Effect of Delay on Selection Dynamics in Long-Term Sphere ...emis.maths.adelaide.edu.au/journals/HOA/DDNS/... · 2 DiscreteDynamicsinNatureandSociety modeltodescribetheinteractivegrowthoftheCSCsand

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 606250 5 pageshttpdxdoiorg1011552013606250

Research ArticleEffect of Delay on Selection Dynamics in Long-Term SphereCulture of Cancer Stem Cells

Peng Tang1 Aijun Fan2 Jianquan Li3 Jun Jiang1 and Kaifa Wang4

1 Breast Disease Center Southwest Hospital Third Military Medical University Chongqing 400038 China2 Chongqing Academy of Science amp Technology Chongqing 401123 China3Department of Applied Mathematics and Physics Air Force Engineering University Xirsquoan 710051 China4 School of Biomedical Engineering Third Military Medical University Chongqing 400038 China

Correspondence should be addressed to Kaifa Wang kaifawangyahoocomcn

Received 26 January 2013 Accepted 15 February 2013

Academic Editor Yonghui Xia

Copyright copy 2013 Peng Tang 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

To quantitatively study the effect of delay on selection dynamics in long-term sphere culture of cancer stem cells (CSCs) a selectiondynamic model with time delay is proposedTheoretical results show that the ubiquitous time delay in cell proliferationmay be oneof the important factors to induce fluctuation and numerical simulations indicate that the proposed selection dynamical modelwith time delay can provide a better fitting effect for the experiment of a long-term sphere culture of CSCs Thus it is valuable toconsider the delay effect in the future study on the dynamics of nongenetic heterogeneity of clonal cell populations

1 Introduction

In the past years research on cancer stem cells (CSCs) hasbecome a focus of cancer research because CSCs have self-renewing and multidirectional differentiation capability andmay result in tumors [1ndash6] Recently cell state dynamics dueto non-genetic heterogeneity of clonal cell populations alsohas received more and more attention [7ndash10]

In order to expand CSCs sphere culture is performed byexperimental cell biologists [11 12] However whether long-term sphere culture can maintain a high ratio of CSCs isunclear For this question it is interesting that [13 14] obtaina similar quantitative result through different mathematicalmodel that is the ratio of CSCs will towards an apparentequilibrium state in a long-term sphere culture Concretely[13] proposed a kinetic model using ordinary differentialequations that considered the symmetric and asymmetricdivision of CSCs as well as the proliferation and transforma-tion of differentiated cancer cells (DCCs) And [14] puts for-ward a Markov model in which cells transition stochasticallybetween statesHowever the time delay due to thematurationof individual cells has been ignored in [13 14]

In the present paper based on the kinetic model in [13]we further explore the effect of time delay on selectiondynamics in long-term sphere culture The results show thatthe ubiquitous time delay in cell proliferation may be one ofthe important factors to induce fluctuation and the proposedselection dynamical model with time delay can provide abetter fitting effect for the experiment of a long-term sphereculture of CSCs in [13] The organization of the paper is asfollows In Section 2 we formulate the selection dynamicalmodel with time delay in long-term sphere culture of CSCsSection 3 first gives the analytic analysis on our proposedmodel and then presents numerical simulations to comparethe effect of time delay Finally some predictive conclusionswith biological implications are given in Section 4

2 Model Description

Let 119909(119905) and 119910(119905) denote the population sizes of CSCs andDCCs at time 119905 in long-term sphere culture respectivelyOur previous work [13] proposed the followingmathematical

2 Discrete Dynamics in Nature and Society

model to describe the interactive growth of the CSCs andDCCs in a long-term sphere culture

d119909 (119905)d119905=119887119909(1 minus 120573

119909) 119909 (119905) + 119887

119910120573119910119910 (119905) ≜ 119909 (119905) 119891

1(119909 (119905) 119910 (119905))

d119910 (119905)d119905=119887119909120573119909119909 (119905) + 119887

119910(1 minus 120573

119910) 119910 (119905) ≜ 119910 (119905) 119891

2(119909 (119905) 119910 (119905))

(1)

Here the constants 119887119909 119887119910are called the net birth rate or intrin-

sic growth rate of population119909 119910 respectively120573119909denotes the

conversion rate from CSCs to DCCs in the process of CSCsproliferation and 120573

119910denotes the conversion rate from DCCs

to CSCs in the process of DCCs proliferationNote that time delay may play an important role in many

biological models As shown in [15] the maturation of indi-vidual cells may need a period of time 120591 that is the numberof these cells at time 119905 may depend on the population at aprevious time 119905 minus 120591 Under the assumption of equal discreteretarded cell proliferation model (1) can be modified to

d119909 (119905)d119905= 119909 (119905 minus 120591) 119891

1(119909 (119905 minus 120591) 119910 (119905 minus 120591))

d119910 (119905)d119905= 119910 (119905 minus 120591) 119891

2(119909 (119905 minus 120591) 119910 (119905 minus 120591))

(2)

Here 120591 is the time delay due to maturation timeFor (2) inspired by [16] we give the following average

fitness of the population

120601 = 119887119909119909 (119905 minus 120591) + 119887

119910119910 (119905 minus 120591) (3)

Thus the selection dynamics in long-term sphere culture ofCSCs can be written as

d119909 (119905)d119905= 119909 (119905 minus 120591) (119891

1(119909 (119905 minus 120591) 119910 (119905 minus 120591)) minus 120601)

d119910 (119905)d119905= 119910 (119905 minus 120591) (119891


(4)

For (4) let119873(119905) = 119909(119905) + 119910(119905) We have

d119873(119905)d119905= (1 minus 119873 (119905 minus 120591)) 120601 (5)

Therefore119873(119905) rarr 1 as 119905 rarr infin that is the total populationsize remains constant Hence 119909(119905) and 119910(119905) in (4) can beunderstood as the frequency of CSCs andDCCs respectivelyFurthermore since 119910(119905) can be replaced by 1 minus 119909(119905) system(4) describes only a single differential equation that is

d119909 (119905)d119905= 1198861+ 1198862119909 (119905 minus 120591) + 119886

31199092

(119905 minus 120591) ≜ 119891 (119909 (119905 minus 120591))

(6)

in which 1198861= 119887119910120573119910 1198862= 119887119909(1 minus 120573

119909) minus 119887119910(1 + 120573

119910) and 119886

3=

119887119910minus 119887119909 Note that (6) is the final selection dynamical model

with time delay in long-term sphere culture of CSCs

3 Results

31 Dynamic Analysis The objective of this subsection is toanalyze the dynamical behavior of (6) In order to explore theeffect of the delay we split this into two cases

311 Case of 120591 = 0 In this case we focus on the dynamicanalysis if the delay is nonexistent that is 120591 = 0 in (6) Westart by studying the existence of nonnegative equilibria inthe interval [0 1] Let 119891(119909) = 119886

1+1198862119909+11988631199092

= 0 Clearly thediscriminant of the quadratic equation is

Δ = 1198862

2minus 411988611198863= (119887119910minus 119887119909+ 119887119909120573119909minus 119887119910120573119910)

2

+ 4119887119909119887119910120573119909120573119910gt 0

(7)

Hence there are two different real roots for 119891(119909) = 0 if 1198863= 0

Furthermore since

119891 (1) = 1198861+ 1198862+ 1198863= minus119887119909120573119909lt 0 (8)

we know that there is a unique positive equilibrium 119909lowast isin(0 1) for (6) if 119886

3= 0 (see Figures 1(a) and 1(b)) When 119886

3= 0

(see Figure 1(c)) it is clear that there is only one positiveequilibrium

119909lowast

= minus

1198861

1198862

=

120573119909

120573119909+ 120573119910

lt 1 (9)

The combination of the above results and the phasediagram (see Figure 1(d)) of system (6) yields the followingresult

Proposition 1 For system (6) when 120591 = 0 a unique positiveequilibrium 119909lowast isin (0 1) always exists and it is globally asym-ptotically stable

312 Case of 120591 gt 0 In this case we focus on the dynamicanalysis if the delay is existent that is 120591 gt 0 in (6) Clearly theunique positive equilibrium 119909lowast isin (0 1) still remains for (6) inspite of the delay To study the stability of the equilibrium 119909lowastwe first translate 119909lowast to the origin Let

119909 = 119909 minus 119909lowast

(10)

Then (6) becomes after replacing 119909 by 119909 again

d119909 (119905)d119905= (1198862+ 21198863119909lowast

) 119909 (119905 minus 120591) + 11988631199092

(119905 minus 120591) (11)

The variational system of (11) at the origin is given by

d119909 (119905)d119905= (1198862+ 21198863119909lowast

) 119909 (119905 minus 120591) (12)

The characteristic equation of linear system (12) is given by

120582 minus (1198862+ 21198863119909lowast

) 119890minus120591120582

= 0 (13)

If we let 120582 = 120572 + 119894120573 then (13) becomes

120572 minus (1198862+ 21198863119909lowast

) 119890minus120591120572 cos120573120591 = 0

120573 + (1198862+ 21198863119909lowast

) 119890minus120591120572 sin120573120591 = 0

(14)

Discrete Dynamics in Nature and Society 3

119891(119909)

119909

11988611

119909lowast119874

(a)

119891(119909)

119909

1198861

1119909lowast119874

(b)

119891(119909)

119909

1198861

1119909lowast119874

(c)

119909lowast

119905

1

119909(119905)

119874

(d)

Figure 1 Illustrations of function image of 119891(119909) = 0 under different cases ((a) (b) and (c)) and the phase diagram (d) of system (6) Here(a) 1198863gt 0 (b) 119886

3lt 0 and (c) 119886

3= 0

By setting 120572 = 0 in (14) we have

cos120573120591 = 0

sin120573120591 = minus120573

1198862+ 21198863119909lowast

(15)

Solving the first algebraic equation in (15) we have

120573120591 =

120587

2

+ 2119899120587 119899 = 0 1 2 (16)

Substituting (16) into the second equation of (15) we have

120573 = minus (1198862+ 21198863119909lowast

) (17)

Now substituting (17) into (16) we have

120591119899= minus

120587

2 (1198862+ 21198863119909lowast)

minus

2119899120587

1198862+ 21198863119909lowast 119899 = 0 1 2 (18)

Next we compute 1205721015840(120591119899) Differentiating (13) with respect

to 120591 we have

d120582 (120591)d120591= minus

(1198862+ 21198863119909lowast

) 120582119890minus120591120582

1 + 120591 (1198862+ 21198863119909lowast) 119890minus120591120582

(19)

thus

1205721015840

(120591119899) = Re( d120582 (120591)

d120591

10038161003816100381610038161003816100381610038161003816120591=120591119899

) =

1205732

1 + 1205912

1198991205732gt 0 (20)

Therefore similar to [17] according to the results in [1819] or [15 Theorem 22] we can obtain the following resultson (6)

Proposition 2 Suppose 120591 gt 0 Then system (6) has a Hopfbifurcation at

120591 = 120591119899= minus

120587

2 (1198862+ 21198863119909lowast)

minus

2119899120587

1198862+ 21198863119909lowast 119899 = 0 1 2

(21)

Furthermore according to the results in [15 20 21] wehave the following

Proposition 3 Suppose 120591 gt 0 in (6) Then the unique positiveequilibrium 119909lowast isin (0 1) is stable if 0 lt 120591 lt 120591

0and unstable if

120591 gt 1205910

32 Numerical Simulations For model (1) we designed along-term sphere culture of humanbreast cancerMCF-7 stemcells [13] Based on the experimental data using an adap-tive Metropolis-Hastings (M-H) algorithm to carry out anextensive Markov-chain Monte-Carlo (MCMC) simulationwe obtained the estimated parameter values as follows

119887119909= 27506 times 10

minus1

119887119910= 32635 times 10

minus1

120573119909= 14407 times 10

minus2

120573119910= 23288 times 10

minus3

(22)

When retarded cell proliferation was considered basedon the induced selection dynamic model (6) and the experi-mental data in [13] using extensiveMCMC simulation againwe can obtain the estimated delay 120591 = 51401 (Figure 2)

Using the estimated values we can plot the best-fitsolution by fitting model (1) and (6) to the experimental data


0 2000 4000 6000 8000 1000002468

10

120591

(a)

1 2 3 4 5 6 7 8 90

200

400

600

120591

(b)

Figure 2MCMCanalysis of parameter 120591 based on (6) (22) and the experimental data in [13] (a) is the random series and (b) is its histogramThe algorithm ran for 104 iterations with a burn-in of 3000 iterations The initial conditions were 120591 = 09

0 20 40 60 80 100 120 140 1600

01

02

03

04

05

06

07

08

09

1

Time (days)

Freq

uenc

y of

CSC

s

Experimental dataData from model (1)Data from model (6)

Figure 3 Simulations of the dynamical behaviors of the frequencyof CSCs Experimental data are represented by open circlesThe bluedashed line denotes the best fit of model (1) and the red solid linedenotes the best fit of model (6)

respectively (Figure 3) From Figure 3 we find that there isa better simulation effect in model (6) than that in model(1) In fact the sum of squares of the deviations (SSD) in(1) is SSD

(1)= 72943 whereas SSD

(6)= 46387 times 10

minus2

in (6) Note that SSD(6)

is far less than SSD(1) These results

quantitatively confirmed that the induced selection dynamicmodel (6) with time delay can provide a better fitting effect inlong-term sphere culture of CSCs

4 Conclusions

In order to demonstrate the interesting facts about the struc-tural heterogeneity of cancer (the stable ratio between CSCsand DCCs) many studies have been reported because it ishelpful for the cancer community to elucidate the controversyabout the CSC hypothesis and the clone evolution theoryof cancer [10 13 14] and references cited therein In thepresent paper a selection dynamic model with time delay is

proposed and its dynamical behavior is studied Based onthe theoretical analysis and numerical simulations we canconclude the following predictive conclusions

(i) The maturation of individual cells may produce asignificant effect on the dynamic behavior of the selectiondynamics When the delay is nonexistent the frequency ofCSCs will tend to a stable size because the unique positiveequilibrium is globally asymptotically stable (Proposition 1)Conversely if the delay is existent the unique positiveequilibrium may not always maintain its stability and a Hopfbifurcation may be induced (Propositions 2 and 3) that is anoscillated phenomenonmay be induced by the maturation ofindividual cells

(ii) Since the induced selection dynamic model (6) withtime delay can provide a better fitting effect in long-termsphere culture of CSCs (Figure 3) it is reasonable to considerthe delay effect in the future study on the dynamics of non-genetic heterogeneity of clonal cell populations

Since mathematical models can be at best approximatethe behavior of real biological process the results presentedheremay extend those studies on the structural heterogeneityof cancer Note that distributed delay may be more tractableand realistic than discrete delay in the applications of biologyHence it is a worthwhile study in future work to betterunderstand these topics based on the idea of [15]

Acknowledgments

This work is partially supported by the National NaturalScience Foundation of China (nos 30872517 and 11271369)and the Natural Science Foundation Project of CQ CSTC(2010BB5020)

References

[1] T Reya S J Morrison M F Clarke and I L Weissman ldquoStemcells cancer and cancer stem cellsrdquo Nature vol 414 no 6859pp 105ndash111 2001

[2] J Marx ldquoMutant stem cells may seed cancerrdquo Science vol 301no 5638 pp 1308ndash1310 2003

[3] MAl-Hajj andM F Clarke ldquoSelf-renewal and solid tumor stemcellsrdquo Oncogene vol 23 no 43 pp 7274ndash7282 2004

[4] B K Abraham P Fritz M McClellan P Hauptvogel MAthelogou and H Brauch ldquoPrevalence of CD44+CD24-119897119900119908cells in breast cancer may not be associated with clinical


outcome but may favor distant metastasisrdquo Clinical CancerResearch vol 11 no 3 pp 1154ndash1159 2005

[5] MBalicH Lin L Young et al ldquoMost early disseminated cancercells detected in bone marrow of breast cancer patients havea putative breast cancer stem cell phenotyperdquo Clinical CancerResearch vol 12 no 19 pp 5615ndash5621 2006

[6] M S Wicha ldquoCancer stem cells and metastasis lethal seedsrdquoClinical Cancer Research vol 12 no 19 pp 5606ndash5607 2006

[7] S Huang ldquoNon-genetic heterogeneity of cells in developmentmore than just noiserdquo Development vol 136 no 23 pp 3853ndash3862 2009

[8] T Quinn and Z Sinkala ldquoDynamics of prostate cancer stemcells with diffusion and organism responserdquo BioSystems vol 96no 1 pp 69ndash79 2009

[9] S J Altschuler and L F Wu ldquoCellular heterogeneity dodifferences make a differencerdquo Cell vol 141 no 4 pp 559ndash5632010

[10] S Huang ldquoTumor progression chance and necessity in Dar-winian and Lamarckian somatic (mutationless) evolutionrdquoProgress in Biophysics and Molecular Biology vol 110 no 1 pp69ndash86 2012

[11] P Cammareri Y Lombardo M G Francipane S BonventreM Todaro and G Stassi ldquoIsolation and culture of colon Cancerstem cellsrdquoMethods in Cell Biology vol 86 pp 311ndash324 2008

[12] Y Zhong K Guan S Guo et al ldquoSpheres derived from thehuman SK-RC-42 renal cell carcinoma cell line are enriched incancer stem cellsrdquo Cancer Letters vol 299 no 2 pp 150ndash1602010

[13] T Peng M Qinghua T Zhenning W Kaifa and J Jun ldquoLong-term sphere culture cannotmaintain a high ratio of Cancer stemcells a mathematical model and experimentrdquo PLoS ONE vol 6no 11 Article ID e25518 2011

[14] P B Gupta C M Fillmore G Jiang et al ldquoStochastic statetransitions give rise to phenotypic equilibrium in populationsof cancer cellsrdquo Cell vol 146 no 4 pp 633ndash644 2011

[15] Y Yuan and J Belair ldquoStability andHopf bifurcation analysis forfunctional differential equation with distributed delayrdquo SIAMJournal on Applied Dynamical Systems vol 10 no 2 pp 551ndash581 2011

[16] M A Nowak Evolutionary Dynamics Exploring the Equationsof Life The Belknap Press of Harvard University Press Cam-bridge Mass USA 2006

[17] K Wang N Zhang and D Niu ldquoPeriodic oscillations in a spa-tially explicit model with delay effect for vegetation dynamics infreshwater marshesrdquo Journal of Biological Systems vol 19 no 2pp 131ndash147 2011

[18] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977

[19] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics Academic Press Boston Mass USA1993

[20] H I Freedman J W-H So and P Waltman ldquoCoexistence ina model of competition in the chemostat incorporating discretedelaysrdquo SIAM Journal on AppliedMathematics vol 49 no 3 pp859ndash870 1989

[21] T Zhao ldquoGlobal periodic solutions for a differential delaysystem modeling a microbial population in the chemostatrdquoJournal of Mathematical Analysis and Applications vol 193 no1 pp 329ndash352 1995

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model to describe the interactive growth of the CSCs andDCCs in a long-term sphere culture

d119909 (119905)d119905=119887119909(1 minus 120573

119909) 119909 (119905) + 119887

119910120573119910119910 (119905) ≜ 119909 (119905) 119891

1(119909 (119905) 119910 (119905))

d119910 (119905)d119905=119887119909120573119909119909 (119905) + 119887

119910(1 minus 120573

119910) 119910 (119905) ≜ 119910 (119905) 119891

2(119909 (119905) 119910 (119905))

(1)

Here the constants 119887119909 119887119910are called the net birth rate or intrin-

sic growth rate of population119909 119910 respectively120573119909denotes the

conversion rate from CSCs to DCCs in the process of CSCsproliferation and 120573

119910denotes the conversion rate from DCCs

to CSCs in the process of DCCs proliferationNote that time delay may play an important role in many

biological models As shown in [15] the maturation of indi-vidual cells may need a period of time 120591 that is the numberof these cells at time 119905 may depend on the population at aprevious time 119905 minus 120591 Under the assumption of equal discreteretarded cell proliferation model (1) can be modified to

d119909 (119905)d119905= 119909 (119905 minus 120591) 119891

1(119909 (119905 minus 120591) 119910 (119905 minus 120591))

d119910 (119905)d119905= 119910 (119905 minus 120591) 119891

2(119909 (119905 minus 120591) 119910 (119905 minus 120591))

(2)

Here 120591 is the time delay due to maturation timeFor (2) inspired by [16] we give the following average

fitness of the population

120601 = 119887119909119909 (119905 minus 120591) + 119887

119910119910 (119905 minus 120591) (3)

Thus the selection dynamics in long-term sphere culture ofCSCs can be written as

d119909 (119905)d119905= 119909 (119905 minus 120591) (119891


d119910 (119905)d119905= 119910 (119905 minus 120591) (119891


(4)

For (4) let119873(119905) = 119909(119905) + 119910(119905) We have

d119873(119905)d119905= (1 minus 119873 (119905 minus 120591)) 120601 (5)

Therefore119873(119905) rarr 1 as 119905 rarr infin that is the total populationsize remains constant Hence 119909(119905) and 119910(119905) in (4) can beunderstood as the frequency of CSCs andDCCs respectivelyFurthermore since 119910(119905) can be replaced by 1 minus 119909(119905) system(4) describes only a single differential equation that is

d119909 (119905)d119905= 1198861+ 1198862119909 (119905 minus 120591) + 119886

31199092

(119905 minus 120591) ≜ 119891 (119909 (119905 minus 120591))

(6)

in which 1198861= 119887119910120573119910 1198862= 119887119909(1 minus 120573

119909) minus 119887119910(1 + 120573

119910) and 119886

3=

119887119910minus 119887119909 Note that (6) is the final selection dynamical model

with time delay in long-term sphere culture of CSCs

3 Results

31 Dynamic Analysis The objective of this subsection is toanalyze the dynamical behavior of (6) In order to explore theeffect of the delay we split this into two cases

311 Case of 120591 = 0 In this case we focus on the dynamicanalysis if the delay is nonexistent that is 120591 = 0 in (6) Westart by studying the existence of nonnegative equilibria inthe interval [0 1] Let 119891(119909) = 119886

1+1198862119909+11988631199092

= 0 Clearly thediscriminant of the quadratic equation is

Δ = 1198862

2minus 411988611198863= (119887119910minus 119887119909+ 119887119909120573119909minus 119887119910120573119910)

2

+ 4119887119909119887119910120573119909120573119910gt 0

(7)

Hence there are two different real roots for 119891(119909) = 0 if 1198863= 0

Furthermore since

119891 (1) = 1198861+ 1198862+ 1198863= minus119887119909120573119909lt 0 (8)

we know that there is a unique positive equilibrium 119909lowast isin(0 1) for (6) if 119886

3= 0 (see Figures 1(a) and 1(b)) When 119886

3= 0

(see Figure 1(c)) it is clear that there is only one positiveequilibrium

119909lowast

= minus

1198861

1198862

=

120573119909

120573119909+ 120573119910

lt 1 (9)

The combination of the above results and the phasediagram (see Figure 1(d)) of system (6) yields the followingresult

Proposition 1 For system (6) when 120591 = 0 a unique positiveequilibrium 119909lowast isin (0 1) always exists and it is globally asym-ptotically stable

312 Case of 120591 gt 0 In this case we focus on the dynamicanalysis if the delay is existent that is 120591 gt 0 in (6) Clearly theunique positive equilibrium 119909lowast isin (0 1) still remains for (6) inspite of the delay To study the stability of the equilibrium 119909lowastwe first translate 119909lowast to the origin Let

119909 = 119909 minus 119909lowast

(10)

Then (6) becomes after replacing 119909 by 119909 again

d119909 (119905)d119905= (1198862+ 21198863119909lowast

) 119909 (119905 minus 120591) + 11988631199092

(119905 minus 120591) (11)

The variational system of (11) at the origin is given by

d119909 (119905)d119905= (1198862+ 21198863119909lowast

) 119909 (119905 minus 120591) (12)

The characteristic equation of linear system (12) is given by

120582 minus (1198862+ 21198863119909lowast

) 119890minus120591120582

= 0 (13)

If we let 120582 = 120572 + 119894120573 then (13) becomes

120572 minus (1198862+ 21198863119909lowast

) 119890minus120591120572 cos120573120591 = 0

120573 + (1198862+ 21198863119909lowast

) 119890minus120591120572 sin120573120591 = 0

(14)


119891(119909)

119909

11988611

119909lowast119874

(a)

119891(119909)

119909

1198861

1119909lowast119874

(b)

119891(119909)

119909

1198861

1119909lowast119874

(c)

119909lowast

119905

1

119909(119905)

119874

(d)


3lt 0 and (c) 119886

3= 0


cos120573120591 = 0

sin120573120591 = minus120573

1198862+ 21198863119909lowast

(15)


120573120591 =

120587

2

+ 2119899120587 119899 = 0 1 2 (16)


120573 = minus (1198862+ 21198863119909lowast

) (17)


120591119899= minus

120587

2 (1198862+ 21198863119909lowast)

minus

2119899120587

1198862+ 21198863119909lowast 119899 = 0 1 2 (18)


to 120591 we have

d120582 (120591)d120591= minus

(1198862+ 21198863119909lowast

) 120582119890minus120591120582

1 + 120591 (1198862+ 21198863119909lowast) 119890minus120591120582

(19)

thus

1205721015840

(120591119899) = Re( d120582 (120591)

d120591

10038161003816100381610038161003816100381610038161003816120591=120591119899

) =

1205732

1 + 1205912

1198991205732gt 0 (20)



120591 = 120591119899= minus

120587

2 (1198862+ 21198863119909lowast)

minus

2119899120587

1198862+ 21198863119909lowast 119899 = 0 1 2

(21)



0and unstable if

120591 gt 1205910


119887119909= 27506 times 10

minus1

119887119910= 32635 times 10

minus1

120573119909= 14407 times 10

minus2

120573119910= 23288 times 10

minus3

(22)




0 2000 4000 6000 8000 1000002468

10

120591

(a)

1 2 3 4 5 6 7 8 90

200

400

600

120591

(b)


0 20 40 60 80 100 120 140 1600

01

02

03

04

05

06

07

08

09

1

Time (days)

Freq

uenc

y of

CSC

s





(6)= 46387 times 10

minus2




4 Conclusions






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




119891(119909)

119909

11988611

119909lowast119874

(a)

119891(119909)

119909

1198861

1119909lowast119874

(b)

119891(119909)

119909

1198861

1119909lowast119874

(c)

119909lowast

119905

1

119909(119905)

119874

(d)


3lt 0 and (c) 119886

3= 0


cos120573120591 = 0

sin120573120591 = minus120573

1198862+ 21198863119909lowast

(15)


120573120591 =

120587

2

+ 2119899120587 119899 = 0 1 2 (16)


120573 = minus (1198862+ 21198863119909lowast

) (17)


120591119899= minus

120587

2 (1198862+ 21198863119909lowast)

minus

2119899120587

1198862+ 21198863119909lowast 119899 = 0 1 2 (18)


to 120591 we have

d120582 (120591)d120591= minus

(1198862+ 21198863119909lowast

) 120582119890minus120591120582

1 + 120591 (1198862+ 21198863119909lowast) 119890minus120591120582

(19)

thus

1205721015840

(120591119899) = Re( d120582 (120591)

d120591

10038161003816100381610038161003816100381610038161003816120591=120591119899

) =

1205732

1 + 1205912

1198991205732gt 0 (20)



120591 = 120591119899= minus

120587

2 (1198862+ 21198863119909lowast)

minus

2119899120587

1198862+ 21198863119909lowast 119899 = 0 1 2

(21)



0and unstable if

120591 gt 1205910


119887119909= 27506 times 10

minus1

119887119910= 32635 times 10

minus1

120573119909= 14407 times 10

minus2

120573119910= 23288 times 10

minus3

(22)




0 2000 4000 6000 8000 1000002468

10

120591

(a)

1 2 3 4 5 6 7 8 90

200

400

600

120591

(b)


0 20 40 60 80 100 120 140 1600

01

02

03

04

05

06

07

08

09

1

Time (days)

Freq

uenc

y of

CSC

s





(6)= 46387 times 10

minus2




4 Conclusions






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




0 2000 4000 6000 8000 1000002468

10

120591

(a)

1 2 3 4 5 6 7 8 90

200

400

600

120591

(b)


0 20 40 60 80 100 120 140 1600

01

02

03

04

05

06

07

08

09

1

Time (days)

Freq

uenc

y of

CSC

s





(6)= 46387 times 10

minus2




4 Conclusions






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



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