Research article

Mathematical exploration on control of bifurcation for a 3D predator-prey model with delay

  • Received: 11 September 2024 Revised: 09 October 2024 Accepted: 11 October 2024 Published: 21 October 2024
  • MSC : 34C23, 34K18, 37GK15, 39A11, 92B20

  • In this current paper, we developed a new predator-prey model accompanying delay based on the earlier works. By applying inequality strategies, fixed point theorem, and a suitable function, we got new necessary conditions for the existence, uniqueness, nonnegativeness, and boundedness of the solution to the developed delayed predator-prey model. The bifurcation behavior and stability nature of the defined delayed predator-prey model were investigated by using stability and bifurcation theory of delayed differential equations. We have modified the Hopf bifurcation's appearance time and stability domain by building two distinct hybrid delayed feedback controllers for the delayed predator-prey model. The time of Hopf bifurcation appearance and stability domain of the model were explored. Matlab experiment diagrams were given to support the learned important results. The derived outcomes in this paper were original and have significant theoretical implications for maintaining equilibrium between the densities of the three species.

    Citation: Yingyan Zhao, Changjin Xu, Yiya Xu, Jinting Lin, Yicheng Pang, Zixin Liu, Jianwei Shen. Mathematical exploration on control of bifurcation for a 3D predator-prey model with delay[J]. AIMS Mathematics, 2024, 9(11): 29883-29915. doi: 10.3934/math.20241445

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  • In this current paper, we developed a new predator-prey model accompanying delay based on the earlier works. By applying inequality strategies, fixed point theorem, and a suitable function, we got new necessary conditions for the existence, uniqueness, nonnegativeness, and boundedness of the solution to the developed delayed predator-prey model. The bifurcation behavior and stability nature of the defined delayed predator-prey model were investigated by using stability and bifurcation theory of delayed differential equations. We have modified the Hopf bifurcation's appearance time and stability domain by building two distinct hybrid delayed feedback controllers for the delayed predator-prey model. The time of Hopf bifurcation appearance and stability domain of the model were explored. Matlab experiment diagrams were given to support the learned important results. The derived outcomes in this paper were original and have significant theoretical implications for maintaining equilibrium between the densities of the three species.



    It is widely accepted that the delayed dynamical model is an important tool for answering many issues in biology. Numerous academics focus extensively on the development of diverse predator-prey models to elucidate the dynamics and internal structure within biological populations. We can successfully regulate the concentrations of predators and preys in the natural world by investigating the various dynamical characteristics of predator-prey models. Numerous studies on predator-prey models have been submitted and released in recent years, and several outstanding studies have been discussed. For example, Din et al. [1] studied the discrete predator-prey interactions with chaos control and bifurcation analysis. Liu and Guo [2] examined the dynamics of a predator-prey system with nonlinear prey-taxis, using the Lyapunov-Schmidt reduction approach, the fold-Hopf singularity, Hopf bifurcation, and steady state bifurcation are examined in depth. Al-Kaff et al. [3] investigated bifurcation and chaos in a discrete predator-prey system using a linked logistic map, a wide range of the system's behavior is thoroughly examined in the study. Pita et al. [4] investigated a review of some recent advances in predator-prey models. For more extensive investigations, see [5,6,7,8,9].

    In particular, Lotka-Volterra models are vital predator-prey models in biology. They play an important role in describing the correlation between predators and prey. During the past decades, many works on this topic have been reported. For example, Hsu et al. [10] studied three different species of omnivorous Lotka-Volterra food web models. Bunin [11] studied the Lotka-Volterra model for ecological communities exhibiting symmetry and connected it to other well-known models. Wu et al. [12] investigated the Grey Lotka-Volterra model and its application, effectively analyzing the link between the two variables and forecasting their values using the gray Lotka-Volterra model. Kloppers et al. [13] estimated the parameters of the Lotka-Volterra model from empirical data. Marasco et al. [14] found the development of market share dynamics through the application of Lotka-Volterra models, and Wang et al. [15] examined the problems with free boundaries in a Lotka-Volterra competition system. Zhou [16] investigated a two-species Lotka-Volterra competition system in one-dimensional advective environments. Cherniha et al. [17] studied a review and new findings regarding the construction and application of exact solutions to the diffusive Lotka-Volterra system.

    In 2022, Prabir et al. [18] suggested the subsequent Lotka-Volterra mutualistic symbiotic relationship:

    {d1dT=r11[11K1α12K122]+δ1313,d2dT=r22[12K2α21K212]a(1p)23b+(1p)2,d3dT=3[e+ac(1p)2b+(1p)2], (1.1)

    where 1 and 2 indicate the density of two logistically increasing competing species, 3 represents the density of a predatory species of 2, r1,r2 are the natural rates of growth for species 1 and 2, K1,K2 are the carrying capacity of species 1 and 2 respectively, α12,α21 represent the fight between two species 1 and 2, δ13 is the commensal coefficient of 1 over 3 species, a is the frequency at which 3 species attack 2 species, p is the refuge rate of 2 species, c refers to the rate of conservation of 2 species, b is the steady value for half saturation for the Holling type Ⅱ function, and e is the rate of extinction for species 3. One might refer to [18] for a more precise explanation of the meaning of system (1.1).

    In order to make the suggested mathematical model nondimensional, we have now added a few variables and constants as t=r1T,u1=1K1,u2=2K2,u3=a3r1K2,γ12=α12K22K1, γ=δ13K2a,r=r2r1,γ21=α21K21K2,v1=bK2,v2=er1,v3=acr1.

    Then model (1.1) can be simplified to the following form

    {du1(t)dt=u1(t)(1u1(t)γ12u22(t))+γu1(t)u3(t),du2(t)dt=ru2(t)(1u2(t)γ21u21(t))(1p)u2(t)u3(t)v1+(1p)u2(t),du3(t)dt=u3(t)(v2+v3(1p)u2(t)v1+(1p)u2(t)), (1.2)

    with initial conditions u1(0)0,u2(0)0,u3(0)0.

    In this instance we want to call your attention to the truth that the evolution of species frequently depends on both historical and present times; hence, delays must be incorporated into biological models. Notice that the development of the density of two logistically increasing competing species 1 and 2 rely on not only the current time but also the past time. In other words, the development of the density of two logistically increasing competing species 1 and 2 shall have feedback delays. Based on this idea, we assume that there are two self-feedback delays in model (1.2): One is the self-feedback delay from the first species u1 to the first species u1, and the other the self-feedback delay from the first species u2 to the first species u2. The points that follow delayed Lotka-Volterra commercially available symbiosis system may then be loosely formulated:

    {du1(t)dt=u1(t)(1u1(tθ)γ12u22(t))+γu1(t)u3(t),du2(t)dt=ru2(t)(1u2(tθ)γ21u21(t))(1p)u2(t)u3(t)v1+(1p)u2(t),du3(t)dt=u3(t)(v2+v3(1p)u2(t)v1+(1p)u2(t)), (1.3)

    where θ>0 is a time delay and all the parameters γ12,γ,r,γ21,v1,v2,v3 are positive constants. The initial conditions of system (1.3) are given by ui(s)=ϕi(s)0,i=1,2,3,s[θ,0] and ϕi(s)C([θ,0],R+),i=1,2,3.

    Mathematically speaking, delays play a key role in determining how different differential systems behave dynamically [19,20,21]. Delays can lead to a variety of consequences, including changes in stability, the formation of bifurcations, and the start of chaotic behavior [22,23,24]. During the past decades, many authors deal with this topic, see [25,26,27]. Specifically, the dynamic phenomena of delays-induced Hopf bifurcation is quite significant [28,29,30]. From a biological perspective, delay-induced Hopf bifurcation nicely captures the equilibrium between densities of different living populations [31]. We contend that in order to understand the dynamics of interactions across various biological populations, it is critical to investigate delay-induced Hopf bifurcation in a variety of biological models. Motivated by this idea, we will concentrate on the Hopf bifurcation and its bifurcation control mechanisms. Specifically, we will address three critical questions: (1) Assess the distinctive features of the solution to system (1.3), including its nonnegativity, existence and uniqueness, and boundedness. (2) Examine the system's stability problem and the formation of the Hopf bifurcation phenomena (1.3). (3) Build both controllers for adjusting the range of stability and the generation time of the Hopf bifurcation in system (1.3).

    The following is an introduction of the study's primary luminous spot: (ⅰ) A completely new delay-independent bifurcation and stability criterion for system (1.3) is established based on the earlier studies. (ⅱ) Using separate controllers, the realm of stability and the time of Hopf bifurcation in system (1.3) may be regulated well. (ⅲ) This study examines how delay affects Hopf bifurcation and the stability of first and second species densities in system (1.3).

    The stated arrangement of the article is as follows. The "Feature of solution" section discusses the unique characteristics of system (1.3), such as boundedness, nonnegativeness, existence, and uniqueness. The "Bifurcation analysis" section discusses the bifurcation phenomena and the system's stability (1.3). The "Bifurcation domination using extended hybrid controller Ⅰ" section centers around the control issue associated with bifurcation phenomena for system (1.3) by formulating a plausible hybrid delayed feedback controller that involves parameter perturbation is accompanied by latency and state feedback. "Bifurcation domination using extended hybrid controller Ⅱ" section addresses the control challenge of the bifurcation phenomena in system (1.3). Creating a suitable hybrid delayed feedback controller with parameter perturbation for delay and state feedback. The "Software experiments" section provides Matlab software simulation results to assess the veracity of the obtained main results. "Conclusions" concludes this study concisely.

    This section will examine the characteristics of the solution to system (1.3), such as its existence and uniqueness, nonnegativeness, and boundedness. To do this, we will employ fixed point theory, inequality methods, and the design of an appropriate function.

    Theorem 2.1. Let Δ={(u1,u2,u3)R3:max{|u1|,|u2|,|u3|}M}, where M>0 denotes a constant. For every (u10,u20,u30)Δ, system (1.3) under the initial value (u10,u20,u30) owns a unique solution U=(u1,u2,u3)Δ.

    Proof. Set

    f(U)=(f1(U),f2(U),f3(U)), (2.1)

    where

    {f1(U)=u1(t)(1u1(tθ)γ12u22(t))+γu1(t)u3(t),f2(U)=ru2(t)(1u2(tθ)γ21u21(t))(1p)u2(t)u3(t)v1+(1p)u2(t),f3(U)=u3(t)(v2+v3(1p)u2(t)v1+(1p)u2(t)). (2.2)

    For arbitrary U,˜UΔ, one gains

    ||f(U)f(˜U)||=|f1(u)f1(ˉu)|+|f2(u)f2(ˉu)|+|f3(u)f3(ˉu)|=|[u1u1u1(tθ)γ12u1u22+γu1u3][ˉu1ˉu1ˉu1(tθ)γ12ˉu1ˉu22+γˉu1ˉu3]|+|[ru2ru2u2(tθ)rγ21u21u2(1p)u2u3v1+(1p)u2][rˉu2rˉu2ˉu2(tθ)rγ21ˉu21ˉu2(1p)ˉu2ˉu3v1+(1p)ˉu2]|+|[v2u3+v3(1p)u2u3v1+(1p)u2)][v2ˉu3+v3(1p)ˉu2ˉu3v1+(1p)ˉu2)]||u1ˉu1|+|u1u1(tθ)ˉu1ˉu1(tθ)|+γ12|u1u22ˉu1ˉu22|+γ|u1u3ˉu1ˉu3|+r|u2ˉu2|+r|u2u2(tθ)ˉu2ˉu2(tθ)|+rγ21|u21u2ˉu21ˉu2|+(1p)|(1p)u2u3v1+(1p)u2(1p)ˉu2ˉu3v1+(1p)ˉu2|+v2|u3ˉu3|+v3(1p)|u2u3v1+(1p)u2ˉu2ˉu3v1+(1p)ˉu2||u1ˉu1|+|u1u1(tθ)ˉu1u1(tθ)+ˉu1u1(tθ)ˉu1ˉu1(tθ)|+γ12|u1u22ˉu1u22+ˉu1u22ˉu1ˉu22|+γ|u1u3ˉu1u3+ˉu1u3ˉu1ˉu3|+r|u2ˉu2|+r|u2u2(tθ)ˉu2u2(tθ)+ˉu2u2(tθ)ˉu2ˉu2(tθ)|+rγ21|u21u2ˉu21u2+ˉu21u2ˉu21ˉu2|+(v3+1)(1p)|u2u3[v1+(1p)ˉu2]ˉu2ˉu3[v1+(1p)u2][v1+(1p)u2][v1+(1p)ˉu2]||u1ˉu1|+M|u1ˉu1|+M|u1(tθ)ˉu1(tθ)|+γ12M2|u1ˉu1|+2γ12M2|u2ˉu2|+γM|u1ˉu1|+γM|u3ˉu3|+r|u2ˉu2|+rM|u2ˉu2|+rM|u2(tθ)ˉu2(tθ)|+2rγ21M2|u1ˉu1|+rγ21M2|u2ˉu2|+v1(v3+1)(1p)M|u2ˉu2|+v1(v3+1)(1p)M|u3ˉu3|+(v3+1)|u3ˉu3|=(1+2M+γ12M2+γM+2rγ21M2)|u1ˉu1|+(2γ12M2+r+rM+rM+rγ21M2+v1(v3+1)(1p)M)|u2ˉu2|+(γM+v1(v3+1)(1p)M+v3+1)|u3ˉu3|=σ1|u1ˉu1|+σ2|u2ˉu2|+σ3|u3ˉu3|L(|u1ˉu1|+|u2ˉu2|+|u3ˉu3|), (2.3)

    where

    {σ1=1+2M+γ12M2+γM+2rγ21M2,σ2=2γ12M2+r+rM+rM+rγ21M2+v1(v3+1)(1p)M,σ3=γM+v1(v3+1)(1p)M+v3+1. (2.4)

    Let

    L=max{σ1,σ2,σ3}. (2.5)

    Then it follows from (2.3) that

    ||f(U)f(˜U)||L||U˜U||. (2.6)

    Consequently, f(U) adheres to the Lipschitz condition for U. By applying the fixed point theorem, it's possible to infer with ease that Theorem 2.1 is right.

    Theorem 2.2. Each solution to system (1.3) beginning with R3+ maintains a nonnegative value.

    Proof. Considering the initial equation of system (1.3), it's possible to achieve

    du1dt=u1[1u1(tθ)γ12u22]+γu1u3, (2.7)

    then

    du1u1=[1u1(tθ)γ12u22+γu3]dt, (2.8)

    which leads to

    t0du1u1=t0[1u1(sθ)γ12u22+γu3]ds, (2.9)

    then one gets

    lnu1(t)u1(0)=t0[1u1(sθ)γ12u22+γu3]ds, (2.10)

    thus

    u1(t)=u1(0)exp{t0[1u1(sθ)γ12u22+γu3]ds}>0. (2.11)

    In the same way, we know

    u2(t)=u2(0)exp{t0[rru2(sθ)rγ21u22(1p)u3v1+(1p)u2]ds}>0. (2.12)
    u3(t)=u3(0)exp{t0[v2+v3(1p)u2v1+(1p)u2]ds}>0. (2.13)

    Therefore, the validity of Theorem 2.2 is affirmed.

    Theorem 2.3. Each solution to system (1.3), starting with R3+, is invariably restricted.

    Proof. From the second equation of system (1.3), we have

    du2dt=ru2(1u2(tθ)γ21u21)(1p)u2u3v1+(1p)u2. (2.14)

    Then

    du2dtru2. (2.15)

    Integrating from (tθ) to t on both sides of the Eq (2.15) leads to

    ttθdu2u2ttθrdt. (2.16)

    We get

    u2(t)u2(tθ)erθ, (2.17)

    then

    u2(tθ)u2(t)erθ, (2.18)

    from (2.14), we can get

    du2dtru2(t)(1u2(tθ))ru2(t)(1u2(t)erθ). (2.19)

    So

    du2dtu2(t)(rrerθu2(t)). (2.20)

    Hence

    u2(t)erθ. (2.21)

    Now, we define the following function

    W1=u2+1v3u3. (2.22)

    Then

    dW1dt=du2dt+1v3du3dt=[ru2(1u2(tθ)γ21u21)(1p)u2u3v1+(1p)u2]+[u3v3(v2+v3(1p)u2v1+(1p)u2)]=ru2(1u2(tθ)γ21u21)v2u3v3ru2[1u2(tθ)]v2u3v3=v2(u2+1v3u3)+v2u2+ru2[1u2(tθ)]v2(u2+1v3u3)+(v2+r)u2v2(u2+1v3u3)+(v2+r)erθ=v2W1(t)+(v2+r)erθ. (2.23)

    Thus

    W1(t)(v2+r)erθv2. (2.24)

    Next we define another function W2 as follows

    W2=u1+u2+1v3u3. (2.25)

    Then, we get

    dW2dt=du1dt+du2dt+1v3du3dt=[u1(1u1(tθ)γ12u22)+γu1u3]+[ru2(1u2(tθ)γ21u21)(1p)u2u3v1+(1p)u2]+[u3v3(v2+v3(1p)u2v1+(1p)u2)]=[u1(1u1(tθ)γ12u22)+γu1u3]+[u1(1u1(tθ)γ12u22)+γu1u3]v2(u1+u2+1v3u3)+v2u1+v2u2+u1[1u1(tθ)]+γu1u3+ru2[1u2(tθ)γ21u21]=v2(u1+u2+1v3u3)γ21ru2u21+(v2+1+γu3)u1+(v2+r)u2v2(u1+u2+1v3u3)γ21ru2u21+(v2+1+γu3)u1+(v2+r)u2v2(u1+u2+1v3u3)γ21rM1u21+(v2+1+γM2)u1+(v2+r)M1v2(u1+u2+1v3u3)+Q, (2.26)

    where

    {M1=erθ,M2=(v2+r)erθv2,Q=4γ21rM12(v2+r)(v2+1+γM2)24γ21rM1.

    Stemming from Eq (2.26), it is deduced that

    W2(t)Qv2. (2.27)

    By Eq (2.28), we obtain

    W2(t)Qv2,whent. (2.28)

    Consequently, every solution to the system (1.3) is consistently limited.

    This part delves into examining the bifurcation and stability aspects of model (1.3). Initially, our assumption is that E(u1,u2,u3) represents the balance point in model (1.3); subsequently, u1,u2,u3 adhere to this stipulation:

    {u1[1u1γ12u22]+γu1u3=0,ru2[1u2γ21u21](1p)u2u3v1+(1p)u2=0,u3[v2+v3(1p)u2v1+(1p)u2]=0. (3.1)

    Let

    {ˉu1(t)=u1(t)u1,ˉu2(t)=u2(t)u2,ˉu3(t)=u3(t)u3. (3.2)

    Substituting system (3.2) into system (1.3), we achieve the linear configuration of model (1.3) at E(u1,u2,u3) (denote ˉu1 as u1, ˉu2 as u2, ˉu3 as u3)

    {du1dt=b1u1+b2u2+b3u3+b4u1(tθ),du2dt=b5u1+b6u2+b7u3+b8u2(tθ),du3dt=b9u2+b10u3, (3.3)

    where

    {b1=1u1γ12u22γu3,b2=2γ12u1u2,b3=γu1,b4=u1,b5=2rγ21u1u2,b6=rru2rγ21u21(1p)u3v1+(1p)u2+(1p)2u2u3[v1+(1p)u2]2,b7=(1p)u2v1+(1p)u2,b8=ru2,b9=v3(1p)u3v1+(1p)u2v3(1p)2u2u3[v1+(1p)u2]2,b10=v3(1p)u2v1+(1p)u2v2. (3.4)

    The characteristic Eq (3.3) of the system owns the following expressions:

    det[λb1b4eλθb2b3b5λb6b8eλθb70b9λb10]=0, (3.5)

    which leads to

    λ3+c1λ2+c2λ+c3+(c4λ2+c5λ+c6)eλθ+(c7λ+c8)e2λθ=0, (3.6)

    where

    {c1=b1b6b10,c2=b1b6+b1b10b2b5+b6b10b7b9,c3=b1b7b9b1b6b10+b2b5b10b3b5b9,c4=b4b8,c5=b1b8+b4b6+b4b10+b8b10,c6=b4b7b9b1b8b10b4b6b10,c7=b4b8,c8=b4b8b10. (3.7)

    If θ=0, then Eq (3.6) becomes

    λ3+(c1+c4)λ2+(c2+c5+c7)λ+(c3+c6+c8)=0. (3.8)

    If

    (A1){c1+c4>0,c3+c6+c8>0,(c1+c4)(c2+c5+c7)>c3+c6+c8,

    is fulfilled, then the three roots λ1,λ2,λ3 of Eq (3.8) have negative real parts. Thus, the equilibrium point E(u1,u2,u3) of system (1.3) with θ=0 is locally asymptotically stable.

    Assume that λ=iε is the root of Eq (3.6), then Eq (3.6) becomes

    c4ε2+c5iε+c6+(ε3ic1ε2c2iε+c3)eiεθ+(c7iε+c8)eiεθ=0. (3.9)

    It follows from (3.9) that

    {F1sin(εθ)+F2cos(εθ)=F3,F4sin(εθ)+F5cos(εθ)=F6, (3.10)

    where

    {F1=ε3+g1ε,F2=c1ε2+g2,F3=c4ε2c6,F4=c1ε2+g3,F5=ε3+g4ε,F6=c5ε, (3.11)

    and

    {g1=c7c2,g2=c3+c8,g3=c3c8,g4=c2+c7. (3.12)

    It follows from (3.10) that

    {sin(εθ)=F2F6F3F5F2F4F1F5,cos(εθ)=F3F4F1F6F2F4F1F5. (3.13)

    Because of sin2(εθ)+cos2(εθ)=1, we can get

    [F2F6F3F5F2F4F1F5]2+[F3F4F1F6F2F4F1F5]2=1. (3.14)

    So,

    F23F25+F22F26+F21F26+F23F24F21F25F22F242F2F3F5F62F1F3F4F6+2F1F2F4F5=0. (3.15)

    Using (3.11) and (3.15), we know

    ε12+D1ε10+D2ε8+D3ε6+D4ε4+D5ε2+D6=0. (3.16)

    Therefore, the results can be obtained as follows:

    ε12D1ε10D2ε8D3ε6D4ε4D5ε2D6=0, (3.17)

    where

    {D1=c242g42g12c21,D2=2c24g42c4c6+c25+c21c24g24+4g1g4g21c41+2c21g4+2c1g3+2c1g22c21g1,D3=c24g24+4c4c6g4+c26+c21c25+2c25g12c1c24g32c21c4c62g1g24+2g21g4+2c31g3+2c31g22c1c4c5g42c4c5g22c1c4c5g1+2c4c5g32c1g3g4+2(g2c1g1)(c1g4g3)+2c1g1g2,D4=2c4c6g242c26g42c1c25g2+c25g21+c24g23+4c1c4c6g3+c21c26g21g24c21g234c21g2g3c21g22+2c5g4(c1c6+c4g2)+2c5c6g2+2c1c5c6g1+2c5g3(c4g1c6)+2g3g4(g2c1g1)+2g1g2(c1g4g3),D5=c26g24+c25g222c4c6g232c1c26g3+2c1g2g23+2c1g22g32c5c6g2g42c5c6g1g3+2g1g2g3g4,D6=c26g23g22g23. (3.18)

    Let

    1(ε)=ε12D1ε10D2ε8D3ε6D4ε4D5ε2D6. (3.19)

    Assume that

    (A2)|c6|>|g2|.

    By virtue of (A2), we know 1(0)=(c26g23g22g22)<0, and since limε1(ε)>0, then we will know Eq (3.17) has at least one positive real root. Therefore Eq (3.6) has at least one pair of pure roots. Without loss of generality, we can assume that Eq (3.17) has twelve positive real roots (say, εj,j=1,2,3,,12). Relying on (3.12), we know

    θ(n)j=1εj[arccos(F3F4F1F6F2F4F1F5)+2nπ], (3.20)

    where j=1,2,,12;n=0,1,2,;

    {F1=ε3j+g1εj,F2=c1ε2j+g2,F3=c4ε2jc6,F4=c1ε2j+g3,F5=ε3j+g4εj,F6=c5εj. (3.21)

    Assume θ0=min{j=1,2,,12;n=0,1,2,}{θ(n)j} and suppose that when θ=θ0, Eq (3.6) has a pair of imaginary roots ±iε0. Next we present the following assumption:

    (A3)G1RG2R+G1IG2I>0,

    where

    {G1R=2c1ε0sin(ε0θ0)(3ε20c2c7)cos(ε0θ0),G1I=(3ε20+c2c7)sin(ε0θ0)+2c1ε0cos(ε0θ0),G2R=(c1ε30+c3ε0+c8ε0)sin(ε0θ0)+(ε40+c2ε20c7ε20)cos(ε0θ0),G2I=(ε40+c2ε20+c7ε20)sin(ε0θ0)+(c1ε30c3ε0+c8ε0)cos(ε0θ0). (3.22)

    Lemma 3.1. Suppose that λ(θ)=ϵ1(θ)+iϵ2(θ) is the root of Eq (3.6) at θ=θ0 such that ϵ1(θ0)=0, ϵ2(θ0)=ε0, then Re(dλdθ)|θ=θ0,ε=ε0>0.

    Proof. By Eq (3.6), we can get

    (3λ2+2c1λ+c2)eλθdλdθ+(λ3+c1λ2+c2λ+c3)eλθ(dλdθθ+λ)+c7eλθdλdθ(c7λ+c8)eλθ(dλdθθ+λ)+(2c4λ+c5)dλdθ=0. (3.23)

    It means that

    (dλdθ)1=G1(λ)G2(λ)θλ, (3.24)

    where

    {G1(λ)=(3λ2+2c1λ+c2)eλθ+c7eλθ,G2(λ)=(c7λ+c8)λeλθ(λ3+c1λ2+c2λ+c3)λeλθ. (3.25)

    Hence

    Re[(dλdθ)1]θ=θ0,ε=ε0=Re[G1(λ)G2(λ)]θ=θ0,ε=ε0=G1RG2R+G1IG2IG22R+G22I. (3.26)

    By the assumption (A3), we get

    Re[(dλdθ)1]θ=θ0,ε=ε0>0. (3.27)

    This concludes the proof. According to the preceding reasoning, the following result is simply deduced.

    Theorem 3.1. Assume that (A1)(A3) hold, then the equilibrium point E(u1,u2,u3) of model (1.3) holds a locally asymptotically stable state if θ[0,θ0) and model (1.3) generates Hopf bifurcations around the equilibrium point E(u1,u2,u3) when θ=θ0.

    Motivated by the works of [30,32,33,34], we design the hybrid controller in Sections 4 and 5. In fact, we add a perturbation by adjusting the rate of change of population and then we can check the controllability via theoretical analysis and computer simulations. In this part, we will look at the Hopf bifurcation control issue in system (1.3) using a suitable hybrid controller that combines state feedback and parameter perturbation with delay. Using the ideas from [30,32], we achieve the following controlled 3D Lotka-Volterra commensal symbiosis system:

    {du1(t)dt=δ1[u1(t)(1u1(tθ)γ12u22(t))+γu1(t)u3(t)]+δ2[u1(t)u1(tθ)],du2dt=ru2(t)[1u2(tθ)γ21u21(t)](1p)u2(t)u3(t)v1+(1p)u2(t),du3dt=u3(t)[v2+v3(1p)u2(t)v1+(1p)u2(t)], (4.1)

    where δ1,δ2 stands for feedback gain parameters. Systems (4.1) and (1.3) own the same equilibrium points E(u1,u2,u3). Let

    {u1(t)=ˉu1(t)+u1,u2(t)=ˉu2(t)+u2,u3(t)=ˉu3(t)+u3, (4.2)

    denote ˉu1 as u1, ˉu2 as u2, ˉu3 as u3, and the linear system of system (4.1) around E(u1,u2,u3) takes the following expression:

    {du1dt=d1u1+d2u2+d3u3+d4u1(tθ),dˉu2dt=d5u1+d6u2+d7u3+d8u2(tθ),du3dt=d9u2+d10u3, (4.3)

    where

    {d1=(1u1γ12u22γu3)δ1+δ2,d2=2γ12u1u2δ1,d3=γu1δ1,d4=δ1u1δ2,d5=2rγ21u1u2,d6=rru2rγ21u21(1p)u3v1+(1p)u2+(1p)2u2u3[v1+(1p)u2]2,d7=(1p)u2v1+(1p)u2,d8=ru2,d9=v3(1p)u3v1+(1p)u2v3(1p)2u2u3[v1+(1p)u2]2,d10=v3(1p)u2v1+(1p)u2v2. (4.4)

    The following expression is owned by the system (4.4),s characteristic equation:

    det[λd1d4eλθd2d3d5λd6d8eλθd70d9λd10]=0, (4.5)

    which leads to

    λ3+h1λ2+h2λ+h3+(h4λ2+h5λ+h6)eλθ+(h7λ+h8)e2λθ=0, (4.6)

    where

    {h1=d1d6d10,h2=d1d6+d1d10d2d5+d6d10d7d9,h3=d1d7d9d1d6d10+d2d5d10d3d5d9,h4=d4d8,h5=d1d8+d4d6+d4d10+d8d10,h6=d4d7d9d1d8d10d4d6d10,h7=d48,h8=d4d8d10. (4.7)

    If θ=0, then Eq (4.6) reads as:

    λ3+(h1+h4)λ2+(h2+h5+h7)λ+(h3+h6+h8)=0. (4.8)

    If

    (A4){Δ1=h1+h4>0,Δ2=|h1+h41h3+h6+h8h2+h5+h7|>0,Δ3=|h1+h410h3+h6+h8h2+h5+h7h1+h400h3+h6+h8|>0,

    holds, then the three roots λ1,λ2,λ3 of Eq (4.8) have negative real portions. Thus, the equilibrium point E(u1,u2,u3) of model (4.1) remains locally asymptotically stable at θ=0. From (4.6), we can get

    h4λ2+h5λ+h6+(λ3+h1λ2+h2λ+h3)eλθ+(h7λ+h8)eλθ=0. (4.9)

    Suppose that λ=iζ is the root of Eq (4.9). Then, Eq (4.9) takes

    h4(iζ)2+h5(iζ)+h6+[(iζ)3+h1(iζ)2+h2(iζ)+h3]eiζθ+[h7(iζ)+h8]eiζθ=0, (4.10)

    which results in

    h4(iζ)2+h5(iζ)+h6+[(iζ)3+h1(iζ)2+h2(iζ)+h3][cos(ζθ)+isin(ζθ)]+[h7(iζ)+h8][cos(ζθ)isin(ζθ)]=0. (4.11)

    It follows from (4.11) that

    {H1sin(ζθ)+H2cos(ζθ)=H3,H4sin(ζθ)+H5cos(ζθ)=H6, (4.12)

    where

    {H1=ζ3+j1ζ,H2=h1ζ2+j2,H3=h4ζ2h6,H4=h1ζ2+j3,H5=ζ3+j4ζ,H6=h5ζ, (4.13)

    and

    {j1=h7h2,j2=h3+h8,j3=h3h8,j4=h2+h7. (4.14)

    It follows from Cramers rule that

    {sin(ζθ)=H2H6H3H5H2H4H1H5,cos(ζθ)=H3H4H1H6H2H4H1H5. (4.15)

    In view of cos2(ζθ)+sin2(ζθ)=1, we get

    [H2H6H3H5H2H4H1H5]2+[H3H4H1H6H2H4H1H5]2=1. (4.16)

    Then,

    H23H25+2HH26+H21H26+H23H24H21H25H22H242H2H3H5H62H1H3H4H6+2H1H2H4H5=0. (4.17)

    Using (4.13) and (4.17), we know

    ζ12I1ζ10I2ζ8I3ζ6I4ζ4I5ζ2I6=0, (4.18)

    where

    {I1=h242j42j12h21,I2=2h24j42h4h6+h25+h21h24j24+4j1j4j21h41+2h21j4+2h1j3+2h1j22h21j1,I3=h24j24+4h4h6j4+h26+h21h25+2h25j12h1h24j32h21h4h62j1j24+2j21j4+2h31j3+2h31j22h1h4h5j42h4h5j22h1h4h5j1+2h4h5j32h1j3j4+2(j2h1j1)(h1j4j3)+2h1j1j2,I4=2h4h6j242h26j42h1h25j2+h25j21+h24j23+4h1h4h6j3+h21h26j21j24h21j234h21j2j3h21j22+2h5j4(h1h6+h4j2)+2h5h6j2+2h1h5h6j1+2h5j3(h4j1h6)+2j3j4(j2h1j1)+2j1j2(h1j4j3),I5=h26j24+h25j222h4h6j232h1h26j3+2h1j2j23+2h1j22j32h5h6j2j42h5h6j1j3+2j1j2j3j4,I6=h26j23j22j23. (4.19)

    Let

    2(ζ)=ζ12I1ζ10I2ζ8I3ζ6I4ζ4I5ζ2I6. (4.20)

    Suppose that

    (A5)|h6|>|j2|

    holds, noticing that limζ+2(ζ)=+>0, then we find that Eq (4.18) owns at least one positive real root. Thus, Eq (4.6) owns at least one pair of pure roots. Without loss of generality, here we assume that Eq (4.18) admits twelve positive real roots (say, ζι,ι=1,2,3,,12). According to (4.15), one gets

    θ(k)l=1ζl[arccos(H3(ζl)H4(ζl)H1(ζl)H6(ζl)H2(ζl)H4(ζl)H1(ζl)H5(ζl))+2kπ], (4.21)

    where ι=1,2,3,12;k=0,1,2;

    {H1(ζl)=ζ3l+j1ζl,H2(ζl)=h1ζ2l+j2,H3(ζl)=h4ζ2lh6,H4(ζl)=h1ζ2l+j3,H5(ζl)=ζ3l+j4ζl,H6(ζl)=h5ζl. (4.22)

    Denote θ=min{ι=1,2,3,,12;k=0,1,2,}{θ(k)ι} and suppose that when θ=θ, (4.6) owns a pair of imaginary roots ±iζ0.

    Now, the following condition is presented:

    (A6)Q1RQ2R+Q1IQ2I>0,

    where

    {Q1R=2h1ζ0sin(ζ0θ)(3ζ20h2h7)cos(ζ0θ),Q1I=(3ζ20+h2h7)sin(ζ1θ)+2h1ζ0cos(ζ0θ),Q2R=(h1ζ30+h3ζ0+h8ζ0)sin(ζ0θ)+(ζ40+h2ζ20h7ζ20)cos(ζ0θ),Q2I=(ζ40+h2ζ20+h7ζ20)sin(ζ0θ)+(h1ζ30h3ζ0+h8ζ0)cos(ζ0θ). (4.23)

    Lemma 4.1. Let λ(θ)=η1(θ)+iη2(θ) be the root of Eq. (4.9) at θ=θ obeying η1(θ)=0,η2(θ)=ζ0, then Re(dλdθ)|θ=θ,ζ=ζ0>0.

    (3λ2+2h1λ+h2)eλθdλdθ+(λ3+h1λ2+h2λ+h3)eλθ(dλdθθ+λ)+h7eλθdλdθ(h7λ+h8)eλθ(dλdθθ+λ)+(2h4λ+h5)dλdθ=0, (4.24)

    which leads to

    (dλdθ)1=Q1(λ)Q2(λ)θλ, (4.25)

    where

    {Q1(λ)=(3λ2+2h1λ+h2)eλθ+h7eλθ,Q2(λ)=(h7λ+h8)λeλθ(λ3+h1λ2+h2λ+h3)λeλθ. (4.26)

    Hence,

    Re[(dλdθ)1]θ=θ,ζ=ζ0=Re[Q1(λ)Q2(λ)]θ=θ,ζ=ζ0=Q1RQ2R+Q1IQ2IQ22R+Q22I. (4.27)

    By (A6), one gets

    Re[(dλdθ)1]θ=θ,ζ=ζ0>0, (4.28)

    which completes the proof.

    Based on the study above, the following conclusion is lightly acquired.

    Theorem 4.1. Suppose that (A4)(A6) hold, then the equilibrium point E(u1,u2,u3) of model (4.1) holds locally asymptotically stable if θ[0,θ) and model (4.1) produces Hopf bifurcations at the equilibrium point E(u1,u2,u3) when θ=θ.

    In this part, we will investigate the Hopf bifurcation problem of system (1.3) using a suitable extended delayed feedback controller consisting of parameter perturbation with delay. In accordance with [33,34], we propose the following controlled predator-prey model:

    {du1(t)dt=ρ1[u1(t)(1u1(tθ)γ12u22(t))+γu1(t)u3(t)]+ρ2[u1(t)u1(tθ)],du2dt=ru2(t)[1u2(tθ)γ21u21](1p)u2u3v1+(1p)u2,du3dt=ρ3u3[v2+v3(1p)u2v1+(1p)u2]+ρ4[u3(t)u3(tθ)], (5.1)

    where ρ1ρ4 stands for feedback gain parameters. System (5.1) owns the same equilibrium point E(u1,u2,u3) as that of system (1.3). Let

    {u1(t)=ˉu1(t)+u1,u2(t)=ˉu2(t)+u2,u3(t)=ˉu3(t)+u3, (5.2)

    denote ˉu1 as u1, ˉu2 as u2, ˉu3 as u3, and the linear system of system (5.1) around E(u1,u2,u3) takes the following expression:

    {du1dt=f1u1+f2u2+f3u3+f4u1(tθ),du2dt=f5u1+f6u2+f7u3+f8u2(tθ),du3dt=f9u2+f10u3+f11u3(tθ), (5.3)

    where

    {f1=(1u1γ12u22γu3)ρ1+ρ2,f2=2γ12u1u2ρ1,f3=γu1ρ1,f4=ρ1u1ρ2,f5=2rγ21u1u2,f6=rru2rγ21u21(1p)u3v1+(1p)u2+(1p)2u2u3[v1+(1p)u2]2,f7=(1p)u2v1+(1p)u2,f8=ru2,f9=(v3(1p)u3v1+(1p)u2v3(1p)2u2u3[v1+(1p)u2]2)ρ3,f10=(ρ4v2+v3(1p)u2v1+(1p)u2)ρ3,f11=ρ4. (5.4)

    The characteristic equation of system (5.3) owns the following expression:

    det[λf1d4eλθf2f3f5λf6f8eλθf70f9λf10f11eλθ]=0, (5.5)

    which leads to

    λ3+m1λ2+m2λ+m3+(m4λ2+m5λ+m6)eλθ+(m7λ+m8)e2λθ+m9e3λθ=0, (5.6)

    that is,

    (λ3+m1λ2+m2λ+m3)e2λθ+(m4λ2+m5λ+m6)eλθ+m9eλθ+m7λ+m8=0, (5.7)

    where

    {m1=f1f6f10,m2=f1f6+f1f10f2f5+f6f10f7f9,m3=f1f7f9f1f6f10+f2f5f10f3f5f9,m4=f4f8f11,m5=f1f8+f4f6+f4f10+f8f10+f1f11+f6f11,m6=f4f7f9f1f8f10f4f6f10+f1f6f11+f2f5f11,m7=f4f8+f4f11+f8f11,m8=f4f8f10f4f6f10f4f6f11f1f8f11,m9=f4f8f11. (5.8)

    If δ=0, then Eq (5.6) reads as:

    λ3+(m1+m4)λ2+m3+(m2+m5+m7)+m3+m6+m8+m9=0. (5.9)

    If

    (A7){1=m1+m4>0,2=|m1+m41m3+m6+m8+m9m2+m5+m7|>0,3=|m1+m410m3+m6+m8+m9m2+m5+m7m1+m400m3+m6+m8+m9|>0,

    is fulfilled, the three roots of Eq (5.6), \(\lambda_1, \lambda_2, \lambda_3\), have negative real components. Therefore, the equilibrium point E(u1,u2,u3) of system (5.1) with θ=0 is locally asymptotically stable.

    Suppose that λ=iω is the root of Eq (5.7), then Eq (5.7) becomes:

    [(iω)3+m1(iω)2+m2(iω)+m3]e2iωθ+[m4(iω)2+m5(iω)+m6)]eiωθ+m9eiωθ+m7(iω)+m8=0. (5.10)

    By (5.10), we have

    {(ω3m2ω)sin(2ωθ)+(m1ω2+m3)cos(2ωθ)m5ωsin(ωθ)+(m6+m9m4ω2)cos(ωθ)+m8=0,(m2ωω3)cos(2ωθ)+(m1ω2+m3)sin(2ωθ)+m5ωcos(ωθ)+(m6m9m4ω2)sin(ωθ)+m7ω=0, (5.11)

    and from (5.11), we can get

    M1cos2(ωθ)+M2cos(ωθ)+M3=(M4+M5cos(ωθ))1cos2(ωθ), (5.12)

    where

    {M1=2m1ω2+2m3,M2=m6+m9m4ω2,M3=m1ω2m3+m8,M4=m5ω,M5=2m2ω2ω3. (5.13)

    So, we can get

    N1cos4(ωθ)+N2cos3(ωθ)+N3cos2(ωθ)+N4cos(ωθ)+N5=0, (5.14)

    where

    {N1=M21+M25,N2=2M1M2+2M4M5,N3=2M1M3+M22+M24M25,N4=2M1M3+2M4M5,N5=M23+M24. (5.15)

    From (5.14), we can suppose that cos(ωθ)=y, and we have

    N1y4+N2y3+N3y2+N4y+N5=0. (5.16)

    According to the computer software, we can get

    cos(ωiθi)=yi(i=1,2,3,4). (5.17)

    It follows from (5.17) that

    θ(n)t=1ωt[arccosyt+2nπ], (5.18)

    where t=1,2,3,4; n=0,1,2,3,.

    Let θ=min{t=1,2,3,4;n=1,2,3,}{θ(n)t}, and assume that when θ=θ, Eq (5.6) has at least one pair of pure real roots ±iω0. Next, the following assumption is needed:

    (A8)T1RT2R+T1IT2I>0,

    where

    {T1R=(3ω20+m2)cos(2ω0θ)2m1ω0sin(2ω0θ)+m5cos(ω0θ)2m4ω0sin(ω0θ)+m7,T1I=(3ω20+m2)sin(2ω0θ)+2m1ω0cos(2ω0θ)+m5sin(ω0θ)+2m4ω0cos(ω0θ),T2R=(2ω40+2m2ω20)cos(2ω0θ)(2m1ω302m3ω0)sin(2ω0θ)+m5ω20cos(ω0θ)+(m9ω0m4ω30+m6ω0)sin(ω0θ),T2I=(2ω40+2m2ω20)sin(2ω0θ)+(2m1ω302m3ω0)cos(2ω0θ)+m5ω20sin(ω0θ)+(m9ω0+m4ω30m6ω0)cos(ω0θ). (5.19)

    Lemma 5.1. Suppose that λ(θ)=ψ1(θ)+iψ2(θ) is the root of Eq (5.7) at θ=θ such that ψ1(θ)=0, ψ2(θ)=ω0, then Re(dλdθ)|θ=θ,ω=ω0>0.

    Proof. By Eq (5.7), one gets

    (3λ2+2m1λ+m2)e2λθdλdθ+(λ3+m1λ2+m2λ+m3)e2λθ(dλdθ2θ+2λ)+(2m4λ+m5)eλθdλdθ+(m4λ2+m5λ+m6)eλθ(dλdθθ+λ)+m7dλdθm9eλθ(dλdθθ+λ)=0, (5.20)

    which implies

    (dλdθ)1=T1(λ)T2(λ)θλ, (5.21)

    where

    {T1(λ)=(3λ2+2m1λ+m2)e2λθ+(2m4λ+m5)eλθ+m7,T2(λ)=λ[2(λ3+m1λ2+m2λ+m3)e2λθ+(m4λ2+m5λ+m6)eλθm9eλθ]. (5.22)

    Hence

    Re[(dλdθ)1]θ=θ,ω=ω0=Re[T1(λ)T2(λ)]θ=θ,ω=ω0=T1RT2R+T1IT2IT22R+T22I. (5.23)

    By (A8), we have

    Re[(dλdθ)1]θ=θ,ω=ω0>0, (5.24)

    which concludes the proof.

    Based on the research presented above, the following conclusion is loosely drawn.

    Theorem 5.1. Suppose that (A7)(A8) hold, then the equilibrium point E(u1,u2,u3) of model (5.1) is locally asymptotically stable if θ[0,θ) and model (5.1) generates Hopf bifurcations at the equilibrium point E(u1,u2,u3) when θ=θ.

    Remark 5.1. In Section 4, the controller is called the hybrid controller that includes state feedback and parameter perturbation with delay. This controller is only added to the first equation of system (1.3). In Section 5, the controller is called the extended hybrid controller that includes state feedback and parameter perturbation with delay. This controller is added to the first equation and the third equation of system (1.3). Hybrid controller Ⅱ owns more control parameters than those in hybrid controller Ⅰ and has greater adjustment flexibility in controlling the stability domain and the onset of Hopf bifurcation of system (1.3).

    Example 6.1. Think about the following Lotka-Volterra commensal symbiosis system:

    {du1(t)dt=u1(t)(1u1(tθ)1.1u22(t))+0.04u1(t)u3(t),du2(t)dt=0.5u2(t)(1u2(tθ)0.3u21(t))(10.1)u2(t)u3(t)0.2+(10.1)u2(t),du3(t)dt=u3(t)(0.35+0.5(10.1)u2(t)0.2+(10.1)u2(t)). (6.1)

    It is straightforward to see that system (6.1) has a single positive equilibrium point E(0.7092,0.5185,0.1224). One can easily verify that the conditions (A1)(A3) of Theorem 3.1 hold true. Using Matlab software, we can obtain θ01.3. To validate the accuracy of Theorem 3.1, we use two distinct delay values: θ=0.8 and θ=1.9. For θ=0.8<θ01.3, simulation graphs are provided in Figure 1. Figure 1 shows that u10.7092,u20.5185,u30.1224 as t+. In this case, the equilibrium point E(0.7092,0.5185,0.1224) of model (6.1) has a locally asymptotically stable state. For θ=1.9>θ01.3, we receive simulation graphs, as shown in Figure 2. Figure 2 shows that u1 maintains a periodic vibrating level around 0.7092, whereas u2 maintains a level around 0.5185 and u3 maintains a periodic vibrating level around 0.1224. That is, a set of periodic solutions (known as Hopf bifurcations) arise at the equilibrium point E(0.7092,0.5185,0.1224).

    Figure 1.  Matlab simulation figures of system (6.1) under the delay θ=0.8<θ01.3, and the equilibrium point E(u1,u2,u3)=E(0.7092,0.5185,0.1224) holds a locally asymptotically stable level.
    Figure 2.  Matlab simulation figures of system (6.1) with delay θ=1.9>θ01.3, and a cluster of periodic solutions (i.e., Hopf bifurcations) arise around the equilibrium point E(E(u1,u2,u3)=E(0.7092,0.5185,0.1224).

    Example 6.2. Think about the following controlled Lotka-Volterra commensal symbiosis system:

    {du1(t)dt=δ1[u1(t)(1u1(tθ)1.1u22(t))+0.04u1(t)u3(t)]+δ2[u1(t)u1(tθ)],du2dt=0.5u2(t)[1u2(tθ)0.3u21(t)](10.1)u2(t)u3(t)0.2+(10.1)u2(t),du3dt=u3(t)[0.35+0.5(10.1)u2(t)0.2+(10.1)u2(t)]. (6.2)

    It is straightforward to see that system (6.2) has a single positive equilibrium point E(0.7092,0.5185,0.1224). Let δ1=0.5,δ2=0.1. One can easily verify that the conditions (A4)(A6) of Theorem 4.1 hold true. Using Matlab software, we can obtain θ1.8. To validate the correctness of the acquired assertions of Theorem 4.1, we use two distinct delay values: θ=1.6 and θ=2.25. For θ=1.6<θ1.8, we get simulation diagrams which are presented in Figure 3. Based on Figure 3, we find that u10.7092,u20.5185,u30.1224 when t+. In other words, the equilibrium point E(0.7092,0.5185,0.1224) of model (6.2) holds a locally asymptotically stable state. For θ=2.25>θ1.8, we get simulation diagrams which are presented in Figure 4. Based on Figure 4, we find that u1 maintains a periodic vibrating level around 0.7092, whereas u2 maintains a level around 0.5185 and u3 maintains a periodic vibrating level around 0.1224. That is to say, a family of periodic solutions (namely, Hopf bifurcations) appear near the equilibrium point E(0.7092,0.5185,0.1224).

    Figure 3.  Matlab simulation figures of system (6.2) under the delay θ=1.6<θ1.8, and the equilibrium point E(u1,u2,u3)=E(0.7092,0.5185,0.1224) holds a locally asymptotically stable level.
    Figure 4.  Matlab simulation figures of system (6.2) under the delay θ=2.25>θ1.8, and a cluster of periodic solutions (i.e., Hopf bifurcations) arise around the equilibrium point E(u1,u2,u3)=E(0.7092,0.5185,0.1224).

    Example 6.3. Think about the following controlled Lotka-Volterra commensal symbiosis system:

    {du1(t)dt=ρ1[u1(t)(1u1(tθ)1.1u22(t))+0.04u1(t)u3(t)]+ρ2[u1(t)u1(tθ)],du2dt=0.5u2(t)[1u2(tθ)0.3u21](10.1)u2u30.2+(10.1)u2,du3dt=ρ3u3[0.35+0.5(10.1)u20.2+(10.1)u2]+ρ4[u3(t)+u3(tθ)]. (6.3)

    It is straightforward to see that system (6.3) has a single positive equilibrium point E(0.7092,0.5185,0.1224). Let ρ1=0.5,ρ2=0.1,ρ3=0.6,ρ4=0.1. One can easily verify that the conditions (A7) and (A8) of Theorem 5.1 hold true. By applying Matlab software, one can get θ2.20. To validate the correctness of the acquired assertions of Theorem 5.1, we choose both different delay values: θ=2.00 and θ=2.75. For θ=2.00<θ2.20, we get simulation diagrams which are presented in Figure 5. Based on Figure 5, we find that u10.7092,u20.5185,u30.1224 when t+. In other words, the equilibrium point E(0.7092,0.5185,0.1224) of model (6.3) holds a locally asymptotically stable state. For θ=2.75>θ2.20, we get simulation diagrams which are presented in Figure 6. Based on Figure 6, we find that u1 maintains a periodic vibrating level around 0.7092, whereas u2 maintains a level around 0.5185 and u3 maintains a periodic vibrating level around 0.1224. That is to say, a family of periodic solutions (namely, Hopf bifurcations) appear near the equilibrium point E(0.7092,0.5185,0.1224).

    Figure 5.  Matlab simulation figures of system (6.2) under the delay θ=2.00<θ2.20, and the equilibrium point E(u1,u2,u3)=E(0.7092,0.5185,0.1224) holds a locally asymptotically stable level.
    Figure 6.  Matlab simulation figures of system (6.3) under the delay θ=2.75>θ2.20, and a cluster of periodic solutions (i.e., Hopf bifurcations) arise around the equilibrium point E(u1,u2,u3)=E(0.7092,0.5185,0.1224).

    Remark 6.1. It follows from the Matlab simulation results of Examples 6.1–6.3 that one can know that the bifurcation value of system (6.1) is equal to 1.3, the bifurcation value of system (6.2) is equal to 1.8 and the bifurcation value of system (6.3) is equal to 2.20, which indicates that we can expand the domain of stability of system (6.1) and postpone the time of emergence of Hopf bifurcation of system (6.1) via the formulated two hybrid delayed feedback controllers.

    It is generally recognized that the delayed dynamical model is an important tool for understanding the interactions of many biological populations in the natural environment [35,36,37]. Many studies on predator-prey models have conducted and yielded numerous results over the last few decades [38,39,40]. In this study, we provide a novel delayed Lotka-Volterra commensal symbiosis model. This paper discusses the uniqueness, nonnegativeness, and boundedness of the delayed Lotka-Volterra commensal symbiosis solution. The Hopf bifurcation issue is addressed. Then, the critical delay value θ0 is retrieved. In order to modify the domain of stability and the time of the bifurcation phenomenon in this model, we have successfully developed two distinct hybrid delayed feedback controllers. Two critical delay values, θ,θ, are acquired. In these two controllers, the role of delay is displayed. Theoretically, the exploration fruits are very useful for managing and balancing the populations of two species. Furthermore, the exploratory concepts may be used for other fractional-order and integer-order dynamical systems in a wide range of disciplines to dominate the bifurcation phenomena, stability, and chaos [41,42,43]. During the past decades, many works on this topis is explored, see [44,45,46]. In this paper, we only deal with the Hopf bifurcation onset and Hopf bifurcation control in this paper. We leave the stability and direction of Hopf bifurcation periodic solutions for future work and we will refer to the works in [47,48,49]. In addition, we will explore the Hopf bifurcation of fractional-order dynamical models [50,51,52].

    Yingyan Zhao: Conceptualization, formal analysis, investigation, methodology, software, writing-original draft, writing-review & editing; Changjin Xu: Conceptualization, formal analysis, investigation, methodology, software, writing-original draft, writing-review & editing; Yiya Xu: Formal analysis, investigation, software, writing-review & editing; Jinting Lin: Conceptualization, formal analysis, investigation, methodology, writing-original draft, writing-review & editing; Yicheng Pang: Conceptualization, investigation, methodology, writing-review & editing; Zixin Liu: Conceptualization, investigation, methodology, writing-review & editing; Jianwei Shen: Conceptualization, investigation, methodology, writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

    This work is supported by National Natural Science Foundation of China (No. 12261015, No. 62062018), Project of High-level Innovative Talents of Guizhou Province ([2016]5651), Guizhou Provincial Science and Technology Projects (ZK[2022]025) and Science Project of Guizhou University of Finance and Economics (2022XSXMB08).

    The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.

    The authors declare that they have no conflict of interest.



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