Mean-field descriptions of collective migration with strong adhesion

PHYSICAL REVIEW E 85 051922 (2012)

Mean-field descriptions of collective migration with strong adhesion

Stuart T Johnston1 Matthew J Simpson12 and Ruth E Baker3

1School of Mathematical Sciences Queensland University of Technology Brisbane Australia2Tissue Repair and Regeneration Program Institute of Health and Biomedical Innovation (IHBI)

Queensland University of Technology Brisbane Australia3Centre for Mathematical Biology Mathematical Institute University of Oxford 24-29 St Gilesrsquo Oxford OX1 3LB United Kingdom

(Received 21 March 2012 revised manuscript received 11 May 2012 published 29 May 2012)

Random walk models based on an exclusion process with contact effects are often used to represent collectivemigration where individual agents are affected by agent-to-agent adhesion Traditional mean-field representationsof these processes take the form of a nonlinear diffusion equation which for strong adhesion does not predict theaveraged discrete behavior We propose an alternative suite of mean-field representations showing that collectivemigration with strong adhesion can be accurately represented using a moment closure approach

DOI 101103PhysRevE85051922 PACS number(s) 8717Rt 8717Jj

I INTRODUCTION

Microscopic transport processes modulated by adhesionare important for many applications including the study ofbiomolecules [1] granular media [2] and biological cells[34] For these applications it is essential to understand howindividual-level details of the adhesion mechanism lead topopulation-level properties that govern system-wide behaviorTherefore accurate mean-field models of these mechanismsare essential Here we study a discrete motility mechanismbased on an exclusion process [5] with contact effects Thesemodels have been used to study the migration of gliomacells [67] breast cancer cells [8] and wound healing processes[9] Anguige and Schmieser [10] were the first to derive amean-field description of such a discrete model with othersreported subsequently [681112] These previous studiesreported mean-field representations in the form of a nonlineardiffusion partial differential equation (pde) [12]

The form of the nonlinear diffusivity function reflectsthe physical behavior in the discrete model [10ndash12] Whencontact enhances migration the nonlinear diffusivity func-tion is always positive [6111314] When contacts reducemigration (ie adhesion) the nonlinear diffusivity functioncan become negative when contact effects dominate [81011]The transition from positive to negative nonlinear diffusivityis associated with clustering in the discrete simulations [13]under these conditions existing mean-field models do notpredict the average behavior of the discrete process [61113]For example both Deroulers et al [6] and Fernando et al[11] showed that the traditional mean-field pde fails to makeaccurate predictions when contact effects became sufficientlystrong Fernando et al [11] provided further insight byproposing a heuristic measure to predict the parameter regimewhere the mean-field pde was either accurate or inaccurateAlthough insightful this previous study provided no means ofmaking accurate mean-field predictions when contact effectswere strong

Currently it is impossible to quantify how and why thetraditional pde representation fails to predict the averageddiscrete behavior as these models provide no way of examiningthe validity of the assumptions underlying the traditionalmean-field pde Here we address these issues by showing thatan adhesive motility mechanism can be described by a suite

of three mean-field models We show that the traditional pdeinvokes two key assumptions namely

(1) that effects of O(3) and smaller are neglected in thelimit that rarr 0 where is the lattice spacing and

(2) that the occupancy status of lattice sites are assumed tobe independent so that correlation effects are ignored

Two alternative mean-field models are developed that relaxboth these assumptions independently Comparing averageddiscrete simulation results to the predictions of the suite ofthree mean-field models highlights the role of correlationeffects and shows that it is possible to make accurate mean-field predictions with strong adhesion using a moment closureapproach

II DISCRETE MECHANISM

We consider a one-dimensional lattice with spacing Sites are indexed by l and have location x = l Timeis uniformly discretized with time step τ and a randomsequential update method is used to simulate the process [15]During each time step agents attempt to step to nearestneighbor sites provided that the target site is vacant Motilityevents that would place an agent on an occupied site areaborted Motility events are regulated by contact effects thatrepresent agent-to-agent adhesion [10] by altering the motilityusing an adhesion parameter σ isin [minus11] For example if weconsider the schematic illustration in Fig 1 the agent at sitel minus 1 would attempt to move to the vacant site l minus 2 withprobability (1 minus σ )2 per time step when site l is occupiedAlternatively this event would occur with probability 12per time step if site l were vacant Setting σ gt 0 representsadhesion whereas setting σ lt 0 represents repulsion [11]

III MEAN-FIELD REPRESENTATIONS

We define the lattice variable φl isin 0l Cl to representthe state of the lth site so that φl = 0l indicates that sitel is vacant and φl = Cl indicates that site l is occupiedAveraging the occupancy of each site over many identicallyprepared realizations gives cl isin [01] [611] In our notationupper case Cl represents the occupancy of the lth site ina single realization whereas lower case cl represents theaverage occupancy where the average is constructed over a

051922-11539-3755201285(5)051922(10) copy2012 American Physical Society

JOHNSTON SIMPSON AND BAKER PHYSICAL REVIEW E 85 051922 (2012)

l l+1l-1l-2 l+2 l+3 l+4 l+5l-3l-4

1212(1-σ)2 0

FIG 1 (Color online) The random walk takes place on a one-dimensional lattice where each site can be occupied by at most oneagent An isolated agent steps in the positive or negative x directionwith probability 12 per computational time step For example theagent at site l + 3 would step to site l + 2 with probability 12or to l + 4 with probability 12 Contact effects alter the motilityprobability for example in the configuration shown the agent atsite l minus 1 would step to l minus 2 with probability (1 minus σ )2 where σ isin[minus11] represents the contact effect The agent at site l minus 1 wouldstep to site l with probability 0 since the target site is occupied

large number of identically prepared realizations of the sameprocess We now introduce three ways to approximate cl bymaking different assumptions about the underlying discreteprocess

A Partial differential equation representation

To connect the discrete mechanism with a pde we form adiscrete conservation statement describing δcl the change inaverage occupancy of site l per time step The conservationequation can be written as

δcl = 12 [clminus1(1 minus cl)(1 minus σclminus2) + cl+1(1 minus cl)(1 minus σcl+2)]

minus 12 [cl(1 minus cl minus 1)(1 minus σcl+1)

+ cl(1 minus cl+1)(1 minus σcl minus 1)] (1)

where positive terms on the right of Eq (1) represent events thatwould place agents at site l and negative terms represent eventsthat would remove agents from site l The discrete conservationstatement is related to a pde as rarr 0 and τ rarr 0 and cl isidentified as a continuous variable c(xt) [61011] Expandingall terms in Eq (1) in a truncated Taylor series about site lneglecting terms of O(3) and higher [61011] and dividingthe resulting expression by τ we take limits as rarr 0 andτ rarr 0 with the ratio (2τ ) held constant [16] to obtain

partc

partt= D0

part

partx

[D(c)

partc

partx

] (2)

where D0 = limτrarr0(2)(2τ ) is the free-agent diffusivityand the nonlinear diffusivity function is given by [1011]

D(c) = 1 minus σc(4 minus 3c) (3)

Two key assumptions lead to Eq (2) First we assume termsof O(3) and smaller can be neglected Second we assumethe average occupancies of sites to be independent so that forexample the net averaged probability of a transition from site l

to l + 1 is proportional to (1 minus cl+1)(1 minus σclminus1) This impliesthat the occupancy of sites l + 1 and l minus 1 are independentwhich in general is untrue [1718] Without further analysisit is impossible to deduce how these two assumptions controlthe net error associated with Eq (2) We now introduce twoalternative mean-field models that systematically relax bothassumptions

B Ordinary differential equation representation

To avoid neglecting terms of O(3) and smaller as rarr 0we retain the spatial structure of the random walk in Eq (1)by identifying discrete values of cl with a continuous variablecl(t) Dividing Eq (1) by τ and considering the limit as τ rarr 0gives a system of ordinary differential equations (odes)

dcl

dt= 1

2[clminus1(1 minus cl)(1 minus σclminus2) + cl+1(1 minus cl)(1 minus σcl+2)]

minus 1

2[cl(1 minus clminus1)(1 minus σcl+1) + cl(1 minus cl+1)(1 minus σclminus1)]

(4)

for each site l We note that Eq (4) still makes the independenceassumption and we now develop a third mean-field model thatremoves this assumption

C Moment closure representation

We use k-point distribution functions ρ(k) (k = 123 )to describe the averaged occupancies of k tuplets of lattice sites[171920] For k = 1 the distribution function is a univariatedistribution describing the average density of agents on site l sothat ρ(1)(Cl) = cl For k = 2 the bivariate distribution functioncan be defined in terms of correlation functions [1719] whichcan be written as

F (lm) = ρ(2)(ClCm)

ρ(1)(Cl)ρ(1)(Cm) (5)

where l = m These correlation functions allow us to relaxthe independence assumptions inherent in Eqs (2) and (4)Setting F (lm) equiv 1 indicates that the occupancies of sites l

and m are independent Instead we avoid this assumption byallowing F (lm) to evolve as part of the solution [17] Withthese definitions we have

dcl

dt= 1

2[ρ(3)(0lminus2Clminus10l) + (1 minus σ )ρ(3)(Clminus2Clminus10l)]

+ 1

2[ρ(3)(0l Cl+10l+2) + (1 minus σ )ρ(3)(0l Cl+1Cl+2)]

minus 1

2[ρ(3)(0lminus1Cl0l+1) minus (1 minus σ )ρ(3)(0lminus1ClCl+1)]

minus 1

2[ρ(3)(0lminus1Cl0l+1) minus (1 minus σ )ρ(3)(Clminus1Cl0l+1)]

(6)

Positive terms on the right of Eq (6) represent events thatwould place an agent at site l whereas negative terms onthe right of Eq (6) represent events that would remove anagent from site l To simplify Eq (6) we apply a summationrule [17] to rewrite the unbiased ρ(3) terms as equivalent ρ(2)

terms The Kirkwood superposition approximation (KSA) isthen used to rewrite the remaining ρ(3) terms as combinationsof ρ(2) terms The KSA is a moment closure approximationthat has been used in many applications including ecology[21ndash23] physical chemistry [24] disease biology [2526] anddiffusion-mediated reactions [27] The KSA can be written as

ρ(3)(φlφmφn) = ρ(2)(φlφm)ρ(2)(φlφn)ρ(2)(φmφn)

ρ(1)(φl)ρ(1)(φm)ρ(1)(φn) (7)

051922-2

MEAN-FIELD DESCRIPTIONS OF COLLECTIVE PHYSICAL REVIEW E 85 051922 (2012)

Combining Eq (7) with the simplified version of Eq (6) givesdcl

dt= 1

2[cl+1 minus 2c2 + clminus1] minus σ

2(1 minus cl)clminus2clminus1[1 minus clF (l minus 2l)][1 minus clF (l minus 1l)]F (l minus 2l minus 1)

minus σ

2(1 minus cl)cl+1cl+2[1 minus clF (ll + 1)][1 minus clF (ll + 2)]F (l + 1l + 2) + σ

2(1 minus clminus1)clcl+1[1 minus clminus1F (l minus 1l)]

times [1 minus clminus1F (l minus 1l + 1)]F (ll + 1) + σ

2(1 minus cl+1)clminus1cl[1 minus cl+1F (l minus 1l + 1)][1 minus cl+1F (ll + 1)]F (ll minus 1)

(8)

To solve Eq (8) we require a model for the evolution of F (ll + 1) and F (ll + 2) which are correlation functions quantifying thedegree to which the occupancy of the pairs of sites (ll + 1) and (ll + 2) are correlated To solve for these terms we considerthe time rate of change of certain two-point distribution functions which are related to higher order distribution functionsleading to an infinite system of equations that we close using the KSA [1718] For example the evolution of ρ(2)(ClCl+1) isgiven by

dρ(2)(ClCl+1)

dt= 1

2[ρ(4)(0lminus2Clminus10l Cl+1) + (1 minus σ )ρ(4)(Clminus2Clminus10l Cl+1)] + 1

2[ρ(4)(Cl0l+1Cl+20l+3)

+ (1 minus σ )ρ(4)(Cl0l+1Cl+20l+3)] minus 1

2[(1 minus σ )ρ(3)(0lminus1ClCl+1) + (1 minus σ )ρ(3)(ClCl+10l+2)] (9)

To simplify Eq (9) we apply a summation rule [17] to rewrite the unbiased ρ(4) terms as equivalent ρ(3) terms Then we usethe summation rule again to write some of the resulting ρ(3) terms as equivalent expressions depending only on ρ(2) terms Thisgives us

dρ(2)(ClCl+1)

dt= 1

2[ρ(2)(Clminus1Cl+1) + ρ(2)(ClCl+2) minus 2ρ(2)(ClCl+1)] minus σ

2[ρ(4)(Clminus2Clminus10l Cl+1) + ρ(4)(Cl0l+1Cl+2Cl+3)]

+ σ

2[ρ(3)(0lminus1ClCl+1) + ρ(3)(ClCl+10l+2)] (10)

We now use the KSA to reduce the ρ(3) and ρ(4) terms in Eq (10) For the ρ(4) terms we use [24]

ρ(4)(φlφmφnφo) = ρ(3)(φlφmφn)ρ(3)(φlφmφo)ρ(3)(φlφnφo)ρ(3)(φmφnφo)ρ(1)(φl)ρ(1)(φm)ρ(1)(φn)ρ(1)(φo)

ρ(2)(φlφm)ρ(2)(φlφn)ρ(2)(φlφo)ρ(2)(φmφn)ρ(2)(φmφo)ρ(2)(φnφo) (11)

The ρ(3) terms appearing in Eq (11) can then be reduced into ρ(2) terms using Eq (7)At this stage there are two possible ways to simplify Eq (10) Either we(1) introduce the KSA directly into Eq (10) to express the ρ(3) and ρ(4) terms as ρ(2) terms or(2) apply the summation rule again to further simplify those terms in Eq (10) that are proportional to σ

Following the second approach we obtain

dρ(2)(ClCl+1)

dt= 1


2[ρ(2)(Clminus1Cl+1) + ρ(2)(ClCl+2) minus 2ρ(2)(ClCl+1)]

+ σ

2[ρ(4)(0lminus2Clminus10l Cl+1) + ρ(4)(Cl0l+1Cl+20l+3)] (12)

We apply the KSA to Eq (12) and rewrite everything in terms of the correlation functions to obtain

dF (ll + 1)

dt= minusF (11 + 1)

[dcl+1

dt

1

cl+1+ dcl

dt

1

cl

]+ 1

2

[clminus1

cl

F (l minus 1l + 1) + cl+2

cl+1F (ll + 2) minus 2F (ll + 1)

]

minus σ

2

[clminus1

cl



]

+ σ

2

[clminus1

cl(1 minus clminus2)2(1 minus cl)2F (l minus 1l + 1) [1 minus clminus2 minus cl + clclminus2F (l minus 2l)]

times [1 minus clminus2F (l minus 2l minus 1)] [1 minus clminus2F (l minus 2l + 1)] [1 minus clF (l minus 1l)] [1 minus clF (ll + 1)]

]

+ σ

2

[cl+2

cl+1(1 minus cl+1)2(1 minus cl+3)2F (ll + 2)[1 minus cl+1 minus cl+3 + cl+1cl+3F (l + 1l + 3)]

times [1 minus cl+1F (ll + 1)][1 minus cl+3F (ll + 3)][1 minus cl+1F (l + 1l + 2)][1 minus cl+3F (l + 2l + 3)]

] (13)

051922-3


To solve the moment closure model we use the same initialcondition c(x0) as in the discrete simulations and set theinitial values of F (lm) equiv 1 for all m = l + 1l + 2l + 3

and for all all lattice sites l [18] While it is possible inprinciple to solve F (lm) for all values of m to cover theperiodic domain it is more practical to solve a truncatedsystem F (lm) for m = l + 1l + 2 M assuming thatF (lM + 1) equiv 1 We did this iteratively by solving for cl F (ll + 1) and setting F (ll + 2) equiv 1 and then separatelysolving for cl F (ll + 1) F (ll + 2) and setting F (ll + 3) equiv1 These two approaches yielded results for c(xt) that wereindistinguishable Therefore we take the simplest possibleapproach and report results corresponding to the solution ofcl and F (ll + 1) with F (ll + 2) equiv 1 We also remark thatas we pointed out earlier it is possible to simplify Eq (10)in an alternative way by applying the KSA directly to the ρ(3)

and ρ(4) terms in that equation without using the summationrule For completeness we also resolved all problems in thiswork using the alternative expression for dF (ll + 1)dt andfound that both approaches yielded c(xt) profiles that wereindistinguishable

IV RESULTS AND DISCUSSION

We consider a lattice with 1 x 1000 and an initialdistribution of agents given by

c(x0) =

⎧⎪⎨⎪⎩

01 1 x lt 480

10 481 x 520

01 521 lt x 1000

(14)

Periodic boundary conditions are imposed and simulationsare performed for a range of σ including (minus100minus095

minus090 090095100) In each case we estimate thedensity profile using 1000 identically prepared realizationsResults in Figs 2 and 3 are given at t = 1000 and t = 5000 re-spectively Snapshots are shown for modest (σ = 065) strong(σ = 080) and extreme (σ = 095) adhesion We show 20identically prepared realizations of the same stochastic processwhich illustrate the effects of adhesion since clustering occurswhen adhesion dominates [Figs 2(b) and 3(b)] The densityprofiles in the central region of the lattice are compared withthe solutions of Eqs (2) (4) and (8) The numerical solutionof Eq (2) is obtained with a finite difference approximationwith constant grid spacing δx and implicit Euler stepping withconstant time steps δt [28] Picard linearization with absoluteerror tolerance ε is used to solve the resulting nonlinearalgebraic systems The numerical solutions of Eqs (4) and (8)are obtained using a fourth order Runge-Kutta method withconstant time step δt [18] All numerical results presented inthis paper are obtained using values of δx δt and ε chosento be sufficiently small so that the numerical results are gridindependent

For all cases of extreme (σ = 095) and strong (σ = 080)adhesion shown in Figs 2 and 3 the solution of Eq (2)is discontinuous [Figs 2(d) 2(j) 3(d) and 3(j)] Thesediscontinuities are associated with D(c) becoming negative fora region of c [101129] In this regime the pde fails to predictthe discrete profiles which appear to be smooth For modest(σ = 065) adhesion the solution of Eq (2) remains smooth

since D(c) gt 0 [Figs 2(p) and 3(p)] For modest adhesionthe accuracy of Eq (2) is much higher relative to the strong(σ = 080) and extreme (σ = 095) adhesion cases AlthoughEq (2) performs better for σ = 065 we still observe thatEq (2) slightly overestimates the peak density at t = 1000[Fig 2(p)]

When D(c) becomes negative for a region of c the solutionof Eq (2) is qualitatively different from the solution whenD(c) is always positive When D(c) is always positiveEq (2) is uniformly parabolic and satisfies the usual maximumprinciple This means that the solution is bounded by the initialcondition so that in our case c(xt) 1 for all t gt 0 [2930]Conversely when D(c) becomes negative for a region ofc Eq (2) is not uniformly parabolic and does not satisfythe usual maximum principle This means that c(xt) maybecome greater than the initial condition as the profile evolves[Figs 2(d) and 3(d)] Similar behavior has been observedpreviously in a different context DiCarlo [31] used a nonlineardiffusion equation called Richardsrsquo equation to study fluidflow through a partially saturated porous medium This previ-ous work showed that the infiltration front was monotone andnever increased above the long-term saturation level wheneverthe nonlinear diffusivity function was always positive Similarto our results DiCarlo showed that when the nonlineardiffusivity function contained a negative region the infiltrationfront became nonmonotone and the saturation level at theleading edge increased above the long-term saturation levelmeaning that the governing equation no longer satisfied theusual maximum principle

Comparing the averaged discrete profiles and the solutionof Eq (4) indicates that this model predicts smooth profileshowever these profiles do not accurately predict the discretedensity data for strong (σ = 080) and extreme (σ = 095)adhesion [Figs 2(e) 2(k) 3(e) and 3(k)] Alternatively thesolution of Eq (8) predicts smooth profiles that are accurateeven for strong (σ = 080) and extreme (σ = 095) adhesion[Figs 2(f) 2(l) 3(f) and 3(l)] These results provide us with aqualitative indication of the relative roles of the assumptionsunderlying Eq (2) We see that Eq (4) without truncationprovides a modest improvement over Eq (2) whereas Eq (8)with no truncation or independence assumptions provides amajor improvement relative to Eq (2) This indicates thatthe key assumption leading to the failure of Eq (2) is theindependence assumption

The moment closure model Eq (8) also provides us witha quantitative measure of the role of correlation effectsthrough the correlation functions shown in Figs 2(s) 2(t)3(s) and 3(t) Our results show that F (ll + 1) increaseswith σ confirming that correlation effects increase withincreasing adhesion and we see that the continuum F (ll + 1)profiles predict the discrete values quite well at both t = 1000[Fig 2(s)] and t = 5000 [Fig 3(s)] We also present discreteestimates of F (ll + 2) [Figs 2(t) and 3(t)] which are neglectedin our moment closure results since we set F (ll + 2) = 1Comparing profiles of F (ll + 1) and F (ll + 2) show thatnearest neighbor correlation effects are more pronounced thannext nearest neighbor correlation effects Our neglect of nextnearest neighbor correlation effects in the moment closuremodel appears reasonable given the quality of the matchbetween the discrete data and the solution of Eq (8)

051922-4


Modest adhesion σ = 065

Strong adhesion σ = 080

Extreme Adhesion σ = 095

σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o)1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

000(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c)

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

000

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c

FIG 2 (Color online) Mean-field and discrete results for a range of adhesive strengths (a)ndash(f) extreme adhesion (σ = 095) (g)ndash(l) strongadhesion (σ = 080) and (m)ndash(r) modest adhesion (σ = 065) (a) (b) (g) (h) (m) (n) For each adhesive strength two snapshots of thediscrete process are shown at t = 0 and t = 1000 respectively All discrete results correspond to = τ = 1 simulations are performedon a lattice with 1 x 1000 and periodic boundary conditions Discrete snapshots show 20 identically prepared realizations of the sameone-dimensional process in the region 401 x 600 (d) (j) (p) Comparisons of averaged density profiles (red) the initial condition (blackdashed) and the solution of Eq (2) (blue) (e) (k) (q) Comparisons of averaged density profiles (red) the initial condition (black dashed)and the solution of Eq (4) (blue) (f) (l) (r) Comparisons of averaged density profiles (red) the initial condition (black dashed) and thesolution of Eq (8) (blue) All discrete simulation results and mean-field solutions were obtained using periodic boundary conditions (c)(i) (o) Show the nonlinear diffusivity function D(c) = 1 minus σc(4 minus 3c) associated with Eq (2) Results for extreme (σ = 095) and strong(σ = 080) adhesion show that D(c) becomes negative in some interval c isin [c1c2] while results for the modest adhesion (σ = 065) show thatD(c) gt 0 for all c isin [01] (s) (t) Continuum (blue) and discrete (red) profiles of F (ll + 1) and F (ll + 2) respectively In each plot profilesof the correlation function are given for extreme (σ = 095) strong (σ = 080) and modest adhesion (σ = 065) with the arrow showing thedirection of increasing σ (u) The error profile E as a function of the adhesion parameter σ isin [minus11] at t = 1000 Error profiles are given forEqs (2) (blue dashed) (4) (blue) and (8) (red) All numerical solutions of Eq (2) correspond to δx = 02 δt = 001 and ε = 1 times 10minus6 Allnumerical solutions of Eqs (4) and (8) correspond to δt = 005

051922-5





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o) 0(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460C 1002 04 06 08

D(C)

-04

10

00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-6


To quantify the accuracy of Eqs (2) (4) and (8) we usean error norm given by

E = 1

100

l=550suml=451

[cl minus MF (xt)]2 (15)

where MF (xt) is the density predicted by one of Eqs (2)and (4) or (8) and cl is the average density at site l fromthe averaged discrete simulations We calculate E using sitesin the region 451 l 550 since the details of the evolveddensity profiles in Figs 2 and 3 are localized in this regionFigures 2(u) and 3(u) compare the accuracy of Eqs (2) (4) and(8) for the entire range of the adhesion parameter σ isin [minus11]showing that the error varies over two orders of magnitudeFor all cases of repulsive motion (σ lt 0) and mildly adhesivemotion (0 lt σ lt 05) Eqs (2) (4) and (8) perform similarlywe see that the solution of each mean-field model accuratelymatches the discrete profiles This is is consistent withprevious research [11] For modest to extreme adhesion(050 σ 10) Eqs (2) and (4) become very inaccuratewhile Eq (8) continues to make accurate predictions for allσ isin [minus11]

Comparing the performance of Eqs (2) (4) and (8) inFig 2 at t = 1000 with the results in Fig 3 at t = 5000indicates that the same qualitative trends are apparent at bothtime points The profiles at t = 1000 (Fig 2) for extremeadhesion (σ = 095) and strong adhesion (σ = 080) showthat the density profiles have not changed much from theinitial distribution while the results for moderate adhesion(σ = 065) show that the density profile has spread out muchfurther along the lattice by t = 1000 The profiles at t = 5000(Fig 3) for strong adhesion (σ = 080) show that the densityprofile has spread much further across the lattice and theresults for moderate adhesion (σ = 065) show that the densityprofile is almost horizontal by t = 5000 Since our workis motivated by studying cell migration assays which aretypically conducted over relatively short time periods it isappropriate for us to focus on relatively short simulations sothat we can examine the transient response of the system andinvestigate how the shape of the initial condition changesOur results for extreme adhesion (σ = 095) indicate thatthese profiles do not change much during the timescale ofthe simulations whereas our results for strong (σ = 080) andmoderate (σ = 065) adhesion show that the profiles changedramatically during the timescale of the simulations It isimportant that we consider this range of behaviors since similarobservations are often made in cell migration experimentswhere certain cell types do not migrate very far over some timeperiods whereas other cell types migrate over much largerdistances during the same time period [32] One hypothesisthat might explain these experimental results is that certaincell types are affected by cell-to-cell adhesion much more thanother cell types [32] The key result of our work is to showthat the usual mean-field model given by Eq (2) is unableto describe the discrete data for strong and extreme adhesionat any time point This is significant because many previousstudies have derived traditional mean-field pde models whichsuffer from the same limitations as Eq (2) None of theseprevious studies have presented any alternative mean-field

models that can predict the averaged discrete profiles whencontact effects dominate [68111214]

Although all density profiles shown in Figs 2 and 3correspond to adhesion (σ gt 0) we also generated similarprofiles over the entire range of the parameter σ isin [minus11]to obtain the error profile in Figs 2(u) and 3(u) Results forσ lt 0 correspond to agent repulsion [11] and the contacteffects act to increase the rate at which the density profilesmooths with time In this context results with σ lt 0 are lessinteresting since D(c) is always positive and agent clusteringdoes not occur Furthermore Eq (2) appears to make accuratepredictions for all cases of repulsion Therefore we choose topresent snapshots and detailed comparisons in Figs 2 and 3for adhesion cases only (σ gt 0)

Our comparisons of Eqs (2) (4) and (8) in Figs 2 and 3were for an initial condition Eq (14) where the averageoccupancy of sites was either c(x0) = 01 or c(x0) = 10with a sharp discontinuity between these two values Wechose this initial condition because Eq (2) is well posedsince the initial condition jumps across the region whereD(c) is negative With σ gt 075 D(c) in Eq (2) containsa region c isin [c1c2] where D(c) lt 0 (0 lt c1 lt c2 lt 1) andit is only possible to solve Eq (2) when the initial conditionis chosen such that c(x0) is not in the interval [c1c2] [29]Had we chosen an initial condition that did not obey theserestrictions Eq (2) would be ill posed with no solution [29]For completeness we now consider a second set of results fora different initial condition given by

c(x0) = 01 + 09 exp

[minus(x minus 500)2

400

] (16)

This initial condition is Gaussian shaped and accesses allvalues of 01 lt c(x0) lt 1 For values of σ gt 075 this initialcondition does not jump across the region where D(c) isnegative which means that Eq (2) is ill posed and we cannotobtain a solution [1329] Regardless of this complicationwith Eq (2) we repeated all simulations shown previouslyin Figs 2 and 3 with the Gaussian-shaped initial conditionand we report the results in Figs 4 and 5 at t = 1000 andt = 5000 respectively

Results in Figs 4 and 5 show the exact same qualitativetrends that were illustrated previously in Figs 2 and 3 Formodest adhesion (σ = 065) we see that Eqs (4) and (8)perform similarly and both mean-field models predict theaveraged discrete data accurately [Figs 4(n) 4(o) 5(n) and5(o)] For strong (σ = 080) and extreme adhesion (σ = 085)we see that Eq (4) which neglects correlation effects isunable to predict the averaged discrete data at either t =1000 or t = 5000 [Figs 4(d) 4(i) 5(d) and 5(i)] whereasEq (8) leads to an accurate mean-field prediction in all casesconsidered here Comparing discrete estimates of F (ll + 1)with those predicted using the moment closure model showsthat the moment closure approach captures nearest neighborcorrelation effects accurately [Figs 4(p) and 5(p)] and wesee that next nearest neighbor correlation effects are lesspronounced than nearest neighbor correlation effects Thedifferences in the performance of Eqs (4) and (8) are quantifiedin terms of the error norm Eq (15) in Figs 4(r) and 5(r)

051922-7


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c

FIG 4 (Color online) Mean-field and discrete results for a range of adhesive strengths (a)ndash(e) extreme adhesion (σ = 095) (f)ndash(j)strong adhesion (σ = 080) and (k)ndash(o) modest adhesion (σ = 065) (a) (b) (f) (g) (k) (l) For each adhesive strength two snapshots ofthe discrete process are shown at t = 0 and t = 1000 respectively All discrete results correspond to = τ = 1 simulations are performedon a lattice with 1 x 1000 and periodic boundary conditions Discrete snapshots show 20 identically prepared realizations of the sameone-dimensional process in the region 401 x 600 (d) (i) (n) Comparisons of averaged density profiles (red) the initial condition (blackdashed) and the solution of Eq (4) (blue) (e) (j) (o) Comparisons of averaged density profiles (red) the initial condition (black dashed) andthe solution of Eq (8) (blue) All discrete simulation results and mean-field solutions were obtained using periodic boundary conditions (c)(h) (m) show the nonlinear diffusivity function D(c) = 1 minus σc(4 minus 3c) associated with Eq (2) Results for extreme (σ = 095) and strong(σ = 080) adhesion show that D(c) becomes negative in some interval c isin [c1c2] while results for the modest adhesion (σ = 065) show thatD(c) gt 0 for all c isin [01] (p) (q) Continuum (blue) and discrete (red) profiles of F (ll + 1) and F (ll + 2) respectively In each plot profilesof the correlation function are given for extreme (σ = 095) strong (σ = 080) and modest adhesion (σ = 065) with the arrow showing thedirection of increasing σ (r) The error profile E as a function of the adhesion parameter σ isin [minus11] at t = 1000 Error profiles are given forEqs (4) (blue) and (8) (red) All numerical solutions of Eq (2) correspond to δx = 02 δt = 001 and ε = 1 times 10minus6 All numerical solutionsof Eqs (4) and (8) correspond to δt = 005

051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c

FIG 5 (Color online) Mean-field and discrete results for a range of adhesive strengths (a)ndash(e) extreme adhesion (σ = 095) (f)ndash(j)strong adhesion (σ = 080) and (k)ndash(o) modest adhesion (σ = 065) (a) (b) (f) (g) (k) (l) For each adhesive strength two snapshots ofthe discrete process are shown at t = 0 and t = 5000 respectively All discrete results correspond to = τ = 1 simulations are performedon a lattice with 1 x 1000 and periodic boundary conditions Discrete snapshots show 20 identically prepared realizations of the sameone-dimensional process in the region 401 x 600 (d) (i) (n) Comparisons of averaged density profiles (red) the initial condition (blackdashed) and the solution of Eq (4) (blue) (e) (j) (o) Comparisons of averaged density profiles (red) the initial condition (black dashed) andthe solution of Eq (8) (blue) All discrete simulation results and mean-field solutions were obtained using periodic boundary conditions (c)(h) (m) Show the nonlinear diffusivity function D(c) = 1 minus σc(4 minus 3c) associated with Eq (2) Results for extreme (σ = 095) and strong(σ = 080) adhesion show that D(c) becomes negative in some interval c isin [c1c2] while results for the modest adhesion (σ = 065) show thatD(c) gt 0 for all c isin [01] (p) (q) Continuum (blue) and discrete (red) profiles of F (ll + 1) and F (ll + 2) respectively In each plot profilesof the correlation function are given for extreme (σ = 095) strong (σ = 080) and modest adhesion (σ = 065) with the arrow showing thedirection of increasing σ (r) The error profile E as a function of the adhesion parameter σ isin [minus11] at t = 5000 Error profiles are given forEqs (4) (blue) and (8) (red) All numerical solutions of Eq (2) correspond to δx = 02 δt = 001 and ε = 1 times 10minus6 All numerical solutionsof Eqs (4) and (8) correspond to δt = 005

051922-9


V CONCLUSION

Our analysis shows it is possible to make accuratemean-field predictions of a discrete exclusion process withstrong adhesion Other mean-field predictions are valid formild contact effects only [6ndash8111214] Identifying andquantifying why traditional mean-field models fail to predicthighly adhesive motion requires new approaches that relaxthe assumptions underlying the traditional approach Our suiteof mean-field models allow us to quantify the accuracy ofassumptions relating to spatial truncation effects and theneglect of correlation effects We find that the traditional pdeis extremely sensitive to the neglect of correlations

The model presented in this paper is a simplified model ofcell migration since it deals only with one-dimensional motionwithout cell birth and death processes Our previous work onmoment closure models has shown how to incorporate cell

birth and death processes as well as showing that it is possibleto develop moment closure models in higher dimensionsThese additional details could also be incorporated intothe current model Other extensions to the discrete modelinclude studying adhesive migration where we explicitlyaccount for agent shape and size effects [33] or the study ofadhesive migration on a growing substrate [34] We anticipatethat accurate mean-field models of these these extensionswill require a similar but more detailed moment closureapproach

ACKNOWLEDGMENTS

We acknowledge support from Emeritus Professor SeanMcElwain the Australian Research Council Project NoDP0878011 and the Australian Mathematical SciencesInstitute for a summer vacation scholarship to STJ

[1] K Kendall Molecular Adhesion and Its Applications (KluwerNew York 2004)

[2] The Physics of Granular Media edited by Haye Hinrichsen andDietrich E Wolf (Wiley-VCH Weinheim 2006)

[3] L Wolpert Principles of Development (Oxford University PressOxford 2002)

[4] R A Weinberg The Biology of Cancer (Garland ScienceNew York 2007)

[5] T M Liggett Stochastic Interacting Systems Contact Voterand Exclusion Processes (Springer New York 1999)

[6] C Deroulers M Aubert M Badoual and B GrammaticosPhys Rev E 79 031917 (2009)

[7] E Khain M Katakowski S Hopkins A Szalad X ZhengF Jiang and M Chopp Phys Rev E 83 031920 (2011)

[8] M J Simpson C Towne D L Sean McElwain and Z UptonPhys Rev E 82 041901 (2010)

[9] E Khain L Sander and C M Schneider-Mizell J Stat Phys128 209 (2007)

[10] K Anguige and C Schmeiser J Math Biol 58 395 (2009)[11] A E Fernando K A Landman and M J Simpson Phys Rev

E 81 011903 (2010)[12] C J Penington B D Hughes and K A Landman Phys Rev

E 84 041120 (2011)[13] M J Simpson K A Landman B D Hughes and A E

Fernando Physica A 389 1412 (2010)[14] K A Landman and A E Fernando Physica A 390 3742 (2011)[15] D Chowdhury S Schadschneider and K Nishinari Phys Life

Rev 2 318 (2005)[16] E A Codling M J Plank and S Benhamou J R Soc

Interface 5 813 (2008)

[17] R E Baker and M J Simpson Phys Rev E 82 041905 (2010)[18] M J Simpson and R E Baker Phys Rev E 83 051922 (2011)[19] J Mai N V Kuzovkov and W von Niessen J Chem Phys 98

10017 (1993)[20] J Mai N V Kuzovkov and W von Niessen Physica A 203

298 (1994)[21] R Law D J Murrell and U Dieckmann Ecology 84 252

(2003)[22] D J Murrell U Dieckmann and R Law J Theor Biol 229

421 (2004)[23] M Raghib N A Hill and U Dieckmann J Math Biol 62

605 (2011)[24] A Singer J Chem Phys 121 3657 (2004)[25] K J Sharkey J Math Biol 57 311 (2008)[26] C E Dangerfield J V Ross and M J Keeling J R Soc

Interface 6 761 (2009)[27] K Seki and M Tachiya Phys Rev E 80 041120 (2009)[28] M J Simpson K A Landman and T P Clement Math

Comput Simulat 70 44 (2005)[29] T P Witelski Appl Math Lett 8 27 (1995)[30] M H Protter and H F Weinberger Maximum Principles in

Differential Equations (Prentice-Hall New Jersey 1967)[31] D A DiCarlo R Juanes T LaForce and T P Witelski Water

Resour Res 44 W02406 (2008)[32] Y Kam C Guess L Estrada B Weidow and V Quaranta

BMC Cancer 8 198 (2008)[33] M J Simpson R E Baker and S W McCue Phys Rev E 83

021901 (2011)[34] B J Binder K A Landman M J Simpson M Mariani and

D F Newgreen Phys Rev E 78 031912 (2008)

051922-10


l l+1l-1l-2 l+2 l+3 l+4 l+5l-3l-4

1212(1-σ)2 0

FIG 1 (Color online) The random walk takes place on a one-dimensional lattice where each site can be occupied by at most oneagent An isolated agent steps in the positive or negative x directionwith probability 12 per computational time step For example theagent at site l + 3 would step to site l + 2 with probability 12or to l + 4 with probability 12 Contact effects alter the motilityprobability for example in the configuration shown the agent atsite l minus 1 would step to l minus 2 with probability (1 minus σ )2 where σ isin[minus11] represents the contact effect The agent at site l minus 1 wouldstep to site l with probability 0 since the target site is occupied

large number of identically prepared realizations of the sameprocess We now introduce three ways to approximate cl bymaking different assumptions about the underlying discreteprocess

A Partial differential equation representation

To connect the discrete mechanism with a pde we form adiscrete conservation statement describing δcl the change inaverage occupancy of site l per time step The conservationequation can be written as

δcl = 12 [clminus1(1 minus cl)(1 minus σclminus2) + cl+1(1 minus cl)(1 minus σcl+2)]

minus 12 [cl(1 minus cl minus 1)(1 minus σcl+1)

+ cl(1 minus cl+1)(1 minus σcl minus 1)] (1)

where positive terms on the right of Eq (1) represent events thatwould place agents at site l and negative terms represent eventsthat would remove agents from site l The discrete conservationstatement is related to a pde as rarr 0 and τ rarr 0 and cl isidentified as a continuous variable c(xt) [61011] Expandingall terms in Eq (1) in a truncated Taylor series about site lneglecting terms of O(3) and higher [61011] and dividingthe resulting expression by τ we take limits as rarr 0 andτ rarr 0 with the ratio (2τ ) held constant [16] to obtain

partc

partt= D0

part

partx

[D(c)

partc

partx

] (2)

where D0 = limτrarr0(2)(2τ ) is the free-agent diffusivityand the nonlinear diffusivity function is given by [1011]

D(c) = 1 minus σc(4 minus 3c) (3)

Two key assumptions lead to Eq (2) First we assume termsof O(3) and smaller can be neglected Second we assumethe average occupancies of sites to be independent so that forexample the net averaged probability of a transition from site l

to l + 1 is proportional to (1 minus cl+1)(1 minus σclminus1) This impliesthat the occupancy of sites l + 1 and l minus 1 are independentwhich in general is untrue [1718] Without further analysisit is impossible to deduce how these two assumptions controlthe net error associated with Eq (2) We now introduce twoalternative mean-field models that systematically relax bothassumptions

B Ordinary differential equation representation

To avoid neglecting terms of O(3) and smaller as rarr 0we retain the spatial structure of the random walk in Eq (1)by identifying discrete values of cl with a continuous variablecl(t) Dividing Eq (1) by τ and considering the limit as τ rarr 0gives a system of ordinary differential equations (odes)

dcl

dt= 1

2[clminus1(1 minus cl)(1 minus σclminus2) + cl+1(1 minus cl)(1 minus σcl+2)]

minus 1

2[cl(1 minus clminus1)(1 minus σcl+1) + cl(1 minus cl+1)(1 minus σclminus1)]

(4)

for each site l We note that Eq (4) still makes the independenceassumption and we now develop a third mean-field model thatremoves this assumption

C Moment closure representation

We use k-point distribution functions ρ(k) (k = 123 )to describe the averaged occupancies of k tuplets of lattice sites[171920] For k = 1 the distribution function is a univariatedistribution describing the average density of agents on site l sothat ρ(1)(Cl) = cl For k = 2 the bivariate distribution functioncan be defined in terms of correlation functions [1719] whichcan be written as

F (lm) = ρ(2)(ClCm)

ρ(1)(Cl)ρ(1)(Cm) (5)

where l = m These correlation functions allow us to relaxthe independence assumptions inherent in Eqs (2) and (4)Setting F (lm) equiv 1 indicates that the occupancies of sites l

and m are independent Instead we avoid this assumption byallowing F (lm) to evolve as part of the solution [17] Withthese definitions we have

dcl

dt= 1

2[ρ(3)(0lminus2Clminus10l) + (1 minus σ )ρ(3)(Clminus2Clminus10l)]

+ 1

2[ρ(3)(0l Cl+10l+2) + (1 minus σ )ρ(3)(0l Cl+1Cl+2)]

minus 1

2[ρ(3)(0lminus1Cl0l+1) minus (1 minus σ )ρ(3)(0lminus1ClCl+1)]

minus 1

2[ρ(3)(0lminus1Cl0l+1) minus (1 minus σ )ρ(3)(Clminus1Cl0l+1)]

(6)

Positive terms on the right of Eq (6) represent events thatwould place an agent at site l whereas negative terms onthe right of Eq (6) represent events that would remove anagent from site l To simplify Eq (6) we apply a summationrule [17] to rewrite the unbiased ρ(3) terms as equivalent ρ(2)

terms The Kirkwood superposition approximation (KSA) isthen used to rewrite the remaining ρ(3) terms as combinationsof ρ(2) terms The KSA is a moment closure approximationthat has been used in many applications including ecology[21ndash23] physical chemistry [24] disease biology [2526] anddiffusion-mediated reactions [27] The KSA can be written as

ρ(3)(φlφmφn) = ρ(2)(φlφm)ρ(2)(φlφn)ρ(2)(φmφn)

ρ(1)(φl)ρ(1)(φm)ρ(1)(φn) (7)

051922-2



dt= 1



minus σ





(8)


dρ(2)(ClCl+1)

dt= 1


2[ρ(4)(Cl0l+1Cl+20l+3)




dρ(2)(ClCl+1)

dt= 1



+ σ







dρ(2)(ClCl+1)

dt= 1



+ σ



dF (ll + 1)

dt= minusF (11 + 1)

[dcl+1

dt

1

cl+1+ dcl

dt

1

cl

]+ 1

2

[clminus1

cl



]

minus σ

2

[clminus1

cl



]

+ σ

2

[clminus1



]

+ σ

2

[cl+2



] (13)

051922-3







c(x0) =

⎧⎪⎨⎪⎩

01 1 x lt 480

10 481 x 520

01 521 lt x 1000

(14)








051922-4





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o)1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

000(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c)

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

000

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-5





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o) 0(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460C 1002 04 06 08

D(C)

-04

10

00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-6



E = 1

100

l=550suml=451







c(x0) = 01 + 09 exp


400

] (16)



051922-7


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10



dt= 1



minus σ





(8)


dρ(2)(ClCl+1)

dt= 1


2[ρ(4)(Cl0l+1Cl+20l+3)




dρ(2)(ClCl+1)

dt= 1



+ σ







dρ(2)(ClCl+1)

dt= 1



+ σ



dF (ll + 1)

dt= minusF (11 + 1)

[dcl+1

dt

1

cl+1+ dcl

dt

1

cl

]+ 1

2

[clminus1

cl



]

minus σ

2

[clminus1

cl



]

+ σ

2

[clminus1



]

+ σ

2

[cl+2



] (13)

051922-3







c(x0) =

⎧⎪⎨⎪⎩

01 1 x lt 480

10 481 x 520

01 521 lt x 1000

(14)








051922-4





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o)1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

000(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c)

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

000

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-5





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o) 0(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460C 1002 04 06 08

D(C)

-04

10

00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-6



E = 1

100

l=550suml=451







c(x0) = 01 + 09 exp


400

] (16)



051922-7


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10







c(x0) =

⎧⎪⎨⎪⎩

01 1 x lt 480

10 481 x 520

01 521 lt x 1000

(14)








051922-4





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o)1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

000(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c)

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

000

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-5





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o) 0(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460C 1002 04 06 08

D(C)

-04

10

00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-6



E = 1

100

l=550suml=451







c(x0) = 01 + 09 exp


400

] (16)



051922-7


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o)1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

000(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c)

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

000

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-5





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o) 0(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460C 1002 04 06 08

D(C)

-04

10

00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-6



E = 1

100

l=550suml=451







c(x0) = 01 + 09 exp


400

] (16)



051922-7


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10





σ

10-3

10-2

10-1

E

-08 -04 00 04 08(u)0

F(ll+1)

(s)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(t)

5

x480 500 520 540460

4

3

2

1

6

(m)

(n)

(o) 0(q)

1

x480 500 520 540460

0(r)

1

x480 500 520 540460

0(p)

1

x480 500 520 540460

(g)

(h)

(i)

1

0(k)x

480 500 520 540460

1

0(l)x

480 500 520 540460

1

0(j)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(f)x

480 500 520 5404600

1

(d)x

480 500 520 540460C 1002 04 06 08

D(C)

-04

10

00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

1002 04 06 08

D(C)

-04

10

C00

x401 600t=0

x401 600t=5000

Eq (2)

Eq (2)

Eq (2)

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c c

c c c

c c c


051922-6



E = 1

100

l=550suml=451







c(x0) = 01 + 09 exp


400

] (16)



051922-7


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10



E = 1

100

l=550suml=451







c(x0) = 01 + 09 exp


400

] (16)



051922-7


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(e)x

480 500 520 5404600

1

(d)x

480 500 520 540460

x401 600t=1000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=1000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=1000

401 600t=0

x

x

00

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-8


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10


σ

10-3

10-2

10-1

E

-08 -04 00 04 08(r)0

F(ll+1)

(p)

5

x480 500 520 540460

4

3

2

1

6

0

F(ll+2)

(q)

5

x480 500 520 540460

4

3

2

1

6




(k)

(l)

(m) 0(o)

1

x480 500 520 540460

0(n)

1

x480 500 520 540460

(f)

(g)

(h)

1

0(j)x

480 500 520 540460

1

0(i)x

480 500 520 540460

(a)

(b)

(c) 0

1

(d)x

480 500 520 540460

x401 600t=5000

x401 600t=0

C 1002 04 06 08

D(C)

-04

10

00

x

x

401 600t=5000

401 600t=0

1002 04 06 08

D(C)

-04

10

C00

1002 04 06 08

D(C)

-04

10

C

401 600t=5000

401 600t=0

x

x

00

0

1

(e)x

480 500 520 540460

Eq (4)

Eq (4)

Eq (4)

Eq (8)

Eq (8)

Eq (8)

c c

c c

c c


051922-9


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10


V CONCLUSION




ACKNOWLEDGMENTS






























051922-10

Documents

Mean-field descriptions of collective migration with strong adhesion