Pa icle algo i hms o popula ion dynamics in lows
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Pa icle algo i hms o popula ion dynamics in lows
P asad Pe leka 1, Robe o Benzi2, Simone Pigolo i3, and
Fede ico Toschi1
1Depa men o Physics and Depa men o Ma hema ics and Compu e Science, Eindho en
Uni e si y o Technology, Eindho en 5600MB, The Ne he lands.
CNR-IAC, Via dei Tau ini 19, 00185 Rome, I aly.
2Dipa imen o di Fisica and INFN, Uni e si `a “To Ve ga a”, Via della Rice ca Scien i ica 1,
I-00133 Roma, I aly.
3Depa amen de Fisica i Enginye ia Nuclea , Uni e si a Poli `ecnica de Ca alunya Edi .
GAIA, Rambla San Neb idi s/n, 08222 Te assa, Ba celona, Spain.
E-mail: . oschi@ ue.nl
Abs ac . We p esen and discuss pa icle based algo i hms o nume ically s udy he
dynamics o popula ion subjec ed o an ad ec ing low condi ion. We discuss ew possible
a ian s o he algo i hms and compa e hem in a model comp essible low. A compa ison agains
app op ia e e sions o he con inuum s ochas ic Fishe equa ion (sFKPP) is also p esen ed and
discussed. The algo i hms can be used o s udy popula ions gene ics in luid en i onmen s.
1. In oduc ion
Bac e ial colonies g owing on he su ace o a Pe i dish o m in iguing pa e ns ha can
be ma hema ically modeled and s udied in e ms o a s epping s one model o popula ion
gene ics [1]. Oceans co e a as amoun o he Ea h’s su ace suppo ing an inc edibly di e si y
o species, playing a key ole since ea ly s age o li e on ou plane . Many mic oo ganisms mus
ha e ound ways o su i e in he lows gene a ed in ocean. The e o e, hough no much explo ed,
i is impo an o unde s and he ole o lows on models o he dynamic o popula ions. To
unde s and his ques ion wo app oaches can be aken: (a) Use con inuum models and couple
hem wi h luid lows [3, 4]. This is an app oach used commonly in oceanog aphy models o
s udy plank on blooms [2]; (b) Use pa icle based models coupled o lows. The ad an age o
his app oach is ha i allows us o pose in a simple way ques ions ela ed o he popula ions
o numbe luc ua ions and he ex inc ion o a mu an species, which in u n is cen al o he
unde s anding o popula ion gene ics [1, 10]. A simila app oach has been employed ea lie o
s udy he ole o disc e e e ec s in he p opaga ion o a Fishe wa e [5].
In his pape we discuss algo i hms o s udy pa icle-based popula ion dynamics in lows,
ollowing he abo e app oach (b). An en i y based algo i hm wi h di e en a ian s is discussed
in sec ion 2. We hen alida e ou algo i hm o he Mo an p ocess in sec ion 3, whe e we
i s p esen he esul s o he case o uni o mly mixed popula ions wi h ze o di usi i y and
compa e ou esul s agains heo e ical p edic ions. Nex we ocus on he spa ial di usi i y. The
con inuum limi o Mo an model wi h spa ial di usi i y is he s ochas ic Fishe -Kolmogo o -
Pe o sky-Piscouno equa ion (sFKPP) [7, 8, 11]. We s udy he ole o popula ion luc ua ions
on he speed o a Fishe wa e and compa e ou esul s agains he asymp o ic p edic ions o
Re . [11].
Pa icles in Tu bulence 2011 IOP Publishing
Jou nal o Physics: Con e ence Se ies 333 (2011) 012013 doi:10.1088/1742-6596/333/1/012013
Published unde licence by IOP Publishing L d
1
Finally, in sec ion 4 we s udy he eac ion A→A+A,A+A→Ain a low, which also has
he Fishe equa ion as a con inuum limi , bu wi h a di e en noise e m [11, 15]. We show ha
in p esence o a simple, de e minis ic, comp essible low he a e age popula ion size (ca ying
capaci y) dec eases, in ag eemen wi h ea lie in es iga ions [4, 15].
2. Nume ical algo i hm
One o he example o popula ion dynamics is he g ow h o bac e ia on a Pe i dish. The
mo ion o he on ie o a bac e ial colony can be unde s ood as a solu ion o he s ochas ic
Fishe equa ion. Ano he way o s udy on ie g ow h is by encoding he bi h and dea h
ules a an indi idual en i y le el. The ad an age o his app oach is ha biologically ele an
ques ions abou he bi h and ex inc ion o species can be easily add essed. We now desc ibe
he wo algo i hms used by us o s udy popula ion gene ics wi h lows.
2.1. Algo i hm 1
The nume ical algo i hm ha is used o sol e he popula ion dynamics is simila o he algo i hm
used o s udy he on p opaga ion o Fishe equa ion in Re . [5]. We conside each indi idual in
he popula ion as a pa icle ha is ad ec ed by he low, di used by he B ownian mo ion, and
ha can ep oduce i sel , die, o compe e wi h o he pa icles. Bi h, dea h and compe i ion
can be modeled in e ms o bina y eac ions. In o de o e icien ly implemen hese bina y
eac ions (which necessa ily happen when he wo pa icles su icien ly close o each o he ) we
in oduce a spa ial mesh. Mo e p ecisely, we di ide ou one dimensional domain o size Lin o
Msubin e als o wid h δ=L/M and whe e δis he in e ac ion dis ance. As a ypical s a ing
condi ion one may conside pa icles uni o mly dis ibu ed a posi ions x∈L. The popula ion
is e ol ed acco ding o he ollowing wo main s eps:
•Ad ec ion and spa ial di usion: he pa icles a e ad ec ed and di used acco ding o
xi( + ∆ ) = xi( ) + u∆ +√2D∆ Γi( ), i= 1,2,...n. He e Dis he spa ial di usion
coe icien , uis he ad ec ing eloci y, ∆ is he ime s ep, nis he o al numbe o pa icles,
and Γ( ) is a Gaussian whi e noise wi h ze o mean and uni a iance.
•Popula ion dynamics: he pa icle popula ion is assumed homogeneous inside he
subdomain δand in e ac wi h each o he using a p esc ibed se o ules de e mining he
popula ion dynamics.
Gi en he amewo k ha models ad ec ion, di usion and locali y o in e ac ion, one mus
speci y he p ope eac ions which encode o he biological p ocesses in he popula ions. He e
below we desc ibe how o implemen h ee o he building blocks a he base o he di e en
eac ion models used in his manusc ip . Supposing ha ou ecosys em is composed by wo
dis inc popula ions, Aand B, we deno e he numbe o pa icles wi hin he in e ac ion adius,
δ, o ype Aas NAand hose o ype Bas NB. The o al numbe o pa icles in he in e al δ
is indica ed wi h N=NA+NBwhe eas he o al numbe o pa icles in Lis indica ed wi h n.
(i) A+Bk
−→ B+B: In his ype o eac ion, a pa icle o one ype, A, in e ac s wi h a pa icle
o he o he ype, B, and a a a e kcon e s i sel in o B. To implemen his ule we i s
coun he numbe o Bpa icles p esen in δ. I kNB∆ < hen he pa icle Acon e s
i sel in o Bo he wise no hing happens ( deno es a andom numbe uni o mly dis ibu ed
in [0,1]).
(ii) Ak
−→ A+A: In his eac ion a pa icle o ype Agi es bi h o ano he pa icle o he
same ype a a a e k. I k∆ < hen he pa icle mul iplies in o wo; o he wise no hing
happens.
Pa icles in Tu bulence 2011 IOP Publishing
Jou nal o Physics: Con e ence Se ies 333 (2011) 012013 doi:10.1088/1742-6596/333/1/012013
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(iii) A+Ak
−→ A: In his eac ion a pa icle o ype Ain e ac s wi h ano he pa icle o i s own
kind and one o he wo dies wi h a a e k. Again we i s coun he numbe o Apa icles
p esen in δ, i k(NA−1)∆ < hen he pa icle annihila es; o he wise no hing happens.
2.2. Algo i hm 2
This ype o algo i hm cons i u es a a ian o e algo i hm 1. He e he ad ec ion and di usion
pa s a e implemen ed in he same way. Two main di e ences a e implemen ed in he popula ion
dynamics pa o he algo i hm:
G id size A each imes ep, he pa icle dis ibu ion is binned on a ine esolu ion. The bin
size is chosen as δ/m, whe e δis he in e ac ion dis ance as be o e and mis an odd na u al
numbe (in he ollowing we will choose m= 11). Bina y eac ions occu a a a e which
depends on he o al numbe o neighbo s in he ow o mcells cen e ed on he bin whe e
each pa icle is.
Ra es The imes ep ∆ is aken small enough ha no mo e han one eac ion o each ype
can occu wi hin each imes ep. Then, a ep oduc ion A→2Aoccu s wi h a p obabili y
kn( )∆ whe e n( ) is he o al numbe o pa icles. A dea h by compe i ion, A+A→A
occu s wi h a a e k∆ PjNj( ), whe e Nj( ) is he o al numbe o pa icles in he ow
cen e ed in he bin whe e pa icle jis and he sum uns on all he pa icles p esen in he
sys em. In he case o ep oduc ion, he ep oducing pa icle is chosen a andom wi h
uni o m p obabili y, while in he case o dea h by compe i ion he choice is weigh ed wi h
he numbe o neighbo s, i.e. pa icle ihas a p obabili y o be chosen equal o Ni/PjNj.
The i s modi ica ion allow o ha e compe i ion in an a ea δmo e p ecisely cen e ed in he
pa icle posi ion. We ound ha he second modi ica ion slows down simula ion conside ably o
ou pa ame e alues. The e o e, i should be implemen ed only in cases whe e he imescales o
popula ion dynamics a e much longe han hose o ad ec ion di usion, so ha one is no o ced
o choose a e y small ime s ep in o de o a oid ha ing mo e han one eac ion pe ime s ep.
3. Valida ion
The alida ion is di ided in wo pa s aimed a es ing, espec i ely, he eac ion algo i hm alone
and he di usion- eac ion implemen a ion. We i s ocus on he eac ion pa and alida e ou
popula ion dynamics algo i hm o he case o Mo an p ocess [6, 9] o popula ion gene ics.
He e we compa e ou esul s wi h he heo e ical p edic ion on he ixa ion p obabili y om
Kimu a [9] (see Subsec ion 3.1).
In subsec ion 3.2 we add he e ec o he spa ial di usion: wi h his addi ion he Mo an p ocess
becomes equi alen o a s ochas ic Fishe equa ion [7, 8, 11]. We compa e he esul s o ou
disc e e simula ions agains he analy ical p edic ions based on he s ochas ic Fishe equa ion,
bo h in s ong and weak noise limi [11, 14].
3.1. Mo an model
We conside a homogeneous popula ion o wo species Aand Bwi h NAindi iduals o ype-A
and NBindi iduals o ype Bin a domain o size δ. When an indi idual o ype A(B) in e ac s
wi h an en i y o he o he ype B(A) hen B(A) con e s in o A(B) wi h a a e kA(kB).
These eac ion p ocesses can be w i en in o m he ollowing o m:
(A+BkA
−−→ A+A
A+BkB
−−→ B+B
Du ing he abo e p ocesses he o al numbe o en i ies N≡NA+NB emain conse ed
o all imes. The con inuum limi o he abo e p ocess is he s ochas ic equa ion dc =
Pa icles in Tu bulence 2011 IOP Publishing
Jou nal o Physics: Con e ence Se ies 333 (2011) 012013 doi:10.1088/1742-6596/333/1/012013
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µc(1 −c) + pσ2c(1 −c)dW (FKPP equa ion wi h noise) whe e c=NA/N is he concen a ion
o he ype-Aspecies, µ≡(kA−kB)N/δ and σ2≡(kA+kB)
δ,dW is a Wiene p ocess.
One o he cen al ques ion in popula ion gene ics is he ixa ion p obabili y o a mu a ion.
Fo wha conce ns he Mo an model his ques ion can be o mula ed as: i Bis he pa en species
and Ais he mu an , wha is he p obabili y ha he mu an con e s all he pa en species B
in o i sel A(i.e. Age s ixa ed)? The wo con ol pa ame e s a e he ini ial mu an popula ion
concen a ion p=cA( = 0) = NA/N and he selec i e ad an age µ. Fo con enience, in his
sec ion we se δ= 1. Fo he case o he Mo an p ocess Kimu a [9, 10] p edic ed ha he
ixa ion p obabili y is:
P ix =1−e−αNp
1−e−αN (1)
whe e α≡kA−kB
kA+kBand pis he ini ial concen a ion o he mu an popula ion. No e ha o
kA=kBo µ= 0 (case co esponding o no selec i e ad an age), he ixa ion p obabili y
P ix =p. The plo in Figu e 1 shows he ixa ion p obabili y o wo di e en popula ions o
size Nwi h p= 1/N and a ying µ. The nega i e alues o he selec i e ad an age implies
ha he pa en species has a selec i e ad an age o e he mu an . We ind ha , in ag eemen
wi h he heo e ical p edic ion, he ixa ion p obabili y inc eases wi h he selec i e ad an age
o he mu an . Fo he special case o ze o selec i e ad an age (kA=kB) we ind, in ag eemen
wi h heo y, P ix =p. In ou simula ions he ixa ion p obabili y, P ix , is de ined as one minus
he ac ion o ealiza ions o which he mu an dies. Ou measu emen s we e ob ained by
a e aging o e an ensemble o 2000 ealiza ions o each da a poin .
3.2. S ochas ic Fishe equa ion
To s udy he e ec o spa ial di usi i y, Fishe , Kolmogo o e al. [7, 8] added a di usion e m
o he Fishe equa ion:
dc = [µc(1 −c) + D∇2c]d +pσ2c(1 −c)dW. (2)
In absence o noise, o he case o a localized an ini ial condi ion, he Fishe equa ion exhibi s
a a eling wa e solu ion wi h a on speed gi en by F= 2√Dµ [7, 8]. Mo e ecen ly, i was
shown ha he speed o he Fishe wa e is educed in p esence o noise. In he egime o weak
noise [11, 12, 13, 14]:
∼pDµ 2−π2
(log N)2(3)
whe e Nis he o al popula ion wi hin he in e ac ion adius δand σ2∝1/N. In he s ong
noise egime he Fishe speed is d as ically educed [11, 14]:
∼2Dµ
σ2.(4)
In o de o alida e ou algo i hm (1) we conduc ed a se ies o disc e e pa icle simula ions
o he Mo an p ocess wi h spa ial di usion, wi h di usi i y D. The con inuum limi is he
sFKPP equa ion o di e en alues o µand σ2. In Figu e 2 we plo he no malized speed,
/ F, o he Fishe wa e e sus he dimensionless noise s eng h σ/(Dµ)1/4as ob ained om
ou disc e e pa icle simula ions. In he simula ions he size o he domain was L= 100 wi h an
in e ac ion adius o δ= 1. We obse e, in ag eemen wi h [11], a c oss-o e in he Fishe wa e
speed a ound he alue uni y o he no malized noise s eng h. The asymp o ic p edic ions o
he weak and s ong noise limi s, see Equa ions (3) and (4), a e also cap u ed wi h ou disc e e
pa icle simula ions.
Pa icles in Tu bulence 2011 IOP Publishing
Jou nal o Physics: Con e ence Se ies 333 (2011) 012013 doi:10.1088/1742-6596/333/1/012013
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0
0.1
0.2
0.3
0.4
0.5
0.6
-1 -0.5 0 0.5 1 1.5 2
FIXATION PROBABILTY
µ
N=10
N=10, analy ical
N=40
N=40, analy ical
Figu e 1. The beha iou o he ixa ion p obabili y as a unc ion o he selec i e ad an age µ,
o popula ion sizes N= 10 and N= 40. A compa ison wi h he analy ical p edic ion is shown
(see ex o mo e de ails). We ha e chosen he ini ial allele equency as p=NA/N = 1/N in
all simula ions.
4. Resul s on he ca ying capaci y in a model oy low
A biologically ele an quan i y is he ca ying capaci y (a e age popula ion size in an
ecosys em). In a ecen pape [3] i was shown ha a one-dimensional u bulen comp essible
eloci y ield leads o a educ ion in he ca ying capaci y. This educ ion o ca ying capaci y
was la e e i ied in a mo e ealis ic wo-dimensional su ace low model o u bulence [4].
To in es iga e he beha io o ou algo i hms in p esence o low ield we conside a e y
simple sinusoidal low u=Usin(x). This low is comp essible and leads o a educ ion in
he ca ying capaci y on inc easing he low s eng h [15]. Below we conside a e y simple
model o popula ion dynamics ( he bi h-coagula ion p ocess) whose mean- ield limi also gi es
he Fishe equa ion.
(A+Aχ
−→ A
Aγ
−→ A+A
whe e χdeno es he dea h (coagula ion) a e and γis he bi h a e. The con inuum limi o
he bi h-coagula ion p ocess gi es:
∂ c+∂x(uc) = D∇2c+µc(1 −c) + pσ2c(1 + c)Γ( ),(5)
whe e cis he local popula ion concen a ion and Γ( ) is a Gaussian whi e noise wi h ze o mean
and uni a iance. We use he algo i hm desc ibed in Sec ion 2 o s udy he popula ion dynamics
Pa icles in Tu bulence 2011 IOP Publishing
Jou nal o Physics: Con e ence Se ies 333 (2011) 012013 doi:10.1088/1742-6596/333/1/012013
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0.1
1
0.1 1 10
No malized on speed
Noise s eng h
Weak noise
S ong noise:
Disc e e pa icle simula ion
Figu e 2. The no malized on speed o he Fishe wa e (made dimensionless by F) e sus
he dimensionless noise s eng h. Da a om ou pa icles ssimula ions a e he ed illed ci cles,
while he asymp o ic conjec u es o he weak noise (blue line, ∼√Dµ h2−π2
(log N)2i) and he
s ong noise (black line, ∼2Dµ
σ2) limi s a e also shown. Each da a poin is ob ained by doing
an a e age o e 50 independen noise ealiza ions.
in his simple low wi h he abo e popula ion ules. An impo an poin o no ice is ha his
model, a a iance wi h he Mo an model, allows o luc ua ions in he o al popula ion size.
In absence o any low he popula ion eaches a s eady s a e wi h a popula ion size N0, while
he popula ion size can changes d as ically due o he p esence o he sinusoidal low ield. We
de ine he ca ying capaci y as:
Zp( )≡Na
N0
(6)
whe e Na is he a e age popula ion size in he p esence o he low. Figu e 3 shows ha he
ca ying capaci y Zpdec eases s ongly a inc easing he o cing s eng h. This is consis en
wi h ea lie 1dand 2dsimula ions o popula ion dynamics in u bulence low ield [3, 4] as well
as wi h he s ochas ic simula ions om [15].
5. Conclusion
We ha e in oduced and discussed simple algo i hms o s udy he dynamics o popula ions
subjec o an ad ec ing low. We benchma ked he nume ical implemen a ion amongs a ian s
o he algo i hm i sel as well as agains heo e ical esul s. We show ha he comp essible low
educes he o al popula ion size and he eby he ca ying capaci y. The p oposed algo i hms
a e a na u al choice o in es iga e popula ion gene ics in lows. Fu u e s udies would look a
he e ec o popula ion gene ics in highe dimensional lows which a e close o ealis ic se ings
Pa icles in Tu bulence 2011 IOP Publishing
Jou nal o Physics: Con e ence Se ies 333 (2011) 012013 doi:10.1088/1742-6596/333/1/012013
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
<Zp>
U
Algo i hm1
Algo i hm2
S ochas ic
Figu e 3. Ca ying capaci y e sus he o cing s eng h o he sinusoidal low ield, U, o
ou one-dimensional comp essible oy low. We obse e ha he ca ying capaci y d ama ically
dec ease wi h inc easing o cing s eng h and sa u a es o U≥0.3. Algo i hm 1 and algo i hm
2 a e he wo algo i hms desc ibed abo e, while s ochas ic e e s o he esul s ob ained om
he nume ical in eg a ion o Equa ion 5.
o e.g., su ace lows. This would aid in ou unde s anding o ixa ion o species in na u al
en i onmen s whe e luid low plays an impo an ole.
6. Acknowledgemen
We hank M. H. Jensen and D.R. Nelson o discussions. We acknowledge he COST Ac ion
MP0806 o suppo . FT and PP acknowledge he Ka li Ins i u e o Theo e ical Physics o
hospi ali y. This esea ch was suppo ed in pa by he Na ional Science Founda ion unde
G an No. NSF PHY05-51164.
7. Re e ence
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[2] Ma hias Sandulescu, C is ´obal L´opez, Emilio He n´andez-Ga c´
ia, and Ul ike Feudel, Biological ac i i y in he
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[3] R. Benzi and D.R. Nelson, Fishe equa ion wi h u bulence in one dimension, Physica D (Ams e dam) 238,
2003 (2009).
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[6] P. Mo an, The S a is ical P ocesses o E olu iona y Theo y. Cla endon P ess (1962).
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[7] R.A. Fishe , The wa e o ad ance o ad an ageous genes, Ann. Eugenics 7, 353 (1967).
[8] A. Kolmogo o , N. Pe o sky, N. Piscouno , E ude de l’equa ion de la di usion a ec c oissance de la quan i e
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[11] C. R. Doe ing, C. Muelle , and P. Sme eka, In e ac ing pa icles, he s ochas ic Fishe -Kolmogo o -Pe o sky-
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(2009).
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a Xi :1106.3506 1.
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Jou nal o Physics: Con e ence Se ies 333 (2011) 012013 doi:10.1088/1742-6596/333/1/012013
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