# Further Results on Bifurcation for a Fractional-Order Predator-Prey System concerning Mixed Time Delays

Authors: Zhenjiang Yao; Bingnan Tang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In the present work, we mainly focus on a new established fractional-order predator-prey system concerning both types of time delays. Exploiting an advisable change of variable, we set up an isovalent fractional-order predator-prey model concerning a single delay. Taking advantage of the stability criterion and bifurcation theory of fractional-order dynamical system and regarding time delay as bifurcation parameter, we establish a new delay-independent stability and bifurcation criterion for the involved fractional-order predator-prey system. The numerical simulation figures and bifurcation plots successfully support the correctness of the established key conclusions.


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## Body

## 1. Introduction

Setting up mathematical models to describe the natural phenomena has become an important topic in real life. The interaction of predator population and prey population plays a significant role in maintaining ecological balance in nature. In order to grasp the change law of predator population and prey population, a large number of predator-prey models have been established and many fruits on dynamical properties of various predator-prey models have been reported. Usually, time delay often exists in biological systems due to the lag of the response of different predators and preys. In many situations, time delay will lead to the loss of stability, periodic oscillation, bifurcation, and chaotic behavior of predator-prey models. Thus, the study on the impact of time delay on dynamical nature of predator-prey models has attracted great interest of many scholars in the fields of biology and mathematics. For a long time in the past, lots of valuable works on predator-prey models have been published. For instance, Dubey et al. [1] investigated the stability behavior, Hopf bifurcation, and chaos of delayed predator-prey system. Ren and Shi [2] dealt with the global boundedness and stability of solutions of a predator-prey system with time delay. Li and Guo [3] introduced a new way to study the permanence and extinction for a stochastic prey-predator system involving functional response. Alsakaji et al. [4] made a detailed discussion on permanence, local and global stabilities, Hopf bifurcation, and a predator-prey model with time delay. For more publications about this topic, one can see [5–8].Here we notice that the works of [1–8] are concerned with the integer-order predator-prey models. In recent years, fractional-order dynamical systems have found potential application in numerous areas such as all sorts of physical waves, neural network systems, biological technique, finance, automatic control, and so on [9–11]. A lot of researchers think that fractional-order dynamical system can more accurately describe the real phenomenon in realistic world than the classical integer-order ones due to its owned memory trait and hereditary nature [12]. Nowadays a great deal of valuable works on fractional-order dynamical systems have been published (see [13–22]). In particular, the study on fractional-order predator-prey systems is also continuously displayed. For example, Yousef and Chandan Maji [23] revealed the effect of fear for a fractional-order predator-prey model. Xie et al. [24] proved the non-negative and boundedness of a fractional-order predator-prey model and established some conditions to ensure the existence and stability of the positive equilibrium point of the fractional-order predator-prey model. In 2019, Zhou et al. [25] considered the bifurcation control issue for a fractional-order predator-prey system involving delays. For more details, one can see [26–28].Hopf bifurcation caused by time delay is a vital dynamical phenomenon in predator-prey systems. Up to now, plenty of publications on Hopf bifurcation of integer-order predator-prey models have been available. The impact of time delay on Hopf bifurcation has been revealed. However, the investigation on Hopf bifurcation for fractional-order predator-prey models is comparatively few. Recently, some scholars are devoted to Hopf bifurcation of fractional-order predator-prey models and some valuable fruits have been derived. For instance, Alidousti [29] investigated the stability and Hopf bifurcation problem of a fractional predator-prey system. Yuan et al. [30] established a set of sufficient conditions to ensure the stability and the onset of Hopf bifurcation for a fractional-order predator-prey model. Wang et al. [31] discussed the stability and bifurcation for a generalized fractional-order predator-prey system involving time delay and interspecific competition. Huang et al. [32] applied a new technique to control Hopf bifurcation of a fractional predator-prey system involving delays. In 2019, Xu et al. [42] did a very valuable work on Hopf bifurcation for delayed neural networks. As to more works about this theme, we refer the readers to [33–36].Up to now, the investigation on Hopf bifurcation of fractional-order delayed predator-prey systems merely involves discrete time delay. To reflect the time lag of response of predator population and prey population during the course of interaction of predator and prey in biological systems, it is very essential to introduce the distributed time delay into predator-prey models. Now there are only very few works on Hopf bifurcation of predator-prey system involving distributed time delay. Thus, a natural problem arises: what is the impact of distributed time delay on Hopf bifurcation of predator-prey system involving distributed time delay? This motives us to deal with the Hopf bifurcation for predator-prey system involving distributed time delay.In 2020, Rahman et al. [37] investigated the following predator-prey system concerning both types of delays:(1)dw1tdt=w1ta1−α11∫−∞tUt−vw1vdv−α12w2t−ς,dw2tdt=w2t−a2+α21w1t−ς−α22∫−∞tUt−vw2vdv,where w1t denotes the population density of prey at time t and w2t stands for the population density of predator at time t, a1>0 stands for the growth rate of the prey populations without predators, a2>0 stands for the death rate of the predator populations without prey, α11 represents the self-regulation rate for the prey, α12 represents the rate of predation of the prey by predators, α21 represents the conversion rate of predators and α22 represents the intraspecific competition among predators, U. denotes the non-negative continuous delay kernel which is defined on 0,∞ and is integrable on 0,∞, and ς≥0 denotes the feedback time delay between the predator and the prey. For details, see [37].Usually, the kernel function owns the following two forms:(i)
∫−∞tUvdv=1,Uv=δe−δt−v,δ>0.(ii)
∫−∞tUt−vdv=1,Uv=δe−δv,δ>0.Rahman et al. [37] chose kernel function as case (ii). By means of stability criterion and Hopf bifurcation theory of delayed differential equation, Rahman et al. [37] established a sufficient criterion ensuring the stability and the appearance of Hopf bifurcation of model (1). Meanwhile, the concrete formula determining bifurcation peculiarities is presented by virtue of center manifold theory and normal form theorem.Inspired by the analysis above, we are to analyze the stability and Hopf bifurcation for fractional-order predator-prey model involving discrete time delay and distributed time delay. On the basis of the research of Rahman et al. [37], in this work, we revise model (1) as the fractional-order form:(2)dw1ϱtdtϱ=w1ta1−α11∫−∞tUt−vw1vdv−α12w2t−ς,dw2ϱtdtϱ=w2t−a2+α21w1t−ς−α22∫−∞tUt−vw2vdv,where 0<ρ<1 is a constant, w1t denotes the population density of prey at time t and w2t stands for the population density of predator at time t, a1>0 stands for the growth rate of the prey populations without predators, a2>0 stands for the death rate of the predator populations without prey, α11 represents the self-regulation rate for the prey, α12 represents the rate of predation of the prey by predators, α21 represents the conversion rate of predators and α22 represents the intraspecific competition among predators, U. denotes the non-negative continuous delay kernel which is defined on 0,∞ and is integrable on 0,∞, and ς≥0 denotes the feedback time delay between the predator and the prey. For more implication of the parameters in system (2), one can see [37]. In this research, we choose the kernel function U. as (ii).This article is organized as follows. Section2 lists several necessary theories about fractional-order dynamical system. Section 3 gives the bifurcation condition for model (2) involving kernel function (ii). Section 4 presents simulation plots to support the validity of the obtained key conclusions. Section 5 ends this article.


## 2. Basic Principle on Fractional-Order Dynamical System

In this section, we present some indispensable basic knowledge about fractional-order dynamical system.Definition 1.
(see [38]). The Caputo-type fractional-order derivative is given by(3)Dϱwτ=1Γk−ϱ∫τ0τwkuτ−uϱ−k+1du,where wτ∈τ0,∞,R,Γu=∫0∞τu−1e−τdτ, τ≥τ0,k∈Z+,k−1≤ϱ<k.Lemma 1 (see [39, 40]).
For the fractional-order model(4)dϱvtdtϱ=wt,vt,v0=v0,where 0<ϱ≤1 and wt,vt:R+×Rn⟶Rn. Let v0 be the equilibrium point of (4). We say that v0 is locally asymptotically stable provided that each eigenvalue λ of ∂wt,v/∂v|v=v0 obeys argλ>ϱπ/2.Lemma 2 (see [41]).
For the fractional-order model(5)dϱ1V1tdtϱ1=f11V1t−ς11+f12V2t−ς12+⋯+f1lVlt−ς1l,dϱ2V2tdtϱ2=f21V1t−ς21+f22V2t−ς22+⋯+f2lVlt−ς2l,⋮dϱlVltdtϱl=fl1V1t−ςl1+fl2V2t−ςl2+⋯+fllVlt−ςll,where ϱj∈0,1j=1,2,…,l, denote(6)Δs=sϱ1−f11e−sς11−f12e−sς12⋯−f1le−sς1l−f21e−sς12s>2−f22e−sς22⋯−f2le−sς2l⋮⋮⋱⋮−fl1e−sςl1−fl2e−sςl2⋯s>l−flle−sςll.
We say that the zero solution of model (5) is asymptotically stable provided that detΔs=0 possesses the roots with negative real parts.
For the fractional-order model(7)dϱ1V1tdtϱ1=f11V1t+f12V2t+⋯+f1lVlt,dϱ2V2tdtϱ2=f21V1t+f22V2t+⋯+f2lVlt,⋮dϱlVltdtϱl=fl1V1t+fl2V2t+⋯+fllVlt,where 0<ϱj≤1j=1,2,…,l, the characteristic equation of model (7) owns the following expression:(8)detsϱ1−f11−f12⋯−f1l−f21sϱ2−f22⋯−f2l⋮⋮⋱⋮−fl1−fl2⋯sϱl−fll=0.
Assume thatϕh=ϵh/εh,ϵh,εh∈Z+,ϵh,εh=1 and let ε be the lowest common multiple of βh of ψh,h=1,2,…,l.Lemma 3 (see [41]).
Assume that each rootλs of the following equation:(9)detλεϕ1−f11−f12⋯−f1l−f21λεϕ2−f22⋯−q2l⋮⋮⋱⋮−fl1−fl2⋯λεϕl−fll=0,conforms to argλ>π/2ε; then, the zero solution to model (7) is locally asymptotically stable.


## 3. Bifurcation Exploration for Predator-Prey Model (2)

In this section, we are to study the stability property and the appearance of Hopf bifurcation of predator-prey model (2). Set(10)w3t=∫−∞tUt−vw1vdv=∫−∞tδe−δt−vw1vdv,w4t=∫−∞tUt−sw2sds=∫−∞tδe−δt−vw2vdv,and then(11)dw3tdt=∫−∞tδe−δt−vw1vdv′=−δw3t+δw1t,dw4tdt=∫−∞tδe−δt−vw2vdv′=−δw4t+δw2t.Thus, system (2) becomes the following equivalent form:(12)dw1ϱtdtϱ=w1ta1−α11w3t−α12w2t−ς,dw2ϱtdtϱ=w2t−a2+α21w1t−ς−α22w4t,dw3tdt=−δw3t+δw1t,dw4tdt=−δw4t+δw2t.Assume that(13)ℋ1a1α21>a2α11.It is easy to obtain that system (12) owns the equilibrium points E100,0,0,0, E20a1/α11,0,a1/α11,0, and E30w10,w20,w30,w40, where(14)w1∗=a1α22+a2α12α11α22+α12α21,w2∗=a1α21−a2α11α11α22+α12α21,w3∗=a1α22+a2α12α11α22+α12α21,w4∗=a1α21−a2α11α11α22+α12α21.Ifℋ1 holds, then the equilibrium point E30w10,w20,w30,w40 is a positive equilibrium point. Considering the biological implication of predator-prey model (2), we only deal with the positive equilibrium point E30w10,w20,w30,w40. The linear system of equation (12) around E30w10,w20,w30,w40 is(15)dw1ϱtdtϱ=b1w1t+b2w2t−ς+b3w3t,dw2ϱtdtϱ=c1w1t−ς+c2w2t+c3w4t,dw3tdt=−δw3t+δw1t,dw4tdt=−δw4t+δw2t,where(16)b1=a1−α11w30−α12w20,b2=−α12w10,b3=−α11w10,c1=α21w20,c2=α21w10−a2−α22w40,c3=α22w20.The characteristic equation of (15) takes the following form:(17)detsϱ−b1−b2e−sς−b30−c1e−sςsϱ−c20−c3−δ0s+δ00−δ0s+δ=0.Setϱ=ϵ/ε, where ϵ,ε∈Z+ and ϵ,ε=1. Let λ=s1/ε. When ς=0, then equation (17) becomes(18)detsϱ−b1−b2−b30−c1sϱ−c20−c3−δ0s+δ00−δ0s+δ=0.Lemma 4.
Assume thatς=0 and all the roots λ of equation (18) obey argλ>π/2ε; then, the positive equilibrium point E30w10,w20,w30,w40 of model (12) is locally asymptotically stable.Proof.
Clearly, whenς=0, then characteristic equation (17) becomes equation (18). By virtue of Lemma 3, one can easily obtain that Lemma 4 holds.
By virtue of equation (17), one obtains(19)s2ϱ+2+σ1s2ϱ+1+σ2s2ϱ+σ3sϱ+2+σ4sϱ+1+σ5sϱ+σ6s2+σ7s+σ8+σ9s2+σ10s+σ11e−2sς=0,where(20)σ1=2δ,σ2=δ2,σ3=−b1+c2,σ4=−c3δ−2δb1+c2−b3δ,σ5=−c3δ2−b1+c2δ2−b3δ2,σ6=b1c2,σ7=δb1c3+2b1c2+b3c2,σ8=δ2b3c3+b1c3+b3c2+b1c2,σ9=−b2c1,σ10=−2δ2b2c1,σ11=−b2c1δ2.
Assume thats=iχ=χcosπ/2+isinπ/2 is the root of equation (19); then, one gets(21)χ2ϱ+2cos2ϱ+2π2+isin2ϱ+2π2+σ1χ2ϱ+1cos2ϱ+1π2+isin2ϱ+1π2+σ2χ2ϱcosϱπ+isinϱπ+σ3χϱ+2cosϱ+2π2+isinϱ+2π2+σ4χϱ+1cosϱ+1π2+isinϱ+1π2+σ5σϱcosϱπ2+isinϱπ2−σ6χ2+iσ7χ+σ8+−σ9χ2+iσ10χ+σ11cos2χς−isin2χς=0.
By means of equation (21), we get(22)A1cos2χς+A2sin2χς=A3,A2cos2χς−A1sin2χς=A4,where(23)A1=σ11−σ9χ2,A2=σ10χ,A3=−χ2ϱ+2cos2ϱ+2π2−σ1χ2ϱ+1cos2ϱ+1π2−σ2χ2ϱcosϱπ−σ3χϱ+2cosϱ+2π2−σ4χϱ+1cosϱ+1π2−σ5σϱcosϱπ2+σ6χ2−σ8,A4=−χ2ϱ+2sin2ϱ+2π2−σ1χ2ϱ+1sin2ϱ+1π2−σ2χ2ϱsinϱπ−σ3χϱ+2sinϱ+2π2−σ4χϱ+1sinϱ+1π2−σ5σϱsinϱπ2−σ7χ.
It follows from (22) that(24)cos2χς=A1A3+A2A4A12+A22,(25)A12+A22=A32+A42.
In equation (23), let(26)γ1=−cos2ϱ+2π2,γ2=−σ1cos2ϱ+1π2,γ3=−σ2cosϱπ,γ4=−σ3cosϱ+2π2,γ5=−σ4cosϱ+1π2,γ6=−σ5cosϱπ2,γ7=σ6,γ8=−σ8,ρ1=−sin2ϱ+2π2,ρ2=−σ1sin2ϱ+1π2,ρ3=−σ2sinϱπ,ρ4=−σ3sinϱ+2π2,ρ5=−σ4sinϱ+1π2,ρ6=−σ5sinϱπ2,ρ7=−σ7,and then (21) becomes(27)A1=σ11−σ9χ2,A2=σ10χ,A3=γ1χ2ρ+2+γ2χ2ρ+1+γ3χ2ρ+γ4χρ+2+γ5χρ+1+γ6χρ+γ7χ2+γ8,A4=ρ1χ2ρ+2+ρ2χ2ρ+1+ρ3χ2ρ+ρ4χρ+2+ρ5χρ+1+ρ6χρ+ρ7χ.
By virtue of (25) and (27), one gets(28)ϑ1χ4ϱ+4+ϑ2χ4ϱ+3+ϑ3χ4ϱ+2+ϑ4χ4ϱ+1+ϑ5χ4ϱ+ϑ6χ3ϱ+4+ϑ7χ3ϱ+3+ϑ8χ3ϱ+2+ϑ9χ3ϱ+1+ϑ10χ3ϱ+ϑ11χ2ϱ+4+ϑ12χ2ϱ+3+ϑ13χ2ϱ+2+ϑ14χ2ϱ+1+ϑ15χ2ϱ+ϑ16χϱ+4+ϑ17χϱ+3+ϑ18χϱ+2+ϑ19χϱ+1+ϑ20χϱ+ϑ21χ4+ϑ22χ2+ϑ23=0,where(29)ϑ1=γ12+ρ12,ϑ2=2γ1γ2+ρ1ρ2,ϑ3=γ22+ρ22+2γ1γ3+ρ1ρ3,ϑ4=2γ2γ3+ρ2ρ3,ϑ5=γ32+ρ32,ϑ6=2γ1γ4+ρ1ρ4,ϑ7=2γ1γ5+ρ1ρ5,ϑ8=2γ1γ6+γ2γ5+γ3γ4+ρ1ρ6+ρ2ρ5+ρ3ρ4,ϑ9=2γ2γ6+γ3γ5+ρ2ρ6+ρ3ρ5,ϑ10=2γ3γ6+ρ3ρ6,ϑ11=γ42+ρ42+2γ1γ7+ρ1ρ7,ϑ12=2γ4γ5+γ2γ7+ρ4ρ5+ρ1ρ7,ϑ13=γ52+ρ52+2γ4γ6+γ3γ7+γ1γ8+ρ4ρ6+ρ2ρ7,ϑ14=2γ5γ6+γ2γ8+ρ5ρ6+ρ3ρ7,ϑ15=γ62+ρ62,ϑ16=2γ4γ7,ϑ17=2γ5γ7+ρ4ρ7,ϑ18=2γ6γ7+γ4γ8+ρ5ρ7,ϑ19=2γ5γ8+ρ6ρ7,ϑ20=2γ6γ8,ϑ21=γ72−σ92,ϑ22=2γ7γ8−ρ72−σ112+2σ9σ11,ϑ23=γ82−σ112.
Denote(30)Qχ=ϑ1χ4ϱ+4+ϑ2χ4ϱ+3+ϑ3χ4ϱ+2+ϑ4χ4ϱ+1+ϑ5χ4ϱ+ϑ6χ3ϱ+4+ϑ7χ3ϱ+3+ϑ8χ3ϱ+2+ϑ9χ3ϱ+1+ϑ10χ3ϱ+ϑ11χ2ϱ+4+ϑ12χ2ϱ+3+ϑ13χ2ϱ+2+ϑ14χ2ϱ+1+ϑ15χ2ϱ+ϑ16χϱ+4+ϑ17χϱ+3+ϑ18χϱ+2+ϑ19χϱ+1+ϑ20χϱ+ϑ21χ4+ϑ22χ2+ϑ23=0.
Now the following assumption is given.
ℋ2ϑ23<0, where ϑ23 is defined by (29).Lemma 5.
Assume thatℋ2 is fulfilled; then, equation (19) possesses at least a pair of purely imaginary roots.Proof.
It is easy to see thatQ0=ϑ23<0 and limχ⟶∞Qχ=+∞. Then, one can conclude that equation (28) owns at least one positive root, which implies that equation (19) owns at least a pair of purely imaginary roots.
In equation (28), because the parameter ϱ is a fractional number, it is not inconvenient to solve the solution of equation (28). So, we are to change equation (28) to an isovalent equation with the powers involving integer number. Let x=χ1/ε; then, χ=xε. It follows from equation (28) that(31)ϑ1x4ϵ+4ε+ϑ2x4ϵ+3ε+ϑ3x4ϵ+2ε+ϑ4x4ϵ+ε+ϑ5x4ϵ+ϑ6x3ϵ+4ε+ϑ7x3ϵ+3ε+ϑ8x3ϵ+2ε+ϑ9y3ϵ+ε+ϑ10x3ϵ+ϑ11x2ϵ+4ε+ϑ12x2ϵ+3ε+ϑ13x2ϵ+2ε+ϑ14x2ϵ+ε+ϑ15y2ϵ+ϑ16xϵ+4ε+ϑ17xϵ+3ε+ϑ18xϵ+2ε+ϑ19xϵ+ε+ϑ20xϵ+ϑ21x4ε+ϑ22x2ε+ϑ23=0.
By virtue of computer, we can easily find the roots of equation (31). Suppose that equation (31) owns the positive root which is denoted by xj; then, equation (28) owns the root χj=xjε>0. Suppose that equation (31) owns h positive roots xj,j=1,2,…,h. It follows from (24) that(32)ςjl=12χjarccosA1A3+A2A4A12+A22+2lπ,j=1,2,…,h;l=0,1,2,….
Set(33)ς0=ςj00minj=1,2,…,hςj0,χ0=χ|ς=ς0.
Next, the following hypothesis is prepared as follows.
ℋ3D11D21+D12D22>0, where(34)D11=2ϱ+2χ02ϱ+1cos2ϱ+1π2+σ12ϱ+1χ02ϱcosϱπ+2ρσ2χ02ϱ−1cos2ϱ−1π2+2σ3ϱ+2σ0ϱ+1cosϱ+1π2+σ4ϱ+1σ0ϱcosϱπ2+σ5ϱσ0ϱ−1cosϱ−1π2+σ7+σ10cos2χ0ς0+2σ9sin2χ0ς0,D12=2ϱ+2χ02ϱ+1sin2ϱ+1π2+σ12ϱ+1χ02ϱsinϱπ+2ρσ2χ02ϱ−1sin2ϱ−1π2+2σ3ϱ+2σ0ϱ+1sinϱ+1π2+σ4ϱ+1σ0ϱsinϱπ2+σ5ϱσ0ϱ−1sinϱ−1π2+2σ6−σ10sin2χ0ς0+2σ9cos2χ0ς0,D21=2χ0σ11−σ9χ02cos2χ0ς0+2χ02σ10sin2χ0ς0,D22=2χ02σ10cos2χ0ς0−2χ0σ11−σ9χ02sin2χ0ς0.Lemma 6.
Letsς=ψ1ς+iψ2ς be the root of equation (19) around ς=ς0 satisfying ψ1ς0=0,ψ2ς0=χ0; then, one gets Reds/dς|ς=ς0,χ=χ0>0.Proof.
By virtue of equation (19), we get(35)2ϱ+2s2ϱ+1+σ12ϱ+1s2ϱ+2ϱσ2s2ϱ−1+σ3ϱ+2sϱ+1+σ4ϱ+1sϱ+σ5ϱsϱ−1+2σ6s+σ7dsdς+2σ9s+σ10e−2sςdsdς−2e−2sςdsdςς+sσ9s2+σ10s+σ11=0.
It follows from (35) that(36)dsdς−1=D1sD2s−ςs,where(37)D1s=2ρ+2s2ρ+1+σ12ρ+1s2ρ+2ρσ2s2ρ−1+2σ3ρ+2sρ+1+σ4ρ+1sρ+σ5ρsρ−1+2σ6s+σ7+2σ9s+σ10e−2sς,D2s=2se−2sςσ9s2+σ10s+σ11.
Then,(38)Redsdςς=ς0,χ=χ0−1=ReD1sD2sς=ς0,χ=χ0=D11D21+D12D22D212+D222.
Byℋ3, one has(39)Redsdςς=ς0,χ=χ0−1>0,which completes the proof.
According to the study above, one gets the following conclusion.Theorem 1.
Suppose thatℋ1–ℋ3 are fulfilled and every root λ for equation (18) satisfies argλ>π/2ε; then, the positive equilibrium point of model (2) is locally asymptotically stable provided that 0≤ς<ς0 and a Hopf bifurcation takes place around the positive equilibrium point provided that ς=ς0.Remark 1.
In 2020, Rahman et al. [37] dealt with the Hopf bifurcation for integer-order predator-prey model involving discrete and distributed delay. In this article, we have dealt with Hopf bifurcation for fractional-order predator-prey model involving discrete and distributed delay. The method of research in [37] cannot be used to investigate the fractional-order case. From this viewpoint, we say that the research is a good complement of the work of [37].Remark 3.
For many works on Hopf bifurcation of fractional-order predator-prey systems, numerous scholars focus on the fractional-order predator-prey models involving discrete time delay and do not involve the distributed time delay. In this article, we are concerned with the fractional-order predator-prey model involving discrete time delay and distributed time delay. After a suitable variable substitution, we obtain an isovalent fractional-order predator-prey model which includes integer-order operator and fractional-order operator. The discussion on the characteristic equation of the isovalent fractional-order predator-prey model has become more complex. So, we think that our works enrich and develop the stability and bifurcation theory of fractional-order dynamical system.


## 4. Simulation Figures

Given the following predator-prey model:(40)dw1ϱtdtϱ=w1t1−0.5w3t−0.6w2t−ς,dw2ϱtdtϱ=w2t−1+1.2w1t−ς−0.5w4t,dw3tdt=−0.3w3t+0.3w1t,dw4tdt=−0.3w4t+0.3w2t,it is not difficult to derive that model (40) owns a unique positive equilibrium point E30w10,w20,w30,w40=1.1340,0.7216,1.1340,0.7216. Select ϱ=0.93=93/100. Then, ϵ=93,ε=100. By means of computer software, one derives χ0=1.8722,ς0=0.05,ϑ23=−0.8347,D11=0.5091,D12=0.6783,D21=−0.4377,D22=0.7901. Then, D11D21+D12D22=0.3131>0. Furthermore, each root λ of equation (18) satisfies argλ>π/200. So, all the assumptions of Theorem 1 are satisfied. Therefore, E301.1340,0.7216,1.1340,0.7216 of model (40) is locally asymptotically stable for 0≤ς<0.5. To illustrate this situation, we select ς=0.04<ς0=0.05. The computer simulation plots are presented in Figure 1. Figure 1 indicates that the variables w1t,w2t,w3t,w4t tend to 1.1340,0.7216,1.1340,0.7216, respectively. When ς passes through ς0=0.05, then a family of Hopf bifurcation caused by time delay ς takes place in the neighborhood of E301.1340,0.7216,1.1340,0.7216. To verify this situation, we select ς=0.067>η0=0.05. The computer simulation plots are shown in Figure 2. Figure 2 suggests that the variables w1t,w2t,w3t,w4t will exhibit periodic oscillatory phenomenon. Moreover, the bifurcation plots are drawn in Figures 3–6. Figures 3–6 show that the bifurcation value of model (40) is roughly equal to 0.05.Figure 1
The stable behavior of system (40) when ς=0.04<ς0=0.05.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 2
Hopf bifurcation phenomenon of system (40) when ς=0.07>ς0=0.05.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 3
Bifurcation graph of system (40): ς−w1.Figure 4
Bifurcation graph of system (40): ς−w2.Figure 5
Bifurcation graph of system (40): ς−w3.Figure 6
Bifurcation graph of system (40): ς−w4.


## 5. Conclusions

The investigation on the stability and bifurcation peculiarity of delayed predator-prey models plays a vital role in maintaining ecological balance in real world. In this work, based on the work of [37], we set up a new fractional-order predator-prey model concerning discrete delay and distributed delay. By virtue of apposite change of variable, we derive an equivalent fractional-order predator-prey model concerning one delay. By analyzing the characteristic equation of the equivalent fractional-order predator-prey model and regarding the time delay as bifurcation parameter, the stability and bifurcation condition for the involved predator-prey model is established. The influence of time delay on the stability and bifurcation of the involved predator-prey model has been revealed. The computer simulation plots and bifurcation diagrams are displayed to sustain the validity of the derived main results.



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*Source: 1535920-2021-06-18.xml*