Research ArticleStability and Bifurcation of Two Kinds of Three-DimensionalFractional Lotka-Volterra Systems

Jinglei Tian Yongguang Yu and Hu Wang

Department of Mathematics Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Yongguang Yu ygyubjtueducn

Received 23 October 2013 Revised 12 January 2014 Accepted 17 January 2014 Published 12 March 2014

Academic Editor Yuncai Wang

Copyright copy 2014 Jinglei Tian et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Two kinds of three-dimensional fractional Lotka-Volterra systems are discussed For one system the asymptotic stability of theequilibria is analyzed by providing some sufficient conditions And bifurcation property is investigated by choosing the fractionalorder as the bifurcation parameter for the other system In particular the critical value of the fractional order is identified at whichthe Hopf bifurcation may occur Furthermore the numerical results are presented to verify the theoretical analysis

1 Introduction

In recent years fractional calculus has attracted much atten-tion of researchers It has been pointed out that fractionalcalculus plays an outstanding role in modelling and simula-tion of systems such as viscoelastic systems dielectric polar-ization electromagnetic waves heat conduction roboticsand biological systems In fact fractional derivatives providean excellent instrument for the description of memory andhereditary properties of various materials and processes incomparison with the classical integer-order counterpartsTherefore it may be more important and useful to investigatethe fractional systems

Traditionally the fractional differential equation definedby mathematicians is a Riemann-Liouville fractional deriva-tive [1] But this definition is less popular because of thefact that it requires initial conditions to be expressed interms of fractional integrals and their derivativesMeanwhilethere is no known physical interpretation for such types ofinitial conditions In contrast the alternative definition of thefractional derivative given by Caputo [2] has the advantageof only requiring the initial conditions given in terms ofinteger-order derivatives These initial conditions of integer-order derivatives can be measured accurately and representwell-understood features of a physical situation In [2] ithas been pointed out that Caputorsquos derivative is equivalent

to the Riemann-Liouville derivative under homogeneous ini-tial conditions and some smoothness conditions ThereforeCaputorsquos definition of fractional derivative is used throughoutin this paper

As is well known in the field of mathematical biologythe traditional Lotka-Volterra systems are very importantmathematical models which describe multispecies popu-lation dynamics in a nonautonomous environment Manyimportant and interesting results on the dynamical behaviorsfor the Lotka-Volterra systems have been found in [3ndash9] such as the existence and uniqueness of solutions thepermanence extinction global asymptotic behavior andbifurcation Because of the good memory and hereditaryproperties of fractional derivatives it is often necessary tostudy the corresponding fractional systems Therefore thedynamical analysis of the fractional Lotka-Volterra systemshas attracted a great deal of attention due to its theoreticaland practical significance

Many important results regarding stability of fractionalsystems have been obtained For instance the stabilityexistence uniqueness and numerical solution of the frac-tional logistic equation are investigated in [10] The stabilityand solutions of fractional predator-prey and rabies modelsare discussed in [11] In addition bifurcation properties offractional systems have been studied in some papers Forexample conditions for the occurrence of Hopf rsquos bifurcation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 695871 8 pageshttpdxdoiorg1011552014695871

2 Mathematical Problems in Engineering

are explored based on numerical simulations in [12] Thecritical values of the fractional order are identified for whichHopf rsquos bifurcation may occur based on the stability analysisin [13]Thus it is significant to study the dynamical behaviorsin the fractional population systems

To the best of our knowledge some papers have con-centrated on the dynamic investigation of the fractionalpopulation systems [10 11] However there are few resultson bifurcation phenomena of the fractional population sys-tems Therefore in the paper we mainly consider stabilityand bifurcation in the three-dimensional fractional Lotka-Volterra systems

Motivated by the above discussions some dynamicalproperties of two kinds of three-dimensional fractionalLotka-Volterra systems are investigated in this paper Exis-tence and uniqueness of solutions are considered Somesufficient conditions are provided for the asymptotic stabilityof equilibria Specifically bifurcation behaviors are analyzedby formulating the critical values of the fractional order atwhich Hopf rsquos bifurcations may take place

The rest of this paper is organized as follows In Section 2a three-dimensional fractional Lotka-Volterra predator-preysystem with interspecific competition is introduced And theasymptotic stability of the system is studied In Section 3a three-dimensional fractional Lotka-Volterra predator-preysystem is provided and bifurcation properties are investi-gated The numerical results in Section 4 are given to verifythe theoretical findings Finally the paper is concluded inSection 5

2 Stability Analysis of a Three-DimensionalFractional Lotka-Volterra Predator-PreySystem with Interspecific Competition

Consider a three-dimensional fractional Lotka-Volterra sys-tem

1198631199021199091 (119905) = 1199091 (119905) (1198871 minus 119886111199091 (119905) minus 119886121199092 (119905) minus 119886131199093 (119905))

1198631199021199092 (119905) = 1199092 (119905) (minus1198872 + 119886211199091 (119905) minus 119886221199092 (119905) minus 119886231199093 (119905))

1198631199021199093 (119905) = 1199093 (119905) (minus1198873 + 119886311199091 (119905) minus 119886321199092 (119905) minus 119886331199093 (119905))

(1)

with the initial values 119909119894(119905)|119905=0 = 119909119894(0) 119894 = 1 2 3 where0 lt 119902 le 1 especially when 119902 = 1 the system (1) is a classicalinteger-order system All constant coefficients 119886119894119895 and 119887119894 (119894 119895 =1 2 3) can be arbitrary positive real numbers 1199091(119905) ge 0

represents the density of prey species at time 119905 and 1199092(119905) ge 01199093(119905) ge 0 represent the densities of predator species at time 119905In this case system (1) can be regarded as a fractional Lotka-Volterra predator-prey systemwith interspecific competition

In the following existence and uniqueness of solutionsfor system (1) are given In addition the important resultsrelated to the stability of the fractional systems are presentedto provide the theoretical bases for the further study

Here the fractional Lotka-Volterra system (1) can berewritten in the form

119863119902119883 (119905) = 119860119883 (119905) + 1199091 (119905) 1198611119883 (119905)

+ 1199092 (119905) 1198612119883(119905) + 1199093 (119905) 1198613119883 (119905)

119883 (0) = 1198830

(2)

where 0 lt 119902 le 1 119905 isin (0 119879] and

119883 (119905) = (

1199091 (119905)

1199092 (119905)

1199093 (119905)

) 1198830 = (

1199091 (0)

1199092 (0)

1199093 (0)

)

119860 = (

1198871 0 0

0 minus1198872 0

0 0 minus1198873

)

1198611 = (

minus11988611 minus11988612 minus11988613

0 0 0

0 0 0

) 1198612 = (

0 0 0

11988621 minus11988622 minus11988623

0 0 0

)

1198613 = (

0 0 0

0 0 0

11988631 minus11988632 minus11988633

)

(3)

Definition 1 For 119883(119905) = (1199091(119905) 1199092(119905) 1199093(119905))119879 let 119862lowast[0 119879] be

the set of continuous column vectors 119883(119905) on the interval[0 119879] The norm of 119883(119905) isin 119862

lowast[0 119879] is given by 119883(119905) =

sum3

119894=1sup119905|119909119894(119905)|

Theorem 2 System (2) has a unique solution if 119883(119905) isin

119862lowast[0 119879]

Proof Let 119865(119883(119905)) = 119860119883(119905) + 1199091(119905)1198611119883(119905) + 1199092(119905)1198612119883(119905) +

1199093(119905)1198613119883(119905) then 119883(119905) isin 119862lowast[0 119879] implies 119865(119883(119905)) isin

119862lowast[0 119879] In addition take 119883(119905) 119884(119905) isin 119862

lowast[0 119879] and

119883(119905) = 119884(119905) the following inequality holds

119865 (119883 (119905)) minus 119865 (119884 (119905))

=1003817100381710038171003817119860 (119883 (119905) minus 119884 (119905)) + 1199091 (119905) 1198611119883 (119905)

minus 1199101 (119905) 1198611119884 (119905) + 1199092 (119905) 1198612119883 (119905)

minus 1199102 (119905) 1198612119884 (119905) + 1199093 (119905) 1198613119883 (119905)

minus1199103 (119905) 1198613119884 (119905)1003817100381710038171003817

=1003817100381710038171003817119860 (119883 (119905) minus 119884 (119905)) + 1199091 (119905) 1198611 (119883 (119905) minus 119884 (119905))

+ (1199091 (119905) minus 1199101 (119905)) 1198611119884 (119905) + 1199092 (119905) 1198612 (119883 (119905) minus 119884 (119905))

+ (1199092 (119905) minus 1199102 (119905)) 1198612119884 (119905) + 1199093 (119905) 1198613 (119883 (119905) minus 119884 (119905))

+ (1199093 (119905) minus 1199103 (119905)) 1198613119884 (119905)1003817100381710038171003817

le 119860 (119883 (119905) minus 119884 (119905)) +10038171003817100381710038171199091 (119905) 1198611 (119883 (119905) minus 119884 (119905))

1003817100381710038171003817

+1003817100381710038171003817(1199091 (119905) minus 1199101 (119905)) 1198611119884 (119905)

1003817100381710038171003817 +10038171003817100381710038171199092 (119905) 1198612 (119883 (119905) minus 119884 (119905))

1003817100381710038171003817

+1003817100381710038171003817(1199092 (119905) minus 1199102 (119905)) 1198612119884 (119905)

1003817100381710038171003817 +10038171003817100381710038171199093 (119905) 1198613 (119883 (119905) minus 119884 (119905))

1003817100381710038171003817

Mathematical Problems in Engineering 3

+1003817100381710038171003817(1199093 (119905) minus 1199103 (119905)) 1198613119884 (119905)

1003817100381710038171003817

le (119860 +10038171003817100381710038171198611

1003817100381710038171003817 (10038161003816100381610038161199091 (119905)

1003816100381610038161003816 + 119884 (119905))

+10038171003817100381710038171198612

1003817100381710038171003817 (10038161003816100381610038161199092 (119905)

1003816100381610038161003816 + 119884 (119905))

+10038171003817100381710038171198613

1003817100381710038171003817 (10038161003816100381610038161199093 (119905)

1003816100381610038161003816 + 119884 (119905))) 119883 (119905) minus 119884 (119905)

le 119871 119883 (119905) minus 119884 (119905)

(4)

where 119871 = 119860 + (1198611 + 1198612 + 1198613)(1198721 +1198722) gt 0 and1198721and1198722 are positive and satisfy 119883(119905) le 1198721 119884(119905) le 1198722 asa result of 119883(119905) 119884(119905) isin 119862

lowast[0 119879] Based on Theorems 21 and

22 in [14] system (2) has a unique solution

Theorem 3 (see [15]) The linear autonomous system 119863119902119909 =

119860119909 is asymptotically stable if and only if

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

119902120587

2 (5)

where119860 isin 119877119899times119899 119902 isin (0 1) and 120582 isin 120590(119860) 120590(119860) denotes the set

of all eigenvalues of the matrix 119860

Theorem 4 Let 119909lowast be an equilibrium of the nonlinear system(1) then the equilibrium 119909

lowast is locally asymptotically stable if

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

119902120587

2 (6)

where 120582 isin 120590(119869(119909lowast)) 120590(119869(119909lowast)) denotes the set of all eigenvalues

of the Jacobian matrix 119869(119909lowast)

Proof The proof follows fromTheorem 3 and [11]

In the following the stability of system (1) is investigatedby giving some appropriate conditions The asymptotic sta-bility of the equilibria is demonstrated based on Theorem 4Through simple calculation the equilibria of system (1) areobtained and denoted as

1198751 = (0 0 0) 1198752 = (0 0 minus1198873

11988633

)

1198753 = (0 minus1198872

11988622

0) 1198754 = (1198871

11988611

0 0)

1198755 = (011988852

1198885

11988853

1198885

) 1198756 = (11988861

1198886

011988863

1198886

)

1198757 = (11988871

1198887

11988872

1198887

0) 1198758 = (11988881

1198888

11988882

1198888

11988883

1198888

)

(7)

where 11988852 = 119887311988623 minus 119887211988633 11988853 = 119887211988632 minus 119887311988622 1198885 = 1198862211988633 minus

1198862311988632 11988861 = 119887111988633 + 119887311988613 11988863 = 119887111988631 minus 119887311988611 1198886 =

1198861111988633 + 1198861311988631 11988871 = 119887111988622 + 119887211988612 11988872 = 119887111988621 minus 119887211988611 1198887 =

1198861111988622+1198861211988621 11988881 = 11988711198862211988633minus11988711198862311988632+11988721198861211988633minus11988721198861311988632+

11988731198861311988622 minus 11988731198861211988623 11988882 = 11988711198862111988633 minus 11988711198862311988631 minus 11988721198861111988633 minus

11988721198861311988631 + 11988731198861111988623 + 11988731198861311988621 11988883 = 11988711198862211988631 minus 11988711198862111988632 +

11988721198861111988632 + 11988721198861211988631 minus 11988731198861111988622 minus 11988731198861211988621 and 1198888 = 119886111198862211988633 minus

119886111198862311988632 + 119886121198862111988633 minus 119886121198862311988631 + 119886131198862211988631 minus 119886131198862111988632

Because of the fact that all constant coefficients of system(1) are positive 1198752 1198753 and 1198755 are in contradiction with theactual situation hence the asymptotical stability of other fiveequilibria will be studied in detail

Theorem 5 For the three-dimensional fractional Lotka-Volterra system (1) the following results can be obtained

(a) 1198751 is unstable(b) 1198754 is locally asymptotically stable if 119887111988611 lt 119887211988621

119887111988611 lt 119887311988631(c) 1198756 is locally asymptotically stable if 119887311988631 lt 119887111988611 lt

119887211988621(d) 1198757 is locally asymptotically stable if 119887211988621 lt 119887111988611 lt

119887311988631(e) 1198758 is locally asymptotically stable if 1198863211988622 lt 1198863111988621 lt

1198863311988623

Proof For 1198751 = (0 0 0) its Jacobian matrix is

119869 (1198751) = (

1198871 0 0

0 minus1198872 0

0 0 minus1198873

) (8)

and the eigenvalues of 119869(1198751) satisfy 1205821 = 1198871 gt 0 1205822 = minus1198872 lt

0 and 1205823 = minus1198873 lt 0 hence the equilibrium 1198751 is unstableFor 1198754 its Jacobian matrix is

119869 (1198754) =((

(

minus1198871 minus119887111988612

11988611

minus119887111988613

11988611

0119887111988621 minus 119887211988611

11988611

0

0 0119887111988631 minus 119887311988611

11988611

))

)

(9)

and the eigenvalues of 119869(1198754) satisfy 1205821 = minus1198871 lt 0 1205822 =

(119887111988621 minus 119887211988611)11988611 lt 0 and 1205823 = (119887111988631 minus 119887311988611)11988611 lt 0hence the equilibrium 1198754 is locally asymptotically stable

For 1198756 use the notations below

119869 (1198756) = (

11986011 11986012 11986013

0 11986022 0

11986031 11986032 11986033

) (10)

and its characteristic equation is

(120582 minus 11986022) (1205822minus (11986011 + 11986033) 120582 + 1198601111986033 minus 1198601311986031) = 0

(11)

Based on the condition from (c) the following formulas canbe easily got

11986011 =11988811

1198880

lt 0 11986013 =11988813

1198880

lt 0

11986031 =11988831

1198880

gt 0 11986022 =11988822

1198880

lt 0

11986033 =11988833

1198880

lt 0

(12)

4 Mathematical Problems in Engineering

where 11988811 = minus11988711198861111988633 minus 11988731198861111988613 11988813 = minus11988711198861311988633 minus

11988731198861311988613 11988831 = 11988711198863111988631 minus 11988731198861111988631 11988822 = 11988633(119887111988621 minus 119887211988611) +

11988623(119887311988611minus119887111988631)+11988613(119887311988621minus119887211988631) 11988833 = minus11988711198863111988633+11988731198861111988633and 1198880 = a1111988633 + 1198861311988631 Then the following results can beobtained

1205821 = 11986022 lt 0 1205822 + 1205823 = 11986011 + 11986033 lt 0

12058221205823 = 1198601111986033 minus 1198601311986031 gt 0

(13)

Hence the equilibrium 1198756 is locally asymptotically stableSimilarly it can be readily derived that the equilibrium 1198757

is locally asymptotically stableFor 1198758 let 1198758 = (119909

lowast

1 119909lowast

2 119909lowast

3) the Jacobian matrix of 1198758 can

be written as

119869 (1198758) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

2minus11988623119909

lowast

2

11988631119909lowast

3minus11988632119909

lowast

3minus11988633119909

lowast

3

) = (

11986111 11986112 11986113

11986121 11986122 11986123

B31 11986132 11986133

)

(14)

and its characteristic equation is

1205823+ 1198621120582

2+ 1198622120582 + 1198623 = 0 (15)

where 1198621 = minus(11986111 + 11986122 + 11986133) 1198622 = 1198611111986122 + 1198611111986133 +

1198612211986133 minus 1198612311986132 minus 1198611211986121 minus 1198611311986131 and 1198623 = minus119861111198612211986133 +

119861111198612311986132 + 119861121198612111986133 + 119861131198612211986131 minus 119861121198612311986131 minus 119861131198612111986132For simplicity the equivalent characteristic equation is intro-duced as follows

1205823+ 11988611205822+ 1198862120582 + 1198863 = (120582 minus 119886) (120582

2minus 119887120582 + 119888) = 0 (16)

On the basis of the above equivalent substitutions thefollowing inequalities can be gained

1198861 = 11988611119909lowast

1+ 11988622119909

lowast

2+ 11988633119909

lowast

3gt 0

1198862 = (1198861111988622 + 1198861211988621) 119909lowast

1119909lowast

2+ (1198861111988633 + 1198861311988631) 119909

lowast

1119909lowast

3

+ (1198862211988633 minus 1198862311988632) 119909lowast

2119909lowast

3gt 0

1198863 = (11988611 (1198862211988633 minus 1198862311988632) + 11988612 (1198862111988633 minus 1198862311988631)

+11988613 (1198862211988631 minus 1198862111988632)) 119909lowast

1119909lowast

2119909lowast

3gt 0

11988611198862 minus 1198863 = 1198891119909lowast

1119909lowast

2119909lowast

3+ 1198892119909

lowast

1119909lowast

2

2+ 1198893119909

lowast

1119909lowast

3

2

+ 1198894119909lowast

1

2119909lowast

2+ 1198895119909

lowast

1

2119909lowast

3

+ 1198896 (11988622119909lowast

2

2119909lowast

3+ 11988633119909

lowast

2119909lowast

3

2) gt 0

119886 + 119887 = minus1198861 lt 0

119886119887 + 119888 = 1198862 gt 0

119886119888 = minus1198863 lt 0

minus1198862119887 minus 119886119887

2minus 119887119888 = 11988611198862 minus 1198863 gt 0

(17)

where 1198891 = 2119886111198862211988633 + 119886121198862311988631 + 119886131198862111988632 1198892 =

119886111198862

22+ 119886121198862111988622 1198893 = 11988611119886

2

33+ 119886131198863111988633 1198894 = 119886

2

1111988622 +

119886111198861211988621 1198895 = 1198862

1111988633 + 119886111198861311988631 and 1198896 = 1198862211988633 minus 1198862311988632

Using the proof by contradiction it can be concluded that theeigenvalues of 119869(1198758) satisfy

1205821 = 119886 lt 0

1205822 + 1205823 = 119887 lt 0

12058221205823 = 119888 gt 0

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt

119902120587

2 119894 = 2 3

(18)

Hence the equilibrium1198758 is locally asymptotically stable

For the further dynamic investigation of the fractionalpopulation systems the other fractional Lotka-Volterra sys-tems will be considered in the following section Particularlybifurcation properties for the system will be studied in detail

3 Bifurcation Analysis ofa Three-Dimensional FractionalLotka-Volterra Predator-Prey System

Consider a three-dimensional fractional Lotka-Volterra sys-tem

1198631199021199091 (119905) = 1199091 (119905) (1198871 minus 119886111199091 (119905) minus 119886121199092 (119905) minus 119886131199093 (119905))

1198631199021199092 (119905) = 1199092 (119905) (minus1198872 + 119886211199091 (119905) minus 119886221199092 (119905))

1198631199021199093 (119905) = 1199093 (119905) (minus1198873 + 119886311199091 (119905) minus 119886331199093 (119905))

(19)

with the initial values 119909119894(119905)|119905=0 = 119909119894(0) 119894 = 1 2 3 where0 lt 119902 lt 1 11988611 lt 0 and the other constant coefficients arepositive 1199091(119905) ge 0 represents the density of prey species attime 119905 and 1199092(119905) ge 0 1199093(119905) ge 0 represent the densities ofpredator species at time 119905 In this case system (19) can beregarded as a fractional Lotka-Volterra predator-prey system

On the basis of Theorem 2 it is not difficult to prove thatsystem (19) has a unique solution in a similar way

It is clear that there are eight equilibria for system(19) Here we focus on the bifurcation investigation of theequilibrium 119909

lowast which can be called a positive equilibriumwhen some conditions are satisfied The equilibrium 119909

lowast isobtained as

119909lowast= (119909lowast

1 119909lowast

2 119909lowast

3) = (

11988911

11988911988922

11988911988933

119889) (20)

where 11988911 = 11988711198862211988633 + 11988721198861211988633 + 11988731198861311988622 11988922 = 11988711198862111988633 minus

1198872(1198861111988633 + 1198861311988631) + 11988731198861311988621 11988933 = 11988711198862211988631 + 11988721198861211988631 minus

1198873(1198861111988622 + 1198861211988621) and 119889 = 119886111198862211988633 + 119886121198862111988633 + 119886131198862211988631And its Jacobian matrix can be expressed as

119869 (119909lowast) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

20

11988631119909lowast

30 minus11988633119909

lowast

3

) (21)

Mathematical Problems in Engineering 5

Furthermore the eigenvalues of 119869(119909lowast) satisfy the characteris-tic equation

(120582 minus 120573) (1205822minus (120572 + 120573) 120582 + 120572120573 minus 120574) = 0 (22)

where 120572 = minus11988611119909lowast

1 120573 = minus11988622119909

lowast

2= minus11988633119909

lowast

3 and 120574 =

minus1198861211988621119909lowast

1119909lowast

2minus 1198861311988631119909

lowast

1119909lowast

3

In the following by choosing the fractional order 119902

as the bifurcation parameter and analyzing the associatedcharacteristic equation (22) of system (19) at the positiveequilibrium we investigate the bifurcation phenomena of thepositive equilibrium of system (19) and obtain the conditionsunder which system (19) undergoes a Hopf bifurcation

Proposition 6 The positive equilibrium 119909lowast of system (19) is

locally asymptotically stable if and only if all the followingconditions are satisfied

(i) 120573 lt 0(ii) 120572120573 minus 120574 gt 0 and(iii) 120572 + 120573 lt 2 cos(1199021205872)radic120572120573 minus 120574

Proof For the characteristic equation (22) the root 1205821 = 120573 lt

0 and 1205822 1205823 satisfy the equation 1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0

It is clear that | arg(12058223)| gt 1199021205872 if and only if the conditions(ii) and (iii) hold Based onTheorem 4 Proposition 6 provesto be true

In addition by analyzing the condition (iii) ofProposition 6 in detail the following results can be gained

Proposition 7 With respect to system (19) if 120573 lt 0 and 120572120573 minus120574 gt 0 the following statements can be obtained

(a) If 120572+120573 le 0 the equilibrium 119909lowast is locally asymptotically

stable for any 119902 isin (0 1)(b) If 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equilibrium 119909

lowast is locallyasymptotically stable if and only if 119902 isin (0 119902

lowast) where

119902lowast= (2120587) arccos ((120572 + 120573)2radic120572120573 minus 120574)

(c) If 120572+120573 ge 2radic120572120573 minus 120574 the equilibrium 119909lowast is unstable for

any 119902 isin (0 1)

Proof The conclusions (a) and (c) are obvious For thestatement (b) due to 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equation1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0 has two complex roots 1205822 1205823

and their real part is (120572 + 120573)2 gt 0 Then | arg(120582119894)| =

arccos((120572 + 120573)2radic120572120573 minus 120574) 119894 = 2 3 Besides according to thecondition arccos((120572 + 120573)2radic120572120573 minus 120574) = 119902

lowast1205872 119902 isin (0 119902

lowast) if

and only if | arg(120582119894)| gt 1199021205872 119894 = 2 3 Based onTheorem 4 itis concluded that Proposition 7 is true

Remark 8 It is apparent that the critical value satisfies 119902lowast isin(0 1) When 119902 isin (0 119902

lowast) 119909lowast is locally asymptotically stable

when 119902 isin (119902lowast 1) and specially 119902 = 1 119909lowast is unstable That is

to say it has verified that fractional differential equations areat least as stable as their integer-order counterparts [4]

Remark 9 Under the situation of statement (b) a bifurcationphenomenon must happen at the critical value 119902lowast However

it is difficult to confirm precise bifurcation type As an inter-esting bifurcation behavior Hopf rsquos bifurcation is expected totake place

According to Proposition 7 if some appropriate condi-tions about the constant coefficients of system (19) can befound so that statement (b) is satisfied system (19) willundergo a bifurcation phenomenon And the critical value119902lowast of the bifurcation parameter 119902 can be expressed by theconstant coefficients of system (19) From this the followingtheorem is specifically proposed

Theorem 10 With respect to system (19) if the followingconditions are satisfied

(i) 1198872 = 1198873 11988622 = 11988633 11988621 = 11988631 = minus11988611(ii) 11988612 + 11988613 minus 11988622 gt 0

then the positive equilibrium 119909lowast is locally asymptotically stable

if and only if 119902 isin (0 119902lowast) where

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(23)

Proof According to the condition (i) the equilibrium 119909lowast can

be expressed as

119909lowast= (

119887111988622 + 119887211988612 + 119887211988613

11988611 (11988622 minus 11988612 minus 11988613)

1198871 + 1198872

11988612 + 11988613 minus 11988622

1198871 + 1198872

11988612 + 11988613 minus 11988622

)

(24)

For (22) the following results can be obtained

120573 = minus11988622119909lowast

2lt 0 120572 + 120573 = 1198872 gt 0

120572120573 minus 120574 =1198901

11988612 + 11988613 minus 11988622

gt 0

4 (120572120573 minus 120574) minus (120572 + 120573)2=

41198901

11988612 + 11988613 minus 11988622

minus 1198872

2

=1198902

11988612 + 11988613 minus 11988622

gt 0

(25)

where 1198901 = (1198871 + 1198872)(119887111988622 + 119887211988612 + 119887211988613) and 1198902 = 11988622(41198872

1+

1198872

2)+31198872

2(11988612+11988613)+411988711198872(11988612+11988613+11988622) Obviously the above

conclusions satisfy statement (b) of Proposition 7 then it canbe derived that

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(26)

6 Mathematical Problems in Engineering

5 8 11 14 17 20

02

46

8100

5

10

x3

x2

x1

q = 1

q = 09

q = 08

Figure 1 The trajectory of system (1) converges to the equilibrium1198758= (537 157 17)

Hence the positive equilibrium 119909lowast of system (19) is locally

asymptotically stable if and only if 119902 isin (0 119902lowast)

According to the statement of Theorem 10 it can beconcluded that the positive equilibrium 119909

lowast is locally asymp-totically stable if and only if 119902 isin (0 119902

lowast) At 119902 = 119902

lowast the Hopfbifurcation is expected to take place As 119902 increases above thecritical value 119902lowast the positive equilibrium 119909

lowast is unstable anda limit cycle is expected to appear in the proximity of 119909lowast dueto the Hopf bifurcation phenomenon

The analysis of periodic solutions in fractional dynamicalsystems is a very recent and promising research topic Asa consequence the nonexistence of exact periodic solutionsin time invariant fractional systems is obtained [16] As anapplication it is emphasized that the limit cycle observed innumerical simulations of a simple fractional neural networkcannot be an exact periodic solution of the system [17]In addition there are some other papers providing thenumerical evidences of limit cycles

Remark 11 Even though exact periodic solutions do not existin autonomous fractional systems [16 17] limit cycles havebeen observed by numerical simulations in many systemssuch as a fractional neural system [13] a fractional Van derPol system [18] fractional Chua and Chenrsquos systems [19 20]and a fractional financial system [21]

4 Numerical Simulation

In this paper an Adams-type predictor-corrector methodis used for the numerical solutions of fractional differentialequations This method has been introduced in [22 23]and further investigated in [24ndash27] In order to verify thetheoretical analysis the following numerical results are given

For system (1) the approximate solutions are displayedin Figure 1 for the step size 0005 and different values of119902 119902 = 1 119902 = 09 119902 = 08 respectively Taking 1198871 =

12 1198872 = 11988611 = 11988613 = 11988621 = 11988623 = 11988631 = 1 11988612 =

11988632 = 11988633 = 2 and 1198873 = 11988622 = 3 and choosing theinitial values 1199091(0) = 20 1199092(0) = 10 and 1199093(0) = 10 the

55 60 65 701

15

2

25

3

35

4

45

5

x

t

1

(a)

5515

2

25

3

35

4

45

5

55

6

60 65 70

x

t

2

(b)

55 60 65 7015

2

25

3

35

4

45

5

55

6

x

t

3

(c)

Figure 2 The solution of system (19) versus time with 119902 = 084

equilibrium 1198758 is (537 157 17) Then Figure 1 shows thatthe equilibrium 1198758 is locally asymptotically stable Namelythe fifth conclusion of Theorem 5 is verified Similarly theother conclusions of Theorem 5 can be confirmed

For system (19) the approximate solutions are displayedin Figures 2 3 and 4 for the step size 0001 and different valuesof 119902 119902 = 082 and 119902 = 084 Taking 1198871 = 11988612 = 11988613 = 11988622 =

11988633 = 1 1198872 = 1198873 = 11988621 = 11988631 = 2 and 11988611 = minus2 andchoosing the initial values 1199091(0) = 3 1199092(0) = 1199093(0) = 4 thepositive equilibrium is 119909lowast = (25 3 3) and the critical value

Mathematical Problems in Engineering 7

15 2 25 3 35

225

335

42

3

4

x3

x2

x1

Figure 3 When 119902 = 082 the trajectory of system (19) converges tothe equilibrium 119909

lowast= (25 3 3)

12

34

5

0

2

4

60510

x3

x2

x 1

Figure 4 When 119902 = 084 the trajectory of system (19) converges toan asymptotically stable limit cycle

is 119902lowast = 08337 Indeed Figures 2ndash4 present the fact that thepositive equilibrium 119909

lowast is locally asymptotically stable when119902 = 082 isin (0 08337) and when 119902 = 084 increases across119902lowast= 08337 an asymptotically stable limit cycle appears in a

neighborhood of the positive equilibrium 119909lowast

5 Conclusion

In this paper two kinds of three-dimensional fractionalLotka-Volterra systems have been studied The main resultsare divided into two parts On the one hand for system(1) the asymptotic stability of the equilibria is investigatedby providing simple and reasonable sufficient conditionsAnd simulation results prove to be quite consistent with thetheoretical findings On the other hand for system (19) theconditions which could lead to bifurcation phenomena areobtained Specifically the fractional order 119902 isin (0 1) is chosenas the bifurcation parameter and the expression of the criticalvalue 119902

lowast is precisely derived Furthermore the numericalresult is presented to illustrate that Hopf rsquos bifurcation cantake place

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Nature ScienceFoundation of China under Grant no 11371049 and theScience Foundation of Beijing Jiaotong University underGrant 2011JBM130

References

[1] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley-IntersciencePublication New York NY USA 1993

[2] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[3] S Ahmad and A C Lazer ldquoAverage conditions for globalasymptotic stability in a nonautonomous Lotka-Volterra sys-temrdquoNonlinear AnalysisTheory Methods and Applications vol40 no 1 pp 37ndash49 2000

[4] S Ahmad and A C Lazer ldquoAverage growth and total per-manence in a competitive Lotka-Volterra Systemrdquo Annali diMatematica Pura ed Applicata vol 185 supplement 5 pp S47ndashS67 2006

[5] P van den Driessche and M L Zeeman ldquoThree-dimensionalcompetitive Lotka-Volterra systems with no periodic orbitsrdquoSIAM Journal on Applied Mathematics vol 58 no 1 pp 227ndash234 1998

[6] Z Teng and L Chen ldquoGlobal asymptotic stability of periodicLotka-Volterra systems with delaysrdquoNonlinear AnalysisTheoryMethods and Applications vol 45 no 8 pp 1081ndash1095 2001

[7] N Fang and X X Chen ldquoPermanence of a discrete multispeciesLotka-Volterra competition predator-prey system with delaysrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2185ndash2195 2008

[8] G Lu and Z Lu ldquoPermanence for two-species Lotka-Volterracooperative systems with delaysrdquoMathematical Biosciences andEngineering vol 5 no 3 pp 477ndash484 2008

[9] X P Yan and W T Li ldquoStability and Hopf bifurcation for adelayed cooperative systemwith diffusion effectsrdquo InternationalJournal of Bifurcation and Chaos vol 18 no 2 pp 441ndash4532008

[10] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[11] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[12] H A El-Saka E Ahmed M I Shehata and A M A El-SayedldquoOn stability persistence and Hopf bifurcation in fractionalorder dynamical systemsrdquo Nonlinear Dynamics vol 56 no 1-2 pp 121ndash126 2009

[13] E Kaslik and S Sivasundaram ldquoNonlinear dynamics and chaosin fractional-order neural networksrdquo Neural Networks vol 32pp 245ndash256 2012

[14] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[15] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe Computational Engineering in Systems Applications vol 2pp 963ndash968 Lille France July 2012

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

are explored based on numerical simulations in [12] Thecritical values of the fractional order are identified for whichHopf rsquos bifurcation may occur based on the stability analysisin [13]Thus it is significant to study the dynamical behaviorsin the fractional population systems

To the best of our knowledge some papers have con-centrated on the dynamic investigation of the fractionalpopulation systems [10 11] However there are few resultson bifurcation phenomena of the fractional population sys-tems Therefore in the paper we mainly consider stabilityand bifurcation in the three-dimensional fractional Lotka-Volterra systems

Motivated by the above discussions some dynamicalproperties of two kinds of three-dimensional fractionalLotka-Volterra systems are investigated in this paper Exis-tence and uniqueness of solutions are considered Somesufficient conditions are provided for the asymptotic stabilityof equilibria Specifically bifurcation behaviors are analyzedby formulating the critical values of the fractional order atwhich Hopf rsquos bifurcations may take place

The rest of this paper is organized as follows In Section 2a three-dimensional fractional Lotka-Volterra predator-preysystem with interspecific competition is introduced And theasymptotic stability of the system is studied In Section 3a three-dimensional fractional Lotka-Volterra predator-preysystem is provided and bifurcation properties are investi-gated The numerical results in Section 4 are given to verifythe theoretical findings Finally the paper is concluded inSection 5

2 Stability Analysis of a Three-DimensionalFractional Lotka-Volterra Predator-PreySystem with Interspecific Competition

Consider a three-dimensional fractional Lotka-Volterra sys-tem

1198631199021199091 (119905) = 1199091 (119905) (1198871 minus 119886111199091 (119905) minus 119886121199092 (119905) minus 119886131199093 (119905))

1198631199021199092 (119905) = 1199092 (119905) (minus1198872 + 119886211199091 (119905) minus 119886221199092 (119905) minus 119886231199093 (119905))

1198631199021199093 (119905) = 1199093 (119905) (minus1198873 + 119886311199091 (119905) minus 119886321199092 (119905) minus 119886331199093 (119905))

(1)

with the initial values 119909119894(119905)|119905=0 = 119909119894(0) 119894 = 1 2 3 where0 lt 119902 le 1 especially when 119902 = 1 the system (1) is a classicalinteger-order system All constant coefficients 119886119894119895 and 119887119894 (119894 119895 =1 2 3) can be arbitrary positive real numbers 1199091(119905) ge 0

represents the density of prey species at time 119905 and 1199092(119905) ge 01199093(119905) ge 0 represent the densities of predator species at time 119905In this case system (1) can be regarded as a fractional Lotka-Volterra predator-prey systemwith interspecific competition

In the following existence and uniqueness of solutionsfor system (1) are given In addition the important resultsrelated to the stability of the fractional systems are presentedto provide the theoretical bases for the further study

Here the fractional Lotka-Volterra system (1) can berewritten in the form

119863119902119883 (119905) = 119860119883 (119905) + 1199091 (119905) 1198611119883 (119905)

+ 1199092 (119905) 1198612119883(119905) + 1199093 (119905) 1198613119883 (119905)

119883 (0) = 1198830

(2)

where 0 lt 119902 le 1 119905 isin (0 119879] and

119883 (119905) = (

1199091 (119905)

1199092 (119905)

1199093 (119905)

) 1198830 = (

1199091 (0)

1199092 (0)

1199093 (0)

)

119860 = (

1198871 0 0

0 minus1198872 0

0 0 minus1198873

)

1198611 = (

minus11988611 minus11988612 minus11988613

0 0 0

0 0 0

) 1198612 = (

0 0 0

11988621 minus11988622 minus11988623

0 0 0

)

1198613 = (

0 0 0

0 0 0

11988631 minus11988632 minus11988633

)

(3)

Definition 1 For 119883(119905) = (1199091(119905) 1199092(119905) 1199093(119905))119879 let 119862lowast[0 119879] be

the set of continuous column vectors 119883(119905) on the interval[0 119879] The norm of 119883(119905) isin 119862

lowast[0 119879] is given by 119883(119905) =

sum3

119894=1sup119905|119909119894(119905)|

Theorem 2 System (2) has a unique solution if 119883(119905) isin

119862lowast[0 119879]

Proof Let 119865(119883(119905)) = 119860119883(119905) + 1199091(119905)1198611119883(119905) + 1199092(119905)1198612119883(119905) +

1199093(119905)1198613119883(119905) then 119883(119905) isin 119862lowast[0 119879] implies 119865(119883(119905)) isin

119862lowast[0 119879] In addition take 119883(119905) 119884(119905) isin 119862

lowast[0 119879] and

119883(119905) = 119884(119905) the following inequality holds

119865 (119883 (119905)) minus 119865 (119884 (119905))

=1003817100381710038171003817119860 (119883 (119905) minus 119884 (119905)) + 1199091 (119905) 1198611119883 (119905)

minus 1199101 (119905) 1198611119884 (119905) + 1199092 (119905) 1198612119883 (119905)

minus 1199102 (119905) 1198612119884 (119905) + 1199093 (119905) 1198613119883 (119905)

minus1199103 (119905) 1198613119884 (119905)1003817100381710038171003817

=1003817100381710038171003817119860 (119883 (119905) minus 119884 (119905)) + 1199091 (119905) 1198611 (119883 (119905) minus 119884 (119905))

+ (1199091 (119905) minus 1199101 (119905)) 1198611119884 (119905) + 1199092 (119905) 1198612 (119883 (119905) minus 119884 (119905))

+ (1199092 (119905) minus 1199102 (119905)) 1198612119884 (119905) + 1199093 (119905) 1198613 (119883 (119905) minus 119884 (119905))

+ (1199093 (119905) minus 1199103 (119905)) 1198613119884 (119905)1003817100381710038171003817

le 119860 (119883 (119905) minus 119884 (119905)) +10038171003817100381710038171199091 (119905) 1198611 (119883 (119905) minus 119884 (119905))

1003817100381710038171003817

+1003817100381710038171003817(1199091 (119905) minus 1199101 (119905)) 1198611119884 (119905)

1003817100381710038171003817 +10038171003817100381710038171199092 (119905) 1198612 (119883 (119905) minus 119884 (119905))

1003817100381710038171003817

+1003817100381710038171003817(1199092 (119905) minus 1199102 (119905)) 1198612119884 (119905)

1003817100381710038171003817 +10038171003817100381710038171199093 (119905) 1198613 (119883 (119905) minus 119884 (119905))

1003817100381710038171003817

Mathematical Problems in Engineering 3

+1003817100381710038171003817(1199093 (119905) minus 1199103 (119905)) 1198613119884 (119905)

1003817100381710038171003817

le (119860 +10038171003817100381710038171198611

1003817100381710038171003817 (10038161003816100381610038161199091 (119905)

1003816100381610038161003816 + 119884 (119905))

+10038171003817100381710038171198612

1003817100381710038171003817 (10038161003816100381610038161199092 (119905)

1003816100381610038161003816 + 119884 (119905))

+10038171003817100381710038171198613

1003817100381710038171003817 (10038161003816100381610038161199093 (119905)

1003816100381610038161003816 + 119884 (119905))) 119883 (119905) minus 119884 (119905)

le 119871 119883 (119905) minus 119884 (119905)

(4)

where 119871 = 119860 + (1198611 + 1198612 + 1198613)(1198721 +1198722) gt 0 and1198721and1198722 are positive and satisfy 119883(119905) le 1198721 119884(119905) le 1198722 asa result of 119883(119905) 119884(119905) isin 119862

lowast[0 119879] Based on Theorems 21 and

22 in [14] system (2) has a unique solution

Theorem 3 (see [15]) The linear autonomous system 119863119902119909 =

119860119909 is asymptotically stable if and only if

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

119902120587

2 (5)

where119860 isin 119877119899times119899 119902 isin (0 1) and 120582 isin 120590(119860) 120590(119860) denotes the set

of all eigenvalues of the matrix 119860

Theorem 4 Let 119909lowast be an equilibrium of the nonlinear system(1) then the equilibrium 119909

lowast is locally asymptotically stable if

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

119902120587

2 (6)

where 120582 isin 120590(119869(119909lowast)) 120590(119869(119909lowast)) denotes the set of all eigenvalues

of the Jacobian matrix 119869(119909lowast)

Proof The proof follows fromTheorem 3 and [11]

In the following the stability of system (1) is investigatedby giving some appropriate conditions The asymptotic sta-bility of the equilibria is demonstrated based on Theorem 4Through simple calculation the equilibria of system (1) areobtained and denoted as

1198751 = (0 0 0) 1198752 = (0 0 minus1198873

11988633

)

1198753 = (0 minus1198872

11988622

0) 1198754 = (1198871

11988611

0 0)

1198755 = (011988852

1198885

11988853

1198885

) 1198756 = (11988861

1198886

011988863

1198886

)

1198757 = (11988871

1198887

11988872

1198887

0) 1198758 = (11988881

1198888

11988882

1198888

11988883

1198888

)

(7)

where 11988852 = 119887311988623 minus 119887211988633 11988853 = 119887211988632 minus 119887311988622 1198885 = 1198862211988633 minus

1198862311988632 11988861 = 119887111988633 + 119887311988613 11988863 = 119887111988631 minus 119887311988611 1198886 =

1198861111988633 + 1198861311988631 11988871 = 119887111988622 + 119887211988612 11988872 = 119887111988621 minus 119887211988611 1198887 =

1198861111988622+1198861211988621 11988881 = 11988711198862211988633minus11988711198862311988632+11988721198861211988633minus11988721198861311988632+

11988731198861311988622 minus 11988731198861211988623 11988882 = 11988711198862111988633 minus 11988711198862311988631 minus 11988721198861111988633 minus

11988721198861311988631 + 11988731198861111988623 + 11988731198861311988621 11988883 = 11988711198862211988631 minus 11988711198862111988632 +

11988721198861111988632 + 11988721198861211988631 minus 11988731198861111988622 minus 11988731198861211988621 and 1198888 = 119886111198862211988633 minus

119886111198862311988632 + 119886121198862111988633 minus 119886121198862311988631 + 119886131198862211988631 minus 119886131198862111988632

Because of the fact that all constant coefficients of system(1) are positive 1198752 1198753 and 1198755 are in contradiction with theactual situation hence the asymptotical stability of other fiveequilibria will be studied in detail

Theorem 5 For the three-dimensional fractional Lotka-Volterra system (1) the following results can be obtained

(a) 1198751 is unstable(b) 1198754 is locally asymptotically stable if 119887111988611 lt 119887211988621

119887111988611 lt 119887311988631(c) 1198756 is locally asymptotically stable if 119887311988631 lt 119887111988611 lt

119887211988621(d) 1198757 is locally asymptotically stable if 119887211988621 lt 119887111988611 lt

119887311988631(e) 1198758 is locally asymptotically stable if 1198863211988622 lt 1198863111988621 lt

1198863311988623

Proof For 1198751 = (0 0 0) its Jacobian matrix is

119869 (1198751) = (

1198871 0 0

0 minus1198872 0

0 0 minus1198873

) (8)

and the eigenvalues of 119869(1198751) satisfy 1205821 = 1198871 gt 0 1205822 = minus1198872 lt

0 and 1205823 = minus1198873 lt 0 hence the equilibrium 1198751 is unstableFor 1198754 its Jacobian matrix is

119869 (1198754) =((

(

minus1198871 minus119887111988612

11988611

minus119887111988613

11988611

0119887111988621 minus 119887211988611

11988611

0

0 0119887111988631 minus 119887311988611

11988611

))

)

(9)

and the eigenvalues of 119869(1198754) satisfy 1205821 = minus1198871 lt 0 1205822 =

(119887111988621 minus 119887211988611)11988611 lt 0 and 1205823 = (119887111988631 minus 119887311988611)11988611 lt 0hence the equilibrium 1198754 is locally asymptotically stable

For 1198756 use the notations below

119869 (1198756) = (

11986011 11986012 11986013

0 11986022 0

11986031 11986032 11986033

) (10)

and its characteristic equation is

(120582 minus 11986022) (1205822minus (11986011 + 11986033) 120582 + 1198601111986033 minus 1198601311986031) = 0

(11)

Based on the condition from (c) the following formulas canbe easily got

11986011 =11988811

1198880

lt 0 11986013 =11988813

1198880

lt 0

11986031 =11988831

1198880

gt 0 11986022 =11988822

1198880

lt 0

11986033 =11988833

1198880

lt 0

(12)

4 Mathematical Problems in Engineering

where 11988811 = minus11988711198861111988633 minus 11988731198861111988613 11988813 = minus11988711198861311988633 minus

11988731198861311988613 11988831 = 11988711198863111988631 minus 11988731198861111988631 11988822 = 11988633(119887111988621 minus 119887211988611) +

11988623(119887311988611minus119887111988631)+11988613(119887311988621minus119887211988631) 11988833 = minus11988711198863111988633+11988731198861111988633and 1198880 = a1111988633 + 1198861311988631 Then the following results can beobtained

1205821 = 11986022 lt 0 1205822 + 1205823 = 11986011 + 11986033 lt 0

12058221205823 = 1198601111986033 minus 1198601311986031 gt 0

(13)

Hence the equilibrium 1198756 is locally asymptotically stableSimilarly it can be readily derived that the equilibrium 1198757

is locally asymptotically stableFor 1198758 let 1198758 = (119909

lowast

1 119909lowast

2 119909lowast

3) the Jacobian matrix of 1198758 can

be written as

119869 (1198758) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

2minus11988623119909

lowast

2

11988631119909lowast

3minus11988632119909

lowast

3minus11988633119909

lowast

3

) = (

11986111 11986112 11986113

11986121 11986122 11986123

B31 11986132 11986133

)

(14)

and its characteristic equation is

1205823+ 1198621120582

2+ 1198622120582 + 1198623 = 0 (15)

where 1198621 = minus(11986111 + 11986122 + 11986133) 1198622 = 1198611111986122 + 1198611111986133 +

1198612211986133 minus 1198612311986132 minus 1198611211986121 minus 1198611311986131 and 1198623 = minus119861111198612211986133 +

119861111198612311986132 + 119861121198612111986133 + 119861131198612211986131 minus 119861121198612311986131 minus 119861131198612111986132For simplicity the equivalent characteristic equation is intro-duced as follows

1205823+ 11988611205822+ 1198862120582 + 1198863 = (120582 minus 119886) (120582

2minus 119887120582 + 119888) = 0 (16)

On the basis of the above equivalent substitutions thefollowing inequalities can be gained

1198861 = 11988611119909lowast

1+ 11988622119909

lowast

2+ 11988633119909

lowast

3gt 0

1198862 = (1198861111988622 + 1198861211988621) 119909lowast

1119909lowast

2+ (1198861111988633 + 1198861311988631) 119909

lowast

1119909lowast

3

+ (1198862211988633 minus 1198862311988632) 119909lowast

2119909lowast

3gt 0

1198863 = (11988611 (1198862211988633 minus 1198862311988632) + 11988612 (1198862111988633 minus 1198862311988631)

+11988613 (1198862211988631 minus 1198862111988632)) 119909lowast

1119909lowast

2119909lowast

3gt 0

11988611198862 minus 1198863 = 1198891119909lowast

1119909lowast

2119909lowast

3+ 1198892119909

lowast

1119909lowast

2

2+ 1198893119909

lowast

1119909lowast

3

2

+ 1198894119909lowast

1

2119909lowast

2+ 1198895119909

lowast

1

2119909lowast

3

+ 1198896 (11988622119909lowast

2

2119909lowast

3+ 11988633119909

lowast

2119909lowast

3

2) gt 0

119886 + 119887 = minus1198861 lt 0

119886119887 + 119888 = 1198862 gt 0

119886119888 = minus1198863 lt 0

minus1198862119887 minus 119886119887

2minus 119887119888 = 11988611198862 minus 1198863 gt 0

(17)

where 1198891 = 2119886111198862211988633 + 119886121198862311988631 + 119886131198862111988632 1198892 =

119886111198862

22+ 119886121198862111988622 1198893 = 11988611119886

2

33+ 119886131198863111988633 1198894 = 119886

2

1111988622 +

119886111198861211988621 1198895 = 1198862

1111988633 + 119886111198861311988631 and 1198896 = 1198862211988633 minus 1198862311988632

Using the proof by contradiction it can be concluded that theeigenvalues of 119869(1198758) satisfy

1205821 = 119886 lt 0

1205822 + 1205823 = 119887 lt 0

12058221205823 = 119888 gt 0

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt

119902120587

2 119894 = 2 3

(18)

Hence the equilibrium1198758 is locally asymptotically stable

For the further dynamic investigation of the fractionalpopulation systems the other fractional Lotka-Volterra sys-tems will be considered in the following section Particularlybifurcation properties for the system will be studied in detail

3 Bifurcation Analysis ofa Three-Dimensional FractionalLotka-Volterra Predator-Prey System

Consider a three-dimensional fractional Lotka-Volterra sys-tem

1198631199021199091 (119905) = 1199091 (119905) (1198871 minus 119886111199091 (119905) minus 119886121199092 (119905) minus 119886131199093 (119905))

1198631199021199092 (119905) = 1199092 (119905) (minus1198872 + 119886211199091 (119905) minus 119886221199092 (119905))

1198631199021199093 (119905) = 1199093 (119905) (minus1198873 + 119886311199091 (119905) minus 119886331199093 (119905))

(19)

with the initial values 119909119894(119905)|119905=0 = 119909119894(0) 119894 = 1 2 3 where0 lt 119902 lt 1 11988611 lt 0 and the other constant coefficients arepositive 1199091(119905) ge 0 represents the density of prey species attime 119905 and 1199092(119905) ge 0 1199093(119905) ge 0 represent the densities ofpredator species at time 119905 In this case system (19) can beregarded as a fractional Lotka-Volterra predator-prey system

On the basis of Theorem 2 it is not difficult to prove thatsystem (19) has a unique solution in a similar way

It is clear that there are eight equilibria for system(19) Here we focus on the bifurcation investigation of theequilibrium 119909

lowast which can be called a positive equilibriumwhen some conditions are satisfied The equilibrium 119909

lowast isobtained as

119909lowast= (119909lowast

1 119909lowast

2 119909lowast

3) = (

11988911

11988911988922

11988911988933

119889) (20)

where 11988911 = 11988711198862211988633 + 11988721198861211988633 + 11988731198861311988622 11988922 = 11988711198862111988633 minus

1198872(1198861111988633 + 1198861311988631) + 11988731198861311988621 11988933 = 11988711198862211988631 + 11988721198861211988631 minus

1198873(1198861111988622 + 1198861211988621) and 119889 = 119886111198862211988633 + 119886121198862111988633 + 119886131198862211988631And its Jacobian matrix can be expressed as

119869 (119909lowast) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

20

11988631119909lowast

30 minus11988633119909

lowast

3

) (21)

Mathematical Problems in Engineering 5

Furthermore the eigenvalues of 119869(119909lowast) satisfy the characteris-tic equation

(120582 minus 120573) (1205822minus (120572 + 120573) 120582 + 120572120573 minus 120574) = 0 (22)

where 120572 = minus11988611119909lowast

1 120573 = minus11988622119909

lowast

2= minus11988633119909

lowast

3 and 120574 =

minus1198861211988621119909lowast

1119909lowast

2minus 1198861311988631119909

lowast

1119909lowast

3

In the following by choosing the fractional order 119902

as the bifurcation parameter and analyzing the associatedcharacteristic equation (22) of system (19) at the positiveequilibrium we investigate the bifurcation phenomena of thepositive equilibrium of system (19) and obtain the conditionsunder which system (19) undergoes a Hopf bifurcation

Proposition 6 The positive equilibrium 119909lowast of system (19) is

locally asymptotically stable if and only if all the followingconditions are satisfied

(i) 120573 lt 0(ii) 120572120573 minus 120574 gt 0 and(iii) 120572 + 120573 lt 2 cos(1199021205872)radic120572120573 minus 120574

Proof For the characteristic equation (22) the root 1205821 = 120573 lt

0 and 1205822 1205823 satisfy the equation 1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0

It is clear that | arg(12058223)| gt 1199021205872 if and only if the conditions(ii) and (iii) hold Based onTheorem 4 Proposition 6 provesto be true

In addition by analyzing the condition (iii) ofProposition 6 in detail the following results can be gained

Proposition 7 With respect to system (19) if 120573 lt 0 and 120572120573 minus120574 gt 0 the following statements can be obtained

(a) If 120572+120573 le 0 the equilibrium 119909lowast is locally asymptotically

stable for any 119902 isin (0 1)(b) If 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equilibrium 119909

lowast is locallyasymptotically stable if and only if 119902 isin (0 119902

lowast) where

119902lowast= (2120587) arccos ((120572 + 120573)2radic120572120573 minus 120574)

(c) If 120572+120573 ge 2radic120572120573 minus 120574 the equilibrium 119909lowast is unstable for

any 119902 isin (0 1)

Proof The conclusions (a) and (c) are obvious For thestatement (b) due to 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equation1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0 has two complex roots 1205822 1205823

and their real part is (120572 + 120573)2 gt 0 Then | arg(120582119894)| =

arccos((120572 + 120573)2radic120572120573 minus 120574) 119894 = 2 3 Besides according to thecondition arccos((120572 + 120573)2radic120572120573 minus 120574) = 119902

lowast1205872 119902 isin (0 119902

lowast) if

and only if | arg(120582119894)| gt 1199021205872 119894 = 2 3 Based onTheorem 4 itis concluded that Proposition 7 is true

Remark 8 It is apparent that the critical value satisfies 119902lowast isin(0 1) When 119902 isin (0 119902

lowast) 119909lowast is locally asymptotically stable

when 119902 isin (119902lowast 1) and specially 119902 = 1 119909lowast is unstable That is

to say it has verified that fractional differential equations areat least as stable as their integer-order counterparts [4]

Remark 9 Under the situation of statement (b) a bifurcationphenomenon must happen at the critical value 119902lowast However

it is difficult to confirm precise bifurcation type As an inter-esting bifurcation behavior Hopf rsquos bifurcation is expected totake place

According to Proposition 7 if some appropriate condi-tions about the constant coefficients of system (19) can befound so that statement (b) is satisfied system (19) willundergo a bifurcation phenomenon And the critical value119902lowast of the bifurcation parameter 119902 can be expressed by theconstant coefficients of system (19) From this the followingtheorem is specifically proposed

Theorem 10 With respect to system (19) if the followingconditions are satisfied

(i) 1198872 = 1198873 11988622 = 11988633 11988621 = 11988631 = minus11988611(ii) 11988612 + 11988613 minus 11988622 gt 0

then the positive equilibrium 119909lowast is locally asymptotically stable

if and only if 119902 isin (0 119902lowast) where

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(23)

Proof According to the condition (i) the equilibrium 119909lowast can

be expressed as

119909lowast= (

119887111988622 + 119887211988612 + 119887211988613

11988611 (11988622 minus 11988612 minus 11988613)

1198871 + 1198872

11988612 + 11988613 minus 11988622

1198871 + 1198872

11988612 + 11988613 minus 11988622

)

(24)

For (22) the following results can be obtained

120573 = minus11988622119909lowast

2lt 0 120572 + 120573 = 1198872 gt 0

120572120573 minus 120574 =1198901

11988612 + 11988613 minus 11988622

gt 0

4 (120572120573 minus 120574) minus (120572 + 120573)2=

41198901

11988612 + 11988613 minus 11988622

minus 1198872

2

=1198902

11988612 + 11988613 minus 11988622

gt 0

(25)

where 1198901 = (1198871 + 1198872)(119887111988622 + 119887211988612 + 119887211988613) and 1198902 = 11988622(41198872

1+

1198872

2)+31198872

2(11988612+11988613)+411988711198872(11988612+11988613+11988622) Obviously the above

conclusions satisfy statement (b) of Proposition 7 then it canbe derived that

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(26)

6 Mathematical Problems in Engineering

5 8 11 14 17 20

02

46

8100

5

10

x3

x2

x1

q = 1

q = 09

q = 08

Figure 1 The trajectory of system (1) converges to the equilibrium1198758= (537 157 17)

Hence the positive equilibrium 119909lowast of system (19) is locally

asymptotically stable if and only if 119902 isin (0 119902lowast)

According to the statement of Theorem 10 it can beconcluded that the positive equilibrium 119909

lowast is locally asymp-totically stable if and only if 119902 isin (0 119902

lowast) At 119902 = 119902

lowast the Hopfbifurcation is expected to take place As 119902 increases above thecritical value 119902lowast the positive equilibrium 119909

lowast is unstable anda limit cycle is expected to appear in the proximity of 119909lowast dueto the Hopf bifurcation phenomenon

The analysis of periodic solutions in fractional dynamicalsystems is a very recent and promising research topic Asa consequence the nonexistence of exact periodic solutionsin time invariant fractional systems is obtained [16] As anapplication it is emphasized that the limit cycle observed innumerical simulations of a simple fractional neural networkcannot be an exact periodic solution of the system [17]In addition there are some other papers providing thenumerical evidences of limit cycles

Remark 11 Even though exact periodic solutions do not existin autonomous fractional systems [16 17] limit cycles havebeen observed by numerical simulations in many systemssuch as a fractional neural system [13] a fractional Van derPol system [18] fractional Chua and Chenrsquos systems [19 20]and a fractional financial system [21]

4 Numerical Simulation

In this paper an Adams-type predictor-corrector methodis used for the numerical solutions of fractional differentialequations This method has been introduced in [22 23]and further investigated in [24ndash27] In order to verify thetheoretical analysis the following numerical results are given

For system (1) the approximate solutions are displayedin Figure 1 for the step size 0005 and different values of119902 119902 = 1 119902 = 09 119902 = 08 respectively Taking 1198871 =

12 1198872 = 11988611 = 11988613 = 11988621 = 11988623 = 11988631 = 1 11988612 =

11988632 = 11988633 = 2 and 1198873 = 11988622 = 3 and choosing theinitial values 1199091(0) = 20 1199092(0) = 10 and 1199093(0) = 10 the

55 60 65 701

15

2

25

3

35

4

45

5

x

t

1

(a)

5515

2

25

3

35

4

45

5

55

6

60 65 70

x

t

2

(b)

55 60 65 7015

2

25

3

35

4

45

5

55

6

x

t

3

(c)

Figure 2 The solution of system (19) versus time with 119902 = 084

equilibrium 1198758 is (537 157 17) Then Figure 1 shows thatthe equilibrium 1198758 is locally asymptotically stable Namelythe fifth conclusion of Theorem 5 is verified Similarly theother conclusions of Theorem 5 can be confirmed

For system (19) the approximate solutions are displayedin Figures 2 3 and 4 for the step size 0001 and different valuesof 119902 119902 = 082 and 119902 = 084 Taking 1198871 = 11988612 = 11988613 = 11988622 =

11988633 = 1 1198872 = 1198873 = 11988621 = 11988631 = 2 and 11988611 = minus2 andchoosing the initial values 1199091(0) = 3 1199092(0) = 1199093(0) = 4 thepositive equilibrium is 119909lowast = (25 3 3) and the critical value

Mathematical Problems in Engineering 7

15 2 25 3 35

225

335

42

3

4

x3

x2

x1

Figure 3 When 119902 = 082 the trajectory of system (19) converges tothe equilibrium 119909

lowast= (25 3 3)

12

34

5

0

2

4

60510

x3

x2

x 1

Figure 4 When 119902 = 084 the trajectory of system (19) converges toan asymptotically stable limit cycle

is 119902lowast = 08337 Indeed Figures 2ndash4 present the fact that thepositive equilibrium 119909

lowast is locally asymptotically stable when119902 = 082 isin (0 08337) and when 119902 = 084 increases across119902lowast= 08337 an asymptotically stable limit cycle appears in a

neighborhood of the positive equilibrium 119909lowast

5 Conclusion

In this paper two kinds of three-dimensional fractionalLotka-Volterra systems have been studied The main resultsare divided into two parts On the one hand for system(1) the asymptotic stability of the equilibria is investigatedby providing simple and reasonable sufficient conditionsAnd simulation results prove to be quite consistent with thetheoretical findings On the other hand for system (19) theconditions which could lead to bifurcation phenomena areobtained Specifically the fractional order 119902 isin (0 1) is chosenas the bifurcation parameter and the expression of the criticalvalue 119902

lowast is precisely derived Furthermore the numericalresult is presented to illustrate that Hopf rsquos bifurcation cantake place

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Nature ScienceFoundation of China under Grant no 11371049 and theScience Foundation of Beijing Jiaotong University underGrant 2011JBM130

References

[1] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley-IntersciencePublication New York NY USA 1993

[2] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[3] S Ahmad and A C Lazer ldquoAverage conditions for globalasymptotic stability in a nonautonomous Lotka-Volterra sys-temrdquoNonlinear AnalysisTheory Methods and Applications vol40 no 1 pp 37ndash49 2000

[4] S Ahmad and A C Lazer ldquoAverage growth and total per-manence in a competitive Lotka-Volterra Systemrdquo Annali diMatematica Pura ed Applicata vol 185 supplement 5 pp S47ndashS67 2006

[5] P van den Driessche and M L Zeeman ldquoThree-dimensionalcompetitive Lotka-Volterra systems with no periodic orbitsrdquoSIAM Journal on Applied Mathematics vol 58 no 1 pp 227ndash234 1998

[6] Z Teng and L Chen ldquoGlobal asymptotic stability of periodicLotka-Volterra systems with delaysrdquoNonlinear AnalysisTheoryMethods and Applications vol 45 no 8 pp 1081ndash1095 2001

[7] N Fang and X X Chen ldquoPermanence of a discrete multispeciesLotka-Volterra competition predator-prey system with delaysrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2185ndash2195 2008

[8] G Lu and Z Lu ldquoPermanence for two-species Lotka-Volterracooperative systems with delaysrdquoMathematical Biosciences andEngineering vol 5 no 3 pp 477ndash484 2008

[9] X P Yan and W T Li ldquoStability and Hopf bifurcation for adelayed cooperative systemwith diffusion effectsrdquo InternationalJournal of Bifurcation and Chaos vol 18 no 2 pp 441ndash4532008

[10] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[11] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[12] H A El-Saka E Ahmed M I Shehata and A M A El-SayedldquoOn stability persistence and Hopf bifurcation in fractionalorder dynamical systemsrdquo Nonlinear Dynamics vol 56 no 1-2 pp 121ndash126 2009

[13] E Kaslik and S Sivasundaram ldquoNonlinear dynamics and chaosin fractional-order neural networksrdquo Neural Networks vol 32pp 245ndash256 2012

[14] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[15] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe Computational Engineering in Systems Applications vol 2pp 963ndash968 Lille France July 2012

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

+1003817100381710038171003817(1199093 (119905) minus 1199103 (119905)) 1198613119884 (119905)

1003817100381710038171003817

le (119860 +10038171003817100381710038171198611

1003817100381710038171003817 (10038161003816100381610038161199091 (119905)

1003816100381610038161003816 + 119884 (119905))

+10038171003817100381710038171198612

1003817100381710038171003817 (10038161003816100381610038161199092 (119905)

1003816100381610038161003816 + 119884 (119905))

+10038171003817100381710038171198613

1003817100381710038171003817 (10038161003816100381610038161199093 (119905)

1003816100381610038161003816 + 119884 (119905))) 119883 (119905) minus 119884 (119905)

le 119871 119883 (119905) minus 119884 (119905)

(4)

where 119871 = 119860 + (1198611 + 1198612 + 1198613)(1198721 +1198722) gt 0 and1198721and1198722 are positive and satisfy 119883(119905) le 1198721 119884(119905) le 1198722 asa result of 119883(119905) 119884(119905) isin 119862

lowast[0 119879] Based on Theorems 21 and

22 in [14] system (2) has a unique solution

Theorem 3 (see [15]) The linear autonomous system 119863119902119909 =

119860119909 is asymptotically stable if and only if

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

119902120587

2 (5)

where119860 isin 119877119899times119899 119902 isin (0 1) and 120582 isin 120590(119860) 120590(119860) denotes the set

of all eigenvalues of the matrix 119860

Theorem 4 Let 119909lowast be an equilibrium of the nonlinear system(1) then the equilibrium 119909

lowast is locally asymptotically stable if

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

119902120587

2 (6)

where 120582 isin 120590(119869(119909lowast)) 120590(119869(119909lowast)) denotes the set of all eigenvalues

of the Jacobian matrix 119869(119909lowast)

Proof The proof follows fromTheorem 3 and [11]

In the following the stability of system (1) is investigatedby giving some appropriate conditions The asymptotic sta-bility of the equilibria is demonstrated based on Theorem 4Through simple calculation the equilibria of system (1) areobtained and denoted as

1198751 = (0 0 0) 1198752 = (0 0 minus1198873

11988633

)

1198753 = (0 minus1198872

11988622

0) 1198754 = (1198871

11988611

0 0)

1198755 = (011988852

1198885

11988853

1198885

) 1198756 = (11988861

1198886

011988863

1198886

)

1198757 = (11988871

1198887

11988872

1198887

0) 1198758 = (11988881

1198888

11988882

1198888

11988883

1198888

)

(7)

where 11988852 = 119887311988623 minus 119887211988633 11988853 = 119887211988632 minus 119887311988622 1198885 = 1198862211988633 minus

1198862311988632 11988861 = 119887111988633 + 119887311988613 11988863 = 119887111988631 minus 119887311988611 1198886 =

1198861111988633 + 1198861311988631 11988871 = 119887111988622 + 119887211988612 11988872 = 119887111988621 minus 119887211988611 1198887 =

1198861111988622+1198861211988621 11988881 = 11988711198862211988633minus11988711198862311988632+11988721198861211988633minus11988721198861311988632+

11988731198861311988622 minus 11988731198861211988623 11988882 = 11988711198862111988633 minus 11988711198862311988631 minus 11988721198861111988633 minus

11988721198861311988631 + 11988731198861111988623 + 11988731198861311988621 11988883 = 11988711198862211988631 minus 11988711198862111988632 +

11988721198861111988632 + 11988721198861211988631 minus 11988731198861111988622 minus 11988731198861211988621 and 1198888 = 119886111198862211988633 minus

119886111198862311988632 + 119886121198862111988633 minus 119886121198862311988631 + 119886131198862211988631 minus 119886131198862111988632

Because of the fact that all constant coefficients of system(1) are positive 1198752 1198753 and 1198755 are in contradiction with theactual situation hence the asymptotical stability of other fiveequilibria will be studied in detail

Theorem 5 For the three-dimensional fractional Lotka-Volterra system (1) the following results can be obtained

(a) 1198751 is unstable(b) 1198754 is locally asymptotically stable if 119887111988611 lt 119887211988621

119887111988611 lt 119887311988631(c) 1198756 is locally asymptotically stable if 119887311988631 lt 119887111988611 lt

119887211988621(d) 1198757 is locally asymptotically stable if 119887211988621 lt 119887111988611 lt

119887311988631(e) 1198758 is locally asymptotically stable if 1198863211988622 lt 1198863111988621 lt

1198863311988623

Proof For 1198751 = (0 0 0) its Jacobian matrix is

119869 (1198751) = (

1198871 0 0

0 minus1198872 0

0 0 minus1198873

) (8)

and the eigenvalues of 119869(1198751) satisfy 1205821 = 1198871 gt 0 1205822 = minus1198872 lt

0 and 1205823 = minus1198873 lt 0 hence the equilibrium 1198751 is unstableFor 1198754 its Jacobian matrix is

119869 (1198754) =((

(

minus1198871 minus119887111988612

11988611

minus119887111988613

11988611

0119887111988621 minus 119887211988611

11988611

0

0 0119887111988631 minus 119887311988611

11988611

))

)

(9)

and the eigenvalues of 119869(1198754) satisfy 1205821 = minus1198871 lt 0 1205822 =

(119887111988621 minus 119887211988611)11988611 lt 0 and 1205823 = (119887111988631 minus 119887311988611)11988611 lt 0hence the equilibrium 1198754 is locally asymptotically stable

For 1198756 use the notations below

119869 (1198756) = (

11986011 11986012 11986013

0 11986022 0

11986031 11986032 11986033

) (10)

and its characteristic equation is

(120582 minus 11986022) (1205822minus (11986011 + 11986033) 120582 + 1198601111986033 minus 1198601311986031) = 0

(11)

Based on the condition from (c) the following formulas canbe easily got

11986011 =11988811

1198880

lt 0 11986013 =11988813

1198880

lt 0

11986031 =11988831

1198880

gt 0 11986022 =11988822

1198880

lt 0

11986033 =11988833

1198880

lt 0

(12)

4 Mathematical Problems in Engineering

where 11988811 = minus11988711198861111988633 minus 11988731198861111988613 11988813 = minus11988711198861311988633 minus

11988731198861311988613 11988831 = 11988711198863111988631 minus 11988731198861111988631 11988822 = 11988633(119887111988621 minus 119887211988611) +

11988623(119887311988611minus119887111988631)+11988613(119887311988621minus119887211988631) 11988833 = minus11988711198863111988633+11988731198861111988633and 1198880 = a1111988633 + 1198861311988631 Then the following results can beobtained

1205821 = 11986022 lt 0 1205822 + 1205823 = 11986011 + 11986033 lt 0

12058221205823 = 1198601111986033 minus 1198601311986031 gt 0

(13)

Hence the equilibrium 1198756 is locally asymptotically stableSimilarly it can be readily derived that the equilibrium 1198757

is locally asymptotically stableFor 1198758 let 1198758 = (119909

lowast

1 119909lowast

2 119909lowast

3) the Jacobian matrix of 1198758 can

be written as

119869 (1198758) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

2minus11988623119909

lowast

2

11988631119909lowast

3minus11988632119909

lowast

3minus11988633119909

lowast

3

) = (

11986111 11986112 11986113

11986121 11986122 11986123

B31 11986132 11986133

)

(14)

and its characteristic equation is

1205823+ 1198621120582

2+ 1198622120582 + 1198623 = 0 (15)

where 1198621 = minus(11986111 + 11986122 + 11986133) 1198622 = 1198611111986122 + 1198611111986133 +

1198612211986133 minus 1198612311986132 minus 1198611211986121 minus 1198611311986131 and 1198623 = minus119861111198612211986133 +

119861111198612311986132 + 119861121198612111986133 + 119861131198612211986131 minus 119861121198612311986131 minus 119861131198612111986132For simplicity the equivalent characteristic equation is intro-duced as follows

1205823+ 11988611205822+ 1198862120582 + 1198863 = (120582 minus 119886) (120582

2minus 119887120582 + 119888) = 0 (16)

On the basis of the above equivalent substitutions thefollowing inequalities can be gained

1198861 = 11988611119909lowast

1+ 11988622119909

lowast

2+ 11988633119909

lowast

3gt 0

1198862 = (1198861111988622 + 1198861211988621) 119909lowast

1119909lowast

2+ (1198861111988633 + 1198861311988631) 119909

lowast

1119909lowast

3

+ (1198862211988633 minus 1198862311988632) 119909lowast

2119909lowast

3gt 0

1198863 = (11988611 (1198862211988633 minus 1198862311988632) + 11988612 (1198862111988633 minus 1198862311988631)

+11988613 (1198862211988631 minus 1198862111988632)) 119909lowast

1119909lowast

2119909lowast

3gt 0

11988611198862 minus 1198863 = 1198891119909lowast

1119909lowast

2119909lowast

3+ 1198892119909

lowast

1119909lowast

2

2+ 1198893119909

lowast

1119909lowast

3

2

+ 1198894119909lowast

1

2119909lowast

2+ 1198895119909

lowast

1

2119909lowast

3

+ 1198896 (11988622119909lowast

2

2119909lowast

3+ 11988633119909

lowast

2119909lowast

3

2) gt 0

119886 + 119887 = minus1198861 lt 0

119886119887 + 119888 = 1198862 gt 0

119886119888 = minus1198863 lt 0

minus1198862119887 minus 119886119887

2minus 119887119888 = 11988611198862 minus 1198863 gt 0

(17)

where 1198891 = 2119886111198862211988633 + 119886121198862311988631 + 119886131198862111988632 1198892 =

119886111198862

22+ 119886121198862111988622 1198893 = 11988611119886

2

33+ 119886131198863111988633 1198894 = 119886

2

1111988622 +

119886111198861211988621 1198895 = 1198862

1111988633 + 119886111198861311988631 and 1198896 = 1198862211988633 minus 1198862311988632

Using the proof by contradiction it can be concluded that theeigenvalues of 119869(1198758) satisfy

1205821 = 119886 lt 0

1205822 + 1205823 = 119887 lt 0

12058221205823 = 119888 gt 0

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt

119902120587

2 119894 = 2 3

(18)

Hence the equilibrium1198758 is locally asymptotically stable

For the further dynamic investigation of the fractionalpopulation systems the other fractional Lotka-Volterra sys-tems will be considered in the following section Particularlybifurcation properties for the system will be studied in detail

3 Bifurcation Analysis ofa Three-Dimensional FractionalLotka-Volterra Predator-Prey System

Consider a three-dimensional fractional Lotka-Volterra sys-tem

1198631199021199091 (119905) = 1199091 (119905) (1198871 minus 119886111199091 (119905) minus 119886121199092 (119905) minus 119886131199093 (119905))

1198631199021199092 (119905) = 1199092 (119905) (minus1198872 + 119886211199091 (119905) minus 119886221199092 (119905))

1198631199021199093 (119905) = 1199093 (119905) (minus1198873 + 119886311199091 (119905) minus 119886331199093 (119905))

(19)

with the initial values 119909119894(119905)|119905=0 = 119909119894(0) 119894 = 1 2 3 where0 lt 119902 lt 1 11988611 lt 0 and the other constant coefficients arepositive 1199091(119905) ge 0 represents the density of prey species attime 119905 and 1199092(119905) ge 0 1199093(119905) ge 0 represent the densities ofpredator species at time 119905 In this case system (19) can beregarded as a fractional Lotka-Volterra predator-prey system

On the basis of Theorem 2 it is not difficult to prove thatsystem (19) has a unique solution in a similar way

It is clear that there are eight equilibria for system(19) Here we focus on the bifurcation investigation of theequilibrium 119909

lowast which can be called a positive equilibriumwhen some conditions are satisfied The equilibrium 119909

lowast isobtained as

119909lowast= (119909lowast

1 119909lowast

2 119909lowast

3) = (

11988911

11988911988922

11988911988933

119889) (20)

where 11988911 = 11988711198862211988633 + 11988721198861211988633 + 11988731198861311988622 11988922 = 11988711198862111988633 minus

1198872(1198861111988633 + 1198861311988631) + 11988731198861311988621 11988933 = 11988711198862211988631 + 11988721198861211988631 minus

1198873(1198861111988622 + 1198861211988621) and 119889 = 119886111198862211988633 + 119886121198862111988633 + 119886131198862211988631And its Jacobian matrix can be expressed as

119869 (119909lowast) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

20

11988631119909lowast

30 minus11988633119909

lowast

3

) (21)

Mathematical Problems in Engineering 5

Furthermore the eigenvalues of 119869(119909lowast) satisfy the characteris-tic equation

(120582 minus 120573) (1205822minus (120572 + 120573) 120582 + 120572120573 minus 120574) = 0 (22)

where 120572 = minus11988611119909lowast

1 120573 = minus11988622119909

lowast

2= minus11988633119909

lowast

3 and 120574 =

minus1198861211988621119909lowast

1119909lowast

2minus 1198861311988631119909

lowast

1119909lowast

3

In the following by choosing the fractional order 119902

as the bifurcation parameter and analyzing the associatedcharacteristic equation (22) of system (19) at the positiveequilibrium we investigate the bifurcation phenomena of thepositive equilibrium of system (19) and obtain the conditionsunder which system (19) undergoes a Hopf bifurcation

Proposition 6 The positive equilibrium 119909lowast of system (19) is

locally asymptotically stable if and only if all the followingconditions are satisfied

(i) 120573 lt 0(ii) 120572120573 minus 120574 gt 0 and(iii) 120572 + 120573 lt 2 cos(1199021205872)radic120572120573 minus 120574

Proof For the characteristic equation (22) the root 1205821 = 120573 lt

0 and 1205822 1205823 satisfy the equation 1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0

It is clear that | arg(12058223)| gt 1199021205872 if and only if the conditions(ii) and (iii) hold Based onTheorem 4 Proposition 6 provesto be true

In addition by analyzing the condition (iii) ofProposition 6 in detail the following results can be gained

Proposition 7 With respect to system (19) if 120573 lt 0 and 120572120573 minus120574 gt 0 the following statements can be obtained

(a) If 120572+120573 le 0 the equilibrium 119909lowast is locally asymptotically

stable for any 119902 isin (0 1)(b) If 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equilibrium 119909

lowast is locallyasymptotically stable if and only if 119902 isin (0 119902

lowast) where

119902lowast= (2120587) arccos ((120572 + 120573)2radic120572120573 minus 120574)

(c) If 120572+120573 ge 2radic120572120573 minus 120574 the equilibrium 119909lowast is unstable for

any 119902 isin (0 1)

Proof The conclusions (a) and (c) are obvious For thestatement (b) due to 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equation1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0 has two complex roots 1205822 1205823

and their real part is (120572 + 120573)2 gt 0 Then | arg(120582119894)| =

arccos((120572 + 120573)2radic120572120573 minus 120574) 119894 = 2 3 Besides according to thecondition arccos((120572 + 120573)2radic120572120573 minus 120574) = 119902

lowast1205872 119902 isin (0 119902

lowast) if

and only if | arg(120582119894)| gt 1199021205872 119894 = 2 3 Based onTheorem 4 itis concluded that Proposition 7 is true

Remark 8 It is apparent that the critical value satisfies 119902lowast isin(0 1) When 119902 isin (0 119902

lowast) 119909lowast is locally asymptotically stable

when 119902 isin (119902lowast 1) and specially 119902 = 1 119909lowast is unstable That is

to say it has verified that fractional differential equations areat least as stable as their integer-order counterparts [4]

Remark 9 Under the situation of statement (b) a bifurcationphenomenon must happen at the critical value 119902lowast However

it is difficult to confirm precise bifurcation type As an inter-esting bifurcation behavior Hopf rsquos bifurcation is expected totake place

According to Proposition 7 if some appropriate condi-tions about the constant coefficients of system (19) can befound so that statement (b) is satisfied system (19) willundergo a bifurcation phenomenon And the critical value119902lowast of the bifurcation parameter 119902 can be expressed by theconstant coefficients of system (19) From this the followingtheorem is specifically proposed

Theorem 10 With respect to system (19) if the followingconditions are satisfied

(i) 1198872 = 1198873 11988622 = 11988633 11988621 = 11988631 = minus11988611(ii) 11988612 + 11988613 minus 11988622 gt 0

then the positive equilibrium 119909lowast is locally asymptotically stable

if and only if 119902 isin (0 119902lowast) where

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(23)

Proof According to the condition (i) the equilibrium 119909lowast can

be expressed as

119909lowast= (

119887111988622 + 119887211988612 + 119887211988613

11988611 (11988622 minus 11988612 minus 11988613)

1198871 + 1198872

11988612 + 11988613 minus 11988622

1198871 + 1198872

11988612 + 11988613 minus 11988622

)

(24)

For (22) the following results can be obtained

120573 = minus11988622119909lowast

2lt 0 120572 + 120573 = 1198872 gt 0

120572120573 minus 120574 =1198901

11988612 + 11988613 minus 11988622

gt 0

4 (120572120573 minus 120574) minus (120572 + 120573)2=

41198901

11988612 + 11988613 minus 11988622

minus 1198872

2

=1198902

11988612 + 11988613 minus 11988622

gt 0

(25)

where 1198901 = (1198871 + 1198872)(119887111988622 + 119887211988612 + 119887211988613) and 1198902 = 11988622(41198872

1+

1198872

2)+31198872

2(11988612+11988613)+411988711198872(11988612+11988613+11988622) Obviously the above

conclusions satisfy statement (b) of Proposition 7 then it canbe derived that

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(26)

6 Mathematical Problems in Engineering

5 8 11 14 17 20

02

46

8100

5

10

x3

x2

x1

q = 1

q = 09

q = 08

Figure 1 The trajectory of system (1) converges to the equilibrium1198758= (537 157 17)

Hence the positive equilibrium 119909lowast of system (19) is locally

asymptotically stable if and only if 119902 isin (0 119902lowast)

According to the statement of Theorem 10 it can beconcluded that the positive equilibrium 119909

lowast is locally asymp-totically stable if and only if 119902 isin (0 119902

lowast) At 119902 = 119902

lowast the Hopfbifurcation is expected to take place As 119902 increases above thecritical value 119902lowast the positive equilibrium 119909

lowast is unstable anda limit cycle is expected to appear in the proximity of 119909lowast dueto the Hopf bifurcation phenomenon

The analysis of periodic solutions in fractional dynamicalsystems is a very recent and promising research topic Asa consequence the nonexistence of exact periodic solutionsin time invariant fractional systems is obtained [16] As anapplication it is emphasized that the limit cycle observed innumerical simulations of a simple fractional neural networkcannot be an exact periodic solution of the system [17]In addition there are some other papers providing thenumerical evidences of limit cycles

Remark 11 Even though exact periodic solutions do not existin autonomous fractional systems [16 17] limit cycles havebeen observed by numerical simulations in many systemssuch as a fractional neural system [13] a fractional Van derPol system [18] fractional Chua and Chenrsquos systems [19 20]and a fractional financial system [21]

4 Numerical Simulation

In this paper an Adams-type predictor-corrector methodis used for the numerical solutions of fractional differentialequations This method has been introduced in [22 23]and further investigated in [24ndash27] In order to verify thetheoretical analysis the following numerical results are given

For system (1) the approximate solutions are displayedin Figure 1 for the step size 0005 and different values of119902 119902 = 1 119902 = 09 119902 = 08 respectively Taking 1198871 =

12 1198872 = 11988611 = 11988613 = 11988621 = 11988623 = 11988631 = 1 11988612 =

11988632 = 11988633 = 2 and 1198873 = 11988622 = 3 and choosing theinitial values 1199091(0) = 20 1199092(0) = 10 and 1199093(0) = 10 the

55 60 65 701

15

2

25

3

35

4

45

5

x

t

1

(a)

5515

2

25

3

35

4

45

5

55

6

60 65 70

x

t

2

(b)

55 60 65 7015

2

25

3

35

4

45

5

55

6

x

t

3

(c)

Figure 2 The solution of system (19) versus time with 119902 = 084

equilibrium 1198758 is (537 157 17) Then Figure 1 shows thatthe equilibrium 1198758 is locally asymptotically stable Namelythe fifth conclusion of Theorem 5 is verified Similarly theother conclusions of Theorem 5 can be confirmed

For system (19) the approximate solutions are displayedin Figures 2 3 and 4 for the step size 0001 and different valuesof 119902 119902 = 082 and 119902 = 084 Taking 1198871 = 11988612 = 11988613 = 11988622 =

11988633 = 1 1198872 = 1198873 = 11988621 = 11988631 = 2 and 11988611 = minus2 andchoosing the initial values 1199091(0) = 3 1199092(0) = 1199093(0) = 4 thepositive equilibrium is 119909lowast = (25 3 3) and the critical value

Mathematical Problems in Engineering 7

15 2 25 3 35

225

335

42

3

4

x3

x2

x1

Figure 3 When 119902 = 082 the trajectory of system (19) converges tothe equilibrium 119909

lowast= (25 3 3)

12

34

5

0

2

4

60510

x3

x2

x 1

Figure 4 When 119902 = 084 the trajectory of system (19) converges toan asymptotically stable limit cycle

is 119902lowast = 08337 Indeed Figures 2ndash4 present the fact that thepositive equilibrium 119909

lowast is locally asymptotically stable when119902 = 082 isin (0 08337) and when 119902 = 084 increases across119902lowast= 08337 an asymptotically stable limit cycle appears in a

neighborhood of the positive equilibrium 119909lowast

5 Conclusion

In this paper two kinds of three-dimensional fractionalLotka-Volterra systems have been studied The main resultsare divided into two parts On the one hand for system(1) the asymptotic stability of the equilibria is investigatedby providing simple and reasonable sufficient conditionsAnd simulation results prove to be quite consistent with thetheoretical findings On the other hand for system (19) theconditions which could lead to bifurcation phenomena areobtained Specifically the fractional order 119902 isin (0 1) is chosenas the bifurcation parameter and the expression of the criticalvalue 119902

lowast is precisely derived Furthermore the numericalresult is presented to illustrate that Hopf rsquos bifurcation cantake place

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Nature ScienceFoundation of China under Grant no 11371049 and theScience Foundation of Beijing Jiaotong University underGrant 2011JBM130

References

[1] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley-IntersciencePublication New York NY USA 1993

[2] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[3] S Ahmad and A C Lazer ldquoAverage conditions for globalasymptotic stability in a nonautonomous Lotka-Volterra sys-temrdquoNonlinear AnalysisTheory Methods and Applications vol40 no 1 pp 37ndash49 2000

[4] S Ahmad and A C Lazer ldquoAverage growth and total per-manence in a competitive Lotka-Volterra Systemrdquo Annali diMatematica Pura ed Applicata vol 185 supplement 5 pp S47ndashS67 2006

[5] P van den Driessche and M L Zeeman ldquoThree-dimensionalcompetitive Lotka-Volterra systems with no periodic orbitsrdquoSIAM Journal on Applied Mathematics vol 58 no 1 pp 227ndash234 1998

[6] Z Teng and L Chen ldquoGlobal asymptotic stability of periodicLotka-Volterra systems with delaysrdquoNonlinear AnalysisTheoryMethods and Applications vol 45 no 8 pp 1081ndash1095 2001

[7] N Fang and X X Chen ldquoPermanence of a discrete multispeciesLotka-Volterra competition predator-prey system with delaysrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2185ndash2195 2008

[8] G Lu and Z Lu ldquoPermanence for two-species Lotka-Volterracooperative systems with delaysrdquoMathematical Biosciences andEngineering vol 5 no 3 pp 477ndash484 2008

[9] X P Yan and W T Li ldquoStability and Hopf bifurcation for adelayed cooperative systemwith diffusion effectsrdquo InternationalJournal of Bifurcation and Chaos vol 18 no 2 pp 441ndash4532008

[10] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[11] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[12] H A El-Saka E Ahmed M I Shehata and A M A El-SayedldquoOn stability persistence and Hopf bifurcation in fractionalorder dynamical systemsrdquo Nonlinear Dynamics vol 56 no 1-2 pp 121ndash126 2009

[13] E Kaslik and S Sivasundaram ldquoNonlinear dynamics and chaosin fractional-order neural networksrdquo Neural Networks vol 32pp 245ndash256 2012

[14] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[15] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe Computational Engineering in Systems Applications vol 2pp 963ndash968 Lille France July 2012

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where 11988811 = minus11988711198861111988633 minus 11988731198861111988613 11988813 = minus11988711198861311988633 minus

11988731198861311988613 11988831 = 11988711198863111988631 minus 11988731198861111988631 11988822 = 11988633(119887111988621 minus 119887211988611) +

11988623(119887311988611minus119887111988631)+11988613(119887311988621minus119887211988631) 11988833 = minus11988711198863111988633+11988731198861111988633and 1198880 = a1111988633 + 1198861311988631 Then the following results can beobtained

1205821 = 11986022 lt 0 1205822 + 1205823 = 11986011 + 11986033 lt 0

12058221205823 = 1198601111986033 minus 1198601311986031 gt 0

(13)

Hence the equilibrium 1198756 is locally asymptotically stableSimilarly it can be readily derived that the equilibrium 1198757

is locally asymptotically stableFor 1198758 let 1198758 = (119909

lowast

1 119909lowast

2 119909lowast

3) the Jacobian matrix of 1198758 can

be written as

119869 (1198758) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

2minus11988623119909

lowast

2

11988631119909lowast

3minus11988632119909

lowast

3minus11988633119909

lowast

3

) = (

11986111 11986112 11986113

11986121 11986122 11986123

B31 11986132 11986133

)

(14)

and its characteristic equation is

1205823+ 1198621120582

2+ 1198622120582 + 1198623 = 0 (15)

where 1198621 = minus(11986111 + 11986122 + 11986133) 1198622 = 1198611111986122 + 1198611111986133 +

1198612211986133 minus 1198612311986132 minus 1198611211986121 minus 1198611311986131 and 1198623 = minus119861111198612211986133 +

119861111198612311986132 + 119861121198612111986133 + 119861131198612211986131 minus 119861121198612311986131 minus 119861131198612111986132For simplicity the equivalent characteristic equation is intro-duced as follows

1205823+ 11988611205822+ 1198862120582 + 1198863 = (120582 minus 119886) (120582

2minus 119887120582 + 119888) = 0 (16)

On the basis of the above equivalent substitutions thefollowing inequalities can be gained

1198861 = 11988611119909lowast

1+ 11988622119909

lowast

2+ 11988633119909

lowast

3gt 0

1198862 = (1198861111988622 + 1198861211988621) 119909lowast

1119909lowast

2+ (1198861111988633 + 1198861311988631) 119909

lowast

1119909lowast

3

+ (1198862211988633 minus 1198862311988632) 119909lowast

2119909lowast

3gt 0

1198863 = (11988611 (1198862211988633 minus 1198862311988632) + 11988612 (1198862111988633 minus 1198862311988631)

+11988613 (1198862211988631 minus 1198862111988632)) 119909lowast

1119909lowast

2119909lowast

3gt 0

11988611198862 minus 1198863 = 1198891119909lowast

1119909lowast

2119909lowast

3+ 1198892119909

lowast

1119909lowast

2

2+ 1198893119909

lowast

1119909lowast

3

2

+ 1198894119909lowast

1

2119909lowast

2+ 1198895119909

lowast

1

2119909lowast

3

+ 1198896 (11988622119909lowast

2

2119909lowast

3+ 11988633119909

lowast

2119909lowast

3

2) gt 0

119886 + 119887 = minus1198861 lt 0

119886119887 + 119888 = 1198862 gt 0

119886119888 = minus1198863 lt 0

minus1198862119887 minus 119886119887

2minus 119887119888 = 11988611198862 minus 1198863 gt 0

(17)

where 1198891 = 2119886111198862211988633 + 119886121198862311988631 + 119886131198862111988632 1198892 =

119886111198862

22+ 119886121198862111988622 1198893 = 11988611119886

2

33+ 119886131198863111988633 1198894 = 119886

2

1111988622 +

119886111198861211988621 1198895 = 1198862

1111988633 + 119886111198861311988631 and 1198896 = 1198862211988633 minus 1198862311988632

Using the proof by contradiction it can be concluded that theeigenvalues of 119869(1198758) satisfy

1205821 = 119886 lt 0

1205822 + 1205823 = 119887 lt 0

12058221205823 = 119888 gt 0

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt

119902120587

2 119894 = 2 3

(18)

Hence the equilibrium1198758 is locally asymptotically stable

For the further dynamic investigation of the fractionalpopulation systems the other fractional Lotka-Volterra sys-tems will be considered in the following section Particularlybifurcation properties for the system will be studied in detail

3 Bifurcation Analysis ofa Three-Dimensional FractionalLotka-Volterra Predator-Prey System

Consider a three-dimensional fractional Lotka-Volterra sys-tem

1198631199021199091 (119905) = 1199091 (119905) (1198871 minus 119886111199091 (119905) minus 119886121199092 (119905) minus 119886131199093 (119905))

1198631199021199092 (119905) = 1199092 (119905) (minus1198872 + 119886211199091 (119905) minus 119886221199092 (119905))

1198631199021199093 (119905) = 1199093 (119905) (minus1198873 + 119886311199091 (119905) minus 119886331199093 (119905))

(19)

with the initial values 119909119894(119905)|119905=0 = 119909119894(0) 119894 = 1 2 3 where0 lt 119902 lt 1 11988611 lt 0 and the other constant coefficients arepositive 1199091(119905) ge 0 represents the density of prey species attime 119905 and 1199092(119905) ge 0 1199093(119905) ge 0 represent the densities ofpredator species at time 119905 In this case system (19) can beregarded as a fractional Lotka-Volterra predator-prey system

On the basis of Theorem 2 it is not difficult to prove thatsystem (19) has a unique solution in a similar way

It is clear that there are eight equilibria for system(19) Here we focus on the bifurcation investigation of theequilibrium 119909

lowast which can be called a positive equilibriumwhen some conditions are satisfied The equilibrium 119909

lowast isobtained as

119909lowast= (119909lowast

1 119909lowast

2 119909lowast

3) = (

11988911

11988911988922

11988911988933

119889) (20)

where 11988911 = 11988711198862211988633 + 11988721198861211988633 + 11988731198861311988622 11988922 = 11988711198862111988633 minus

1198872(1198861111988633 + 1198861311988631) + 11988731198861311988621 11988933 = 11988711198862211988631 + 11988721198861211988631 minus

1198873(1198861111988622 + 1198861211988621) and 119889 = 119886111198862211988633 + 119886121198862111988633 + 119886131198862211988631And its Jacobian matrix can be expressed as

119869 (119909lowast) = (

minus11988611119909lowast

1minus11988612119909

lowast

1minus11988613119909

lowast

1

11988621119909lowast

2minus11988622119909

lowast

20

11988631119909lowast

30 minus11988633119909

lowast

3

) (21)

Mathematical Problems in Engineering 5

Furthermore the eigenvalues of 119869(119909lowast) satisfy the characteris-tic equation

(120582 minus 120573) (1205822minus (120572 + 120573) 120582 + 120572120573 minus 120574) = 0 (22)

where 120572 = minus11988611119909lowast

1 120573 = minus11988622119909

lowast

2= minus11988633119909

lowast

3 and 120574 =

minus1198861211988621119909lowast

1119909lowast

2minus 1198861311988631119909

lowast

1119909lowast

3

In the following by choosing the fractional order 119902

as the bifurcation parameter and analyzing the associatedcharacteristic equation (22) of system (19) at the positiveequilibrium we investigate the bifurcation phenomena of thepositive equilibrium of system (19) and obtain the conditionsunder which system (19) undergoes a Hopf bifurcation

Proposition 6 The positive equilibrium 119909lowast of system (19) is

locally asymptotically stable if and only if all the followingconditions are satisfied

(i) 120573 lt 0(ii) 120572120573 minus 120574 gt 0 and(iii) 120572 + 120573 lt 2 cos(1199021205872)radic120572120573 minus 120574

Proof For the characteristic equation (22) the root 1205821 = 120573 lt

0 and 1205822 1205823 satisfy the equation 1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0

It is clear that | arg(12058223)| gt 1199021205872 if and only if the conditions(ii) and (iii) hold Based onTheorem 4 Proposition 6 provesto be true

In addition by analyzing the condition (iii) ofProposition 6 in detail the following results can be gained

Proposition 7 With respect to system (19) if 120573 lt 0 and 120572120573 minus120574 gt 0 the following statements can be obtained

(a) If 120572+120573 le 0 the equilibrium 119909lowast is locally asymptotically

stable for any 119902 isin (0 1)(b) If 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equilibrium 119909

lowast is locallyasymptotically stable if and only if 119902 isin (0 119902

lowast) where

119902lowast= (2120587) arccos ((120572 + 120573)2radic120572120573 minus 120574)

(c) If 120572+120573 ge 2radic120572120573 minus 120574 the equilibrium 119909lowast is unstable for

any 119902 isin (0 1)

Proof The conclusions (a) and (c) are obvious For thestatement (b) due to 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equation1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0 has two complex roots 1205822 1205823

and their real part is (120572 + 120573)2 gt 0 Then | arg(120582119894)| =

arccos((120572 + 120573)2radic120572120573 minus 120574) 119894 = 2 3 Besides according to thecondition arccos((120572 + 120573)2radic120572120573 minus 120574) = 119902

lowast1205872 119902 isin (0 119902

lowast) if

and only if | arg(120582119894)| gt 1199021205872 119894 = 2 3 Based onTheorem 4 itis concluded that Proposition 7 is true

Remark 8 It is apparent that the critical value satisfies 119902lowast isin(0 1) When 119902 isin (0 119902

lowast) 119909lowast is locally asymptotically stable

when 119902 isin (119902lowast 1) and specially 119902 = 1 119909lowast is unstable That is

to say it has verified that fractional differential equations areat least as stable as their integer-order counterparts [4]

Remark 9 Under the situation of statement (b) a bifurcationphenomenon must happen at the critical value 119902lowast However

it is difficult to confirm precise bifurcation type As an inter-esting bifurcation behavior Hopf rsquos bifurcation is expected totake place

According to Proposition 7 if some appropriate condi-tions about the constant coefficients of system (19) can befound so that statement (b) is satisfied system (19) willundergo a bifurcation phenomenon And the critical value119902lowast of the bifurcation parameter 119902 can be expressed by theconstant coefficients of system (19) From this the followingtheorem is specifically proposed

Theorem 10 With respect to system (19) if the followingconditions are satisfied

(i) 1198872 = 1198873 11988622 = 11988633 11988621 = 11988631 = minus11988611(ii) 11988612 + 11988613 minus 11988622 gt 0

then the positive equilibrium 119909lowast is locally asymptotically stable

if and only if 119902 isin (0 119902lowast) where

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(23)

Proof According to the condition (i) the equilibrium 119909lowast can

be expressed as

119909lowast= (

119887111988622 + 119887211988612 + 119887211988613

11988611 (11988622 minus 11988612 minus 11988613)

1198871 + 1198872

11988612 + 11988613 minus 11988622

1198871 + 1198872

11988612 + 11988613 minus 11988622

)

(24)

For (22) the following results can be obtained

120573 = minus11988622119909lowast

2lt 0 120572 + 120573 = 1198872 gt 0

120572120573 minus 120574 =1198901

11988612 + 11988613 minus 11988622

gt 0

4 (120572120573 minus 120574) minus (120572 + 120573)2=

41198901

11988612 + 11988613 minus 11988622

minus 1198872

2

=1198902

11988612 + 11988613 minus 11988622

gt 0

(25)

where 1198901 = (1198871 + 1198872)(119887111988622 + 119887211988612 + 119887211988613) and 1198902 = 11988622(41198872

1+

1198872

2)+31198872

2(11988612+11988613)+411988711198872(11988612+11988613+11988622) Obviously the above

conclusions satisfy statement (b) of Proposition 7 then it canbe derived that

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(26)

6 Mathematical Problems in Engineering

5 8 11 14 17 20

02

46

8100

5

10

x3

x2

x1

q = 1

q = 09

q = 08

Figure 1 The trajectory of system (1) converges to the equilibrium1198758= (537 157 17)

Hence the positive equilibrium 119909lowast of system (19) is locally

asymptotically stable if and only if 119902 isin (0 119902lowast)

According to the statement of Theorem 10 it can beconcluded that the positive equilibrium 119909

lowast is locally asymp-totically stable if and only if 119902 isin (0 119902

lowast) At 119902 = 119902

lowast the Hopfbifurcation is expected to take place As 119902 increases above thecritical value 119902lowast the positive equilibrium 119909

lowast is unstable anda limit cycle is expected to appear in the proximity of 119909lowast dueto the Hopf bifurcation phenomenon

The analysis of periodic solutions in fractional dynamicalsystems is a very recent and promising research topic Asa consequence the nonexistence of exact periodic solutionsin time invariant fractional systems is obtained [16] As anapplication it is emphasized that the limit cycle observed innumerical simulations of a simple fractional neural networkcannot be an exact periodic solution of the system [17]In addition there are some other papers providing thenumerical evidences of limit cycles

Remark 11 Even though exact periodic solutions do not existin autonomous fractional systems [16 17] limit cycles havebeen observed by numerical simulations in many systemssuch as a fractional neural system [13] a fractional Van derPol system [18] fractional Chua and Chenrsquos systems [19 20]and a fractional financial system [21]

4 Numerical Simulation

In this paper an Adams-type predictor-corrector methodis used for the numerical solutions of fractional differentialequations This method has been introduced in [22 23]and further investigated in [24ndash27] In order to verify thetheoretical analysis the following numerical results are given

For system (1) the approximate solutions are displayedin Figure 1 for the step size 0005 and different values of119902 119902 = 1 119902 = 09 119902 = 08 respectively Taking 1198871 =

12 1198872 = 11988611 = 11988613 = 11988621 = 11988623 = 11988631 = 1 11988612 =

11988632 = 11988633 = 2 and 1198873 = 11988622 = 3 and choosing theinitial values 1199091(0) = 20 1199092(0) = 10 and 1199093(0) = 10 the

55 60 65 701

15

2

25

3

35

4

45

5

x

t

1

(a)

5515

2

25

3

35

4

45

5

55

6

60 65 70

x

t

2

(b)

55 60 65 7015

2

25

3

35

4

45

5

55

6

x

t

3

(c)

Figure 2 The solution of system (19) versus time with 119902 = 084

equilibrium 1198758 is (537 157 17) Then Figure 1 shows thatthe equilibrium 1198758 is locally asymptotically stable Namelythe fifth conclusion of Theorem 5 is verified Similarly theother conclusions of Theorem 5 can be confirmed

For system (19) the approximate solutions are displayedin Figures 2 3 and 4 for the step size 0001 and different valuesof 119902 119902 = 082 and 119902 = 084 Taking 1198871 = 11988612 = 11988613 = 11988622 =

11988633 = 1 1198872 = 1198873 = 11988621 = 11988631 = 2 and 11988611 = minus2 andchoosing the initial values 1199091(0) = 3 1199092(0) = 1199093(0) = 4 thepositive equilibrium is 119909lowast = (25 3 3) and the critical value

Mathematical Problems in Engineering 7

15 2 25 3 35

225

335

42

3

4

x3

x2

x1

Figure 3 When 119902 = 082 the trajectory of system (19) converges tothe equilibrium 119909

lowast= (25 3 3)

12

34

5

0

2

4

60510

x3

x2

x 1

Figure 4 When 119902 = 084 the trajectory of system (19) converges toan asymptotically stable limit cycle

is 119902lowast = 08337 Indeed Figures 2ndash4 present the fact that thepositive equilibrium 119909

lowast is locally asymptotically stable when119902 = 082 isin (0 08337) and when 119902 = 084 increases across119902lowast= 08337 an asymptotically stable limit cycle appears in a

neighborhood of the positive equilibrium 119909lowast

5 Conclusion

In this paper two kinds of three-dimensional fractionalLotka-Volterra systems have been studied The main resultsare divided into two parts On the one hand for system(1) the asymptotic stability of the equilibria is investigatedby providing simple and reasonable sufficient conditionsAnd simulation results prove to be quite consistent with thetheoretical findings On the other hand for system (19) theconditions which could lead to bifurcation phenomena areobtained Specifically the fractional order 119902 isin (0 1) is chosenas the bifurcation parameter and the expression of the criticalvalue 119902

lowast is precisely derived Furthermore the numericalresult is presented to illustrate that Hopf rsquos bifurcation cantake place

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Nature ScienceFoundation of China under Grant no 11371049 and theScience Foundation of Beijing Jiaotong University underGrant 2011JBM130

References

[1] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley-IntersciencePublication New York NY USA 1993

[2] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[3] S Ahmad and A C Lazer ldquoAverage conditions for globalasymptotic stability in a nonautonomous Lotka-Volterra sys-temrdquoNonlinear AnalysisTheory Methods and Applications vol40 no 1 pp 37ndash49 2000

[4] S Ahmad and A C Lazer ldquoAverage growth and total per-manence in a competitive Lotka-Volterra Systemrdquo Annali diMatematica Pura ed Applicata vol 185 supplement 5 pp S47ndashS67 2006

[5] P van den Driessche and M L Zeeman ldquoThree-dimensionalcompetitive Lotka-Volterra systems with no periodic orbitsrdquoSIAM Journal on Applied Mathematics vol 58 no 1 pp 227ndash234 1998

[6] Z Teng and L Chen ldquoGlobal asymptotic stability of periodicLotka-Volterra systems with delaysrdquoNonlinear AnalysisTheoryMethods and Applications vol 45 no 8 pp 1081ndash1095 2001

[7] N Fang and X X Chen ldquoPermanence of a discrete multispeciesLotka-Volterra competition predator-prey system with delaysrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2185ndash2195 2008

[8] G Lu and Z Lu ldquoPermanence for two-species Lotka-Volterracooperative systems with delaysrdquoMathematical Biosciences andEngineering vol 5 no 3 pp 477ndash484 2008

[9] X P Yan and W T Li ldquoStability and Hopf bifurcation for adelayed cooperative systemwith diffusion effectsrdquo InternationalJournal of Bifurcation and Chaos vol 18 no 2 pp 441ndash4532008

[10] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[11] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[12] H A El-Saka E Ahmed M I Shehata and A M A El-SayedldquoOn stability persistence and Hopf bifurcation in fractionalorder dynamical systemsrdquo Nonlinear Dynamics vol 56 no 1-2 pp 121ndash126 2009

[13] E Kaslik and S Sivasundaram ldquoNonlinear dynamics and chaosin fractional-order neural networksrdquo Neural Networks vol 32pp 245ndash256 2012

[14] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[15] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe Computational Engineering in Systems Applications vol 2pp 963ndash968 Lille France July 2012

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Furthermore the eigenvalues of 119869(119909lowast) satisfy the characteris-tic equation

(120582 minus 120573) (1205822minus (120572 + 120573) 120582 + 120572120573 minus 120574) = 0 (22)

where 120572 = minus11988611119909lowast

1 120573 = minus11988622119909

lowast

2= minus11988633119909

lowast

3 and 120574 =

minus1198861211988621119909lowast

1119909lowast

2minus 1198861311988631119909

lowast

1119909lowast

3

In the following by choosing the fractional order 119902

as the bifurcation parameter and analyzing the associatedcharacteristic equation (22) of system (19) at the positiveequilibrium we investigate the bifurcation phenomena of thepositive equilibrium of system (19) and obtain the conditionsunder which system (19) undergoes a Hopf bifurcation

Proposition 6 The positive equilibrium 119909lowast of system (19) is

locally asymptotically stable if and only if all the followingconditions are satisfied

(i) 120573 lt 0(ii) 120572120573 minus 120574 gt 0 and(iii) 120572 + 120573 lt 2 cos(1199021205872)radic120572120573 minus 120574

Proof For the characteristic equation (22) the root 1205821 = 120573 lt

0 and 1205822 1205823 satisfy the equation 1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0

It is clear that | arg(12058223)| gt 1199021205872 if and only if the conditions(ii) and (iii) hold Based onTheorem 4 Proposition 6 provesto be true

In addition by analyzing the condition (iii) ofProposition 6 in detail the following results can be gained

Proposition 7 With respect to system (19) if 120573 lt 0 and 120572120573 minus120574 gt 0 the following statements can be obtained

(a) If 120572+120573 le 0 the equilibrium 119909lowast is locally asymptotically

stable for any 119902 isin (0 1)(b) If 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equilibrium 119909

lowast is locallyasymptotically stable if and only if 119902 isin (0 119902

lowast) where

119902lowast= (2120587) arccos ((120572 + 120573)2radic120572120573 minus 120574)

(c) If 120572+120573 ge 2radic120572120573 minus 120574 the equilibrium 119909lowast is unstable for

any 119902 isin (0 1)

Proof The conclusions (a) and (c) are obvious For thestatement (b) due to 0 lt 120572 + 120573 lt 2radic120572120573 minus 120574 the equation1205822minus (120572 + 120573)120582 + 120572120573 minus 120574 = 0 has two complex roots 1205822 1205823

and their real part is (120572 + 120573)2 gt 0 Then | arg(120582119894)| =

arccos((120572 + 120573)2radic120572120573 minus 120574) 119894 = 2 3 Besides according to thecondition arccos((120572 + 120573)2radic120572120573 minus 120574) = 119902

lowast1205872 119902 isin (0 119902

lowast) if

and only if | arg(120582119894)| gt 1199021205872 119894 = 2 3 Based onTheorem 4 itis concluded that Proposition 7 is true

Remark 8 It is apparent that the critical value satisfies 119902lowast isin(0 1) When 119902 isin (0 119902

lowast) 119909lowast is locally asymptotically stable

when 119902 isin (119902lowast 1) and specially 119902 = 1 119909lowast is unstable That is

to say it has verified that fractional differential equations areat least as stable as their integer-order counterparts [4]

Remark 9 Under the situation of statement (b) a bifurcationphenomenon must happen at the critical value 119902lowast However

it is difficult to confirm precise bifurcation type As an inter-esting bifurcation behavior Hopf rsquos bifurcation is expected totake place

According to Proposition 7 if some appropriate condi-tions about the constant coefficients of system (19) can befound so that statement (b) is satisfied system (19) willundergo a bifurcation phenomenon And the critical value119902lowast of the bifurcation parameter 119902 can be expressed by theconstant coefficients of system (19) From this the followingtheorem is specifically proposed

Theorem 10 With respect to system (19) if the followingconditions are satisfied

(i) 1198872 = 1198873 11988622 = 11988633 11988621 = 11988631 = minus11988611(ii) 11988612 + 11988613 minus 11988622 gt 0

then the positive equilibrium 119909lowast is locally asymptotically stable

if and only if 119902 isin (0 119902lowast) where

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(23)

Proof According to the condition (i) the equilibrium 119909lowast can

be expressed as

119909lowast= (

119887111988622 + 119887211988612 + 119887211988613

11988611 (11988622 minus 11988612 minus 11988613)

1198871 + 1198872

11988612 + 11988613 minus 11988622

1198871 + 1198872

11988612 + 11988613 minus 11988622

)

(24)

For (22) the following results can be obtained

120573 = minus11988622119909lowast

2lt 0 120572 + 120573 = 1198872 gt 0

120572120573 minus 120574 =1198901

11988612 + 11988613 minus 11988622

gt 0

4 (120572120573 minus 120574) minus (120572 + 120573)2=

41198901

11988612 + 11988613 minus 11988622

minus 1198872

2

=1198902

11988612 + 11988613 minus 11988622

gt 0

(25)

where 1198901 = (1198871 + 1198872)(119887111988622 + 119887211988612 + 119887211988613) and 1198902 = 11988622(41198872

1+

1198872

2)+31198872

2(11988612+11988613)+411988711198872(11988612+11988613+11988622) Obviously the above

conclusions satisfy statement (b) of Proposition 7 then it canbe derived that

119902lowast

=2

120587arccos(1198872(2

radic(1198871 + 1198872) (119887111988622 + 119887211988612 + 119887211988613)

11988612 + 11988613 minus 11988622

)

minus1

)

(26)

6 Mathematical Problems in Engineering

5 8 11 14 17 20

02

46

8100

5

10

x3

x2

x1

q = 1

q = 09

q = 08

Figure 1 The trajectory of system (1) converges to the equilibrium1198758= (537 157 17)

Hence the positive equilibrium 119909lowast of system (19) is locally

asymptotically stable if and only if 119902 isin (0 119902lowast)

According to the statement of Theorem 10 it can beconcluded that the positive equilibrium 119909

lowast is locally asymp-totically stable if and only if 119902 isin (0 119902

lowast) At 119902 = 119902

lowast the Hopfbifurcation is expected to take place As 119902 increases above thecritical value 119902lowast the positive equilibrium 119909

lowast is unstable anda limit cycle is expected to appear in the proximity of 119909lowast dueto the Hopf bifurcation phenomenon

The analysis of periodic solutions in fractional dynamicalsystems is a very recent and promising research topic Asa consequence the nonexistence of exact periodic solutionsin time invariant fractional systems is obtained [16] As anapplication it is emphasized that the limit cycle observed innumerical simulations of a simple fractional neural networkcannot be an exact periodic solution of the system [17]In addition there are some other papers providing thenumerical evidences of limit cycles

Remark 11 Even though exact periodic solutions do not existin autonomous fractional systems [16 17] limit cycles havebeen observed by numerical simulations in many systemssuch as a fractional neural system [13] a fractional Van derPol system [18] fractional Chua and Chenrsquos systems [19 20]and a fractional financial system [21]

4 Numerical Simulation

In this paper an Adams-type predictor-corrector methodis used for the numerical solutions of fractional differentialequations This method has been introduced in [22 23]and further investigated in [24ndash27] In order to verify thetheoretical analysis the following numerical results are given

For system (1) the approximate solutions are displayedin Figure 1 for the step size 0005 and different values of119902 119902 = 1 119902 = 09 119902 = 08 respectively Taking 1198871 =

12 1198872 = 11988611 = 11988613 = 11988621 = 11988623 = 11988631 = 1 11988612 =

11988632 = 11988633 = 2 and 1198873 = 11988622 = 3 and choosing theinitial values 1199091(0) = 20 1199092(0) = 10 and 1199093(0) = 10 the

55 60 65 701

15

2

25

3

35

4

45

5

x

t

1

(a)

5515

2

25

3

35

4

45

5

55

6

60 65 70

x

t

2

(b)

55 60 65 7015

2

25

3

35

4

45

5

55

6

x

t

3

(c)

Figure 2 The solution of system (19) versus time with 119902 = 084

equilibrium 1198758 is (537 157 17) Then Figure 1 shows thatthe equilibrium 1198758 is locally asymptotically stable Namelythe fifth conclusion of Theorem 5 is verified Similarly theother conclusions of Theorem 5 can be confirmed

For system (19) the approximate solutions are displayedin Figures 2 3 and 4 for the step size 0001 and different valuesof 119902 119902 = 082 and 119902 = 084 Taking 1198871 = 11988612 = 11988613 = 11988622 =

11988633 = 1 1198872 = 1198873 = 11988621 = 11988631 = 2 and 11988611 = minus2 andchoosing the initial values 1199091(0) = 3 1199092(0) = 1199093(0) = 4 thepositive equilibrium is 119909lowast = (25 3 3) and the critical value

Mathematical Problems in Engineering 7

15 2 25 3 35

225

335

42

3

4

x3

x2

x1

Figure 3 When 119902 = 082 the trajectory of system (19) converges tothe equilibrium 119909

lowast= (25 3 3)

12

34

5

0

2

4

60510

x3

x2

x 1

Figure 4 When 119902 = 084 the trajectory of system (19) converges toan asymptotically stable limit cycle

is 119902lowast = 08337 Indeed Figures 2ndash4 present the fact that thepositive equilibrium 119909

lowast is locally asymptotically stable when119902 = 082 isin (0 08337) and when 119902 = 084 increases across119902lowast= 08337 an asymptotically stable limit cycle appears in a

neighborhood of the positive equilibrium 119909lowast

5 Conclusion

In this paper two kinds of three-dimensional fractionalLotka-Volterra systems have been studied The main resultsare divided into two parts On the one hand for system(1) the asymptotic stability of the equilibria is investigatedby providing simple and reasonable sufficient conditionsAnd simulation results prove to be quite consistent with thetheoretical findings On the other hand for system (19) theconditions which could lead to bifurcation phenomena areobtained Specifically the fractional order 119902 isin (0 1) is chosenas the bifurcation parameter and the expression of the criticalvalue 119902

lowast is precisely derived Furthermore the numericalresult is presented to illustrate that Hopf rsquos bifurcation cantake place

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Nature ScienceFoundation of China under Grant no 11371049 and theScience Foundation of Beijing Jiaotong University underGrant 2011JBM130

References

[1] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley-IntersciencePublication New York NY USA 1993

[2] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[3] S Ahmad and A C Lazer ldquoAverage conditions for globalasymptotic stability in a nonautonomous Lotka-Volterra sys-temrdquoNonlinear AnalysisTheory Methods and Applications vol40 no 1 pp 37ndash49 2000

[4] S Ahmad and A C Lazer ldquoAverage growth and total per-manence in a competitive Lotka-Volterra Systemrdquo Annali diMatematica Pura ed Applicata vol 185 supplement 5 pp S47ndashS67 2006

[5] P van den Driessche and M L Zeeman ldquoThree-dimensionalcompetitive Lotka-Volterra systems with no periodic orbitsrdquoSIAM Journal on Applied Mathematics vol 58 no 1 pp 227ndash234 1998

[6] Z Teng and L Chen ldquoGlobal asymptotic stability of periodicLotka-Volterra systems with delaysrdquoNonlinear AnalysisTheoryMethods and Applications vol 45 no 8 pp 1081ndash1095 2001

[7] N Fang and X X Chen ldquoPermanence of a discrete multispeciesLotka-Volterra competition predator-prey system with delaysrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2185ndash2195 2008

[8] G Lu and Z Lu ldquoPermanence for two-species Lotka-Volterracooperative systems with delaysrdquoMathematical Biosciences andEngineering vol 5 no 3 pp 477ndash484 2008

[9] X P Yan and W T Li ldquoStability and Hopf bifurcation for adelayed cooperative systemwith diffusion effectsrdquo InternationalJournal of Bifurcation and Chaos vol 18 no 2 pp 441ndash4532008

[10] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[11] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[12] H A El-Saka E Ahmed M I Shehata and A M A El-SayedldquoOn stability persistence and Hopf bifurcation in fractionalorder dynamical systemsrdquo Nonlinear Dynamics vol 56 no 1-2 pp 121ndash126 2009

[13] E Kaslik and S Sivasundaram ldquoNonlinear dynamics and chaosin fractional-order neural networksrdquo Neural Networks vol 32pp 245ndash256 2012

[14] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[15] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe Computational Engineering in Systems Applications vol 2pp 963ndash968 Lille France July 2012

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

5 8 11 14 17 20

02

46

8100

5

10

x3

x2

x1

q = 1

q = 09

q = 08

Figure 1 The trajectory of system (1) converges to the equilibrium1198758= (537 157 17)

Hence the positive equilibrium 119909lowast of system (19) is locally

asymptotically stable if and only if 119902 isin (0 119902lowast)

According to the statement of Theorem 10 it can beconcluded that the positive equilibrium 119909

lowast is locally asymp-totically stable if and only if 119902 isin (0 119902

lowast) At 119902 = 119902

lowast the Hopfbifurcation is expected to take place As 119902 increases above thecritical value 119902lowast the positive equilibrium 119909

lowast is unstable anda limit cycle is expected to appear in the proximity of 119909lowast dueto the Hopf bifurcation phenomenon

The analysis of periodic solutions in fractional dynamicalsystems is a very recent and promising research topic Asa consequence the nonexistence of exact periodic solutionsin time invariant fractional systems is obtained [16] As anapplication it is emphasized that the limit cycle observed innumerical simulations of a simple fractional neural networkcannot be an exact periodic solution of the system [17]In addition there are some other papers providing thenumerical evidences of limit cycles

Remark 11 Even though exact periodic solutions do not existin autonomous fractional systems [16 17] limit cycles havebeen observed by numerical simulations in many systemssuch as a fractional neural system [13] a fractional Van derPol system [18] fractional Chua and Chenrsquos systems [19 20]and a fractional financial system [21]

4 Numerical Simulation

In this paper an Adams-type predictor-corrector methodis used for the numerical solutions of fractional differentialequations This method has been introduced in [22 23]and further investigated in [24ndash27] In order to verify thetheoretical analysis the following numerical results are given

For system (1) the approximate solutions are displayedin Figure 1 for the step size 0005 and different values of119902 119902 = 1 119902 = 09 119902 = 08 respectively Taking 1198871 =

12 1198872 = 11988611 = 11988613 = 11988621 = 11988623 = 11988631 = 1 11988612 =

11988632 = 11988633 = 2 and 1198873 = 11988622 = 3 and choosing theinitial values 1199091(0) = 20 1199092(0) = 10 and 1199093(0) = 10 the

55 60 65 701

15

2

25

3

35

4

45

5

x

t

1

(a)

5515

2

25

3

35

4

45

5

55

6

60 65 70

x

t

2

(b)

55 60 65 7015

2

25

3

35

4

45

5

55

6

x

t

3

(c)

Figure 2 The solution of system (19) versus time with 119902 = 084

equilibrium 1198758 is (537 157 17) Then Figure 1 shows thatthe equilibrium 1198758 is locally asymptotically stable Namelythe fifth conclusion of Theorem 5 is verified Similarly theother conclusions of Theorem 5 can be confirmed

For system (19) the approximate solutions are displayedin Figures 2 3 and 4 for the step size 0001 and different valuesof 119902 119902 = 082 and 119902 = 084 Taking 1198871 = 11988612 = 11988613 = 11988622 =

11988633 = 1 1198872 = 1198873 = 11988621 = 11988631 = 2 and 11988611 = minus2 andchoosing the initial values 1199091(0) = 3 1199092(0) = 1199093(0) = 4 thepositive equilibrium is 119909lowast = (25 3 3) and the critical value

Mathematical Problems in Engineering 7

15 2 25 3 35

225

335

42

3

4

x3

x2

x1

Figure 3 When 119902 = 082 the trajectory of system (19) converges tothe equilibrium 119909

lowast= (25 3 3)

12

34

5

0

2

4

60510

x3

x2

x 1

Figure 4 When 119902 = 084 the trajectory of system (19) converges toan asymptotically stable limit cycle

is 119902lowast = 08337 Indeed Figures 2ndash4 present the fact that thepositive equilibrium 119909

lowast is locally asymptotically stable when119902 = 082 isin (0 08337) and when 119902 = 084 increases across119902lowast= 08337 an asymptotically stable limit cycle appears in a

neighborhood of the positive equilibrium 119909lowast

5 Conclusion

In this paper two kinds of three-dimensional fractionalLotka-Volterra systems have been studied The main resultsare divided into two parts On the one hand for system(1) the asymptotic stability of the equilibria is investigatedby providing simple and reasonable sufficient conditionsAnd simulation results prove to be quite consistent with thetheoretical findings On the other hand for system (19) theconditions which could lead to bifurcation phenomena areobtained Specifically the fractional order 119902 isin (0 1) is chosenas the bifurcation parameter and the expression of the criticalvalue 119902

lowast is precisely derived Furthermore the numericalresult is presented to illustrate that Hopf rsquos bifurcation cantake place

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Nature ScienceFoundation of China under Grant no 11371049 and theScience Foundation of Beijing Jiaotong University underGrant 2011JBM130

References

[1] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley-IntersciencePublication New York NY USA 1993

[2] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[3] S Ahmad and A C Lazer ldquoAverage conditions for globalasymptotic stability in a nonautonomous Lotka-Volterra sys-temrdquoNonlinear AnalysisTheory Methods and Applications vol40 no 1 pp 37ndash49 2000

[4] S Ahmad and A C Lazer ldquoAverage growth and total per-manence in a competitive Lotka-Volterra Systemrdquo Annali diMatematica Pura ed Applicata vol 185 supplement 5 pp S47ndashS67 2006

[5] P van den Driessche and M L Zeeman ldquoThree-dimensionalcompetitive Lotka-Volterra systems with no periodic orbitsrdquoSIAM Journal on Applied Mathematics vol 58 no 1 pp 227ndash234 1998

[6] Z Teng and L Chen ldquoGlobal asymptotic stability of periodicLotka-Volterra systems with delaysrdquoNonlinear AnalysisTheoryMethods and Applications vol 45 no 8 pp 1081ndash1095 2001

[7] N Fang and X X Chen ldquoPermanence of a discrete multispeciesLotka-Volterra competition predator-prey system with delaysrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2185ndash2195 2008

[8] G Lu and Z Lu ldquoPermanence for two-species Lotka-Volterracooperative systems with delaysrdquoMathematical Biosciences andEngineering vol 5 no 3 pp 477ndash484 2008

[9] X P Yan and W T Li ldquoStability and Hopf bifurcation for adelayed cooperative systemwith diffusion effectsrdquo InternationalJournal of Bifurcation and Chaos vol 18 no 2 pp 441ndash4532008

[10] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[11] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[12] H A El-Saka E Ahmed M I Shehata and A M A El-SayedldquoOn stability persistence and Hopf bifurcation in fractionalorder dynamical systemsrdquo Nonlinear Dynamics vol 56 no 1-2 pp 121ndash126 2009

[13] E Kaslik and S Sivasundaram ldquoNonlinear dynamics and chaosin fractional-order neural networksrdquo Neural Networks vol 32pp 245ndash256 2012

[14] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[15] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe Computational Engineering in Systems Applications vol 2pp 963ndash968 Lille France July 2012

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

15 2 25 3 35

225

335

42

3

4

x3

x2

x1

Figure 3 When 119902 = 082 the trajectory of system (19) converges tothe equilibrium 119909

lowast= (25 3 3)

12

34

5

0

2

4

60510

x3

x2

x 1

Figure 4 When 119902 = 084 the trajectory of system (19) converges toan asymptotically stable limit cycle

is 119902lowast = 08337 Indeed Figures 2ndash4 present the fact that thepositive equilibrium 119909

lowast is locally asymptotically stable when119902 = 082 isin (0 08337) and when 119902 = 084 increases across119902lowast= 08337 an asymptotically stable limit cycle appears in a

neighborhood of the positive equilibrium 119909lowast

5 Conclusion

In this paper two kinds of three-dimensional fractionalLotka-Volterra systems have been studied The main resultsare divided into two parts On the one hand for system(1) the asymptotic stability of the equilibria is investigatedby providing simple and reasonable sufficient conditionsAnd simulation results prove to be quite consistent with thetheoretical findings On the other hand for system (19) theconditions which could lead to bifurcation phenomena areobtained Specifically the fractional order 119902 isin (0 1) is chosenas the bifurcation parameter and the expression of the criticalvalue 119902

lowast is precisely derived Furthermore the numericalresult is presented to illustrate that Hopf rsquos bifurcation cantake place

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Nature ScienceFoundation of China under Grant no 11371049 and theScience Foundation of Beijing Jiaotong University underGrant 2011JBM130

References

[1] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations Wiley-IntersciencePublication New York NY USA 1993

[2] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[3] S Ahmad and A C Lazer ldquoAverage conditions for globalasymptotic stability in a nonautonomous Lotka-Volterra sys-temrdquoNonlinear AnalysisTheory Methods and Applications vol40 no 1 pp 37ndash49 2000

[4] S Ahmad and A C Lazer ldquoAverage growth and total per-manence in a competitive Lotka-Volterra Systemrdquo Annali diMatematica Pura ed Applicata vol 185 supplement 5 pp S47ndashS67 2006

[5] P van den Driessche and M L Zeeman ldquoThree-dimensionalcompetitive Lotka-Volterra systems with no periodic orbitsrdquoSIAM Journal on Applied Mathematics vol 58 no 1 pp 227ndash234 1998

[6] Z Teng and L Chen ldquoGlobal asymptotic stability of periodicLotka-Volterra systems with delaysrdquoNonlinear AnalysisTheoryMethods and Applications vol 45 no 8 pp 1081ndash1095 2001

[7] N Fang and X X Chen ldquoPermanence of a discrete multispeciesLotka-Volterra competition predator-prey system with delaysrdquoNonlinear Analysis Real World Applications vol 9 no 5 pp2185ndash2195 2008

[8] G Lu and Z Lu ldquoPermanence for two-species Lotka-Volterracooperative systems with delaysrdquoMathematical Biosciences andEngineering vol 5 no 3 pp 477ndash484 2008

[9] X P Yan and W T Li ldquoStability and Hopf bifurcation for adelayed cooperative systemwith diffusion effectsrdquo InternationalJournal of Bifurcation and Chaos vol 18 no 2 pp 441ndash4532008

[10] AMA El-Sayed A EM El-Mesiry andHA A El-Saka ldquoOnthe fractional-order logistic equationrdquo Applied MathematicsLetters vol 20 no 7 pp 817ndash823 2007

[11] E Ahmed A M A El-Sayed and H A A El-Saka ldquoEqui-librium points stability and numerical solutions of fractional-order predator-prey and rabies modelsrdquo Journal of Mathemati-cal Analysis and Applications vol 325 no 1 pp 542ndash553 2007

[12] H A El-Saka E Ahmed M I Shehata and A M A El-SayedldquoOn stability persistence and Hopf bifurcation in fractionalorder dynamical systemsrdquo Nonlinear Dynamics vol 56 no 1-2 pp 121ndash126 2009

[13] E Kaslik and S Sivasundaram ldquoNonlinear dynamics and chaosin fractional-order neural networksrdquo Neural Networks vol 32pp 245ndash256 2012

[14] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[15] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedings ofthe Computational Engineering in Systems Applications vol 2pp 963ndash968 Lille France July 2012

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[16] M S Tavazoei and M Haeri ldquoA proof for non existence ofperiodic solutions in time invariant fractional order systemsrdquoAutomatica vol 45 no 8 pp 1886ndash1890 2009

[17] E Kaslik and S Sivasundaram ldquoNon-existence of periodic solu-tions in fractional-order dynamical systems and a remarkabledifference between integer and fractional-order derivatives ofperiodic functionsrdquo Nonlinear Analysis Real World Applica-tions vol 13 no 3 pp 1489ndash1497 2012

[18] R S Barbosa J A T MacHado B M Vinagre and A JCalderon ldquoAnalysis of the van der Pol oscillator containingderivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[19] D Cafagna and G Grassi ldquoBifurcation and chaos in thefractional-order Chen system via a time-domain approachrdquoInternational Journal of Bifurcation and Chaos vol 18 no 7 pp1845ndash1863 2008

[20] D Cafagna and G Grassi ldquoFractional-order Chuarsquos circuittime-domain analysis bifurcation chaotic behavior and test forchaosrdquo International Journal of Bifurcation and Chaos vol 18no 3 pp 615ndash639 2008

[21] M S Abd-Elouahab N E Hamri and J Wang ldquoChaos controlof a fractional-order financial systemrdquo Mathematical Problemsin Engineering vol 2010 Article ID 270646 18 pages 2010

[22] K Diethelm and A Freed ldquoOn the solution of nonlinearfractional order differential equations used in the modelling ofviscoplasticityrdquo in Scientific Computing in Chemical EngineeringII-Computational Fluid Dynamics Reaction Engineering andMolecular Properties F Keil W Mackens H Voszlig and JWerther Eds pp 217ndash224 Springer Heidelberg Germany1999

[23] K Diethelm and A Freed ldquoThe FracPECE subroutine for thenumerical solution of differential equations of fractional orderrdquoin Forschung und Wissenschaftliches Rechnen 1998 S Heinzeland T Plesser Eds pp 57ndash71 Gesellschaft fr WisseschaftlicheDatenverarbeitung Gottingen Germany 1999

[24] R ZhaoDynamical Analysis of Fractional-Order SpeciesModelsCollege of Science Beijing Jiaotong University Beijing China2011

[25] E Ahmed A M A El-Sayed A E M El-Mesiry and H AA El-Saka ldquoNumerical solution for the fractional replicatorequationrdquo International Journal of Modern Physics C vol 16 no7 pp 1017ndash1025 2005

[26] E Ahmed A M A El-Sayed and H A A El-Saka ldquoOnsomeRouth-Hurwitz conditions for fractional order differentialequations and their applications in Lorenz Rossler Chua andChen systemsrdquo Physics Letters A vol 358 no 1 pp 1ndash4 2006

[27] A E M El-Mesiry A M A El-Sayed and H A A El-Saka ldquoNumerical methods for multi-term fractional (arbitrary)orders differential equationsrdquoAppliedMathematics andCompu-tation vol 160 no 3 pp 683ndash699 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of