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I)C onlribution No .56.N alT a pn~tt Ma rine La boratory.G radu""SdI ool o r O«ao .
oJr.l phy,U ni"enily 01 Rb ode h l. nd,U.S.A.
Raymond A. Shappy
Computer Labor.lory.Un iYc",ily or Rhode Jdlild.
K inpton,Rh ode h land
Nart'pRien Mar ine l aboratory,Uniycnhy or Rhode hllnd,
Kinplon,Rhode hlllld,.,
lDtroduction
A substa nt ial bod y o f evide nce, from the in itial ta gging e xperiment s of
GIUlElIl.T a nd RICH (1926)through recent e xtensive s tudies by th e International
N orth Pa cific F isher ies Comm ission (Canada. J apan and the U nited States).
dem on strates beyo nd reasonable d oubt t hat many indi viduals of the fin species
of Pa cifi c:salmon (Oll c(lr!lJ'nchus )and th e steelhead trout Salmo gaird"eri.
hat ched in a given stream return to the same area as adult s for sra wn i ng a fter
var iable per iods in the o pen sea .Reviews by HARTT (1960,1962)and HASLER
(1960)gi ve some indicat ion o f the di stribution patt ern at sea by species as a
result o f t agging studies.Th e ebove-mcnncned review authors also summarize
s pecificd ata demon str ating th at Pacific sal monid fishes may be found in feeding
area s m ore th an 1000 m il e ~from th eir nat al streams. Indi\lidu als ta!!8ed at
sea i n th ese feeding ar ea s hue successfully returned a s b reed ing a dults to
the ir b o rne streams.
It is the objec t o f th is paper to pr esent a h ypothesis which c:\plains tbe
obse rved migrat ur)'behaviour or 'homing'p henomen u n by mean s o f r andom
sea rch ing co mbined with a low de gree of o r ientat io n to a n o utside st im ul us.
This i n \'cstigat io n is concerned e:\d usi vd)'wit h migrati o n in th e open sea an d
all>n g t he coa st up to the 'dcinity o f thc na tal stream a nd docs not include
mi grati on in t he streams.Th e ro le of o lfaction.first postulated by B ASLER a nd
Wl-lOR¥(1951),a nd la ter de mons trated by WIS BY a nd HASLER.(1954)i n per-
mitt ing r ecognlt ion o f a given st ream when a fish is in its vicinity,is a ccepted.
For the pr esent purposes,a sea sea rch is co nsidered successful when the
vicinity o f th e h ome stream is encountered becau se olfactory senses arc c cn-
sidered effectively operative at this point aud retenti on o f fish in this area is
a ssumed .
Although Pacific salm on id fishes ar e used as a pr imary source of empirical
data for this paper,the hY5=(.~hes i s expl icitly discus sed ma y be applied to o ther
mi gratory a n im3 1s by a suitable choi ce of parameters.It ma y be seen fr om
By
SaaJ B.SaU.
RandomMovement and Orientation in Salmon Migration'
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what fo llo .....s t hat t he hypo the sis is gene rally com pat ible w it h t he salrnc n id
da ta a t hand .It is a p preciated that m o re fi eld ex perime ntatio n is nee d ed in
o rd e r to inc rea se the stat ist ical p rec i..io n o f the pa rameters u sed in t he m od el .
Su bsequent compu tat io n wit h more com prehensive a nd /o r pr ecise da ta w,11
allow f unh cr gene ralizatio n o r mod ifica tion o f th e hy po t hesis p resen ted here in .
It seems a ppr opriat e 01 1 th is ,i "le,howev er. to a ttempt a ..yn thes is o f availa ble
in formation in to a ma thematical model i n o rder 10 u nde rstand be ner t he
mi~r a lio n phenom enon .F urthe r elabo rat ion o f t he n ume r ical p roba bility
model d escribe d her ein is i n p rogress,and efforts .IrC be ing m ade 10 conside r
th e sa me p ro ble m fr om an a nal ytica l v ie wpoint ,
Data a nd ,\"5ump tio n'i
0 31 ;1 for the s l udy have bee n compiled from di verse ..c urccs and a rc u sed
t o demons tru-c t he pbus ibi lilY o f t he h}pt'llhe sis by p rovi ding re ason able
e mpi r ica l value...for th e re quir ed param e ters .I n cases where no evidence t o
s u ppor t a v.ump t i"n\rs a vailable .the m ore co nse rvative a ssum pti on am ong
rea so na ble alte rnatives !I'IS been c ho ..e n.
Onenr attcn lind ..tep len;:th
E vid ence in dica tes th at man y ..pc...·i.:s o f fish.:s po ...<.c iS a n i nna te mechanism
for direction o rien ta tion.IhMTT (1961.1961)ha s recently indicated that t he
res ults o f repeated pu ree wine ~a \Ul g..::s t Iha l salmon m ove in definit e and
consisten t pat tern s i n come are a s 301 sea .A sign ifi canlly hig hcr return Wd S
achieved i n !>e lS with t he sein e o pe n toward the west in t he ..icinity of the
Ale u t ian h la nd ...however,an east wa rd m o vemen t i n t he Berin g Sea wa s also
s ho wn.T h is ~u g g cl l s ca ..twa rd o r ien tatio n o f l o m e ...a lm o n o n t he h igh se as.
h is pre s umed t ha t t hese fish a rcele stined for t h e Pacific C0 3 S1 o f North Am e ri ca .
A ..un compa ss h as been d emon strated for seve re!s pecies evc tu ..ivc o f salmo n
by fh.sU R.HORR ALL.WISHY.an d BR"n UR (l95K).H AS L[R a nd SCIIWASS SI ASS
(I960J .and Sc Il\\'A~m :(19bO).H "Sl [R (1960) indicated t hat ~i l ver sal mon
po sscss a vun-compa m echanism.RM,,,n TUI.(1960)has sho wn fe r co h o
salmon .O .ki sut ch,t ha t a co mpa ..s d irect ion ca n be ma intained d u r ing the d ay
a s w ell 3!>a t n ight.Some co nfli ct i ng evid en ce ,h owev er,is f o und co nce rn ing
nigh t movemen t o f s almon.JOH !"SO S (1 960)fou nd fr om so nic ta g tr ack i ng
ex per imems t hat a dult sa lmo n e ither slo w d e ...'n o r slop m o vem ent at n ight
i n t he C o lumbia Rive r.H AS l[lI.(l Cj 60) s ta les t h at th ere ha ve been direct
o bse rva nc ns o f sa lmo n movemen t a t n igh t i n the sea ,a nd n ight g ill n eui ng
a t sea for adult sa lmon i!>co mmo n p rac t ice,
Other guiding m echanis ms in the c pe n se a .such 3S wa t er cu rr e nt s.ha ve
bee n po stulated by BE\'ElI.TOS a nd H OL T (195 7).TAIT (1952)s ugges ts that
mig ratio n m 3Y be i n fl uenced b y water tempe rature.The changing temper ature
a nd sa lini ty s t r uc t ure o ver lim e i n t he no r th -east Paci fic Ocea n.a s well a s t he
i rr egular d istributi on o f wat er ma sses a nd varia ble cu rre nt pa tterns de scribed
b y severa l a utho rs (OoOIMUO.1958;Do DIMU O a nd H OLLI STER.1958;DoE,
1955 :M ISC'tIIMA a nd SISK IZAW A.1955)do no t ap pea r 10 be co ntin uo usly
errecnve aids f o r o rienta tion t owa rd th e Pacific COlaS t o f N o rth America.The
p.esent mod el d OC'S no t ma ke p revision fo r t hese drlet~of ocea n cu r rents a nd
ca n ,the refore,be conside red a gener alized m odel in th e se nse th at it is n ot
Salm on M igrat ion 155
restr icted t o an y spec ific geo graph ic a rea be cause o f current c onfi gurat ion s
pe culia r 10 t hat area.
C e le s t ial navigat ion.d eli ned by W ,'L LRAH (1960)a s a fo rm o f g oal o rienta-
tio n in which no d irect se ns o r y con tact with th e goal itself o r with known
land mark s i n i ts vicinity a re u sed .has n ot been co ncl usively d em ons tra ted
in a ny animal t o d ate (WA LLRAFF .1960 ).C elest ial na vigation as d efined a b ove
implies that th e d ir ect io n o f the goal is i nd irectly d etermined fr om o ther
stimulus configurations o f the envir onment.T his form o f goal or ientat ion is
not c onsidered necessary in the h yp othesis.
It is suggested that t he p ostulated tendency to swim greater d istances in th e
direc tion of 'home'by m a ture fish may b e d erived fro m t he i nfluence of th e
s un .a nd th at th is tendenc y is ret a ined du r ing th e n ight.It wi ll b e assu med
t hat a ny given fish m ay s win,in a ny d irecti on wi th eq ual p robability at ea ch
t u rn in g po in t.It is al so as sumed that .a t each turning p oi nt,the dista nce
moved in any direct ion al ong a straight pa th is randoml y determ ined wi th in
a sele cted range «(}..20 miles).T h is rand omly de te rm ined d ista nce is multi plied
by a d irect ion s ensi tive factor which introduces a sm all a nd precisely defi ned
bias t o ward the d irect ion of t he n atal s t re a m ,Several d ifferent ranges were
s t u died b UI t he re sults p re se nted herein a re b ased on a r ange of zero 10 t wenty
m ile s.T he g eom etry o f th e sea rch pa ttern w ill be de scribed in the d iscussion
o f th e migratio n m odel.
S peed
S o rr.e ind ica t io n of I ll'a ve rage sp eed o f m igrat ing salmon is requ ired .F rom
an empirica l formula presen te d b y B.'I"RR IOG E (1958), w i th coeffic ients of 3
a nd 4 bea t s per se cond fo r ta ilbeat fre quencies a nd 30 i nche s (76 c m) fo r t he
a ve ra ge fish s ize ,a c alc ula t e d mean s peed o f abou t 3 miles per h our is obtai ne d.
The se frequenci es are t h e lowest value s observed by BAI S RR IDGE.The a bove-
ment ioned formu la p rovides an aver age sp eed of about twice t he value of
1·5 m iles per h our fo r adu lt sa lmo n d ir ectly o bse r ved b y J Oll " SO"(1960)in
t he C o l u mb ia R ive r .It is conje c t ur ed t h at th e s wimming performance o f adult
sa lmo n may b e h igh er i n sa lt wa ter th an in fre sh wa ter i nasmuch as G ROH S
(1954)indica te s a h ig he r pe r for ma nce o f juvenile sa lmon in salt water th an in
fre sh wa te r .O SROI I"E (1961)sugge sts t h a t th e actual pe rforma nce o f a dults
in f resh wa ter e xceeds t he c alc u lated pe rfo rmance an d th at sa lmon arc efficient
veh icles in e nergy t ra nsformations.
T he calculated val ue o f sw imm i ng speed e xceed s t he av e rage ve locit ies of
salm o n obta in ed b y tag r ec aptu re data .H o wever.it sho uld b e recogn ized th at
t ag recapture data gi ve m in imal est ima t es because dista nce is c o mpu ted as a
st raight line o r a gr eat c ircle di stance .Values for the a ppa rent rates of t ravel
on the h igh seas h ave been com puted from data prese nted b y t he International
North Pacific F ish er ies C omm ission (1958. 1959) a nd th e F ishery Re search
In stitute ,U niversity o f W ash i ngton (1959).These range fr om 47 m iles per d ay
for pink sal mo n 10 12 m iles pe r d ay for ch um s almo n. lt al so appears from a n
analysis of the abo ve-menti oned da ta as well as data g iven by PARK ER a nd
K IRKS ES S (1 956) th at swimming s peed m ay be re d uced when se xually m atu re
salmon move in t he vici nity o f the coast o r when t hey arrive i n t he vici nity
of the h ome s tream .
A val ue of 2 ·5 m ile>per h our is a s s u med to be a r easonable e stimate of
'0·
156 S"UL B .SAI L'and RAY MO ND A .SH A.' PY
sustained cru ising spee d o n th e h igh seas .In asmuch a s redu ction of swimming
speed a lo ng th e co ast has bee n s ugges ted. a va lue of 1·25 miles per hou r is
a ssum ed fo r coa stwise searching.O ther val ues for swimming speed were a lso
util ized an d will be b riefly co nside red .It is o bvious.h owever ,that.for a given
level o f en d urance,a r eduction o f a verage searc h speed i s equi valen t to a
pr opo rt ion al red uction i n th e a mount of t ime a va ilabl e to a mig rant for
co nducting t he search.
Endurance
It is di fficult to determ ine exac tly ho w long salmo n take to m igr a te from
so me dis tant a rea a t sea to the home stream .The impul se to mig rate by
ma turing salmo n is pre sumed to be h ormonal.That is, the ho rmone of the
ant er ior part o f the p ituitary gland (the gon ad otrophic hormon e)ult imately
exerts a strong en oug h e ffect o n th e n er vous system t o i n itiate the migrato ry
instinct.Deta ils o f t he h isto-physiolo g ical and experiment al stud ies involved
in the ho rm onal th eory ar e given by G ERBI LS KY (1958).Movement i n the
feeding a rea is a ssumed to be ran do m pri or t o th is gon ad otro ph ic influence.
M igra tion by sa lmon is o bviously as sociated wit h sexual matu rit y.A m ax-
imum e ndurance value o f o ne year is evident beca use t agged sexually mature
sal mon a rc invariably recove red d ur ing the year o f ta gging (H ARTT . 1960).
Empirical va lues for en durance c a n a lso be obtained fr om ta g recapture d ata.
F or exam ple, H ASLE R (1960) describes th e movem ent o f a steclhead tr out
r ep ort ed by the O rego n Sla te G ame Comm ission in whic h 143 d a ys were
i nvolved i n m igration to the vic init y o f Kodiak Island.Ala sk a (abo ut 1200
mil es)a nd 153d ays were r equired fo r r eturn to the home stream a fter recapture
a t sea . D ata pre sented by th e Int ernat ional Nor th Pacific F isheries Comm ission
(19 59)indicate t hat m igration o f immature pink salmon 10 the Kod iak Island
feeding a rea ta kes a bout 175 d ays (spring t o late autumn ).Fr om the e nergy
expenditure s o f fish in m otion rep orted by Wt ~B ERG (1960),i t was estimated
th at a fish can swim up to 40 km (22 mi les)at the expense o f reducing its weight
b y I %.Cl early sairnon feed du ring m igration a s evidenced by successful troll
fisheries a long the Pacific coast of North A merica .Therefore,loss o f weight
is pr o ba bly insign ifica n'during t his pe riod .T hus endurance is pr obably not
as sociat ed with e nergy reser ves durin g migration at sea .
For t he pu rp ose a t han d. 175 da ys i s po stulated as t he max imum length of
t ime a g iven fish will sea rch before d y ing o r giving up .The ch oice of a t ime
limit is ar bitr ary. a nd the as s ump t io n th a t all fish not reac hing th e vicinity o f
th e natal stream within 175 da ys ar e lost or d ead is art ificial.This ki nd of
const raint o n th e model .howev er. helps to i ns ur e t hat the re sults a re more
pessimis tic t han th ose whic h mi ght actually occur i n nature.Ob viou sly. if a
finite t ime limit wer e no t impo sed , a ll fish would ult imately r each horne by
random search a lone!
Distance
A co n venie nt s ummary o f th e di stance from 'home'at wh ich t he various
species of Pacific salmo nid s have been r egularly captured is av ail able from
HA SLER (1960).H ARTI (1960),and t he Int ernational No rt h Pacific F isheries
Comm ission (19 59). Data from t hese so urces for the vari ous s pecies ar e listed
below :-
Salmon M igranc n 157
Sp.d"
D is ilnot i ft mllca
(Fecdlft '~a f rom naLlI Stru m)
S od :ey e
Pink
Chu m
Ch inook
Coh o
Ste elhead
O .nrrka .
O 'l o,b lllcha .
O.keta .
O .uIrQ ltI~'l't j(ha ... . .••.
O.k isu lclt .
S .ga i,dnu i .
1200
1200
1700
zsoo
1200
1200
In th is model ,it was a ssumed th at a ll hypothet ical migrants began their
M onte Carlo search from a po int on th e glob e hav ing the same va lue of lat itude
as the mouth of the hom e stream but lo ngitudinally d isplaced 1200 m iles to
t he west .
Random movement
It is assume d for t his investigat ion that individual fis h search independently.
H owever,t he r esults would not be altered subst antially if t he fish trav elled i n
school s of limited size.A fur ther assumption is tha t searc h ing is ra ndom i n
t he sense th at a fish is n ot abl e to recognize a particular location a t sea o r on
t he coast (exclusive of h ome)even if it l>a d been pre viously tr aversed .
Hom e stream and coast recog :tition
As stated pre viou sly, th e olfactory se nses of m igrating salmon are pr esumed
to be sufficiently acute to guide fish when t hey en counter t he vicinity o f t he
n atal stream.A radi us o f 40 miles from t he stream mouth is arb itrarily chose n
as the are a within which a successful search a t sea is terminated .A small and
practi cally insignifi cant han dicap is imposed upon th e fish a t th is point becau se
t he 40 mile r adius d oes not extend int o the sea .It ap plies o nly to t he coast.
Hence a ny individual w hose sea rch t ime expires when he is even o ne m ile
d ue west of h ome (no t yct imp ingent up on th e co as t)is consi dered lost. Th is
was done to facilitate pr ogramming the comp uter.Other rad ii of the same
ord er of magn itud e would not a ppreciably affect the computed return p rob-
ab ilit y.Aga in,the ch o sen rad ius is cons idered a plau sible figure for computa-
tion s in th is first approximation.
T he coast in the m igra tion mode l is idealized in to a stra ight l ine of i nfinite
length.Thi s provides a pessimistic ret urn probabil ity when cont rasted with t he
actu al coastwise search which has fi n ite boun ds.It is assumed in the mod -I
that fi sh st r iking the idealized coast will rema in in its vicinity.PARKER a nd
KIRK I'ESS (1956)ha ve dem on strated for c hinook salmon.O .t s hawytscha,t hat
coastwise m igrat ion occurs,a nd further t hat the re is evidence to indicate some
directi onal component in migration from nort h to sout h,i nasmuch a s ta gged
fish were inva r iably captured south of the release point .
In order to compare th e model with a specific situat ion,we shall consider
the migration of Pacific salmon to the Pacific coast of No rth A merica (fo r
example, the Columbia R iver f rom a po int 1200 miles to th e west of th e r iver
mouth).It is a ssumed that completely random searching take s pl ace wh en
fish encounter the coast sout h of t he na tal stream an d that ra ndom searching
comb ined with one-half th e d irectional component of movement experienc ed
a t sea takes place when fish strike the coast north of the ho me stre am ar ea.
By this,i t is m eant that the effects pr oduced by d eviation s from p ure ran dom
158 S.u:L B.SAIL"and R "YMO' O A .SH "PPY
search are only hal f as great in th e northern coastal model a s tho se prod uced
i n th e open ocean search.As stated ,revio usly ,search spee d is r educed to
o ne-ha lf its value a t sea when t he coast is r eached.Th is decrea se in speed
pr o port ionately red uces the am ount o f search dis tance ava ilable to a fish for
coastal sea rching bec ause a finite t ime limit is as s umed.The r estr ictions imposed
by t he m odel o n c oastwise searching a re ex tremely severe.R ando m searching
south o f t he 'h ome'a re a d oes not includ e a so uthern r eflecting barr ier,such
a s in creased water temperature.Als o ,a rand om search combined with one-
half th e c oefficient o f d irected vers us und irected movement means t hat th e
r andomly chosen step,i f rand omly de te rmined t<'be in t he d irection o f h ome,
is a ugmented L)o nly a sm all fraction of the step le ngth .T he ra te o f n et
m o vement towa rd h ome is,th er efore,very sm all.An inc rease in th e a mount
o f di rected movement t oward h ome woul d significan t ly in crea se the calculated
return proba bil ities.C onserva tive a ss umptions have again been applied to the
model i n this r espect .
:\li gration :\lodel and Re sults
M igrat ion a s defi ned in t his in vestigation de scribes a beh aviour w hich i nvolves
leaving a particular a rea (feeding g round s)by se xually matu ring fish and the
gai n ing of an o ther d efined a rea (the vicinity o f the home stream),the two
areas being sep arated b y a co nsiderable d istance.Th is definition is compatible
with G ERKIl<G 'S (1959 )u se o f 'homing' wh ich refer.to the ch oice tha t a fish
makes when returning t o a place formerl y oc cupied b y that fish instead o f going
to other eq ually probable p laces. H owever,'homing'generall y refe rs to t he
re turn of a nimals wh ich have bee n a rtifi cially d isp laced,and it is s ubmitted
that th e d istinct ion between a rtificial di s placement a nd na tural movement is
s ufficiently impc.ta nt to d emand n ot o nly sepa rate te rminology but a lso
sepa rate ass umption s regarding t he mechanisms involved.Th e te rm m igration
as used in this investigatio n corresponds t o WILK Il<SOS'S (1952) defi n ition of
a nast rophic m igration .
From wha t has been described in the data and assumptions ,it is obvious
t hat th e analytic form o f model nec essary for t he calc ulation of retu rn p re '>-
abilities coul d e..'ily become involved w ith weighted o r b iassed ra ndom walk s
o n.a plane followed by two d ifferently weighted random walk models along a
straight line .A model o f this nature would inv olve some complex calculations.
For the p urposes at hand ,a numerical probability o r Monte Carlo mod el o f
salmon m igration has been devised.Monte Carlo methods have been a pplied
to dive rse problems with good results as ind ica ted by M EYER (1956).
The pa ttern o f steps (sw im lengths)ch osen fo r t he migrat ion model IS
described by a cardioid whe re r-«
(I)R =r «Q cos 0
(2) R =P (I ;.A cos 0),where A =Q /P
T he variation of step vector ,R .specified b y the parametrics P and Q is
d irect ion sensitive beca u se it is affec ted b y t he rand omly c hosen value o f O.
Figure I illustrates a range of values of A ,the coefficient o f directe d vers us
undirected m ovement.When A =0,the search is comp le tely random and t he
geometry of the search pattern is descr ibed by a ci rcle (s tep vect or is the same
159
R ,P •0 cos e
R ,P (l .AcO Se )
A ,0 1 P
A ,Q
A ,1
va r iabl e
on e -half o f sp eed at sea
175 da ys
1200 m ile s 1222.1 km!
F igure I.Vario us m ovement pattern s illustrating r ang es of '..c'values.
Sw im length is prop o rtional to t he len gth o f.vect or d raw n
from the or igin.....to any point o n the cardioid.
length in an y d irection).When ,oj =I ,the t ypical cardioid is apparen t.Wh en
A =0 ,25 ,as it wa s fo r m ost o f the co mputations, th e resulting cardioid is a
very small dep arture fr om a pure circl e (i n Figure I a circle is superimposed
up on the O'3 car dioid for visual comparison).In thi s case ,if we con sider the
circle i n th e figur e a s havin g a unit rad ius ,the maximum step length va r ies
fr om -1"7 to 1·3 .Th e length o f a vect or from the ori gin to a ny po int on the
card io id is p rop ortional to th e d istance a fish swims in each s te p of h is ran dom
walk.T he co nser vativ e 'A'va lues wh ich have been appl ied to thi s m odel
rep resent ver y weak o rienting influences (postulated a s a sun co mpass).
.Exp licitly,th e foll o wi ng in put data h ave been used for t he model.In all
in stances where un it s o f m easurement i n m iles are ind icated ,the specific un it
is a nautical mile .Appropriate co nversio ns of naut ical mi les to kilometres
are indicated .
Spe ed (a t sea).
S peed ("Io ns coast ).
M axi mum endurance .
D istance (fe e d ing a r ea to ho me strea m)..
A (d irected versu s un direct ed move ment
inde x)v a r iab le
R adi us of 'home'a rea 40 mil es (74 km}
Maximum ste p (s wim)len gth before
turning var iable
Search mode a long coast s outh o f home ..random
Se arch mod e a lo ng co ast no rth of home . .A =:II Ion e-hatf o f "A'on high seas)
It is obv io us that testing all po ssible combinations of the variables incre-
mented ov er a rea sonable range would be proh ib itively t ime consuming in the
mo del an alysis.In s pite o f limited te sting,cert ain generalizations appear
evident.
160 SA UL 8 .SA ILA and R AYMOND A .SHAPPY
i nitialize locat ion of fish
1200 due west of home
ge nerate random value o f 8
gene rate random val ue o f P
~u ch thai 0 :s P :s maximum step len gth
comput e R -PO +A cos 8
a ccumula te hor izontal c omponent o f s tep
1:X =1:X +R eo. S
accumulate veru ca t compon ent of step
L'Y Z:E I Y +R si n 9
a ccumulate tota l d istance trave led
1:5 =1:5 +R
NO
d oes ~X equal or exceed
s t rai ght line d istan ce b etween
s ta rti ng point a nd home (1200 m iles)?~o
Y ES
NO
compute dis tance availab le for c oastal s earch
D COo"=(4200 h ou",(2'S m.p.h.)-1:5
reduce D coast by h alf because c oas tal se arch
speed is o ne-half of open oc ean spe ed
D 'co ast =D coa st/2
GO TO NORTH ERN MODEL
did fish strike coast north o r south of home '1NOrTH--~~~~~~~~~~~~;';2J---SOU~T H
G O T O SOUT HER N M ODEL
Figure 2. Programme logic (or seach o n open ocea n.
A programme u tilizing a r andom nu mber generator and th e abov e para-
meters was written for the IBM 709 Data Processin g System. Figures 2,3,
and 4 show the flow diagrams for the three main sections o f th e com put er
p rogram me. In the a ctual p rogramme t he num ber o f individual m igrants is
specified and the values of the searc h parameters a re accep ted an d sto red.
The si mulation then proceeds by sending each migrant through the log ical
flow shown in th e diagr ams.The performance of each indi vidual is pr oduced
as o utput by the com puter ,and ,at the end o f the run , a fi nal statistical
summary is p roduced.
Salmon Migrat ion
I
F rom o pen ocean mode l
generate r and o m value o ~,roo weh t .~;u
(-1)(m.....s eep lenglh a t ~eal -::P :iU (rna,.sl c;l le nglh a t Sc:.l )
accumu late n ort hward drin
I N ",E X .;.P
ac cum ulat e to tal di sta nce t rav e led al ong co ast
IX -I X 'Po
NO
YES
Figure J.Programme log ic for sea rch along s outhern co ast .
161
NO
R eference to F igure 2 will show how the programme was designed to allow
each hy pothetical migr atory fish to choo se an azimuthal dire ction of travel
from a point at sea 1200 mileswest of t he coa st.The angle between the randomly
ch osen dire ction and the east-west axis is entered in the par ametric fo nn o f the
card ioid to produce th e step vector:R =P +Q cos O.T his process is repeated
un til :(a)the all otted sea rch t i me for th e fi sh expires, or (b) t he acc umulated
eastward mov ement exceeds the s pecified straight line d istance between the
start ing point and the north-south s osorbing barrier,the coast line.If the
seco nd possibility actually occ urs ,the simul ation co ntinues in a one dime nsional
r and o m walk along the coa st.T his coastal searching continues until t he a llo tted
search time is completely used or u ntil t he animal ar rives within -t he specified
h ome regio n.Reference to F igure 3 will shov th at searching is pu rely random
on th e coast south of home.In Figure 4, one can see how a homeward b ias
equal to one half 'A'is applied in t he northern coastal mode l.
A series o f computer runs were made to get some indication of the return
probability a s a function of the magnitude of A,t he coefficient of directed
versus undi rected mo vement.From Table I it is clear that retu rn probability
increases significantly with relatively small changes in 'A'holding o ther para-
meters constant .Fro m thi s table it is also evident th at a return probability
of 0 ·08 was ach ieved with a value of 0·20 for 'A'.T his va lue is very cl ose to
t he aver age of a 10%recovery rate for mature salmon tagged o n the h igh
seas as rep o r ted by HAR TT (1960).Tag returns u p to 22 %ha ve been repor ted
for m ature red salmon (Report of O perations .. .1958,F isheries Research
I nstitute, Un iversity of Washington).An 'A'value of 0 ·3 gives a return prob-
ability of 0·37 which con siderablyexceeds th e ob served return o ftagged salmon .
T his value o f 'A'still d oes not d emand precise o rientation o n th e part o f th e
fish. The values of the other parameters in T able I a re co nsidered to be con-
se rvative. Small sample size a ccounts fo r the ap parent an omaly in r eturn
probability for 'A'values of 0 ·15 and 1·0.
162 S ...L:L B .S"IL4 a nd R A Y MO ND A .S H4 P PY
I
From o pen oc ean mo del
eener ..te rando m v alu e of P \u.:h (hill
(-J I (ma \..u;p I cn ~lh 31 sea):S P S (1)(m 'l:(.s tep length a t s~a )
Y ES is P »O"!
(po\ili,,~P is consi dered a l1orlh"'"ard H cpJ
NO
i nc rease m agmrud s of P
b )'fact or of I -r-.-4i 2 for so ut h wa rd \I C:P
P =PO +A 12)
acc umulate dnfl ..lo ng coa st
1..d rift -=-~:dr ift +P
'-- - +-\.I ccumul.nc to tal distance tra veled a long c oast
1.·C =~·C +I P i
NO
fish is lost
on coast
north of home
Y ES
1'0
Fi gure 4 .Programme lo gic fo r sea rch along nort hern c o ast.
It is evident t hat tag returns p rovide a biassed estima te (low) of a ctual
re turns of salmo n becau se o f immediate l agging mortality a nd incomplete tag
re port ing.Both of these sou rces of e rror are pr obably variable according to
the species and taggi ng cond itio ns.Nat ural mortality is un known fo r t he
•period of migrat ion in the sea.I nasmuch as t here is at present no way t o
estimate these losses, a coefficient o f dire cted versus u ndirected movement
correspo nding to a significantly higher return t han is actually observed is
con sidered mOSI realistic.T he 'A'value o f 0 ·3 is taken to be the b est ap proxi-
mati on of ori entation requirements.
It can be argued that a very high degr -e of o rientati on which is not cou pled
with random searching would ac tually result in no returns if the fish were
subject to even a small drift with wate r m ovements.Thus the simplest as sump-
tion s regard ing orie ntation also appear to be most log ical.It should also be
reempha sized th at these calcu latio ns are based o n a conservative model as
indicated by th e data and a ssumptions.
The results o f the simulated migr ations were pro grammed to permit deta iled
ana lysis o f the o utco me o f each individ ual hypothetical fish.O scilloscope traces
showing t he ac tual paths o f four h ypothet ical fi shes have been pub lished
(SAt LAand SHAPPY,1962 \.Th ese deta ils are not con sidered important except
Salmon M igrati..-n
Table 1
Return probabilit..as a function or the coefficient o f directed \e"u\
un directed movement .Other parameters a re as indicated below :
163
o
0 ·1
0 -15
0·20
0 -15
0 -30
0 ·50
1-0
R n c r n
p",,~t'l ,I II )
o
o
0 ·40
0 ·08
0 -2:!
0 ·37
0 ·70
0 ·60
~U I'\"I ~r of
tn po l MHnl 1""
I lYl
10
10
10('
100
100
10
10
Speed (a t sea)'.,.
Spe ed (alo ng co ast).
M aximum endurance .
D is ta nce ,,.
R adius c f 'home'a rea ' .
Ma ximu m step length ,,
Se arch modc alo ng coa st so uth o f 'home', .
Search mode along c oast north of .horn e'•.••..•• •.
2·5 m.p.h.(4 ,6 k m/h r)
1'15 m.p.h.C ·)k m/h r)
175 d al'
I:!OO miles (1 114 knn
4 0 miles (74 kmJ
10 milcs (37 km }
random
'...t'=on e-half of 'A "above
Table 2
Re turn pr obability as a fun ction of searc hing speed.
Oth er parameters ar e a s indicated below:
Scv~lI i n ;:\pe'N
(m l ln rer IIr
ami Lm pn'h rl
5 (9 ·31
4 (7 '4,
3 15-6 ,
2'514 -6'
Re lu rn
pr "b4 b1 h l ~
0 -60
0-54
0 -36
0 ·22
!'O ur.1N r ('(
t1 ~pul hc'llo;..1 flll'I
100
100
100
100
Maximum en dura nce "".
D istance .
Radi us o f "home'a rea ".""".. "."""" "".."
Max imum step len gt h """",.".
Search mo de along co ast so ut h of 'ho me'.
Sear ch mo de al ong coast n or th of 'home'.".".".
Coeffic ient of di recte d ver sus u ndirect ed movement,
A,o n h igh seas , ,.
171i dan
I:!OO m iles (111';krm
40 m iles (74 l..m)
:!O m iles 07 krm
ra ndo m
A .~0 '115
to point o ut s ome generalizations o bserved when "A"was a ssigned values of
0 -20. 0 '25_and 0-30.(I) Ab out twice a s ma ny fish found "home fr om the
north than from the sou th i n co astw ise searching,A s.however.about twice a s
m any fish made co ntac t with the coast t o the north o f "hom e'tha n to t he s outh
of "home',it appears that th e superimposed direc tional component in t he
no rthern coastal m odel is not s trong e nough t o incr ease n oticeably the p rob-
a bility of s uccessful search fo r fishes s triking the coast n orth of home.T he
restrictions on t he di rectional element o f coa stwise searching were p reviously
staled "an d it,in effectiveness is apparenl.(2) Th e nu mber o f h ypothetical !'s h
lo st at sea from thi s model is ab out 40 %.Of the re maining 60 ~~.a ppro xi-
mately two -third s are lost in the coastal sea rch .(3) Abo ut 7 %of the hypo-
thet ical fishes a rrive h ome d irectly with no coas twise sea rching.
164 S A l"L B .SAIL A a nd RAYMOND A .SHAPPY
Table 3
Return prob abilit y as a function of max imum step length,
Other parameters are as indica ted below:
~....i ml,l,"~cp kn,tb Refilm !"lumba of
(mlln an.1 Ir.m)rrobabilil7 bypo lMtical 1I$b
100 (I8S)0 '13 100
7S (1 39)0 ·24 100
50 (93)0 '24 100
40 (7 4,0 ·29 100
)0 (56)0 '1)100
Spee d (at s ea).
S peed (along coast).
Ma ~imum end urance .
D istanc e .
Rad ius o f 'ho me' a rca .
Search mod e alo ng coa st sout h of 'home'.
Search mode alo ng coast north o f 'home'.
Co ef ficient o f directed ven us u nd irected movement.
A,o n high seas .
2·S m .p .h .(4·6 km /h r)
1·2S m .p.h .(~'3 k m/hr)
115 days
1200 m iles (2224 ko m)
4 0 miles (7 4 k m )
ran dom
A =0 '12S
0 '2S
Re sults were o btained fr om several computer runs b y va rying th e speed of
t l-.-fish but h olding o ther parameters co nstant.It sho uld be rem embered th at
coastwise search speed is reduced 10 o ne-half its value s hown in col umn I o f
T able 2.It is evi d ent f ro m the results shown i n T a bl e 2 t ha t th ere is a n i ncrease
in re turn p roba bility wi th increas ing speed .The reason fo r t his h as been
pr eviously slated.
S imilar r uns w ere ma de in a n effort 10 d eterm ine the relat ionship betwee n
r eturn probability and maximum step length.As stated pre viously ,s tep lengths
were randomly determ ined within an a r bitra ry range.The figu res listed in
column I of Table 3 a re the maximum values for the s pecified range.It is
e vident th a t t he av erage step lengt h randomly chosen be tween 0 and t he
ma ximum i!a bout o ne-half the ma x imum val ue .The return probabilities
a chieved i n T able 3 suggest a complex relat ionship between the m aximum
ste p length a nd re turn probability.It is submitted that this may be due to the
natu re o f t he nu meri cal pr obability model.H owever,preci se empi rical i nfo r-
mation o n t he movement of individ ual fish on t he h igh seas by son ic tracking0;other met hods is h ighly d esir able at th is po int.
Ackn,.wledgement
Part o f the co mputations r equired in this study were performed a t the
Computation C enter,Ma ssachusetts In sti tute o f Technology,Cambri dge,
Massachusetts ,
Summary
A num erical probability m odel (Monte C arlo mel hod)of the migration o f
salmonid fishes was de ve loped .Em p irical values for the mo del parameters
a vailable f rom published li terature wer e util ized 10 demon strate th e plausibility
o f the m odel as an explan ation for m igration in the sea,a nd to provide a
qu antitative i ndicat io n o f th c degree o f orientation requ ired 10 pr ovide a high
r et urn pr obability.Sp ecifically,a random number gen erator i n a high speed
d igital computer was u sed t o a llow a hypothetical m igra t ory fish 10 c hoose a n
S almo n Mig rat io n 165
azim uthal d irectio n of t ravel.T he a ngle between the randomly cho sen d irection
and the ca st-west a xis was e ntered in to t he pa rametric fo rm o f a card ioid to
pr oduce a 's tep vect o r'a s follows :R =P +Q cos O.T his was repea ted unti l
(a)t he a llotted search li me fo r th e an imal exp i red, o r (b)the accum ulated
ea st-west d isplacement exceeded the specified st raight line distance between
the s tarting point and a n orth -sou th barrier (the coast).A small strip of coa st
was d es ignated as 'home'.If an individual reached th e coast .the ca rdioid
model was dropped and the random walk continued along the co ast until the
rema ining search time was completely used or th e an ima l ar rived 'home'.
A return probability of 37 %(sign ifi cantly higher than ob served return s based
o n tag r eco veries)was achieved with a model u t ilizing ran dom mo vement
combined with only a s mall amount of ori entation.It was concluded that
neither na vigation n or precise o r ientation was necessary to e xplain the large
scale m igration of salmon to the vicinity of the ir natal streams from distant
feeding a reas.
It is s ubmined tha t construction and manipulation o f a mathe matical model
is helpfu l in ach ieving a better understand ing of a ny phenomenon .T he model
discussed above is a dmitt edly a n o versimplifi cation of a complex b iological
phen omenon .Howeve r,because of t he paucity of empi rical data regarding the
migra nt's endurance and t he u nknown na ture of the frequency di stribution of
lengths o f ste p vecto rs, a m ore complex set of assump tions does not appear to
be j ustifi ed. Nevertheless,t he results indica ted by th is m ode! do no t s uggest
the necessity of navigational abil ity o r even precise di rectional o rientation on
th e part o f migratin g salmonid fi shes.We a wait the finn foundat ion of d iverse
b io logical da ta to s upport o r modify ou r hypothe sis.
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