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Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model
Journal Pre-proof Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model Chengdai Huang, Heng Liu, Xiaoping Chen, Minsong Zhang, Jinde Cao, Ahmed Alsaedi
PII: DOI: Reference:
S0378-4371(20)30003-0 https://doi.org/10.1016/j.physa.2020.124136 PHYSA 124136
To appear in:
Physica A
Received date : 8 August 2019 Revised date : 18 December 2019 Please cite this article as: C. Huang, H. Liu, X. Chen et al., Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2020.124136. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
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Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator-prey model Chengdai Huanga,∗ , Heng Liub , Xiaoping Chenc , Minsong Zhangd , Jinde Caoe , Ahmed Alsaedif School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China b
School of Science, Guangxi University for Nationalities, Nanning 530006, China
c
Department of Mathematics, Taizhou University, Taizhou 225300, China
School of Mathematics and Statistics, Hubei University of Arts and Science Xiangyang 441053 China
e
Research Center for Complex Systems and Network Sciences, and School of Mathematics, Southeast University, Nanjing 210096, China
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d
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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a
Abstract
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This paper meditates the topic of bifurcation control for a delayed fractional-order predator-prey model in accordance with an enhancing feedback controller. Initially, the bifurcation points of devised model are incisively established via analytic extrapolation by regarding time delay as a bifurcation parameter. Then, a range of contrastive analysis on the influence of bifurcation control are numerically discussed including enhancing feedback, dislocated feedback and eliminating feedback approaches. It views that the stability performance of the developed model can be vastly intensified by enhancing feedback approach. Numerical simulations are finally performed to check the feasibility of the devised scheme.
1. Introduction
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Keywords: Fractional order, Enhancing feedback control, Dislocated feedback control, Bifurcation control, Predator-prey models.
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Predator-prey models are a class of considerable models in the sphere of ecological systems, which are one of the basic topics in ecology as a result of the pervasive significance and existence. The original predator-prey model was formulated by [1, 2]. It is universally known time delays have been merged into biological systems to elaborately delineate the authentic dynamical predator-prey by taking into account the time required for resource regeneration time, maturation period, reaction time, feeding time, gestation period [3, 4]. Numerous good results have been attained on the investigation of delayed predator-prey models [5, 6]. Modelling and control based on the theory of fractional calculus of intricate systems can greatly enhance the capability of discrimination, design and control for dynamic systems since fractional calculus possesses infinite memory [7, 8, 9]. In [10], the author pointed out that fractional calculus admits greater degrees of freedom in modeling dynamical systems in contrast with conventional descriptions of biological ∗
Corresponding author. E-mail addresses: [email protected] (C. Huang)
Preprint submitted to Elsevier
January 4, 2020
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systems. It further detected that the stability of the solutions for a delayed predator-prey system can be largely enhanced [11]. Modelling and control of fractional order dynamical systems has recently grown a hot research topic, and lots of colorful results have been captured [12, 13, 14, 15, 16, 17]. As a matter of fact, a great deal of biological models exhibit fractional dynamics thanks to possessing memory effects. Recently, fractional calculus has successfully introduced into predator-prey models, and some interesting phenomena has been discovered. In [18], the author discovered the number of stability switching can be accurately described in term of fractional modelling. In [19], Mondal et al detected that the solutions of fractional order predator-prey system converge to the respective equilibrium more slowly as fractional order decreases. In [20], Chinnathambi and Rihan found that fractional order can strengthen the stability of prey-predator system and hamper the occurrence of oscillation behaviors. Fractional dynamics of delayed predator-prey without delays has been studied [21, 22, 23, 24].
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Delicate stability results of nonlinear systems can be acquired in the light of puissant bifurcation analysis [25, 26, 27, 28, 29, 30]. Cao et al considered the issue of bifurcation for controlled complex networks model with two nonidentical delays in [25]. It should be stated directly that bifurcations of in traditional delayed integer-order models have been sufficiently investigated because of the maturation of theoretical tools. Nevertheless, the bifurcations of fractional-order dynamical models is on the upsurge [31, 32, 33, 34, 35]. In [32], the authors studied the bifurcation of delayed generalized fractionalorder prey-predator model with interspecific competition, and accurate bifurcation results were derived. In [35], the bifurcation of a fractional-order quaternion-valued neural network with time delay was examined, and it revealed that the onset of bifurcation can be advanced with the increment of fractional order. Very recently, the problem of bifurcation control of delayed fractional models has aroused great interest [36, 37, 38]. In [36], the authors devised a parametric delay feedback controller to control the bifurcation for a delayed fractional dual congestion integer-order model. In [37], it uncovered that the bifurcation phonomania can be controlled by regulating extended feedback delay or fractional order. In [38], a neoteric fractional-order PD scheme was proposed to unperturb the innate bifurcation for integer-order small-world network if setting up appropriative control gains.
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Normally, numerous bifurcation control schemes can be adopted to handle bifurcation dynamics, such as dislocated feedback control, speed feedback control and enhancing feedback control [39, 40]. In [39], the author detected that the feedback coefficients were smaller than the ones of ordinary feedback control in controlling Lorenz system. In [40], it revealed that the system can be efficiently control via enhancing feedback approach by selecting simple external inputs and small feedback coefficient. Actually, it is difficult to completely control the dynamical properties of a complex system relying on a unique feedback variable. In this instance, a larger feedback gain needs to be selected for procuring the anticipated dynamical behaviors of a nonlinear system. Two conspicuous features of enhancing feedback control can be found. i) The stability performance of the controlled system can be immensely improved by taking small control gains once enhancing feedback controllers appear in the controlled system. ii) Enhancing feedback control methods can extremely lessen the control cost in control engineering. Nevertheless, the bifurcation control of fractional order predator-prey systems with delays based on enhancing feedback control has been not felicitously hashed out before. Impelled by the aforementioned discussions, we shall render a theoretical exploration on bifurcation control for a delayed fractional-order predator-prey model in accordance with enhancing feedback control technique in this paper. The key features of this paper are listed as follows: 1) Enhancing feedback control strategy is devised to control the bifurcation in a fractional order predator-prey model with time delay, and the precise bifurcation control results are obtained in terms of theoretical analysis. 2) It further discovers that a larger feedback gain is selected for controlling the onset of bifurcation of the devised system via dislocated feedback scheme. 3) It detects that the control effects of the proposed system can be largely hoisted by enhancing feedback approach than dislocated feedback one. It manifests that the devised enhancing feedback method can attenuate the control cost compared with
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dislocated feedback ones.
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The framework of the current paper is constructed as follows. Section 2 presents some definition and Lemmas regarding fractional calculus. Section 3 addresses the investigated model. Section 4 establishes the essential bifurcation control results via enhancing feedback control method. Section 5 verifies the efficiency of the proposed control scheme through a simulation example. Section 6 finalizes the paper with a conclusion. 2. Preliminaries
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This section addresses some practical definition and Lemmas for the next theoretical analysis and numerical simulations. Definition 1. [41] The Caputo fractional-order derivative can be defined as ∫ t 1 Dtp f (t) = (t − s)l−p−1 f (l) (s)ds, Γ(l − p) 0 ∫∞ where l − 1 ≤ p < l ∈ Z + , Γ(·) is the Gamma function, Γ(s) = 0 ts−1 e−t dt.
L{Dtp f (t); s} If
f k (0)
p
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On account of Laplace transform rule, it follows from the Caputo fractional-order derivatives that
is
= s F (s) −
= 0, k = 1, 2, . . . , n, then
l−1 ∑
sp−k−1 f (k) (0),
k=0
L{Dtp f (t); s}
l − 1 ≤ p < l ∈ Z +.
= sp F (s).
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Lemma 1. [42] The following n-dimensional linear fractional-order system is explored p1 D l1 (t) = a11 l1 (t) + a12 l2 (t) + · · · + k1n ln (t), Dp2 l (t) = a l (t) + a l (t) + · · · + k l (t), 2 21 1 22 2 2n n . .. pn D ln (t) = an1 l1 (t) + an2 l2 (t) + · · · + ann ln (t),
(1)
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where 0 < pi < 1(i = 1, 2, . . . , n). Supposing that p is the lowest common multiple of the denominators φi ψi of pi , where pi = ψ , (φi , ψi ) = 1, φi , ψi ∈ Z + , for i = 1, 2, . . . , n. It is described as i p λ 1 − a11 −a12 ··· −a1n −a21 λp2 − a22 · · · −a2n △(λ) = . .. .. .. .. . . . . −an2
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−an1
· · · λpn − ann
Then the zero solution of system (1) is globally asymptotically stable in the Lyapunov sense if all roots of the equation det(△(λ)) = 0 satisfy | arg(λ)| > pi π/2.
Lemma 2. [42] The following n-dimensional linear fractional-order system with delays is examined p1 D l1 (t) = a11 l1 (t − τ11 ) + a12 l2 (t − τ12 ) + · · · + a1n ln (t − τ1n ), Dp2 l2 (t) = a21 l1 (t − τ21 ) + a22 l2 (t − τ22 ) + · · · + a2n ln (t − τ2n ), (2) .. . pn D ln (t) = an1 l1 (t − τn1 ) + an2 l2 (t − τn2 ) + · · · + ann ln (t − τnn ), 3
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−an1 e−sτn1
−an2 e−sτn2
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where pi ∈ (0, 1)(i = 1, 2, . . . , n), the initial values Vi (t) = Ψi (t) are given for − maxi,j , τi,j = − maxi,j ≤ t ≤ 0 and i = 1, 2, . . . , n. For system, (2) time-delay matrix τ = (τi,j ) ∈ (R+ )n×n , coefficient matrix H = (ai,j )n×n , state variables li (t), li (t − τi,j ) ∈ R, and initial values Ψi (t) ∈ C 0 [−τmax , 0]. Its fractional order is defined as p = (l1 , l2 , . . . , ln ). It is defined as p −a12 e−sτ12 ··· −a1n e−sτ1n s 1 − a11 e−sτ11 −a21 e−sτ21 sp2 − a22 e−sτ22 · · · −a2n e−sτ2n △(s) = . .. .. .. .. . . . . · · · spn − ann e−sτnn
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Then the zero solution of system (2) is Lyapunov globally asymptotically stable if all the roots of the characteristic equation det(△(s)) = 0 have negative real parts. 3. Model formulation
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The following fractional order ratio-dependent predator-prey system with time delay is developed in this paper. x1 (t − τ )x2 (t) p D x1 (t) = x1 (t − τ )[1 − x1 (t − τ )] − x (t − τ ) + αx (t) , 1 2 (3) [ ] x (t − τ ) 2 p D x2 (t) = βx2 (t − τ ) δ − , x1 (t − τ )
where the variables and parameters of systems (3) are displayed in Table.1.
Table 1: The concrete notations of relevant variables and parameters of system (3) Symbols
Description
x2 (t) p
The densities of predator at time t Fractional order, p ∈ (0, 1] Positive constant
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α
The densities of prey at time t
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x1 (t)
β
Positive constant
δ
Positive constant
τ
Time delay for both the densities of the prey and the predator
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The bifurcation control of the following fractional-order version model (3) is mainly concerned in this paper x1 (t − τ )x2 (t) p D x1 (t) = x1 (t − τ )[1 − x1 (t − τ )] − x (t − τ ) + αx (t) + K1 [(x1 (t) − x1 (t − τ )], 1 2 (4) ] [ x (t − τ ) 2 Dp x2 (t) = βx2 (t − τ ) δ − + K2 [(x2 (t) − x2 (t − τ )], x1 (t − τ )
where K1 , K2 denote feedback control gains, K1 ≤ 0, K2 ≤ 0.
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Remark 1. Observing that system (4) degenerates into the integer-order model in [43] when selecting p = 1, K1 = K2 = 0. If K1 = 0, K2 ̸= 0 or K1 ̸= 0, K2 = 0, then model (4) develops into dislocated timedelayed feedback control system. If K1 ̸= 0, K2 ̸= 0, then model (4) becomes enhancing time-delayed feedback control system.
This means that x∗1 =
1+αδ−δ 1+αδ ,
x∗2 =
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Based on the condition 1 + αδ > δ, then system (3) occupies a unique positive equilibrium E ∗ = (x∗1 , x∗2 ), which complies with the following equations x∗2 ∗ 1 − x − = 0, 1 x∗1 + αx∗2 ( x∗ ) β δ − 2∗ = 0. x1 δ(1+αδ−δ) . 1+αδ
Remark 2. E ∗ of system (4) is consistent with system (4), which does not rely on the values of control parameters K1 and K2 . This implies that E ∗ is immutable in the devised controllers.
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To capture the brilliant control effects, the following essential assumption is presented in this paper: (H1) K1 ≤ 0, K2 ≤ 0. Based on [42], this paper is devoted to find out the conditions of bifurcation for model (4) by using time delay a bifurcation parameter. Then, some comparative studies on bifurcation control are carried out. It finds that the stability performance of the controlled system can be excessively ameliorated on the basis of enhancing feedback control in comparison with the dislocated feedback control. 4. Main results
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In this section, time delay shall be selected as a bifurcation parameter to investigate the problem of bifurcation control for the predator-prey model (4) by utilizing enhancing feedback approach. The existence bifurcation and bifurcation point for the proposed model shall be determined.
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Performing transformations u1 (t) = x1 (t)−x∗1 , u2 (t) = x2 (t)−x∗2 , then system (4) can be transformed into the following form (u1 (t − τ ) + x∗1 )(u2 (t) + x∗2 ) p ∗ ∗ D u (t) = [u (t − τ ) + x ][1 − (u (t − τ ) + x )] − 1 1 1 1 1 x1 (t − τ ) + αx2 (t) + K1 [(u1 (t) − u1 (t − τ )], (5) [ ] ∗ p (u2 (t − τ ) + x2 ) + K2 [(u2 (t) − u2 (t − τ )]. D u2 (t) = β(u2 (t − τ ) + x∗2 ) δ − (u1 (t − τ ) + x∗1 )
It gains from system (5) that the linearized form is {
Dp u1 (t) = (γ11 − K1 )u1 (t − τ ) + K1 x1 (t) + γ12 u2 (t),
Dp u2 (t) = γ21 u1 (t − τ ) + K2 u2 (t) + (γ22 − K2 )u2 (t − τ ),
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(6)
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where x∗2 x∗2 x∗1 + , x∗1 + αx∗2 (x∗1 + αx∗2 )2 αx∗ x∗ x∗ = − ∗ 1 ∗ + ∗ 2 1∗ 2 , x1 + αx2 (x1 + αx2 ) βx∗2 = ∗ 2, (x1 ) 2βx∗ = βδ − ∗ 22 . (x1 )
γ12 γ21 γ22
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γ11 = 1 − 2x∗1 −
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The associated characteristic equation of (6) is
ℓ1 (s) + ℓ2 (s)e−sτ + ℓ3 (s)e−2sτ = 0, where ℓ1 (s) = s2p − (K1 + k2 )sp + K1 K2 ,
(7)
ℓ2 (s) = −[(m11 + m22 − K1 − K2 )sp + 2K1 K2 − K1 m22 − K2 m11 + m12 m21 ],
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ℓ3 (s) = (m11 − K1 )(m22 − K2 ).
The real and imaginary parts of ℓq (s)(q = 1, 2, 3) are labeled by ℓrq , ℓiq . Then it calculates that pπ + K1 K2 , 2 pπ = w2p sin pπ − (K1 + K2 )wp sin , 2 pπ p = −[(m11 + m22 − K1 − K2 )w cos + 2K1 K2 − K1 m22 − K2 m11 + m12 m21 ], 2 pπ = −(m11 + m22 − K1 − K2 )wp sin , 2 = (m11 − K1 )(m22 − K2 ),
ℓr1 = w2p cos pπ − (K1 + K2 )wp cos ℓi1 ℓr2 ℓi2 ℓr3
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ℓi3 = 0.
Both sides of Eq.(7) are multiplied by esτ , then it derives that
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ℓ1 (s)esτ + ℓ2 (s) + ℓ3 (s)e−sτ = 0.
Assume that s = w(cos π2 + i sin π2 )(w > 0) is a purely imaginary root of Eq.(8), then it results in { r (ℓ1 + ℓr3 ) cos wτ − ℓi1 cos wτ = −ℓr2 , ℓi1 cos wτ + (ℓr1 − ℓr3 ) cos wτ = −ℓi2 ,
It is further labeled as
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G1 (w) = −ℓr2 (ℓr1 − ℓr3 ) − ℓi1 ℓi2 ,
G2 (w) = −ℓi2 (ℓr1 + ℓr3 ) + ℓi1 ℓr2 ,
G3 (w) = (ℓr1 )2 + (ℓi1 )2 − (ℓr3 )2 , d1 = −(K1 + K2 ), d2 = K1 K2 ,
d3 = m11 + m22 − K1 − K2 ,
d4 = K1 (K2 − m22 ) + K2 (K1 − m11 ) + m12 m21 , d5 = (m11 − K1 )(m22 − K2 ). 6
(8)
(9)
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As far as Eq.(9), it concludes that G1 (w) cos wτ = G (w) , 3 G2 (w) sin wτ = . G3 (w) G3 (w) = G21 (w) + G22 (w). It can be defined from Eq.(11) that
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By means of Eq.(10), it procures that
(10)
In terms of Eq.(12), it obtains that
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H(w) = G23 (w) − G21 (w) − G22 (w) = 0.
H(w) = w8p + ϵ1 w7p + ϵ2 w6p + ϵ3 w5p + ϵ4 w4p + ϵ5 w3p + ϵ6 w2p + ϵ7 wp + ϵ8 = 0, where ϵi (i = 1, 2, . . . , 8) are computed in Appendix. We further give the additional assumption:
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(H2) Eq.(13) has at least positive real roots.
It follows from the first equation of Eq.(10) that ] 1[ G1 (w) + 2kπ , τ (k) = arccos w G3 (w)
Define the bifurcation point
τ0 = min{τ (k) }, where τ (k) is defined by Eq.(14).
k = 0, 1, 2, . . . .
(11)
(12)
(13)
(14)
k = 0, 1, 2, . . . ,
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In the following, we will consider the stability of system (5) when τ = 0. If τ is removed, the characteristic equation (8) develops into λ2 + h1 λ + h2 = 0,
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where
(15)
h1 = −m11 − m22 ,
h2 = k1 m22 + k2 m11 − k1 k2 − m12 − m21 .
It is obvious from h1 > 0, h2 > 0 that the two roots of Eq.(15) have negative parts which satisfying Lemma 1. Hence, the positive equilibrium of the fractional system (4) is asymptotically stable.
(H3)
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To obtain the conditions of bifurcation, we further give the following assumptions: M1 N1 +M2 N2 N12 +N22
̸= 0,
where M1 , M2 , N1 , N2 are described by Eq.(18). Lemma 3. Let s(τ ) = ξ(τ ) + iw(τ ) be the root of Eq.(8) near τ = τj satisfying ξ(τj ) = 0, w(τj ) = w0 , then the following transversality condition holds [ ds ] ̸= 0, Re dτ (w=w0 ,τ =τ0 ) where w0 , τ0 represent the critical frequency and bifurcation point of model (4). 7
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′
′
Proof. The real and imaginary parts of ℓ′p (s)p = 1, 2, 3 are labeled by ℓpr , ℓpi . Using implicit function theorem to differentiate (8) concerning τ , then the following equation can be concluded ℓ′1 (s)
( )] [ ( )] ds [ ′ ds ds ds ds + ℓ2 (s) e−sτ + ℓ2 (s)e−sτ − τ − s + ℓ′3 (s) e−2sτ + ℓ3 (s)e−2sτ − 2τ − s = 0. dτ dτ dτ dτ dτ (16)
M (s) ds = , dτ N (s)
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where
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Eq.(7) implies that ℓ′3 (s) = 0. Mathematically, straightforward eduction from Eq.(18) yields (17)
M (s) = s[ℓ2 (s)e−sτ + 2ℓ3 (s)e−2sτ ],
N (s) = ℓ′1 (2) + [ℓ′2 (s) − τ ℓ2 (s)]e−sτ − 2τ ℓ3 (s)e−2sτ . It educes from Eq.(19) that
where
[ ds ] M1 N1 + M2 N2 = , dτ (w=w0 ,τ =τ0 ) N12 + N22
(18)
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Re
M1 = w0 (ℓr2 sin w0 τ0 − ℓi2 cos w0 τ0 + 2ℓr3 sin 2w0 τ0 − 2ℓi3 cos 2w0 τ0 ),
M2 = w0 (ℓr2 cos w0 τ0 + ℓi2 sin w0 τ0 + 2ℓr3 cos 2w0 τ0 + 2ℓi3 sin 2w0 τ0 ), ′
′
′
′
′
N1 = ℓ1r + (ℓ2r − τ0 ℓr2 ) cos w0 τ0 + (ℓ2i − τ0 ℓi2 ) sin w0 τ0 − 2τ0 ℓr3 cos 2w0 τ0 , ′
N2 = ℓ1i + (ℓ2i − τ0 ℓi2 ) cos w0 τ0 − (ℓ2r − τ0 ℓr2 ) sin 2w0 τ0 + 2τ0 ℓr3 sin 2w0 τ0 .
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(H3) indicates that transversality condition hold. We accomplish the proof of Lemma 3. Based on the previous analysis, the following Theorem is abtainable.
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Theorem 1. If (H1)-(H3) are met, then the following results are available: 1) E ∗ of the system (4) is asymptotically stable when τ ∈ 0, τ0 );
2) System (4) undergoes a Hopf bifurcation at E ∗ when τ = τ0 , i.e., it has a branch of periodic solutions bifurcating from E ∗ near τ = τ0 .
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Remark 3. As a result of the higher order of Eq.(13), it is hard to establish theoretically all the positive real roots of Eq.(13). Nevertheless, it is straightforward to procure the concrete of these positive real roots of Eq.(13) by Maple numerical software. Hence, the values of τ0 can be accurately figured out. Remark 4. It revealed that a lesser feedback gain can not control the onset of bifurcation of a delayed fractional predator-prey system based on dislocated feedback strategy in [44]. On the basis of the dislocated feedback approach, an extended delayed feedback method was designed to control bifurcation of a delayed fractional predator-prey model in spite of selecting a small feedback gain in [37]. It should be noted that the extended feedback delay play an essential role in postponing Hopf bifurcation for such system. In this paper, if adopting enhancing feedback method, the bifurcation of the proposed system can be easily controlled provided that a set of smaller feedback gains are selected. Reversely, the bifurcation of the devised system can be controlled by using dislocated feedback approach only if 8
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a larger feedback gain is requested. It exhibits that the onset of the bifurcation for fractional delayed predator-prey system can be lagged and satisfactory bifurcation control effects are realized compared with the dislocated feedback approaches in this paper. This indicates that the devised enhancing feedback method can abate the control cost compared with dislocated feedback ones.
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Remark 5. The effects of fractional order on the bifurcation point are adequately discussed by calculation. It implies that the better effects in delaying the onset of bifurcation can be achieved as fractional order decreases if feedback gain are established. Simultaneously, it detects that the better control effects can be gained in delaying bifurcation of the proposed system by enhancing feedback method than dislocated feedback one. 5. Numerical examples
where α = 0.7, β = 0.9, δ = 0.6.
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In this section, numerical simulations are employed to divulge the appropriateness and applications of the proposed theory. For the purposes of comparison, the parameters identically derive from [43]. E ∗ can be obtained as (x∗1 , x∗2 ) = (0.5775, 0.3465). The initial value are all designated as (x1 (0), x2 (0)) = (0.6, 0.4). Investigate the controlled system x1 (t − τ )x2 (t) Dp x1 (t) = x1 (t − τ )[1 − x1 (t − τ )] − + K1 [(x1 (t) − x1 (t − τ )], x1 (t − τ ) + αx2 (t) (19) [ x (t − τ ) ] Dp x2 (t) = βx2 (t − τ ) δ − 2 + K2 [(x2 (t) − x2 (t − τ )], x1 (t − τ )
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The enhancing feedback control scheme is firstly considered. Selecting p = 0.96, K1 = −0.12, K2 = −0.24 in system (19), then it derives that w0 = 0.5112, τ0 = 2.7546. In terms of Theorem 1, E ∗ of controlled system (19) is asymptotically stable when τ = 2.5 < τ0 , which, depicted in Fig.1, while Fig.2 display the instability of E ∗ for system (19), Hopf bifurcation arises when τ = 2.8 > τ0 . The bifurcation results of system (19) of with p = 1 are attained as w0∗ = 0.5298, τ0∗ = 2.4954. It is apparent that E ∗ of controlled system (19) is asymptotically stable when choosing τ = 2.5, see Fig.1. Nevertheless, Hopf bifurcation occurs for system (19) with p = 1 when selecting τ = 2.5 due to time delay 2.5 > τ0∗ = 2.4954. This indicates that the better performance of fractional order system (19) can be procured than that of the corresponding integer-order one. This phenomena is commendably corroborated in Fig.3. Subsequently, the effects of the proposed dislocated control and uncontrolled schemes are explored based on comparative investigations.
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Case 1. Selecting p = 0.96, K1 = 0, K2 = −0.24, then it derives that w0 = 0.5713, τ0 = 2.2057. Choosing τ = 2.5 > τ0 = 2.2057, then Fig.4 simulates the volatility of system (19). If we select larger gain K2 = −0.8, then τ0 = 3.4841. Fig.5 well verifies the steadiness of E ∗ for system (19) with 2.5 < τ0 = 3.4841. Case 2. Taking p = 0.96, K1 = −0.12, K2 = 0, then it concludes that w0 = 0.6814, τ0 = 1.8372. Choosing τ = 2.5 > τ0 = 1.8372, then Fig.6 clearly depict the steadiness system (19). If we select larger gain K2 = −0.8, then τ0 = 3.4841. Fig.7 perfectly displays the steadiness of E ∗ for system (19) when 2.5 < τ0 = 3.4841. Case 3. Choosing p = 0.96, K1 = 0, K2 = 0, then it obtains that w0 = 0.7083, τ0 = 1.6634. Choosing τ = 2.5 > τ0 = 1.6634, the instability of system (19) described in Fig.8.
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0.62
0.4 0.38
0.58 0.56 0.54
0.36 0.34
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x2(t)
x1(t)
0.6
0.32 0
100
200
300
0
100
200
300
t
p ro
t 0.4 0.38
x2(t)
2
x (t)
0.4 0.35
0.65
0.34 0.32
400
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0.6 x1(t) 0.55 0
0.36
200 t
0.5
0.55 x1(t)
0.6
Figure 1: Time series and portrait plots of system (19) with p = 0.96, K1 = −0.12, K2 = −0.24, τ = 2.5 < τ0 = 2.7546.
0.7
0.45
x2(t)
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0.6 0.55 0.5
0
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x1(t)
0.65
100
200
0.4 0.35 0.3 0.25
300
0
t
0.6 x (t) 1
0.5 0
300
0.45 0.4
0.3
0.2 0.7
200 t
x2(t)
2
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x (t)
0.4
100
0.35 0.3
400 200
0.25 0.5
t
0.6 x (t)
0.7
1
Figure 2: Time series and portrait plots of system (19) with p = 0.96, K1 = −0.12, K2 = −0.24, τ = 2.8 > τ0 = 2.7546.
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0.62
0.4 0.38
x2(t)
0.58 0.56 0.54
0.36 0.34 0.32
0
100
200
300
0
100
t
p ro
0.4
300
0.38
x2(t)
2
200
t
0.4
x (t)
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x1(t)
0.6
0.35
0.65
0.32
400 200 t
0.34
0.55
Pr e-
0.6 x1(t) 0.55 0
0.36
0.6 x1(t)
0.65
Figure 3: Time series and portrait plots of system (19) with p = 1, K1 = −0.12, K2 = −0.24, τ = 2.5.
6. Conclusion
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Additionally, by varying the values of p, the corresponding τ0 of enhancing feedback control, dislocated feedback control and without control approaches are addressed, which displayed in Fig.9. This implies that enhancing feedback control transcends than others cases, which are authenticated in simulation results in Figs.10-12. If establishing p, K2 = −0.24 or K2 = 0, the values of τ0 can be determined as K1 varies, which depicted in Fig.13. Fig.13 also means that enhancing feedback control oversteps dislocated feedback control, which is illustrated in Figs.14-16. If predetermining p = 0.96, K1 = −0.24 or K1 = 0 and varying K2 in system (19), and the values of τ0 can be figured out, which is listed in Fig.17. It can also be noted from Fig.17 that enhancing feedback control overmatches dislocated feedback control, which is very agreement with numerical simulations in Figs.18-20.
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The problem of bifurcation control for a delayed fractional-order predator-prey model has been smartly investigated by virtue of enhancing feedback approach in this paper. The bifurcation point of controlled system has been totally established. Analysis and simulations further reveal that the better efficiency of bifurcation control has been obtained in term of enhancing feedback approach than dislocated and uncontrolled ones with partially or completely removing the branch for feedback gains. It has detected that the onset of bifurcation can be controlled for the dislocated feedback methods, yet the greater feedback gains should be taken, which increases the cost of control system. On the contrary, the stability performance of the controlled model can be extremely ameliorated on account of the designed enhancing feedback methodology by selecting slighter feedback gains. Numerical results have been provided to confirm the efficiency of the derived theoretical results.
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1.5
0.6 0.4
0.5
0
0
20
40 t
60
−0.2
80
0
20
40 t
60
80
p ro
0
0.2
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x2(t)
x1(t)
1
0.6 0.4
x2(t)
2
x (t)
0.5 0 −0.5 2
0.2 0
1 x1(t)
50 0 0
t
Pr e-
100
−0.2
0
0.5
1
1.5
x1(t)
Figure 4: Time series and portrait plots of system (19) with p = 0.96, K1 = 0, K2 = −0.24, τ = 2.5.
0.7
0.4
x2(t)
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0.6
0.5
20
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2
x (t)
0.4
0
40 t
60
0.6 x (t) 1
0.5 0
0.36 0.34
80
0
20
40 t
60
80
0.4 0.38
0.35
0.7
0.38
0.32
x2(t)
0.55
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x1(t)
0.65
0.36 0.34 0.32
100 50
0.5
t
0.6 x (t)
0.7
1
Figure 5: Time series and portrait plots of system (19) with p = 0.96, K1 = 0, K2 = −0.8, τ = 2.5.
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0
0
x2(t)
5
x1(t)
10
0
50 t
−10
100
0
50 t
100
p ro
−20
−5
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−10
5
0
x2(t)
x2(t)
10 0
−5
−10 20 50 t
Pr e-
100
0 x1(t) −20 0
−10 −20
−10
0
10
x1(t)
Figure 6: Time series and portrait plots of system (19) with p = 0.96, K1 = −0.12, K2 = 0, τ = 2.5.
0.61
0.4
x2(t)
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0.59
0
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2
x (t)
0.4
50 t
x (t) 0.55 0 1
0.36 0.34
100
0
50 t
100
0.4 0.38
0.35
0.6
0.38
0.32
x2(t)
0.58
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x1(t)
0.6
0.36 0.34 0.32
100 50
0.57
t
0.58 x (t)
0.59
0.6
1
Figure 7: Time series and portrait plots of system (19) with p = 0.96, K1 = −4.5, K2 = 0, τ = 2.5.
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1
10
x2(t)
20
x1(t)
2
−1
0
0
10
20
−10
30
0
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0
10
20
30
t
p ro
t 20
10
x2(t)
x2(t)
20 0
0
−20 2
40
Pr e-
0 x1(t)
20 −2 0
t
−10 −1
0
1
2
x1(t)
Figure 8: Time series and portrait plots of system (19) with p = 0.96, K1 = K2 = 0, τ = 2.5.
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K1=−0.12,K2=−0.24
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10
8
τ0
7 6 5
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4
K1=−0.12,K2=0 K =0,K =0 1
2
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9
K1=0,K2=−0.24
3 2
1 0.5
0.6
0.7
φ
0.8
0.9
Figure 9: Comparison on the values of τ0 versus p for system (19).
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1
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0.75 K =−0.12,K =−0.24 1
2
K =0,K =−0.24 1
0.7
2
K1=−0.12,K2=0
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K1=0,K2=0
0.6
p ro
x1(t)
0.65
0.55
0.45
0
10
Pr e-
0.5
20
30
40
50
t
Figure 10: Time series of system (19) with p = 0.72, τ = 2.8.
0.45
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K1=−0.12,K2=−0.24 K1=−0.12,K2=0 K =0,K =0 1
2
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0.4
K1=0,K2=−0.24
x2(t)
0.35
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0.3
0.25
0.2
0
10
20
30
40
t
Figure 11: Time series of system (19) with p = 0.72, τ = 2.8.
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0.45 K =−0.12,K =−0.24 1
2
K =0,K =−0.24 1
2
K1=−0.12,K2=0
0.4
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K1=0,K2=0
p ro
x2(t)
0.35
0.3
0.2 0.45
Pr e-
0.25
0.5
0.55
0.6 x1(t)
0.65
0.7
0.75
Figure 12: Portrait plots of system (19) with p = 0.72, τ = 2.8.
3.6
K2=−0.24
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3.4
3
τ0
2.8 2.6 2.4
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2.2
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3.2
K2=0
2
1.8
1.6 −0.25
−0.2
−0.15
−0.1
−0.05
K
1
Figure 13: Comparison on the values of τ0 versus K1 for system (19) with p = 0.96.
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1.3 K =−0.2,K =−0.24 1
1.2
2
K1=−0.2,K2=0
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1.1
x1(t)
1 0.9
p ro
0.8 0.7 0.6
0.4
0
20
Pr e-
0.5
40
60
80
100
t
Figure 14: Time series of system (19) with p = 0.96, τ = 2.4.
0.6
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K1=−0.2,K2=−0.24 0.5
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0.4
K1=−0.2,K2=0
x2(t)
0.3
0.2
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0.1
0
−0.1
0
20
40
60
80
t
Figure 15: Time series of system (19) with p = 0.96, τ = 2.4.
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0.6 K =−0.2,K =−0.24 1
2
K1=−0.2,K2=0
0.5
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0.4
p ro
x2(t)
0.3
0.2
0.1
−0.1 0.4
0.5
0.6
Pr e-
0
0.7
0.8
0.9
1
1.1
1.2
1.3
x1(t)
Figure 16: Portrait plots of system (19) with p = 0.96, τ = 2.4.
3
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K1=−0.12 2.8
τ0
2.4
2.2
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2
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2.6
K1=0
1.8
1.6 −0.25
−0.2
−0.15
−0.1
−0.05
K
2
Figure 17: Comparison on the values of τ0 versus K2 for system (19) with ϕ = 0.96.
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0
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0.75 K =−0.12,K =−0.23 1
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0.7
0.6
p ro
x1(t)
0.65
0.55
0
20
Pr e-
0.5
0.45
2
K1=0,K2=−0.23
40
60
80
100
t
Figure 18: Time series of system (19) with p = 0.96, τ = 2.3.
0.42
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K1=−0.12,K2=−0.23 0.4
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0.38 0.36 0.34 0.32 0.3
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x2(t)
K1=0,K2=−0.23
0.28 0.26 0.24
0
20
40
60
80
t
Figure 19: Time series of system (19) with p = 0.96, τ = 2.3.
19
100
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0.42 K1=−0.12,K2=−0.23 0.4
K1=0,K2=−0.23
0.38
of
x2(t)
0.36 0.34
0.3 0.28 0.26
0.5
0.55
0.6 x1(t)
0.65
Pr e-
0.24 0.45
p ro
0.32
0.7
0.75
Figure 20: Portrait plots of system (19) with p = 0.96, τ = 2.3.
Acknowledgements
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The work was supported by the National Natural Science Foundation of China under Grant No.11701409, the Natural Science Foundation of Jiangsu Province of China under Grant No.BK20170591, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No.17KJB110018, and the China Postdoctoral Science Foundation under Grant No.2018M642130. This work was also supported by the Key Scientific Research Project for Colleges and Universities of Henan Province under Grant No.20A110004 and the Nanhu Scholars Program for Young Scholars of Xinyang Normal University.
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Appendix A
ϵ1 = 4d1 cos
pπ , 2(
( pπ ) pπ pπ ) pπ + cos2 pπ cos2 + (2d21 − d23 ) sin2 pπ cos2 + cos2 pπ sin2 2 2 2 2 + 4d21 sin2 pπ cos pπ + 4d2 cos pπ, pπ pπ = 2[d1 (d21 − d23 ) + (2d1 d2 − d3 d4 ) sin2 pπ + (6d1 d2 − d3 d4 ) cos2 pπ] cos + 8d1 d2 sin pπ cos pπ sin , 2 2 = d21 (d21 − d23 ) + (2d22 + 4d2 d21 + 2d5 d23 − 2d1 d3 d4 − d24 − 2d25 ) sin2 pπ + (6d22 − d24 − 2d25 ) cos2 pπ pπ pπ + 4(d2 d21 − 2d5 d23 ) cos pπ sin2 + 2(d5 d23 + 6d2 d21 − d2 d23 − 2d1 d3 d4 ) cos pπ cos2 , 2 2 pπ = 2(2d1 d22 + 2d3 d4 d5 − d1 d24 − 2d1 d25 ) sin pπ sin + d1 (4d2 d21 + 2d5 d23 − 2d1 d3 d4 − d2 d23 ) cos pπ 2 pπ pπ · sin2 + 2d1 (d5 d23 + 2d2 d21 − d1 d3 d4 − d2 d23 ) cos3 + (12d1 d22 + 4d3 d4 d5 − 4d1 d25 − 2d1 d24 2 2 pπ − 4d2 d3 d4 ) cos pπ cos , 2 pπ 2 2 + (2d2 d5 d23 + 4d1 d3 d4 d5 = (2d1 d2 + 4d1 d3 d4 d5 − d21 d24 − d22 d23 − d23 d25 − 2d21 d25 − 2d2 d5 d23 ) sin2 2 pπ + 6d21 d22 − d21 d24 − d22 d23 − d23 d25 − 2d21 d25 − 4d1 d2 d3 d4 ) cos2 + (4d32 + 2d5 d24 − 2d2 d24 − 4d2 d25 ) cos pπ, 2 pπ = 2(2d1 d32 + d1 d5 d24 + 2d2 d3 d4 d5 − d1 d2 d24 − d3 d4 d25 − d3 d4 d22 − 2d1 d2 d25 ) cos , 2 = (d22 − d25 )2 − d24 (d22 + d25 ) + 2d2 d5 d24 .
ϵ5
ϵ6
ϵ7 ϵ8
p ro
ϵ4
Pr e-
ϵ3
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ϵ2 = (6d21 − d23 ) sin2 pπ sin2
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1) Enhancing feedback control strategy is devised to control the bifurcation in a delayed fractional order predator-prey model. 2) It further discovers that a larger feedback gain is selected for controlling the bifurcation via dislocated feedback scheme. 3) The control effects of the proposed system can be largely hoisted by enhancing feedback approach. 4) The developed enhancing feedback method can attenuate the control cost compared with dislocated feedback ones.
Journal Pre-proof Declaration of Interest statement:
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We declare that we have no conflict of any financial and personal relationships with other people or organizations.
PII: DOI: Reference:
S0378-4371(20)30003-0 https://doi.org/10.1016/j.physa.2020.124136 PHYSA 124136
To appear in:
Physica A
Received date : 8 August 2019 Revised date : 18 December 2019 Please cite this article as: C. Huang, H. Liu, X. Chen et al., Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator–prey model, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2020.124136. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
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Dynamic optimal control of enhancing feedback treatment for a delayed fractional order predator-prey model Chengdai Huanga,∗ , Heng Liub , Xiaoping Chenc , Minsong Zhangd , Jinde Caoe , Ahmed Alsaedif School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China b
School of Science, Guangxi University for Nationalities, Nanning 530006, China
c
Department of Mathematics, Taizhou University, Taizhou 225300, China
School of Mathematics and Statistics, Hubei University of Arts and Science Xiangyang 441053 China
e
Research Center for Complex Systems and Network Sciences, and School of Mathematics, Southeast University, Nanjing 210096, China
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d
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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Abstract
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This paper meditates the topic of bifurcation control for a delayed fractional-order predator-prey model in accordance with an enhancing feedback controller. Initially, the bifurcation points of devised model are incisively established via analytic extrapolation by regarding time delay as a bifurcation parameter. Then, a range of contrastive analysis on the influence of bifurcation control are numerically discussed including enhancing feedback, dislocated feedback and eliminating feedback approaches. It views that the stability performance of the developed model can be vastly intensified by enhancing feedback approach. Numerical simulations are finally performed to check the feasibility of the devised scheme.
1. Introduction
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Keywords: Fractional order, Enhancing feedback control, Dislocated feedback control, Bifurcation control, Predator-prey models.
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Predator-prey models are a class of considerable models in the sphere of ecological systems, which are one of the basic topics in ecology as a result of the pervasive significance and existence. The original predator-prey model was formulated by [1, 2]. It is universally known time delays have been merged into biological systems to elaborately delineate the authentic dynamical predator-prey by taking into account the time required for resource regeneration time, maturation period, reaction time, feeding time, gestation period [3, 4]. Numerous good results have been attained on the investigation of delayed predator-prey models [5, 6]. Modelling and control based on the theory of fractional calculus of intricate systems can greatly enhance the capability of discrimination, design and control for dynamic systems since fractional calculus possesses infinite memory [7, 8, 9]. In [10], the author pointed out that fractional calculus admits greater degrees of freedom in modeling dynamical systems in contrast with conventional descriptions of biological ∗
Corresponding author. E-mail addresses: [email protected] (C. Huang)
Preprint submitted to Elsevier
January 4, 2020
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systems. It further detected that the stability of the solutions for a delayed predator-prey system can be largely enhanced [11]. Modelling and control of fractional order dynamical systems has recently grown a hot research topic, and lots of colorful results have been captured [12, 13, 14, 15, 16, 17]. As a matter of fact, a great deal of biological models exhibit fractional dynamics thanks to possessing memory effects. Recently, fractional calculus has successfully introduced into predator-prey models, and some interesting phenomena has been discovered. In [18], the author discovered the number of stability switching can be accurately described in term of fractional modelling. In [19], Mondal et al detected that the solutions of fractional order predator-prey system converge to the respective equilibrium more slowly as fractional order decreases. In [20], Chinnathambi and Rihan found that fractional order can strengthen the stability of prey-predator system and hamper the occurrence of oscillation behaviors. Fractional dynamics of delayed predator-prey without delays has been studied [21, 22, 23, 24].
Pr e-
p ro
Delicate stability results of nonlinear systems can be acquired in the light of puissant bifurcation analysis [25, 26, 27, 28, 29, 30]. Cao et al considered the issue of bifurcation for controlled complex networks model with two nonidentical delays in [25]. It should be stated directly that bifurcations of in traditional delayed integer-order models have been sufficiently investigated because of the maturation of theoretical tools. Nevertheless, the bifurcations of fractional-order dynamical models is on the upsurge [31, 32, 33, 34, 35]. In [32], the authors studied the bifurcation of delayed generalized fractionalorder prey-predator model with interspecific competition, and accurate bifurcation results were derived. In [35], the bifurcation of a fractional-order quaternion-valued neural network with time delay was examined, and it revealed that the onset of bifurcation can be advanced with the increment of fractional order. Very recently, the problem of bifurcation control of delayed fractional models has aroused great interest [36, 37, 38]. In [36], the authors devised a parametric delay feedback controller to control the bifurcation for a delayed fractional dual congestion integer-order model. In [37], it uncovered that the bifurcation phonomania can be controlled by regulating extended feedback delay or fractional order. In [38], a neoteric fractional-order PD scheme was proposed to unperturb the innate bifurcation for integer-order small-world network if setting up appropriative control gains.
Jo
urn
al
Normally, numerous bifurcation control schemes can be adopted to handle bifurcation dynamics, such as dislocated feedback control, speed feedback control and enhancing feedback control [39, 40]. In [39], the author detected that the feedback coefficients were smaller than the ones of ordinary feedback control in controlling Lorenz system. In [40], it revealed that the system can be efficiently control via enhancing feedback approach by selecting simple external inputs and small feedback coefficient. Actually, it is difficult to completely control the dynamical properties of a complex system relying on a unique feedback variable. In this instance, a larger feedback gain needs to be selected for procuring the anticipated dynamical behaviors of a nonlinear system. Two conspicuous features of enhancing feedback control can be found. i) The stability performance of the controlled system can be immensely improved by taking small control gains once enhancing feedback controllers appear in the controlled system. ii) Enhancing feedback control methods can extremely lessen the control cost in control engineering. Nevertheless, the bifurcation control of fractional order predator-prey systems with delays based on enhancing feedback control has been not felicitously hashed out before. Impelled by the aforementioned discussions, we shall render a theoretical exploration on bifurcation control for a delayed fractional-order predator-prey model in accordance with enhancing feedback control technique in this paper. The key features of this paper are listed as follows: 1) Enhancing feedback control strategy is devised to control the bifurcation in a fractional order predator-prey model with time delay, and the precise bifurcation control results are obtained in terms of theoretical analysis. 2) It further discovers that a larger feedback gain is selected for controlling the onset of bifurcation of the devised system via dislocated feedback scheme. 3) It detects that the control effects of the proposed system can be largely hoisted by enhancing feedback approach than dislocated feedback one. It manifests that the devised enhancing feedback method can attenuate the control cost compared with
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dislocated feedback ones.
of
The framework of the current paper is constructed as follows. Section 2 presents some definition and Lemmas regarding fractional calculus. Section 3 addresses the investigated model. Section 4 establishes the essential bifurcation control results via enhancing feedback control method. Section 5 verifies the efficiency of the proposed control scheme through a simulation example. Section 6 finalizes the paper with a conclusion. 2. Preliminaries
p ro
This section addresses some practical definition and Lemmas for the next theoretical analysis and numerical simulations. Definition 1. [41] The Caputo fractional-order derivative can be defined as ∫ t 1 Dtp f (t) = (t − s)l−p−1 f (l) (s)ds, Γ(l − p) 0 ∫∞ where l − 1 ≤ p < l ∈ Z + , Γ(·) is the Gamma function, Γ(s) = 0 ts−1 e−t dt.
L{Dtp f (t); s} If
f k (0)
p
Pr e-
On account of Laplace transform rule, it follows from the Caputo fractional-order derivatives that
is
= s F (s) −
= 0, k = 1, 2, . . . , n, then
l−1 ∑
sp−k−1 f (k) (0),
k=0
L{Dtp f (t); s}
l − 1 ≤ p < l ∈ Z +.
= sp F (s).
al
Lemma 1. [42] The following n-dimensional linear fractional-order system is explored p1 D l1 (t) = a11 l1 (t) + a12 l2 (t) + · · · + k1n ln (t), Dp2 l (t) = a l (t) + a l (t) + · · · + k l (t), 2 21 1 22 2 2n n . .. pn D ln (t) = an1 l1 (t) + an2 l2 (t) + · · · + ann ln (t),
(1)
urn
where 0 < pi < 1(i = 1, 2, . . . , n). Supposing that p is the lowest common multiple of the denominators φi ψi of pi , where pi = ψ , (φi , ψi ) = 1, φi , ψi ∈ Z + , for i = 1, 2, . . . , n. It is described as i p λ 1 − a11 −a12 ··· −a1n −a21 λp2 − a22 · · · −a2n △(λ) = . .. .. .. .. . . . . −an2
Jo
−an1
· · · λpn − ann
Then the zero solution of system (1) is globally asymptotically stable in the Lyapunov sense if all roots of the equation det(△(λ)) = 0 satisfy | arg(λ)| > pi π/2.
Lemma 2. [42] The following n-dimensional linear fractional-order system with delays is examined p1 D l1 (t) = a11 l1 (t − τ11 ) + a12 l2 (t − τ12 ) + · · · + a1n ln (t − τ1n ), Dp2 l2 (t) = a21 l1 (t − τ21 ) + a22 l2 (t − τ22 ) + · · · + a2n ln (t − τ2n ), (2) .. . pn D ln (t) = an1 l1 (t − τn1 ) + an2 l2 (t − τn2 ) + · · · + ann ln (t − τnn ), 3
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−an1 e−sτn1
−an2 e−sτn2
of
where pi ∈ (0, 1)(i = 1, 2, . . . , n), the initial values Vi (t) = Ψi (t) are given for − maxi,j , τi,j = − maxi,j ≤ t ≤ 0 and i = 1, 2, . . . , n. For system, (2) time-delay matrix τ = (τi,j ) ∈ (R+ )n×n , coefficient matrix H = (ai,j )n×n , state variables li (t), li (t − τi,j ) ∈ R, and initial values Ψi (t) ∈ C 0 [−τmax , 0]. Its fractional order is defined as p = (l1 , l2 , . . . , ln ). It is defined as p −a12 e−sτ12 ··· −a1n e−sτ1n s 1 − a11 e−sτ11 −a21 e−sτ21 sp2 − a22 e−sτ22 · · · −a2n e−sτ2n △(s) = . .. .. .. .. . . . . · · · spn − ann e−sτnn
p ro
Then the zero solution of system (2) is Lyapunov globally asymptotically stable if all the roots of the characteristic equation det(△(s)) = 0 have negative real parts. 3. Model formulation
Pr e-
The following fractional order ratio-dependent predator-prey system with time delay is developed in this paper. x1 (t − τ )x2 (t) p D x1 (t) = x1 (t − τ )[1 − x1 (t − τ )] − x (t − τ ) + αx (t) , 1 2 (3) [ ] x (t − τ ) 2 p D x2 (t) = βx2 (t − τ ) δ − , x1 (t − τ )
where the variables and parameters of systems (3) are displayed in Table.1.
Table 1: The concrete notations of relevant variables and parameters of system (3) Symbols
Description
x2 (t) p
The densities of predator at time t Fractional order, p ∈ (0, 1] Positive constant
urn
α
The densities of prey at time t
al
x1 (t)
β
Positive constant
δ
Positive constant
τ
Time delay for both the densities of the prey and the predator
Jo
The bifurcation control of the following fractional-order version model (3) is mainly concerned in this paper x1 (t − τ )x2 (t) p D x1 (t) = x1 (t − τ )[1 − x1 (t − τ )] − x (t − τ ) + αx (t) + K1 [(x1 (t) − x1 (t − τ )], 1 2 (4) ] [ x (t − τ ) 2 Dp x2 (t) = βx2 (t − τ ) δ − + K2 [(x2 (t) − x2 (t − τ )], x1 (t − τ )
where K1 , K2 denote feedback control gains, K1 ≤ 0, K2 ≤ 0.
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Remark 1. Observing that system (4) degenerates into the integer-order model in [43] when selecting p = 1, K1 = K2 = 0. If K1 = 0, K2 ̸= 0 or K1 ̸= 0, K2 = 0, then model (4) develops into dislocated timedelayed feedback control system. If K1 ̸= 0, K2 ̸= 0, then model (4) becomes enhancing time-delayed feedback control system.
This means that x∗1 =
1+αδ−δ 1+αδ ,
x∗2 =
p ro
of
Based on the condition 1 + αδ > δ, then system (3) occupies a unique positive equilibrium E ∗ = (x∗1 , x∗2 ), which complies with the following equations x∗2 ∗ 1 − x − = 0, 1 x∗1 + αx∗2 ( x∗ ) β δ − 2∗ = 0. x1 δ(1+αδ−δ) . 1+αδ
Remark 2. E ∗ of system (4) is consistent with system (4), which does not rely on the values of control parameters K1 and K2 . This implies that E ∗ is immutable in the devised controllers.
Pr e-
To capture the brilliant control effects, the following essential assumption is presented in this paper: (H1) K1 ≤ 0, K2 ≤ 0. Based on [42], this paper is devoted to find out the conditions of bifurcation for model (4) by using time delay a bifurcation parameter. Then, some comparative studies on bifurcation control are carried out. It finds that the stability performance of the controlled system can be excessively ameliorated on the basis of enhancing feedback control in comparison with the dislocated feedback control. 4. Main results
al
In this section, time delay shall be selected as a bifurcation parameter to investigate the problem of bifurcation control for the predator-prey model (4) by utilizing enhancing feedback approach. The existence bifurcation and bifurcation point for the proposed model shall be determined.
Jo
urn
Performing transformations u1 (t) = x1 (t)−x∗1 , u2 (t) = x2 (t)−x∗2 , then system (4) can be transformed into the following form (u1 (t − τ ) + x∗1 )(u2 (t) + x∗2 ) p ∗ ∗ D u (t) = [u (t − τ ) + x ][1 − (u (t − τ ) + x )] − 1 1 1 1 1 x1 (t − τ ) + αx2 (t) + K1 [(u1 (t) − u1 (t − τ )], (5) [ ] ∗ p (u2 (t − τ ) + x2 ) + K2 [(u2 (t) − u2 (t − τ )]. D u2 (t) = β(u2 (t − τ ) + x∗2 ) δ − (u1 (t − τ ) + x∗1 )
It gains from system (5) that the linearized form is {
Dp u1 (t) = (γ11 − K1 )u1 (t − τ ) + K1 x1 (t) + γ12 u2 (t),
Dp u2 (t) = γ21 u1 (t − τ ) + K2 u2 (t) + (γ22 − K2 )u2 (t − τ ),
5
(6)
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where x∗2 x∗2 x∗1 + , x∗1 + αx∗2 (x∗1 + αx∗2 )2 αx∗ x∗ x∗ = − ∗ 1 ∗ + ∗ 2 1∗ 2 , x1 + αx2 (x1 + αx2 ) βx∗2 = ∗ 2, (x1 ) 2βx∗ = βδ − ∗ 22 . (x1 )
γ12 γ21 γ22
of
γ11 = 1 − 2x∗1 −
p ro
The associated characteristic equation of (6) is
ℓ1 (s) + ℓ2 (s)e−sτ + ℓ3 (s)e−2sτ = 0, where ℓ1 (s) = s2p − (K1 + k2 )sp + K1 K2 ,
(7)
ℓ2 (s) = −[(m11 + m22 − K1 − K2 )sp + 2K1 K2 − K1 m22 − K2 m11 + m12 m21 ],
Pr e-
ℓ3 (s) = (m11 − K1 )(m22 − K2 ).
The real and imaginary parts of ℓq (s)(q = 1, 2, 3) are labeled by ℓrq , ℓiq . Then it calculates that pπ + K1 K2 , 2 pπ = w2p sin pπ − (K1 + K2 )wp sin , 2 pπ p = −[(m11 + m22 − K1 − K2 )w cos + 2K1 K2 − K1 m22 − K2 m11 + m12 m21 ], 2 pπ = −(m11 + m22 − K1 − K2 )wp sin , 2 = (m11 − K1 )(m22 − K2 ),
ℓr1 = w2p cos pπ − (K1 + K2 )wp cos ℓi1 ℓr2 ℓi2 ℓr3
al
ℓi3 = 0.
Both sides of Eq.(7) are multiplied by esτ , then it derives that
urn
ℓ1 (s)esτ + ℓ2 (s) + ℓ3 (s)e−sτ = 0.
Assume that s = w(cos π2 + i sin π2 )(w > 0) is a purely imaginary root of Eq.(8), then it results in { r (ℓ1 + ℓr3 ) cos wτ − ℓi1 cos wτ = −ℓr2 , ℓi1 cos wτ + (ℓr1 − ℓr3 ) cos wτ = −ℓi2 ,
It is further labeled as
Jo
G1 (w) = −ℓr2 (ℓr1 − ℓr3 ) − ℓi1 ℓi2 ,
G2 (w) = −ℓi2 (ℓr1 + ℓr3 ) + ℓi1 ℓr2 ,
G3 (w) = (ℓr1 )2 + (ℓi1 )2 − (ℓr3 )2 , d1 = −(K1 + K2 ), d2 = K1 K2 ,
d3 = m11 + m22 − K1 − K2 ,
d4 = K1 (K2 − m22 ) + K2 (K1 − m11 ) + m12 m21 , d5 = (m11 − K1 )(m22 − K2 ). 6
(8)
(9)
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As far as Eq.(9), it concludes that G1 (w) cos wτ = G (w) , 3 G2 (w) sin wτ = . G3 (w) G3 (w) = G21 (w) + G22 (w). It can be defined from Eq.(11) that
of
By means of Eq.(10), it procures that
(10)
In terms of Eq.(12), it obtains that
p ro
H(w) = G23 (w) − G21 (w) − G22 (w) = 0.
H(w) = w8p + ϵ1 w7p + ϵ2 w6p + ϵ3 w5p + ϵ4 w4p + ϵ5 w3p + ϵ6 w2p + ϵ7 wp + ϵ8 = 0, where ϵi (i = 1, 2, . . . , 8) are computed in Appendix. We further give the additional assumption:
Pr e-
(H2) Eq.(13) has at least positive real roots.
It follows from the first equation of Eq.(10) that ] 1[ G1 (w) + 2kπ , τ (k) = arccos w G3 (w)
Define the bifurcation point
τ0 = min{τ (k) }, where τ (k) is defined by Eq.(14).
k = 0, 1, 2, . . . .
(11)
(12)
(13)
(14)
k = 0, 1, 2, . . . ,
al
In the following, we will consider the stability of system (5) when τ = 0. If τ is removed, the characteristic equation (8) develops into λ2 + h1 λ + h2 = 0,
urn
where
(15)
h1 = −m11 − m22 ,
h2 = k1 m22 + k2 m11 − k1 k2 − m12 − m21 .
It is obvious from h1 > 0, h2 > 0 that the two roots of Eq.(15) have negative parts which satisfying Lemma 1. Hence, the positive equilibrium of the fractional system (4) is asymptotically stable.
(H3)
Jo
To obtain the conditions of bifurcation, we further give the following assumptions: M1 N1 +M2 N2 N12 +N22
̸= 0,
where M1 , M2 , N1 , N2 are described by Eq.(18). Lemma 3. Let s(τ ) = ξ(τ ) + iw(τ ) be the root of Eq.(8) near τ = τj satisfying ξ(τj ) = 0, w(τj ) = w0 , then the following transversality condition holds [ ds ] ̸= 0, Re dτ (w=w0 ,τ =τ0 ) where w0 , τ0 represent the critical frequency and bifurcation point of model (4). 7
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′
′
Proof. The real and imaginary parts of ℓ′p (s)p = 1, 2, 3 are labeled by ℓpr , ℓpi . Using implicit function theorem to differentiate (8) concerning τ , then the following equation can be concluded ℓ′1 (s)
( )] [ ( )] ds [ ′ ds ds ds ds + ℓ2 (s) e−sτ + ℓ2 (s)e−sτ − τ − s + ℓ′3 (s) e−2sτ + ℓ3 (s)e−2sτ − 2τ − s = 0. dτ dτ dτ dτ dτ (16)
M (s) ds = , dτ N (s)
p ro
where
of
Eq.(7) implies that ℓ′3 (s) = 0. Mathematically, straightforward eduction from Eq.(18) yields (17)
M (s) = s[ℓ2 (s)e−sτ + 2ℓ3 (s)e−2sτ ],
N (s) = ℓ′1 (2) + [ℓ′2 (s) − τ ℓ2 (s)]e−sτ − 2τ ℓ3 (s)e−2sτ . It educes from Eq.(19) that
where
[ ds ] M1 N1 + M2 N2 = , dτ (w=w0 ,τ =τ0 ) N12 + N22
(18)
Pr e-
Re
M1 = w0 (ℓr2 sin w0 τ0 − ℓi2 cos w0 τ0 + 2ℓr3 sin 2w0 τ0 − 2ℓi3 cos 2w0 τ0 ),
M2 = w0 (ℓr2 cos w0 τ0 + ℓi2 sin w0 τ0 + 2ℓr3 cos 2w0 τ0 + 2ℓi3 sin 2w0 τ0 ), ′
′
′
′
′
N1 = ℓ1r + (ℓ2r − τ0 ℓr2 ) cos w0 τ0 + (ℓ2i − τ0 ℓi2 ) sin w0 τ0 − 2τ0 ℓr3 cos 2w0 τ0 , ′
N2 = ℓ1i + (ℓ2i − τ0 ℓi2 ) cos w0 τ0 − (ℓ2r − τ0 ℓr2 ) sin 2w0 τ0 + 2τ0 ℓr3 sin 2w0 τ0 .
al
(H3) indicates that transversality condition hold. We accomplish the proof of Lemma 3. Based on the previous analysis, the following Theorem is abtainable.
urn
Theorem 1. If (H1)-(H3) are met, then the following results are available: 1) E ∗ of the system (4) is asymptotically stable when τ ∈ 0, τ0 );
2) System (4) undergoes a Hopf bifurcation at E ∗ when τ = τ0 , i.e., it has a branch of periodic solutions bifurcating from E ∗ near τ = τ0 .
Jo
Remark 3. As a result of the higher order of Eq.(13), it is hard to establish theoretically all the positive real roots of Eq.(13). Nevertheless, it is straightforward to procure the concrete of these positive real roots of Eq.(13) by Maple numerical software. Hence, the values of τ0 can be accurately figured out. Remark 4. It revealed that a lesser feedback gain can not control the onset of bifurcation of a delayed fractional predator-prey system based on dislocated feedback strategy in [44]. On the basis of the dislocated feedback approach, an extended delayed feedback method was designed to control bifurcation of a delayed fractional predator-prey model in spite of selecting a small feedback gain in [37]. It should be noted that the extended feedback delay play an essential role in postponing Hopf bifurcation for such system. In this paper, if adopting enhancing feedback method, the bifurcation of the proposed system can be easily controlled provided that a set of smaller feedback gains are selected. Reversely, the bifurcation of the devised system can be controlled by using dislocated feedback approach only if 8
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a larger feedback gain is requested. It exhibits that the onset of the bifurcation for fractional delayed predator-prey system can be lagged and satisfactory bifurcation control effects are realized compared with the dislocated feedback approaches in this paper. This indicates that the devised enhancing feedback method can abate the control cost compared with dislocated feedback ones.
p ro
of
Remark 5. The effects of fractional order on the bifurcation point are adequately discussed by calculation. It implies that the better effects in delaying the onset of bifurcation can be achieved as fractional order decreases if feedback gain are established. Simultaneously, it detects that the better control effects can be gained in delaying bifurcation of the proposed system by enhancing feedback method than dislocated feedback one. 5. Numerical examples
where α = 0.7, β = 0.9, δ = 0.6.
Pr e-
In this section, numerical simulations are employed to divulge the appropriateness and applications of the proposed theory. For the purposes of comparison, the parameters identically derive from [43]. E ∗ can be obtained as (x∗1 , x∗2 ) = (0.5775, 0.3465). The initial value are all designated as (x1 (0), x2 (0)) = (0.6, 0.4). Investigate the controlled system x1 (t − τ )x2 (t) Dp x1 (t) = x1 (t − τ )[1 − x1 (t − τ )] − + K1 [(x1 (t) − x1 (t − τ )], x1 (t − τ ) + αx2 (t) (19) [ x (t − τ ) ] Dp x2 (t) = βx2 (t − τ ) δ − 2 + K2 [(x2 (t) − x2 (t − τ )], x1 (t − τ )
urn
al
The enhancing feedback control scheme is firstly considered. Selecting p = 0.96, K1 = −0.12, K2 = −0.24 in system (19), then it derives that w0 = 0.5112, τ0 = 2.7546. In terms of Theorem 1, E ∗ of controlled system (19) is asymptotically stable when τ = 2.5 < τ0 , which, depicted in Fig.1, while Fig.2 display the instability of E ∗ for system (19), Hopf bifurcation arises when τ = 2.8 > τ0 . The bifurcation results of system (19) of with p = 1 are attained as w0∗ = 0.5298, τ0∗ = 2.4954. It is apparent that E ∗ of controlled system (19) is asymptotically stable when choosing τ = 2.5, see Fig.1. Nevertheless, Hopf bifurcation occurs for system (19) with p = 1 when selecting τ = 2.5 due to time delay 2.5 > τ0∗ = 2.4954. This indicates that the better performance of fractional order system (19) can be procured than that of the corresponding integer-order one. This phenomena is commendably corroborated in Fig.3. Subsequently, the effects of the proposed dislocated control and uncontrolled schemes are explored based on comparative investigations.
Jo
Case 1. Selecting p = 0.96, K1 = 0, K2 = −0.24, then it derives that w0 = 0.5713, τ0 = 2.2057. Choosing τ = 2.5 > τ0 = 2.2057, then Fig.4 simulates the volatility of system (19). If we select larger gain K2 = −0.8, then τ0 = 3.4841. Fig.5 well verifies the steadiness of E ∗ for system (19) with 2.5 < τ0 = 3.4841. Case 2. Taking p = 0.96, K1 = −0.12, K2 = 0, then it concludes that w0 = 0.6814, τ0 = 1.8372. Choosing τ = 2.5 > τ0 = 1.8372, then Fig.6 clearly depict the steadiness system (19). If we select larger gain K2 = −0.8, then τ0 = 3.4841. Fig.7 perfectly displays the steadiness of E ∗ for system (19) when 2.5 < τ0 = 3.4841. Case 3. Choosing p = 0.96, K1 = 0, K2 = 0, then it obtains that w0 = 0.7083, τ0 = 1.6634. Choosing τ = 2.5 > τ0 = 1.6634, the instability of system (19) described in Fig.8.
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0.62
0.4 0.38
0.58 0.56 0.54
0.36 0.34
of
x2(t)
x1(t)
0.6
0.32 0
100
200
300
0
100
200
300
t
p ro
t 0.4 0.38
x2(t)
2
x (t)
0.4 0.35
0.65
0.34 0.32
400
Pr e-
0.6 x1(t) 0.55 0
0.36
200 t
0.5
0.55 x1(t)
0.6
Figure 1: Time series and portrait plots of system (19) with p = 0.96, K1 = −0.12, K2 = −0.24, τ = 2.5 < τ0 = 2.7546.
0.7
0.45
x2(t)
al
0.6 0.55 0.5
0
urn
x1(t)
0.65
100
200
0.4 0.35 0.3 0.25
300
0
t
0.6 x (t) 1
0.5 0
300
0.45 0.4
0.3
0.2 0.7
200 t
x2(t)
2
Jo
x (t)
0.4
100
0.35 0.3
400 200
0.25 0.5
t
0.6 x (t)
0.7
1
Figure 2: Time series and portrait plots of system (19) with p = 0.96, K1 = −0.12, K2 = −0.24, τ = 2.8 > τ0 = 2.7546.
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0.62
0.4 0.38
x2(t)
0.58 0.56 0.54
0.36 0.34 0.32
0
100
200
300
0
100
t
p ro
0.4
300
0.38
x2(t)
2
200
t
0.4
x (t)
of
x1(t)
0.6
0.35
0.65
0.32
400 200 t
0.34
0.55
Pr e-
0.6 x1(t) 0.55 0
0.36
0.6 x1(t)
0.65
Figure 3: Time series and portrait plots of system (19) with p = 1, K1 = −0.12, K2 = −0.24, τ = 2.5.
6. Conclusion
urn
al
Additionally, by varying the values of p, the corresponding τ0 of enhancing feedback control, dislocated feedback control and without control approaches are addressed, which displayed in Fig.9. This implies that enhancing feedback control transcends than others cases, which are authenticated in simulation results in Figs.10-12. If establishing p, K2 = −0.24 or K2 = 0, the values of τ0 can be determined as K1 varies, which depicted in Fig.13. Fig.13 also means that enhancing feedback control oversteps dislocated feedback control, which is illustrated in Figs.14-16. If predetermining p = 0.96, K1 = −0.24 or K1 = 0 and varying K2 in system (19), and the values of τ0 can be figured out, which is listed in Fig.17. It can also be noted from Fig.17 that enhancing feedback control overmatches dislocated feedback control, which is very agreement with numerical simulations in Figs.18-20.
Jo
The problem of bifurcation control for a delayed fractional-order predator-prey model has been smartly investigated by virtue of enhancing feedback approach in this paper. The bifurcation point of controlled system has been totally established. Analysis and simulations further reveal that the better efficiency of bifurcation control has been obtained in term of enhancing feedback approach than dislocated and uncontrolled ones with partially or completely removing the branch for feedback gains. It has detected that the onset of bifurcation can be controlled for the dislocated feedback methods, yet the greater feedback gains should be taken, which increases the cost of control system. On the contrary, the stability performance of the controlled model can be extremely ameliorated on account of the designed enhancing feedback methodology by selecting slighter feedback gains. Numerical results have been provided to confirm the efficiency of the derived theoretical results.
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1.5
0.6 0.4
0.5
0
0
20
40 t
60
−0.2
80
0
20
40 t
60
80
p ro
0
0.2
of
x2(t)
x1(t)
1
0.6 0.4
x2(t)
2
x (t)
0.5 0 −0.5 2
0.2 0
1 x1(t)
50 0 0
t
Pr e-
100
−0.2
0
0.5
1
1.5
x1(t)
Figure 4: Time series and portrait plots of system (19) with p = 0.96, K1 = 0, K2 = −0.24, τ = 2.5.
0.7
0.4
x2(t)
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0.6
0.5
20
Jo
2
x (t)
0.4
0
40 t
60
0.6 x (t) 1
0.5 0
0.36 0.34
80
0
20
40 t
60
80
0.4 0.38
0.35
0.7
0.38
0.32
x2(t)
0.55
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x1(t)
0.65
0.36 0.34 0.32
100 50
0.5
t
0.6 x (t)
0.7
1
Figure 5: Time series and portrait plots of system (19) with p = 0.96, K1 = 0, K2 = −0.8, τ = 2.5.
12
Journal Pre-proof
0
0
x2(t)
5
x1(t)
10
0
50 t
−10
100
0
50 t
100
p ro
−20
−5
of
−10
5
0
x2(t)
x2(t)
10 0
−5
−10 20 50 t
Pr e-
100
0 x1(t) −20 0
−10 −20
−10
0
10
x1(t)
Figure 6: Time series and portrait plots of system (19) with p = 0.96, K1 = −0.12, K2 = 0, τ = 2.5.
0.61
0.4
x2(t)
al
0.59
0
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2
x (t)
0.4
50 t
x (t) 0.55 0 1
0.36 0.34
100
0
50 t
100
0.4 0.38
0.35
0.6
0.38
0.32
x2(t)
0.58
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x1(t)
0.6
0.36 0.34 0.32
100 50
0.57
t
0.58 x (t)
0.59
0.6
1
Figure 7: Time series and portrait plots of system (19) with p = 0.96, K1 = −4.5, K2 = 0, τ = 2.5.
13
Journal Pre-proof
1
10
x2(t)
20
x1(t)
2
−1
0
0
10
20
−10
30
0
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0
10
20
30
t
p ro
t 20
10
x2(t)
x2(t)
20 0
0
−20 2
40
Pr e-
0 x1(t)
20 −2 0
t
−10 −1
0
1
2
x1(t)
Figure 8: Time series and portrait plots of system (19) with p = 0.96, K1 = K2 = 0, τ = 2.5.
11
K1=−0.12,K2=−0.24
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10
8
τ0
7 6 5
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4
K1=−0.12,K2=0 K =0,K =0 1
2
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9
K1=0,K2=−0.24
3 2
1 0.5
0.6
0.7
φ
0.8
0.9
Figure 9: Comparison on the values of τ0 versus p for system (19).
14
1
Journal Pre-proof
0.75 K =−0.12,K =−0.24 1
2
K =0,K =−0.24 1
0.7
2
K1=−0.12,K2=0
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K1=0,K2=0
0.6
p ro
x1(t)
0.65
0.55
0.45
0
10
Pr e-
0.5
20
30
40
50
t
Figure 10: Time series of system (19) with p = 0.72, τ = 2.8.
0.45
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K1=−0.12,K2=−0.24 K1=−0.12,K2=0 K =0,K =0 1
2
urn
0.4
K1=0,K2=−0.24
x2(t)
0.35
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0.3
0.25
0.2
0
10
20
30
40
t
Figure 11: Time series of system (19) with p = 0.72, τ = 2.8.
15
50
Journal Pre-proof
0.45 K =−0.12,K =−0.24 1
2
K =0,K =−0.24 1
2
K1=−0.12,K2=0
0.4
of
K1=0,K2=0
p ro
x2(t)
0.35
0.3
0.2 0.45
Pr e-
0.25
0.5
0.55
0.6 x1(t)
0.65
0.7
0.75
Figure 12: Portrait plots of system (19) with p = 0.72, τ = 2.8.
3.6
K2=−0.24
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3.4
3
τ0
2.8 2.6 2.4
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2.2
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3.2
K2=0
2
1.8
1.6 −0.25
−0.2
−0.15
−0.1
−0.05
K
1
Figure 13: Comparison on the values of τ0 versus K1 for system (19) with p = 0.96.
16
0
Journal Pre-proof
1.3 K =−0.2,K =−0.24 1
1.2
2
K1=−0.2,K2=0
of
1.1
x1(t)
1 0.9
p ro
0.8 0.7 0.6
0.4
0
20
Pr e-
0.5
40
60
80
100
t
Figure 14: Time series of system (19) with p = 0.96, τ = 2.4.
0.6
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K1=−0.2,K2=−0.24 0.5
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0.4
K1=−0.2,K2=0
x2(t)
0.3
0.2
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0.1
0
−0.1
0
20
40
60
80
t
Figure 15: Time series of system (19) with p = 0.96, τ = 2.4.
17
100
Journal Pre-proof
0.6 K =−0.2,K =−0.24 1
2
K1=−0.2,K2=0
0.5
of
0.4
p ro
x2(t)
0.3
0.2
0.1
−0.1 0.4
0.5
0.6
Pr e-
0
0.7
0.8
0.9
1
1.1
1.2
1.3
x1(t)
Figure 16: Portrait plots of system (19) with p = 0.96, τ = 2.4.
3
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K1=−0.12 2.8
τ0
2.4
2.2
Jo
2
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2.6
K1=0
1.8
1.6 −0.25
−0.2
−0.15
−0.1
−0.05
K
2
Figure 17: Comparison on the values of τ0 versus K2 for system (19) with ϕ = 0.96.
18
0
Journal Pre-proof
0.75 K =−0.12,K =−0.23 1
of
0.7
0.6
p ro
x1(t)
0.65
0.55
0
20
Pr e-
0.5
0.45
2
K1=0,K2=−0.23
40
60
80
100
t
Figure 18: Time series of system (19) with p = 0.96, τ = 2.3.
0.42
al
K1=−0.12,K2=−0.23 0.4
urn
0.38 0.36 0.34 0.32 0.3
Jo
x2(t)
K1=0,K2=−0.23
0.28 0.26 0.24
0
20
40
60
80
t
Figure 19: Time series of system (19) with p = 0.96, τ = 2.3.
19
100
Journal Pre-proof
0.42 K1=−0.12,K2=−0.23 0.4
K1=0,K2=−0.23
0.38
of
x2(t)
0.36 0.34
0.3 0.28 0.26
0.5
0.55
0.6 x1(t)
0.65
Pr e-
0.24 0.45
p ro
0.32
0.7
0.75
Figure 20: Portrait plots of system (19) with p = 0.96, τ = 2.3.
Acknowledgements
Jo
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The work was supported by the National Natural Science Foundation of China under Grant No.11701409, the Natural Science Foundation of Jiangsu Province of China under Grant No.BK20170591, the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant No.17KJB110018, and the China Postdoctoral Science Foundation under Grant No.2018M642130. This work was also supported by the Key Scientific Research Project for Colleges and Universities of Henan Province under Grant No.20A110004 and the Nanhu Scholars Program for Young Scholars of Xinyang Normal University.
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Appendix A
ϵ1 = 4d1 cos
pπ , 2(
( pπ ) pπ pπ ) pπ + cos2 pπ cos2 + (2d21 − d23 ) sin2 pπ cos2 + cos2 pπ sin2 2 2 2 2 + 4d21 sin2 pπ cos pπ + 4d2 cos pπ, pπ pπ = 2[d1 (d21 − d23 ) + (2d1 d2 − d3 d4 ) sin2 pπ + (6d1 d2 − d3 d4 ) cos2 pπ] cos + 8d1 d2 sin pπ cos pπ sin , 2 2 = d21 (d21 − d23 ) + (2d22 + 4d2 d21 + 2d5 d23 − 2d1 d3 d4 − d24 − 2d25 ) sin2 pπ + (6d22 − d24 − 2d25 ) cos2 pπ pπ pπ + 4(d2 d21 − 2d5 d23 ) cos pπ sin2 + 2(d5 d23 + 6d2 d21 − d2 d23 − 2d1 d3 d4 ) cos pπ cos2 , 2 2 pπ = 2(2d1 d22 + 2d3 d4 d5 − d1 d24 − 2d1 d25 ) sin pπ sin + d1 (4d2 d21 + 2d5 d23 − 2d1 d3 d4 − d2 d23 ) cos pπ 2 pπ pπ · sin2 + 2d1 (d5 d23 + 2d2 d21 − d1 d3 d4 − d2 d23 ) cos3 + (12d1 d22 + 4d3 d4 d5 − 4d1 d25 − 2d1 d24 2 2 pπ − 4d2 d3 d4 ) cos pπ cos , 2 pπ 2 2 + (2d2 d5 d23 + 4d1 d3 d4 d5 = (2d1 d2 + 4d1 d3 d4 d5 − d21 d24 − d22 d23 − d23 d25 − 2d21 d25 − 2d2 d5 d23 ) sin2 2 pπ + 6d21 d22 − d21 d24 − d22 d23 − d23 d25 − 2d21 d25 − 4d1 d2 d3 d4 ) cos2 + (4d32 + 2d5 d24 − 2d2 d24 − 4d2 d25 ) cos pπ, 2 pπ = 2(2d1 d32 + d1 d5 d24 + 2d2 d3 d4 d5 − d1 d2 d24 − d3 d4 d25 − d3 d4 d22 − 2d1 d2 d25 ) cos , 2 = (d22 − d25 )2 − d24 (d22 + d25 ) + 2d2 d5 d24 .
ϵ5
ϵ6
ϵ7 ϵ8
p ro
ϵ4
Pr e-
ϵ3
of
ϵ2 = (6d21 − d23 ) sin2 pπ sin2
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Journal Pre-proof Highlights:
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1) Enhancing feedback control strategy is devised to control the bifurcation in a delayed fractional order predator-prey model. 2) It further discovers that a larger feedback gain is selected for controlling the bifurcation via dislocated feedback scheme. 3) The control effects of the proposed system can be largely hoisted by enhancing feedback approach. 4) The developed enhancing feedback method can attenuate the control cost compared with dislocated feedback ones.
Journal Pre-proof Declaration of Interest statement:
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We declare that we have no conflict of any financial and personal relationships with other people or organizations.