## 2. Model Establishment

In the ecosystem of lakes, phytoplankton is considered to be the most favorable source of food for fish or other aquatic animals. Wang et al. [33] propose a predator model of continuously harvested phytoplankton and herbivorous fish:(1)dudt=ru1−uk−βuvα+u−c1Eu,dvdt=β1uvα+u−dv−ρuvα+u−c2Ev,where u and v represent the population density of phytoplankton and fish at the moment, respectively, and E is the effort for continuous harvesting. For details of other parameters, see [34–36].System (1) shows that, without considering the number of phytoplankton and herbivore, continuous capture will lead to resource depletion. We will seek an integrated capture strategy to achieve ecological stability, for which we make the following assumption:(i)
Assuming that phytoplankton and fish stocks are evenly distributed within the lake(ii)
Letr be the intrinsic growth rate of phytoplankton, α be the absorption rate of phytoplankton by fish, β be the conversion rate of biomass, and d denote the mortality rate of fish(iii)
Formulaau/b+u represents the death number of fish due to the distribution of phytoplankton toxicants, where b represents semisaturation constant and a represents the rate at which phytoplankton releases toxinsLetH denote the threshold at which the fish are allowed to be caught, that is, when the density of the fish is lower than the threshold H, it is unreasonable to catch fish; however, only when the density of fish reaches, the predetermined value H can catch the fish. The amount of phytoplankton is affected when the fish is caught. The numbers of the fish and the amount of phytoplankton are updated to vt−τ/1+θvt and ut1−δut/ut+γ, respectively.Here,δ represents the maximum fishing rate, γ represents the half-saturation constant, τ indicates the number of phytoplankton reductions, and θ represents the morphology parameters, where 0<δ<1, γ>0, τ>0, and θ>0 [37].We have established the following impulse feedback control model based on the abovementioned assumptions:(2)dudt=r−αvtutdvdt=βut−d−autb+utvtvt<H,ut+=ut1−δutut+γvt+=vt−τ1+θvtvt=H.System (2) is called a semicontinuous dynamic system [38–40].When there is no impulse, system (2) becomes(3)dudt=r−αvtut,dvdt=βut−d−autb+utvt.Lemma 1 (see [37]).
System (3) has two equilibrium points, O(0,0) and E∗u∗,v∗, where the boundary equilibrium point 0,0 is the saddle point and the internal balance point u∗,v∗ is a stable center:(4)u∗=d+a−βb+d+a−βb2−4βbd2β,v∗=rα.It is easy to calculate, the first integral of system (3) is(5)Hu,v=βu−dlnu−alnb+u−rlnv+αv+H0,where H0 is the constant related to the initial value. For ease of expression, two isoclines are defined as(6)L1:ν=rα;L2:u=d+a−βb+d+a−βb2−4βbd2β.To study the dynamic properties of system (2), the following research methods are given.


## 3. Poincaré Map of System (2) and Its Properties

We intersect the dynamic properties of model (2) in R2+=u,v,u≥0,v≥0 based on biological significance. In order to accurately define the impulse set and phase set of the pulsed semidynamic system (2), the following set is given first:(7)ΣN=u,vu≥0,v=H−τ1+θH,ΣM=u,vu≥0,v=H.Apparently, withinR2+, ΣN and ΣM represent two straight lines of the vertical v-axis, respectively, and we assume ΣN and ΣM intersect the line L2 at point A+ and point A−, respectively, since the internal balance point E∗ is the center point. According to the size of H, we can divide the pulse set and phase set into the following cases:Case I:H<v∗WhenH<v∗, the trajectory ΓA+ must intersect the ΣM at point A11u11,H (see Figure 1(a)). In this case, impulse sets(8)M1=u,vv=H,u≥u11, and the corresponding phase set is(9)N1=u,vv=H−τ1+θH,u≥u111−δu11u11+γ.Case II:H>v∗WhenH>v∗, if the trajectory ΓA+ intersects the ΣM at a point, we assume this intersect is A22u22,H (see Figure 1(b)). In this case, the impulse set and the phase set are the same as those in Case I. It is easy to infer the impulse set(10)M2=u,v,v=H,u≥u22,and the corresponding phase set(11)N2=u,vv=H−τ1+θH,u≥u221−δu22u22+γ.Figure 1
The domain of phase set and pulse set in three cases. The parameter values are as follows:r=1.444, α=0.1, β=0.15, d=0.5, a=0.5, b=1, δ=0.8, and γ=1. (a) H=12, θ=0.02, and τ=6.2. (b) H=17, θ=0.01, and τ=11.7. (c) H=24, θ=0.01, and τ=17.36.
(a)
(b)
(c)Figure 2
The Poincare´ map Gmu related to the impulsive point series u. The parameter values are as follows: r=1.444, α=0.1, β=0.15, d=0.5, a=0.5, b=1, δ=0.8, and γ=1. (a) H=12, θ=0.02, and τ=6.2. (b) H=15, θ=0.01, and τ=15.4.
(a)
(b)Figure 3
The Poincare´ map Gmu related to the impulsive point series u. The parameter values are as follows: r=1.444, α=0.1, β=0.15, d=0.5, a=0.5, b=1, δ=0.8, and γ=1. (a) H=18, θ=0.02, and τ=16.2. (b) H=24, θ=0.01, and τ=17.36.
(a)
(b)If the trajectoryΓA+ and ΣM does not intersect (see Figure 1(c)), the trajectory ΓA− intersects the ΣN at point A31u31,H−τ/1+θH and point A32u32,H−τ/1+θH, respectively, where u32>u31.In this case, the impulse set(12)M3=u,vv=H,u≥u∗,and the corresponding phase set(13)N3=u,vv=H−τ1+θH,u≥u∗1−δu∗u∗+γ.Based on the abovementioned discussion, we define the Poincaré map as follows.Let(14)Ak+=uk+,vk+=uk+,H−τ1+θH∈ΣN,where 0<uk+<+∞; the trajectory(15)πt,t0,uk+,H−τ1+θHΔ¯¯ut,t0,uk+,H−τ1+θH,vt,t0,uk+,H−τ1+θH,which goes through point Ak+ will reach the ΣM at point(16)Ak+1t1,t0,uk+1,H=ΔAk+1uk+1,H,after time t1, and here(17)vt1,t0,uk+,H−τ1+θH=H.Then, there is(18)uk+1+=ut1,t0,uk+,H−τ1+θHΔ¯¯uuk+,H−τ1+θHΔ¯¯=Puk+,which indicates that the ordinate uk+1+ is determined by uk+.Since pointAk+1 is on the impulse set, Ak+1 jumps to point(19)Ak+1+=uk+1+,H−τ1+θH,where(20)uk+1+=1−δuk+1uk+1+γuk+1=1−δPuk+Puk++γPuk+Δ¯¯=Gmuk+.Consider the scalar differential equation of model (3):(21)dudv=r−αvuβu−d−au/b+uv≜wu,v,uH−τ1+θH=u0+.Letv0+=H−τ/1+θH and u0+=S, then we have u0+,v0+ in ΣN.We define(22)μx=ux;H−τ1+θH,SΔ¯¯μx,S,H−τ1+θH≤x≤H.According to model (21),(23)μx,S=S+∫H−τ/1+θHxωs,us,Sds.From (20) and (23), the Poincare´ map expression of system (2) is(24)GmS=1−δμH−τ/1+θH,SμH−τ/1+θH,S+γμH−τ1+θH,S.The properties of the Poincare´ map Gms is discussed below.Theorem 1.
LetH<v∗, trajectory ΓA+ intersects with line ΣM at point A11u11,H and point A12u12,H, respectively, where u11>u12. Poincare´ map Gms has the following properties:(i)
The domain ofGms is 0,+∞, Gms is monotonically decreasing on 0,u∗, and monotonically increasing on u∗,+∞.(ii)
Gms is continuously differentiable on 0,+∞.(iii)
WhenGmu∗>u∗, there is a unique fixed point on u∗,+∞ (see Figure 2(b)). When Gmu∗<u∗, there is a unique fixed point on 0,u∗ (see Figure 2(a)). When Gmu∗=u∗, u∗ is the fixed point of Gms.Proof.
(i)
SinceE∗u∗,v∗ is the center point and H<v∗, take any point Ak+uk+,vk+ on ΣN, and the trajectory of point Ak+ will reach the impulse set M1 at point Ak+1uk,H, so the domain of Gms is 0,+∞.
For anyuk1+,uk2+∈u∗,+∞ and uk1+<uk2+, from the uniqueness of the solution of the differential equation, it can be known that(25)μuk1+,v<μuk2+,v,H−τ1+θH≤v≤H,
and by the definition of the Poincare´ map we can get Gmuk1+<Gmuk2+. Therefore, Gms is monotonically increasing on u∗,+∞.
Whenuk1+, uk2+∈0,u∗, where uk1+<uk2+. The trajectories start from point Ak1+uk1+,H−τ/1+θH, and point Ak2+uk2+,H−τ/1+θH will pass through the isocline L2 and intersect the set ΣN at point Ak11+uk11+,H−τ/1+θH and point Ak21+uk21+,H−τ/1+θH, respectively, where uki1+i=1,2∈u∗,+∞ and uk11+>uk21+. It can be seen from the uniqueness of the solution of the differential equation that Gmuk11+>Gmuk21+ because Gmuk11+=Gmuk1+ and Gmuk21+=Gmuk2+, so Gmuk1+>Gmuk2+; therefore, Gms is monotonically decreasing on 0,u∗.
Note 1. BecauseGms is monotonically decreasing on 0,u∗ and monotonically increasing on u∗,+∞, it is easy to know that Gms≥Gmu∗ is true for any s∈0,+∞, and Gms=Gmu∗ if and only if s=u∗.
(ii)
It is easy to know thatwu,v is continuously differentiable in the first quadrant from (21), and from the continuous differentiability theorem of differential equations, that is, the Cauchy and Lipschitz theorem with parameters, we know that the Poincare´ map Gms is continuously differentiable in the first quadrant.
(iii)
Considering the Poincare´ map at the position of image Gmu∗ of u∗, there are three cases:(a)
WhenGmu∗=u∗, then u∗ is the fixed point of function Gms.(b)
WhenGmu∗>u∗, that is, Gmu∗−u∗>0 because 0<δ<1, then(26)lims⟶∞Gms=s1−δss+γ=s1−δ−δγs+γ<s1−δ<s.
That is,lims⟶∞Gms−s<0. So, it has at least one u˜∈u∗,+∞ satisfies Gmu˜=u˜, according to the continuous differentiable of the closed interval.
(c)
WhenGmu∗<u∗, let Gmu∗=u1<u∗, we know that Gms is monotonically decreasing on 0,u∗, so Gmu1>Gmu∗=u1, and because Gmu∗<u∗, so there is at least one u˜∈u1,u∗ which satisfies Gmu˜=u˜, according to the continuous differentiable of the closed interval.
In conclusion,Gms has at least one fixed point. Then, we prove the fixed point is unique.
We assume that system (2) has two fixed points, u˜1 and u˜2, that is to say Gmu˜1=u˜1 and Gmu˜2=u˜2. Let u˜1<u˜2, we define(27)du˜1u˜2u=uv,u˜2−uv,u˜1,H−τ1+θH≤v≤H.
Then, take the derivative of the abovementioned formula:(28)du˜1u˜2′u=u′v,u˜2−u′v,u˜1=r−αvvu˜2βu˜2−d−au˜2/b+u˜2−u˜1βu˜1−d−au˜1/b+u˜1.
Let(29)gu=uβu−d−au/b+u.
Then,(30)g′u=−d−au/b+u2βu−d−au/b+u2<0,so(31)gu˜2<gu˜1,that is,(32)du˜1u˜2′u<0,and(33)du˜1u˜2H−τ1+θH>du˜1u˜2H.
From system (2),(34)u˜1=μu˜11−δμu˜1μu˜1+γ=μu˜11−δ+δγμu˜1+γ=μu˜2−du˜1u˜2H1−δ+δγμu˜1+γ=μu˜21−δ+δγμu˜1+γ−du˜1u˜2H1−δ+δγμu˜1+γ>μu˜21−δ+δγμu˜2+γ−u˜1du˜1u˜2Hμu˜1=u˜2−u˜1du˜1u˜2Hμu˜1,that is,(35)u˜1du˜1u˜2Hμu˜1>u˜2−u˜1=du˜1u˜2H−τ1+θH.
It is easy to know thatu˜1/μu˜1<1 if H<v∗, so(36)du˜1u˜2H>u˜1du˜1u˜2Hμu˜1>du˜1u˜2H−τ1+θH.
It is contradictory withdu˜1u˜2H−τ/1+θH>du˜1u˜2H, so the fixed point is unique.
Note 2. WhenH>v∗ and trajectory ΓA+ is an intersect to line ΣM, the Poincare´ map Gms of system (2) has similar properties to case (a).Theorem 2.
LetH>v∗, and trajectory ΓA+ does not intersect line ΣM and trajectory ΓA− intersects the straight line ΣN at point A31u31,H−τ/1+θH and point A32u32,H−τ/1+θH, respectively, where u32>u31. Poincare´ map Gms has the following properties:(i)
The domain ofGms is 0,u31∪u32,+∞, where Gms is monotonically decreasing on 0,u31 and monotonically increasing on u32,+∞.(ii)
Gms is continuously differentiable on 0,u31 and u32,+∞, respectively.(iii)
WhenGmu31≤u31, there is a unique fixed point on 0,u31 (see Figure 3(a)). When Gmu31>u31, there is no fixed point (see Figure 3(b)).Proof.
(i)
SinceE∗u∗,v∗ is the center point. If H>v∗ and trajectory ΓA+ does not intersect line ΣM, trajectory ΓA− intersects the straight line ΣN at two points, that is, A31u31,H−τ/1+θH and A32u32,H−τ/1+θH, respectively, where u32>u31, then we take any point Ak+uk+,vk+ in ΣN. If uk+∈0,u31∪u32,+∞, the trajectory of point Ak+ will reach the impulse set M3 at point Ak+1uk,H, if uk+∈u31,u32 the trajectory of point Ak+ has no intersection with the impulse set M3. So, the domain of Gms is 0,u31∪u32,+∞.
For anyuk1+,uk2+∈u32,+∞ and uk1+<uk2+, from the uniqueness of the solution of the differential equation, it can be known that(37)μuk1+,v<μuk2+,v,H−τ1+θH≤v≤H,
and by the definition of the Poincare´ map we get Gmuk1+<Gmuk2+. Therefore, Gms is monotonically increasing on u32,+∞.
Whenuk1+, uk2+∈0,u31, and where uk1+<uk2+. The trajectories start from point Ak1+H−τ/1+θH,uk1+ and point Ak2+H−τ/1+θH,uk2+ will pass through the isocline L2 which intersects the ΣN at point Ak11+H−τ/1+θH,uk11+ and point Ak21+H−τ/1+θH,uk21+, respectively. Then, uki1+i=1,2∈u32,+∞ and uk11+>uk21+. It can be seen from the uniqueness of the solution of the differential equation that Gmuk1+>Gmuk2+; therefore, Gms is monotonically decreasing on 0,u31.
(ii)
From (21), we can know that wu,v is continuously differentiable in the first quadrant; from the continuous differentiability theorem of differential equations, that is, the Cauchy and Lipschitz theorem with parameters, we know that the Gms is continuously differentiable on 0,u31 and u32,+∞, respectively.
(iii)
Considering the Poincare´ map at the position of image Gmu31 of u31, there are two cases:(a)
WhenGmu31≤u31 (see Figure 3(a)), we assume Gmu31=u1≤u31, and we know that Gms is monotonically decreasing on 0,u31, so Gmu1≥Gmu31=u1, and because Gmu31≤u31, so there is a point u˜∈u1,u31, and it satisfies Gmu˜=u˜, according to the continuous differentiability of the closed interval.(b)
WhenGmu31>u31 (see Figure 3(b)), there is no u˜∈0,u31 which satisfies Gmu˜=u˜.
For anyuk∈u32,+∞, the trajectory of point Akuk,H−τ/1+θH is tangent to the straight line ΣM at point Ak+uk+,H; Ak+uk+,H−τ/1+θH will be pulsed to Ak+1uk+1,H−τ/1+θH. Easy to get uk+1<uk+<uk, that is, uk+1≠uk, so there is no u∈u32,+∞ and satisfies Gmu˜=u˜.
In a word, whenGmu31≤u31, there is a unique fixed point on 0,u31. When Gmu31>u31, there is no fixed point.


## 4. The Order-kk≥1 Periodic Solution of the Semicontinuous Dynamic System (2) and Its Stability

From Theorem1, we know that system (2) has a unique fixed point under certain conditions. That is, system (2) has a unique order-1 periodic solution, and the following are the dynamic properties of system (2).Theorem 3.
H
<
v
∗ and Gmu∗>u∗ are established, then the order-1 periodic solution of system (2) is globally asymptotically stable.Proof.
From (iii) of Theorem1, we know that when Gmu∗>u∗⊗, Gms has a unique fixed point u˜ on u∗,+∞, i.e., Gmu˜=u˜. For any point p0+H−τ/1+θH,u0+ in ΣN, where u0+>u∗, the trajectory of the point p0+ will intersect the impulse set, then reach the point p1+H−τ/1+θH,u1+, which is Gmu0+=u1+, repeating the abovementioned process:(38)GmGmu0+=Gm2u0+,that is,(39)Gmu1+=u2+.
Furthermore,(40)un+=Gmnu0+,n=1,2,….
The following two cases are discussed below according to the size ofu0+:1)
Whenu∗<u0+≤u˜, since Gmu∗>u∗ and Gms is monotonously increasing on u∗,+∞, let Gmui+=ui+1+, then(41)u0+<Gmu0+=u1+≤Gmu˜=u˜,u0+<Gmu0+<Gmu1+=Gm2u0+≤Gmu˜=u˜.Repeating the abovementioned process, we can obtain(42)u0+<Gmu0+<⋯<Gmnu0+<⋯<u˜.From the monotonous boundedness of the sequence, we can obtain(43)limn⟶+∞Gmnu0+=u˜.(2)
Whenu˜<u0+<+∞, from the known conditions,(44)u˜=Gmu˜<Gu0+.Furthermore,(45)u˜=Gmu˜<Gm2u0+<Gu0+.By mathematical induction,(46)u˜=Gmu˜<⋯<Gmn∈u0+<Gmn−1u0+<⋯.Thus,(47)limn⟶+∞Gmnu0+=u˜.(3)
When0<u0+<u∗, since Gmu∗>u∗ and Gms is monotonously decreasing on 0,u∗, so for any u0+∈0,u∗ there is Gmu0+>Gmu∗>u∗. So, we can conclude that Gmu0+>u∗, and we can convert this to two cases based on the size of Gmu0+. When u∗<Gmu0+<u˜, this is the same as case (1) above; when Gmu0+>u˜, this is the same as case (2) above. In both cases,(48)limn⟶+∞Gmnu0+=u˜.
The order-1 periodic solution of system (2) is globally asymptotically stable.Theorem 4.
WhenH<v∗, Gmu∗<u∗ and Gm2u∗<u∗ are true, and then the semicontinuous dynamical system (2) has a stable order-1 periodic solution or order-2 periodic solution.Proof.
Take a pointp0+u0+,H−τ/1+θH on the phase set, here u0+>0, since E∗ is the center of system (2). The trajectory that goes through point p0+ must intersect Tex translation failed at point p1u1,H, where u1=μu0+,H. p1 will reach point p1+u1+,H−τ/1+θH by an impulse, so Gmu0+=u1+, repeating this process to get the sequence(49)un+=Gmnu0+=GmGmn−1u0+,n=1,2…,
Whenu∗≤u0+<+∞, we can know that Gms is monotonically increasing on u∗,+∞ and there is no fixed point on u∗,+∞ from (i) of Theorem 1. Therefore, there must be a positive integer i, which satisfies(50)ui−1+=Gmui−2+<u∗,ui+=Gmui−1+<Gmu∗<u∗.
When0<u0+<u∗, the trajectory through the initial point p0+u0+,H−τ/1+θH will pass through the isocline L2 and intersects Tex translation failed at point p0′u0′,H−τ/1+θH, where 0<u0′<u∗; this translates into the abovementioned situation.
This means that, for anyu0+∈0,+∞, there is always be a positive integer i which satisfies(51)Gmu∗<Gmiu0+<u∗i≥1.
So, we only need to take account of the initial pointp0+u0+,H−τ/1+θH, where u∗≤u0+<Gmu∗. Since Gms is monotonically decreasing on Gmu∗,u∗, so(52)GmGmu∗,u∗⊂Gmu∗,u∗.
LetGmu0+≠u0+ and Gm2u0+≠u0+, that is, the trajectory with the initial point p0+ is not the order-1 periodic solution or order-2 periodic solution of system (2). We consider the following four situations:Case I:u∗≥u0+>Gm2u0+>Gmu0+≥Gmu∗.According to the monotonicity of the Poincare´ map,(53)u2+=Gmu1+>Gmu2+=u3+>Gmu0+=u1+.Furthermore,(54)u4+=Gmu3+<Gmu0+=u0+.Thus, there is(55)u∗≥u0+>u2+>u4+>u3+>u1+≥Gmu∗.Proved by mathematical induction,(56)u∗≥u0+>u2+>⋯>u2n+>u2n+2+>⋯>u2n+1+>u2n−1+>⋯>u1+≥Gmu∗.Case II:u∗≥u1+>u0+>u2+≥Gmu∗.BecauseGms is monotonically decreasing on Gmu∗,u∗. We can obtain(57)u3+=Gmu2+>Gmu0+=u1+>Gmu1+=u2+,u4+=Gmu3+<Gmu1+=u2+<Gmu2+=u3+.Then,(58)u∗≥u3+>u1+>u0+>u2+>u4+≥Gu∗,so(59)u∗≥⋯>u2n+1+>u2n−1+>⋯>u3+>u1+>u0+>u2+>u4+>⋯>u2n+>⋯≥Gu∗.Case III:u∗≥u1+>u2+>u0+≥Gmu∗. In the same way, we can deduce(60)u∗≥u1+>u3+>…>u2n−1+>…>u2n+1+>u2n+>…>u2+>u0+≥Gmu∗.Case IV:u∗≥u2+>u0+>u1+≥Gmu∗.
We can perform the procedure similar to Case I, which can yield\(61)Gmu∗≤u2n+1+<u2n−1+<⋯<u1+<u0+<u2+<⋯<u2n+<u2n+1+<⋯≤u∗.
Considering Cases I and III, there must existu˜∈u∗,Gmu∗ so that(62)limn⟶∞u2n+=limn⟶∞u2n−1+=u˜,which means that system (2) has a stable order-1 periodic solution.
For Cases I and IV, there are two pointsu1˜≠u2˜ and(63)limn⟶∞u2n−1+=u˜1,limn⟶∞u2n+=u˜2.
This shows that system (2) has a stable order-2 periodic solution with the initial points u1˜,H−τ/1+θH andu2˜,H−τ/1+θH.
Theorem4 illustrates that system (2) has a stable order-1 or order-2 periodic solution under certain conditions. However, sufficient and necessary conditions for global stability are not given. Then, we give below theorem.Theorem 5.
LetH1<v∗ and Gmu∗<u∗, then the necessary and sufficient conditions for the global stability of the order-1 periodic solution of system (2) is Gm2u+<u+, for any u+∈0,u∗.Proof.
Sufficiency: It can be seen from Theorem4 that when Gmu∗<u∗, there exists u˜∈Gmu∗,u∗ which satisfies Gmu˜=u˜.
For anyu+∈u˜,u∗, let u1+=Gmu+ and u2+=Gmu1+=Gm2u+ because of Gm2u+<u+<u∗, and from the monotonicity of Gmu, we can get u˜>u1+>Gmu∗; furthermore, u∗>u2+>u4+>u˜>u3+>u1+>Gmu∗, so u∗>u2n+>u˜>u2n+1+>Gmu∗.
By the monotonic boundness of the sequence,(64)limn⟶∞u2n+=limn⟶∞u2n+1+=u˜.
Therefore, the order-1 periodic solution of system (2) is globally asymptotically stable.
Necessity:
We assume that the order-1 periodic solution of system (2) is globally asymptotically stable. Then, the following part proves that Gm2u+<u+ is right for any u+∈u˜,u∗, which is proved by the counterevidence method below.
IfGm2u+<u+ is not true for any u+∈u˜,u∗, then there exists a maximum u0∈u˜,u∗, which satisfies Gm2u0≥u0. For any ε>0, there exist u1 and Gm2u1<u1, which is true for any u˜−ε<u1<u˜+ε from Theorem 4. From the continuity of Gm2u on the interval u0,u1, it follows that there is at least one number u⟶∈u0,u1 and Gm2u⟶=u⟶, which indicate that the trajectory with u⟶,H−τ/1+θH as the initial point is the order-2 periodic solution, and this is contradictory.Theorem 6.
IfGmu∗<u∗ and there exist um+=minu+:Gmu+=u∗, when Gm2u∗<um+, then system (2) has an order-3 periodic solution.Proof.
IfGmu∗<u∗, there is a unique order-1 periodic solution in Gmu∗,u∗, i.e., Gmu˜=u˜, where u˜∈Gmu∗,u∗. Since the Poincare´ map is continuous over the closed interval 0,u˜, and Gmu˜=u˜, then there exist um+∈0,u˜ and Gmum+=u∗. Furthermore, Gm3um+=Gm2u∗<um+; on the contrary, limx⟶0Gm3x>x; according to the nature of the continuous function on the closed interval, at least one value of u⟶ satisfies Gm3u⟶=u⟶. This means that system (3) has an order-3 periodic solution with an initial point of u⟶,H−τ/1+θH.
In a similar way, ifGmk−1u∗<um+, where Gmum+=u∗, then system (2) has an order-kk≥2 periodic solution.


## 5. Simulations

Figures4(a) and 4(b), respectively, show the order-1 periodic solutions simulated under the two conditions of Figures 2(a) and 2(b), in which the blue line represents the trajectory when the order periodic solution is not reached and the red line represents the trajectory when the order periodic solution is reached. This suggests that populations of phytoplankton and fish can be kept within a stable range with the impulsive feedback control.Figure 4
(a) The path curve of Figure2(a) starting from the points (13,7); H=12, θ=0.02, and τ=6.2. (b) The path curve of Figure 2(b) starting from the points (17,7), H=22, θ=0.01, and τ=17.36. The parameters fixed as r=1.444, α=0.1, β=0.15, d=0.5, a=0.5, b=1, δ=0.8, and γ=1.
(a)
(b)We simulated the order-1 periodic solution and order-2 periodic solution in the case of Figure3(a) (see Figure 5). In the simulated Poincare´ map, we found the order-1 periodic solution (the intersection point of the red and yellow line and blue line in Figure 5(a)) and the order-2 periodic solution (the intersection point of the black line and red and yellow line in Figure 5(a)). Figure 5(b) shows the motion trajectories of the order-1 periodic solution and the order-2 periodic solution, in which the blue line is the motion trajectory of the order-2 periodic solution and the red and yellow line is the motion trajectory of the order-1 periodic solution.Figure 5
The positions of the order-1 periodic solution and order-2 periodic solution were found in the case of Figure3(a).
(a)
(b)Figures6(a) and 6(b) are schematic diagrams of the number of phytoplankton and the number of fishes and time t in the order-2 periodic solution in Figure 5(b), respectively. Figures 6(c) and 6(d) are schematic diagrams of the planktonic quantity and the fish quantity and time t in the order-1 periodic solution in Figure 5(b), respectively. As it is revealed in the figure, the number of fish in the order-2 periodic solution and the order-1 periodic solution both change periodically, with a period of one. The number of phytoplankton in the order-2 periodic solution and the order-1 periodic solution are both changing periodically, in which the number of plankton in the order-2 periodic solution has a period of two. The period of the number of phytoplankton in the order-1 periodic solution is one.Figure 6
The time series of phytoplankton density (a) and fish density (b) under order-2 periodic solution of Figure5(b), and the time series of phytoplankton density (c) and fish density (d) under order-1 periodic solution of Figure 5(b).
(a)
(b)
(c)
(d)In Figure7, the blue and red lines indicate the trajectory of the system with or without pulses. This shows that the number of phytoplankton and fish can be kept in a stable range.Figure 7
The time series of phytoplankton density (a), fish density (b), and phase portrait (c) starting from (7,2):r=1.444, α=0.1, β=0.15, d=0.5, a=0.5, b=1, δ=0.8, γ=1, H=10, θ=0.01, and τ=17.36.
(a)
(b)
(c)As can be revealed in Figure8, different initial points will eventually converge to the same order-1 periodic solution and tend to be stable. This indicates the global asymptotic stability of the order-1 periodic solution.Figure 8
The path curve of system (2) starting from the points (1,2), (16,2), and (20,2), where r=0.6, α=0.1, β=0.15, d=0.5, a=0.5, b=1, δ=0.8, γ=1, H=10, θ=0.01, and τ=17.36.


## 6. Conclusion

How to rationally develop and utilize fishery resources has become an essential issue for the sustainable development of lake resources. In this paper, we introduce the impulse feedback control into the phytoplankton-fish model. It is of great significance to study the dynamics of phytoplankton and fish.Compared with [1], we use the Poincare´ map as a tool to give a more comprehensive qualitative analysis and to prove the dynamics of the phytoplankton-fish model, for example, the necessary and sufficient conditions for the global asymptotic stability of the order-1 periodic solution and the existence conditions of the order-kk≥2 periodic solution.For the biological significance, the order-kk≥1 periodic solution of system with state impulsive feedback control indicates that the number of phytoplankton and fish populations can maintain periodic oscillations under certain conditions with proper capture of fish, that is, the number of phytoplankton and fish can be kept in a stable range. These results have some reference values to the dynamic changes in aquatic ecosystem research of fish and phytoplankton.



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*Source: 8974763-2020-04-15.xml*