# Neimark-Sacker Bifurcation and Chaos Control in a Fractional-Order Plant-Herbivore Model

Authors: Qamar Din; A. A. Elsadany; Hammad Khalil
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

This work is related to dynamics of a discrete-time 3-dimensional plant-herbivore model. We investigate existence and uniqueness of positive equilibrium and parametric conditions for local asymptotic stability of positive equilibrium point of this model. Moreover, it is also proved that the system undergoes Neimark-Sacker bifurcation for positive equilibrium with the help of an explicit criterion for Neimark-Sacker bifurcation. The chaos control in the model is discussed through implementation of two feedback control strategies, that is, pole-placement technique and hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the model.


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## Body

## 1. Introduction and Preliminaries

In [1], authors proposed a mathematical model governed by ordinary differential equations related to the interaction between a plant and an insect. We extend this model by interchanging ordinary differential equations into fractional-order differential equations. Arguing as in [1], we assume that the larval stage of an insect develops and grows at the expense of nonreproductive tissues of the plant; on the other hand, these insects at their adult stage convey pollen to or deposit pollen on the flowers of the same plant, so allowing fertilization. Moreover, we denote P, L, A, and F as densities of plant species, larval species of insect, adult species of insect, and flower species involving in pollination, respectively. The interaction among P, L, A, and F is described by the following system of nonlinear differential equations:(1)dPdτ=aP1-bP+cdFA-a1PL,dFdτ=b1P-c1F-dFA,dLdτ=d1FA+a2A-b2a1PL-c2L,dAdτ=b2a1PL-d2A.Then, we recall the definition of Caputo fractional-order derivative [2] for any function f, and it is given by(2)Dαft=Im-αfmt,α>0,where m is the least positive integer satisfying m≥α and Iϑ denotes integral operator of Riemann-Liouville type with order ϑ, and it is defined by(3)Iϑgt=1Γϑ∫0tt-sϑ-1gsds,ϑ>0,where Γ(·) represents Euler’s Gamma function. Furthermore, the fractional-order counterpart of (5) is given by the following system:(4)dαPdτα=aP1-bP+cdFA-a1PL,dαFdτα=b1P-c1F-dFA,dαLdτα=d1FA+a2A-b2a1PL-c2L,dαAdτα=b2a1PL-d2A,where 0<α≤1,dα/dτα is in the sense of the Caputo fractional derivative defined in (2), a is plant intrinsic growth rate, b is plant intraspecific self-regulation coefficient (also the inverse is its carrying capacity), d denotes pollination rate, a1 is called herbivory rate, b1 is flower production rate, c1 is flower decay rate, c2 and d2 are larva and adult mortality rates, c is plant pollination efficiency ratio, d1 denotes adult consumption efficiency ratio, and b2 is called the maturation rate for brevity. Moreover, parameter a2 represents a reproduction rate resulting from the pollination of other plants species. We now consider the fact that flowers last for a very short time as compared to the life cycles of plants and insects. This means that the variables P, L, and A have slower dynamics, and, on the other hand, the variable F has fast dynamics [3]. In case of steady states of plants and insects, one can find a steady state of flowers by putting right hand side of second equation of system (4) that is equal to zero, that is, b1P-c1F-dFA=0; then it follows that F=b1P/c1+dA [1]. Putting F=b1P/c1+dA in system (4), we obtain the following 3-dimensional fractional-order system:(5)dαPdτα=aP1-bP+db1cAPc1+dA-a1PL,dαLdτα=d1b1APc1+dA+a2A-b2a1PL-c2L,dαAdτα=b2a1PL-d2A.For lenient mathematical analysis, one can reduce the number of parameters in system (5) by using the following transformations: (6)x=bP,y=bL,z=bA,t=aτ.We have the following dimensionless system:(7)dαxdtα=x1-x+λcxzη+z-βxy,dαydtα=λd1xzη+z+ϕz-b2βxy-μy,dαzdtα=βxy-νz,where λ=b1/a, η=c1b/d, β=a1/ab, μ=c2/a, ν=d2/a, and ϕ=a2/a. Since population densities cannot be negative, the state space of system (7) is given by (8)x,y,z∈R3:x≥0,y≥0,z≥0. Due to efficient computational results, discrete dynamical systems are much better than related systems in differential equations. Particularly, in case of nonoverlapping generations, difference equations are more suitable to study the behavior of population models [4–8]. For more details on some interesting population models both in differential equations and in difference equations, we refer the interested reader to [9–12]. It is very interesting to investigate the parametric conditions for existence of Neimark-Sacker bifurcation and to discuss chaos control techniques due to emergence of Neimark-Sacker bifurcation for discrete-time population models. For some interesting results related to Neimark-Sacker bifurcation and chaos control of discrete-time population models, we refer the reader to [13–17].Now we consider the counterpart of (7) with piecewise constant arguments as follows:(9)dαxdtα=xt/kk1-xt/kk+λcxt/kkzt/kkη+zt/kk-βxt/kkyt/kk,dαydtα=λd1xt/kkzt/kkη+zt/kk+ϕzt/kk-b2βxt/kkyt/kk-μyt/kk,dαzdtα=βxt/kkyt/kk-νzt/kk,with initial conditions x(0)=x0, y(0)=y0, z(0)=z0. Furthermore, assume that t∈[0,k); then it follows that t/k∈[0,1). So for t∈[0,k), system (9) gives(10)dαxdtα=x01-x0+λcx0z0η+z0-βx0y0,dαydtα=λd1x0z0η+z0+ϕz0-b2βx0y0-μy0,dαzdtα=βx0y0-νz0.The solution of (9) is given by(11)x1t=x0+Iαx01-x0+λcx0z0η+z0-βx0y0,y1t=y0+Iαλd1x0z0η+z0+ϕz0-b2βx0y0-μy0,z1t=z0+Iαβx0y0-νz0,where Iα is the Riemann-Liouville integral operator of order α which is defined in (3). From (3) and (11), it follows that(12)x1t=x0+tαΓα+1x01-x0+λcx0z0η+z0-βx0y0,y1t=y0+tαΓα+1λd1x0z0η+z0+ϕz0-b2βx0y0-μy0,z1t=z0+tαΓα+1βx0y0-νz0.Similarly, for t∈[k,2k), so that t/k∈[1,2), we obtain(13)dαxdtα=x11-x1+λcx1z1η+z1-βx1y1,dαydtα=λd1x1z1η+z1+ϕz1-b2βx1y1-μy1,dαzdtα=βx1y1-νz1.The solution of (13) is given by(14)x2t=x1k+t-kαΓα+1x1k1-x1k+λcx1kz1kη+z1k-βx1ky1k,y2t=y1k+t-kαΓα+1λd1x1kz1kη+z1k+ϕz1k-b2βx1ky1k-μy1k,z2t=z1k+t-kαΓα+1βx1ky1k-νz1k.Repeating the above process n-times, the solution of (9) for t∈nk,(n+1)k is given by(15)xn+1t=xnnk+t-nkαΓα+1xnnk1-xnnk+λcxnnkznnkη+znnk-βxnnkynnk,yn+1t=ynnk+t-nkαΓα+1λd1xnnkznnkη+znnk+ϕznnk-b2βxnnkynnk-μynnk,zn+1t=znnk+t-nkαΓα+1βxnnkynnk-νznnk.Next taking t→(n+1)k and adopting the notation of difference equations, system (15) yields(16)xn+1=xn+kαΓα+1xn1-xn+λcxnznη+zn-βxnyn,yn+1=yn+kαΓα+1λd1xnznη+zn+ϕzn-b2βxnyn-μyn,zn+1=zn+kαΓα+1βxnyn-νzn.Our aim in this paper is to study the local asymptotic stability of equilibrium points of system (16). Moreover, Neimark-Sacker bifurcation for positive equilibrium of system (16) is also investigated. In order to control the chaos due to emergence of Neimark-Sacker bifurcation, pole-placement and hybrid control strategies are implemented on system (16). Similar methods of discretization for fractional-order systems are also used in [18–22].


## 2. Linearized Stability of System (16)

First, we consider possible steady-state (equilibrium point)(x∗,y∗,z∗) of system (16), which can be obtained by solving the following system:(17)x∗1-x∗+λcx∗z∗η+z∗-βx∗y∗=0,λd1x∗z∗η+z∗+ϕz∗-b2βx∗y∗-μy∗=0,βx∗y∗-νz∗=0.Then it is easy to see that (x∗,y∗,z∗)=(0,0,0) and (x∗,y∗,z∗)=(1,0,0) are two solutions of system (17). It follows that P0=(0,0,0) and P1=(1,0,0) are two equilibria of system (16). Neglecting the trivial equilibrium, we are left with(18)1-x∗+λcz∗η+z∗-βy∗=0,λd1x∗z∗η+z∗+ϕz∗-b2βx∗y∗-μy∗=0,βx∗y∗-νz∗=0.It must be noted that the equilibrium point of system (16), that is, the solution of system (18), may not be unique, but we do not care how many. For biological reasons, we are only interested in positive solutions of (18). From system (18), we obtain (19)y∗=λd1βνx∗2+ϕβηνx∗-ην2b2βx∗+μβνx∗b2βx∗+μ-ϕβ2x∗2,z∗=λd1βx∗2+ϕβηx∗-ηνb2βx∗+μνb2βx∗+μ-ϕβx∗,where x∗ is one of the roots of the following quartic polynomial:(20)Pt=At4+Bt3+Ct2+Dt+E,such that (21)A=β2d1b2ν-ϕ,B=βd1βcλϕ+βd1λν+βϕ+μν-b2βcλν-b2βν,C=ββb22cην2+βcηϕ2+βd1ηνϕ-2b2βcηνϕ-b2βd1ην2-cd1λμν-d1μν,D=βμνη2b2cν-2cϕ-d1ν,E=cημ2ν2.We are looking for the unique positive equilibrium point of system (18); for this, we have the following Descartes’s rule of signs.Lemma 1 (see [23]).
Letf(x)=anxn+an-1xn-1+⋯+a1x+a0 be a polynomial function with real coefficients. Then the number of positive real roots of f is either the same as the number of sign changes of f(x) or less than the number of sign changes of f(x) by a positive even integer. Moreover, if f(x) has only one variation in sign, then f has exactly one positive real root.Using Lemma1, we have the following result for existence of unique positive real root of polynomial P(t) given in (20).Lemma 2.
PolynomialP(t) in (20) has unique positive real root if one of the following conditions hold: (i)
A<0,B<0,C<0,D<0,E>0.(ii)
A<0,B>0,C>0,D>0,E>0.Due to above analysis, we have the following result about the existence and uniqueness of positive equilibrium point of system (16).Lemma 3.
Under the conditions of Lemma2, system (16) has unique positive equilibrium point if the following condition holds:(22)ηνb2-ηϕ2λd1+12βη2ϕ2-2βη2νϕb2+βη2ν2b22+4ηλμνd1βλ2d12<x∗<μνβϕ-βνb2, where x∗ is unique positive real root of polynomial (20).Next, the Jacobian matrix for system (16) evaluated at (x,y,z) is given by (23)Jx,y,z=1+kα1-2x-yβ+czλ/z+ηΓ1+α-kαxβΓ1+αckαxηλz+η2Γ1+αkα-yβb2+zλd1/z+ηΓ1+α1+kα-μ-xβb2Γ1+αkαz+η2ϕ+xηλd1z+η2Γ1+αkαyβΓ1+αkαxβΓ1+α1-kανΓ1+α.Theorem 4.
For system (16), the following statements hold true: (i)
The trivial equilibrium pointP0 is unstable.(ii)
The equilibrium pointP1 is locally asymptotically stable if and only if sα/Γ(1+α)<2 and (24)kαημ+ν+βb2+ημ-ν2+4βϕ+βb22μ-2ν+βb2+4βλd1<4ηΓ1+α.Proof.
(i) The Jacobian matrix for system (16) evaluated at trivial equilibrium P0=(0,0,0) is given by(25)J0,0,0=1+kαΓ1+α0001-kαμΓ1+αkαϕΓ1+α001-kανΓ1+α. Now it is easy to see that eigenvalues of Jacobian matrix J(0,0,0) are λ1=1+kα/Γ(1+α), λ2=1-kαμ/Γ(1+α), and λ3=1-kαν/Γ(1+α). Since 0<α≤1 and s>0 implies that kα/Γ(1+α)>0, then it follows that λ1>1. Hence P0 is unstable.
(ii) The Jacobian matrix for system (16) evaluated at trivial equilibrium P1=(1,0,0) is given by (26)J1,0,0=1-kαΓ1+α-sαβΓ1+αsαcληΓ1+α01-sαμ+b2βΓ1+αsαηϕ+λd1ηΓ1+α0sαβΓ1+α1-kανΓ1+α.The eigenvalues of Jacobian matrix J(1,0,0) are given by λ1=1-kα/Γ(1+α) and (27)λ2,3=1-kαημ+ν+βb2±ημ-ν2+4βϕ+βb22μ-2ν+βb2+4βλd12ηΓ1+α.Now it is easy to see that λ1<1 if and only if sα/Γ(1+α)<2 and λ2,3<1 if and only if(28)kαημ+ν+βb2+ημ-ν2+4βϕ+βb22μ-2ν+βb2+4βλd1<4ηΓ1+α.Theorem 5.
The unique positive equilibrium(x∗,y∗,z∗) of system (16) is locally asymptotically stable if the following condition is satisfied: (29)A1+A3<1+A2,A1-3A3<3-A2,A32+A2-A1A3<1,where(30)A1=Mx∗+μ+ν+Mx∗βb2-3,A2=3-2Mx∗+μ+ν+M2μy∗β+ν-cLy∗β+x∗μ+ν-x∗+z∗βϕ+Mx∗β-2+Mx∗+νb2-LMd1,A3=Mν+x∗Mμ-1Mν-1-M2x∗2βϕ+Mx∗β1+My∗βϕ+cLM2βy∗-2My∗μ+Mz∗ϕ+MMν-1My∗βμ-Mz∗βϕ-μ+Mx∗β1-Mx∗+cLMy∗β+MMx∗-1νb2+LM1-Mx∗+My∗βd1-1,L=x∗ηλz∗+η2,M=sαΓ1+α.Proof.
The Jacobian matrix for system (16) evaluated at unique positive equilibrium (x∗,y∗,z∗) is given by (31)Jx∗,y∗,z∗=1-kαx∗Γ1+α-kαx∗βΓ1+αckαx∗ηλz∗+η2Γ1+αkαμy∗-ϕz∗x∗Γ1+α1-kαμ+x∗βb2Γ1+αkαz∗+η2ϕ+x∗ηλd1z∗+η2Γ1+αkαy∗βΓ1+αkαx∗βΓ1+α1-kανΓ1+α.The characteristic polynomial of Jacobian matrix J(x∗,y∗,z∗) evaluated at positive equilibrium is given by(32)PT=T3+A1T2+A2T+A3,where(33)A1=Mx∗+μ+ν+Mx∗βb2-3,A2=3-2Mx∗+μ+ν+M2μy∗β+ν-cLy∗β+x∗μ+ν-x∗+z∗βϕ+Mx∗β-2+Mx∗+νb2-LMd1,A3=Mν+x∗Mμ-1Mν-1-M2x∗2βϕ+Mx∗β1+My∗βϕ+cLM2βy∗-2My∗μ+Mz∗ϕ+MMν-1My∗βμ-Mz∗βϕ-μ+Mx∗β1-Mx∗+cLMy∗β+MMx∗-1νb2+LM1-Mx∗+My∗βd1-1,L=x∗ηλz∗+η2,M=sαΓ1+α.Now applying the Jury condition [11], the unique positive equilibrium point (x∗,y∗,z∗) is locally asymptotically stable if the following conditions are satisfied: (34)A1+A3<1+A2,A1+3A3<3-A2,A32+A2-A1A3<1.In order to study the Neimark-Sacker bifurcation in system (16), we need the following explicit criterion of Hopf bifurcation.Lemma 6 (see [24]).
Consider ann-dimensional discrete dynamical system Xk+1=fμ(Xk), where μ∈R is bifurcation parameter. Let X∗ be a fixed point of fμ and the characteristic polynomial for Jacobian matrix J(X∗)=aijn×n of n-dimensional map fμ is given by(35)Pμλ=λn+a1λn-1+⋯+an-1λ+an,where ai=ai(μ,u), i=1,2,…,n and u is control parameter or another parameter to be determined. Let Δ0±(μ,u)=1, Δ1±(μ,u),…,Δn±(μ,u) be a sequence of determinants defined by Δi±(μ,u)=det⁡(M1±M2), i=1,2,…,n, where (36)M1=1a1a2⋯ai-101a1⋯ai-2001⋯ai-3⋯⋯⋯⋯⋯000⋯1,M2=an-i+1an-i+2⋯an-1anan-i+2an-i+3⋯an0⋯⋯⋯⋯⋯an-1an⋯00an0⋯00. Moreover, the following conditions hold:, (H1)
Eigenvalue assignment:Δn-1-(μ0,u)=0, Δn-1+(μ0,u)>0, Pμ0(1)>0, -1nPμ0(-1)>0, Δi±(μ0,u)>0, i=n-3,n-5,…,1(or2), when n is even or odd, respectively.(H2)
Transversality condition:d/dμΔn-1-(μ,u)μ=μ0≠0.(H3)
Nonresonance condition:cos⁡2π/m≠ψ, or resonance condition cos⁡2π/m=ψ, where m=3,4,5,…, and ψ=-1+0.5Pμ0(1)Δn-3-(μ0,u)/Δn-2+(μ0,u). Then Neimark-Sacker bifurcation occurs at μ0.The following result shows that system (16) undergoes Neimark-Sacker bifurcation if we take μ as bifurcation parameter.Theorem 7.
The unique positive equilibrium point of system (16) undergoes Neimark-Sacker bifurcation if the following conditions hold: (37)1-A2+A3A1-A3=0,1+A2-A3A1+A3>0,1+A1+A2+A3>0,1-A1+A2-A3>0,where A1, A2, and A3 are given in (32).Proof.
According to Lemma6, for n=3, we have in (32) the characteristic polynomial of system (16) evaluated at its unique positive equilibrium, then we obtain the following equalities and inequalities: (38)Δ2-μ=1-A2+A3A1-A3=0,Δ2+μ=1+A2-A3A1+A3>0,Pμ1=1+A1+A2+A3>0,-13Pμ-1=1-A1+A2-A3>0.


## 3. Chaos Control

Controlling chaos in discrete-time models is a topic of great interest for many researchers in recent times, and practical methods can be used in many fields such as biochemistry, cardiology, communications, physics laboratories, and turbulence [25]. Chaos control in discrete-time models can be obtained by using various methods. In this section, we concentrate on two procedures only, that is, pole-placement technique which is based on feedback control methodology and the other one is hybrid control based on feedback control strategy and parameter perturbation.First, we study chaos controlling technique based on pole-placement methodology introduced by Romeiras et al. [26] (also see [27]), which may be treated as generalized OGY method studied for the first time by Ott et al. [28]. In order to apply pole-placement technique to system (16), we rewrite system (16) as follows:(39)xn+1=xn+kαΓα+1xn1-xn+λcxnznη+zn-βxnyn=fxn,yn,zn,β,yn+1=yn+kαΓα+1λd1xnznη+zn+ϕzn-b2βxnyn-μyn=gxn,yn,zn,β,zn+1=zn+kαΓα+1βxnyn-νzn=hxn,yn,zn,β,where β is taken as control parameter. Moreover, β is restricted to lie in some small interval |β-β0|<δ with δ>0 and β0 denotes the nominal value belonging to chaotic region. We apply the stabilizing feedback control strategy in order to move the trajectory towards the desired orbit. Assume that x∗,y∗,z∗ is unstable equilibrium point of system (16) in chaotic region produced by Neimark-Sacker bifurcation; then system (39) can be approximated in the neighborhood of the unstable equilibrium point x∗,y∗,z∗ by the following linear map:(40)xn+1-x∗yn+1-y∗zn+1-z∗≈Axn-x∗yn-y∗zn-z∗+Bβ-β0,where (41)A=∂fx∗,y∗,z∗,β0∂x∂fx∗,y∗,z∗,β0∂y∂fx∗,y∗,z∗,β0∂z∂gx∗,y∗,z∗,β0∂x∂gx∗,y∗,z∗,β0∂y∂gx∗,y∗,z∗,β0∂y∂hx∗,y∗,z∗,β0∂x∂hx∗,y∗,z∗,β0∂y∂hx∗,y∗,z∗,β0∂y,B=∂fx∗,y∗,z∗,β0∂β∂gx∗,y∗,z∗,β0∂β∂hx∗,y∗,z∗,β0∂β=-kαx∗y∗Γα+1-kαb2x∗y∗Γα+1kαx∗y∗Γα+1. It is easy to see that system (39) is controllable provided that the following matrix(42)C=B:AB:A2Bhas rank 3. Next, we assume that β-β0=-Kxn-x∗yn-y∗zn-z∗, where K=k1k2k3; then system (40) can be written as(43)xn+1-x∗yn+1-y∗zn+1-z∗≈A-BKxn-x∗yn-y∗zn-z∗.Moreover, equilibrium point x∗,y∗,z∗ is locally asymptotically stable if and only if all three eigenvalues of the matrix A-BK, say, μ1, μ2, and μ3, lie in an open unit disk. These eigenvalues are known as regulator poles, and problem of placing these regulator poles at suitable position is known as pole-placement problem. Assume that rank of the matrix C is 3; therefore pole-placement problem has a unique solution. Next, we assume that λ3+α1λ2+α2λ+α3=0 is characteristic equation of A and μ3+τ1μ2+τ2μ+τ3=0 is characteristic equation of A-BK; then unique solution of the pole-placement problem is given by(44)K=τ3-α3τ2-α2τ1-α1T-1,where T=CM and (45)M=α2α11α110100.Next in order to control Neimark-Sacker bifurcation in system (16), we apply hybrid control feedback methodology [29, 30]. Assuming that system (16) undergoes Neimark-Sacker bifurcation at equilibrium point (x∗,y∗,z∗), then corresponding controlled system can be written as(46)xn+1=ρxn+ρkαΓα+1xn1-xn+λcxnznη+zn-βxnyn+1-ρxn,yn+1=ρyn+ρkαΓα+1λd1xnznη+zn+ϕzn-b2βxnyn-μyn+1-ρyn,zn+1=ρzn+ρkαΓα+1βxnyn-νzn+1-ρzn,where 0<ρ<1 and controlled strategy in (45) is a combination of both parameter perturbation and feedback control. Moreover, by suitable choice of controlled parameter α, the Neimark-Sacker bifurcation of the equilibrium point (x∗,y∗,z∗) of controlled system (45) can be advanced (delayed) or even completely eliminated. The Jacobian matrix of controlled system (45) evaluated at positive equilibrium x∗,y∗,z∗ is given by(47)1-ρkαx∗Γ1+α-ρkαx∗βΓ1+αρckαx∗ηλz∗+η2Γ1+αρkαμy∗-ϕz∗x∗Γ1+α1-ρkαμ+x∗βb2Γ1+αρkαz∗+η2ϕ+x∗ηλd1z∗+η2Γ1+αρkαy∗βΓ1+αρkαx∗βΓ1+α1-ρkανΓ1+α.Then, positive equilibrium (x∗,y∗,z∗) of the controlled system (45) is locally asymptotically stable if roots of the characteristic polynomial of (47) lie in an open unit disk.


## 4. Numerical Simulations

Example 8.
First, we takeα=0.38, k=0.005, c=0.8, λ=2.7, η=1.1, d1=0.7, b2=1.4, ϕ=2.7, μ=4, ν=1.5, x0=0.48, y0=0.1, z0=0.32, and β∈[8,14] in system (16); then system (16) undergoes Neimark-Sacker bifurcation when β is taken as bifurcation parameter. In this case, at β=10.148, the unique positive equilibrium point of system (16) is (x∗,y∗,z∗) = (0.4784797606357651, 0.0996595753165892, 0.32260552713822044). The characteristic polynomial equation evaluated at this positive equilibrium is given by(48)PT=T3-1.0798313086203364T2-0.8127303335375109T+0.9108087500517679=0.The roots of (48) are T1=-0.9107182237825987 and T2,3=0.9952747662014675±0.097609122147403i with T2,3=1. Moreover, we have(49)Δ2-β=1-A2+A3A1-A3=0,Δ2+β=1+A2-A3A1+A3=0.341217>0,Pβ1=1+A1+A2+A3=0.0182471>0,-13Pβ-1=1-A1+A2-A3=0.356292>0.Hence, according to Theorem 7, the conditions of Neimark-Sacker bifurcation are obtained near the positive equilibrium point (x∗,y∗,z∗) = (0.4784797606357651, 0.0996595753165892, 0.32260552713822044) at the critical value of Neimark-Sacker bifurcation β=10.148.
Furthermore, Figure1 shows that all three populations undergo Neimark-Sacker bifurcation (see Figures 1(a), 1(b), and 1(c)) and corresponding maximum Lyapunov exponents (MLEs) are shown in Figure 1(d). These MLEs confirm the existence of the chaotic sets. In general, the positive Lyapunov exponent is considered to be one of the characteristics implying the existence of chaos.Figure 1
Bifurcation diagrams and MLE for system (16) with α=0.38, k=0.005, c=0.8, λ=2.7, η=1.1, d1=0.7, b2=1.4, ϕ=2.7, μ=4, ν=1.5, x0=0.48, y0=0.1, z0=0.32, and β∈[8,14].
(a)
Bifurcation diagram forxn
(b)
Bifurcation diagram foryn
(c)
Bifurcation diagram forzn
(d)
Maximum Lyapunov exponentsExample 9.
Next we takeα=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, x0=0.37, y0=0.23, z0=0.2, and β∈[2,5] in system (16); then system (16) undergoes Neimark-Sacker bifurcation when β is taken as bifurcation parameter. In this case, at β=3.13, the unique positive equilibrium point of system (16) is (x∗,y∗,z∗) = (0.3691168623529498, 0.2335056090113552, 0.20752106517420343). The characteristic polynomial equation evaluated at this positive equilibrium is given by(50)PT=T3-1.137303699628256T2-0.5469293363877263T+0.7989950157191528=0.The roots of (48) are T1=-0.798954 and T2,3=0.9681288296244075±0.2505552964214181i with T2,3=1. Moreover, we verify the conditions of Theorem 7 as follows:(51)Δ2-β=1-A2+A3A1-A3=0,Δ2+β=1+A2-A3A1+A3=0.723378>0,Pβ1=1+A1+A2+A3=0.114762>0,-13Pβ-1=1-A1+A2-A3=0.791379>0.Hence, according to Theorem 7, the conditions of Neimark-Sacker bifurcation are obtained near the positive equilibrium point (x∗,y∗,z∗) = (0.3691168623529498, 0.2335056090113552, 0.20752106517420343) at the critical value of Neimark-Sacker bifurcation β=3.13.
Furthermore, Figure2 shows that all three populations undergo Neimark-Sacker bifurcation (see Figures 2(a), 2(b), and 2(c)) and corresponding maximum Lyapunov exponents (MLEs) are shown in Figure 2(d).
Next, we fix the value ofβ=2.98. In Figure 3, the plot of xn is shown in Figure 3(a), the plot of yn is shown in Figure 3(b), the plot of zn is shown in Figure 3(c), and phase portrait of system (16) is shown in Figure 3(d). It is clear from Figure 3 that system (16) has unique positive equilibrium point which is locally asymptotically stable.
Furthermore, the phase portraits of system (16) for different values of β are shown in Figure 4.Figure 2
Bifurcation diagrams and MLE for system (16) with α=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, x0=0.37, y0=0.23, z0=0.2, and β∈[2,5].
(a)
Bifurcation diagram forxn
(b)
Bifurcation diagram foryn
(c)
Bifurcation diagram forzn
(d)
Maximum Lyapunov exponentsFigure 3
Plots for system (16) with α=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, x0=0.37, y0=0.23, z0=0.2, and β=2.98.
(a)
Plot ofxn for system (16)
(b)
Plot ofyn for system (16)
(c)
Plot ofzn for system (16)
(d)
Phase portrait of system (16)Figure 4
Phase portraits of system (16) with different values of β and taking α=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, x0=0.37, y0=0.23, and z0=0.2.
(a)
Phase portrait of system (16) for β=3.133
(b)
Phase portrait of system (16) for β=3.18
(c)
Phase portrait of system (16) for β=3.2
(d)
Phase portrait of system (16) for β=3.5Example 10.
We choose the parametric valuesα=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, and β=3.2 and initial conditions x0=0.37, y0=0.23, and z0=0.2 in system (16); then system (16) has a unique positive equilibrium point (x∗,y∗,z∗) = (0.363861,0.229828,0.205847) which is unstable and Figure 4(c) shows a chaotic attractor for system (16) for these parametric values. So, one can take β0=3.2 in order to apply pole-placement technique for system (16). In this case, the corresponding controlled system is given by(52)xn+1=xn+kαΓα+1xn1-xn+λcxnznη+zn-3.2-k1xn-x∗-k2yn-y∗-k3zn-z∗xnyn,yn+1=yn+kαΓα+1λd1xnznη+zn+ϕzn-b23.2-k1xn-x∗-k2yn-y∗-k3zn-z∗xnyn-μyn,zn+1=zn+kαΓα+13.2-k1xn-x∗-k2yn-y∗-k3zn-z∗xnyn-νzn,where α=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, K=k1k2k3, and (x∗,y∗,z∗)=(0.363861,0.229828,0.205847). Furthermore, we have(53)A=0.8584036593529867-0.453108290070442450.05754406574949606-0.18168256820150672-0.218075348691300071.29895594356827870.286199238576560040.453108290070442450.49410608975301007,B=-0.03254276321257405-0.045559868497603660.03254276321257405,C=-0.03254276321257405-0.005418630010373265-0.03178440171863723-0.045559868497603660.05811955270088685-0.0297164798998744570.03254276321257405-0.013877690682676330.01792659188284191.Then it is easy to check that matrix C is of rank 3. Thus system (52) is controllable and its Jacobian matrix at (x∗,y∗,z∗)=(0.363861,0.229828,0.205847) is given by(54)A-BK=0.858404+0.0325428k1-0.453108+0.0325428k20.0575441+0.0325428k3-0.181683+0.0455599k1-0.218075+0.0455599k21.29896+0.0455599k30.286199-0.0325428k10.453108-0.0325428k20.494106-0.0325428k3. The characteristic equation of A-BK is given by(55)PT=T3-1.13443+0.0325428k1+0.0455599k2-0.0325428k3T2-0.558165-0.031499k1-0.109804k2+0.0507953k3T+0.807993-0.00747308k1-0.0702194k2+0.0155057k3=0.Then a necessary and sufficient condition for all roots of the equation (55) to lie inside the unit disk is(56)A2+A0<1+A1,A2-3A0<3-A1,A02+A1-A0A2<1,where A2=-1.13443-0.0325428k1-0.0455599k2+0.0325428k3, A1=-0.558165+0.031499k1+0.109804k2-0.0507953k3, and A0=-0.807993+0.00747308k1+0.0702194k2-0.0155057k3. For simplicity, if we choose k1=0.1 and k2=-0.1, then the unique positive equilibrium point of the controlled system (52) is locally asymptotically stable if and only if 0.569236<k3<7.61676. Next, we choose k1=0.1, k2=-0.1 and selecting k3∈[2,14] as bifurcation parameter for the controlled system (52). Moreover, the bifurcation diagrams for the controlled system are shown in Figure 5. It can be noticed from Figure 5 that the unique positive equilibrium point (x∗,y∗,z∗)=(0.363861,0.229828,0.205847) of the controlled system (52) undergoes period-doubling bifurcation at k3≈7.61676; then both periodic orbits inter the region in which these orbits undergo Neimark-Sacker bifurcation separately. Thus we succeeded to advance the Neimark-Sacker bifurcation by introducing the pole-placement control technique in this case.
Finally, choosing the parametric valuesα=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, and β=3.133 and initial conditions x0=0.37, y0=0.23, and z0=0.2 in system (16), then system (16) undergoes Neimark-Sacker bifurcation for these parametric values. Then unique positive equilibrium point (x∗,y∗,z∗)=(0.368888,0.233346,0.207449) of system (16) is unstable and Figure 4(a) shows that closed invariant circle appears at β=3.133 enclosing this unstable equilibrium. For these parametric values, the controlled system (45) can be written as(57)xn+1=xn+ρkαΓα+1xn1-xn+λcxnznη+zn-βxnyn,yn+1=yn+ρkαΓα+1λd1xnznη+zn+ϕzn-b2βxnyn-μyn,zn+1=zn+ρkαΓα+1βxnyn-νzn,where α=0.3, k=0.03, c=0.3, λ=2.1, η=1.1, d1=1.7, b2=1.4, ϕ=2.5, μ=1.5, ν=1.3, β=3.133, and 0<ρ<1. Then Jacobian matrix of system (57) evaluated at (x∗,y∗,z∗)=(0.368888,0.233346,0.207449) is given by(58)1-0.143552ρ-0.44975ρ0.0581961ρ-0.177865ρ1-1.21337ρ1.30265ρ0.284496ρ0.44975ρ1-0.505894ρ.The characteristic polynomial of (58) is given by(59)T3-3-1.863ρT2+3-3.726ρ+0.178ρ2T-1+1.863ρ-0.178ρ2+0.115ρ3=0.According to the Jury condition, the roots of (58) lie in the unit open disk if and only if 0<ρ<0.996829. Moreover, for ρ=0.95, the plots for the controlled system (57) are shown in Figure 6.Figure 5
Bifurcation diagrams for the controlled system (52) with k1=0.1, k2=-0.1, and 2≤k3≤14.
(a)
Bifurcation diagram ofxn for system (52)
(b)
Bifurcation diagram ofyn for system (52)
(c)
Bifurcation diagram ofzn for system (52)Figure 6
Plots for the controlled system (57) with ρ=0.95.
(a)
Plot ofxn for system (57)
(b)
Plot ofyn for system (57)
(c)
Plot ofzn for system (57)
(d)
Phase portrait of system (57)


## 5. Concluding Remarks

We study the qualitative behavior of a 3-dimensional discrete-time fractional-order plant-herbivore model, and thus we obtain the mathematical results related to existence and uniqueness of positive steady state and conditions for local asymptotic stability of positive equilibrium. In order to confirm complexity in system (16), the existence of Neimark-Sacker bifurcation for positive equilibrium point is proved mathematically and numerical simulations are provided. Through numerical simulation, we show that system (16) undergoes Neimark-Sacker for wide range of bifurcation parameter β. Neimark-Sacker bifurcation is successfully controlled with two different control strategies. From our numerical investigation, it is clear that pole-placement technique based on feedback control methodology restores the stability for wide range of parameters, whereas the hybrid control based on feedback control and parameter perturbation works more effectively. Although hybrid control strategy is basically formulated for control of period-doubling bifurcation, it can also stabilize the unstable steady state for maximum possible range of control parameter in case of Neimark-Sacker bifurcation as well. Moreover, through numerical simulations, it is observed that the original system (16) undergoes Neimark-Sacker bifurcation only and there is no chance of period-doubling bifurcation for this system. But an application of pole-placement control strategy gives an evidence of period-doubling bifurcation for corresponding controlled system.



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