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Some aspects of environmental variability in relation to stock recruitment systems Tautz, A. F. 1970

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S O M E A S P E C T S O F E N V I R O N M E N T A L V A R I A B I L I T Y I N R E L A T I O N TO S T O C K R E C R U I T M E N T S Y S T E M S by A R T H U R F R E D E R I C K T A U T Z B . S c . , U n i v e r s i t y of B r i t i s h Columbia , 1968 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in the Department of Zoology We accept this thesis as conforming to the r equ i r ed standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August, 1970 In p resent ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date A B S T R A C T F o r a va r i e ty of s t o c k - r e c r u i t systems in which environmenta l v a r i a -b i l i t y i s s imula ted by r andom n o r m a l deviates used as m u l t i p l i e r s or d i v i s o r s R i c k e r (1958) and L a r k i n and R i c k e r (1964) demonstrated the benefits of c o m -plete ' s t ab i l i za t ion of escapement as opposed to r e m o v a l of a f ixed propor t ion of the stock each year . P a r t I is p r i m a r i l y concerned with the response of these same systems to a pattern of s tochast ic modi f ica t ion which is more regular i n fo rm, a pattern such as might be imagined to r e su l t f r o m long -t e r m trends in envi ronmenta l condit ions. In addition, some mathemat ica l proper t ies of these systems are d i scussed . P a r t II cons iders the s t o c k - r e c r u i t re la t ionship f r o m a more reduc-t ionis t or mechanis t ic point of v iew. Evidence for dif ferent ia l u t i l i za t ion of spawning areas is presented and spawner d is t r ibut ions i n three different environments are compared . These resu l t s are d i scussed i n t e rms of their re levance to exis t ing s t o c k - r e c r u i t theory. A l s o , observat ions on egg retent ion and s o c i a l fac i l i t a t ion are presented. i i i T A B L E O F C O N T E N T S | Page T I T L E P A G E ; i A B S T R A C T i i T A B L E O F C O N T E N T S i i i L I S T O F F I G U R E S . . . i v L I S T O F T A B L E S v G E N E R A L I N T R O D U C T I O N ' 1 P A R T I 4 The B a s i c Systems 4 S imula t ion P rocedure 8 D e t e r m i n i s t i c S in Effects 11 The Effect of Cycle Length and Ampl i tude Modi f ica t ions 14 Stochast ic Modi f ica t ions 18 The Nature of the Modi f ica t ions 20 D i s c u s s i o n . . . 25 P A R T II 28 Introduct ion 28 Site Selec t ion and Ex t rapensa to ry M o r t a l i t y 32 The Study A r e a s , 33 I F o u r M i l e Creek 33 Me thods . . . . . . . . 34 R e s u l t s . "... 37 P h y s i c a l c h a r a c t e r i s t i c s of the sections 37 The spawning run 39 S t r e a m mig ra to ry behavior 42 Egg retention „.' ~ 44 II P inkut and Weaver Creeks 45 I M P L I C A T I O N S O F T H E S T U D I E S 50 C O N C L U S I O N S 54 A C K N O W L E D G E M E N T S 56 B I B L I O G R A P H Y 57 i v L I S T O F F I G U R E S F i g u r e Page 1 Stock rec ru i tment curves A , B , C of R i c k e r (1958) in wh ich z = w e a ( 1 - w ) for a- = 1.000, 2.000, and 2.678 r e s p e c t i v e l y . . . „ , 5 2 - Stock rec ru i tment curves F , G, H of R i c k e r (1958). See text for explanation 7 3 Catch i n de t e rmin i s t i c s imula t ion of a f ixed percentage f i shery on curve C systems with 12- and 24-year s in wave cycles for amplitude scal ing factor P=2.0 15 4 Catch i n de t e rmin i s t i c s imula t ion of a f ixed percentage f i shery on a curve C s y s t e m with a 12-year s in cyc le for ampli tude sca l ing factor at two levels , P = 0.5 and P .= 2.0 "... 1 7 5 Catch i n de t e rmin i s t i c s imula t ion of a complete ly s tab i l i zed f i shery on curve A and curve B systems with a 12-year s in cyc le and amplitude sca l ing factor P = 2.0 19 6 Re la t ionsh ip between mean catch (c) and standard deviat ion of catch (a c) in complete ly s tab i l i zed f i sher ies for 200 year s imula t ions for curves A , B , and C for which envi ronmenta l effects at var ious l eve l s were s imula ted by scaled random n o r m a l deviates used in the method of R i c k e r (1958) 26 7 Stock r ec ru i tmen t curve A and average curve when the sy s t em i s exposed to stochastic va r i a t ion A ' 30 8 F o u r M i l e Creek showing loca t ion of g r ids A - J and the meter ing site (*) 36 9 T e m p o r a l d i s t r ibu t ion of spawning run and a re la t ive m e a -sure of changes i n d ischarge 40 10 R e g r e s s i o n l ines fitted to mean number of sightings per g r i d plotted against the depth and ve loc i ty of the g r i d 41 11 M e a n depth of the redds i n g r i d A plotted against per cent of m a x i m u m spawner density (18 redds) 43 1 2 D a i l y number of f i sh plotted against a c ross sect ion of the upper g r i d at P inkut Creek 47 LIST O F T A B L E S Table ' • ' Page I Mean catch and escapement i n de te rmin i s t i c models of s ix stock r e c r u i t re la t ionships 12 II - M e a n catch and escapement i n 200 year s imulat ions of f ixed percentage exploi tat ion for each of s ix reproduct ive curve systems, with l o n g - t e r m environmental fluctuations, represented by s in modi f ie r s , at cycle lengths of 6, 11, 12 and 24 years , and at two leve ls of ampli tude. See text for explanation 13 III Mean catch and escapement i n 200 year s imulat ions of a f i she ry with complete ly s tab i l i zed escapement for each of s ix reproduct ive curve systems, when l o n g - t e r m env i ron -, mental fluctuations are represented by s in modi f ie r s , at cyc le lengths of 6, 11, 12, and 24 years and at two l eve l s of ampli tude. See text for explanation IV P h y s i c a l c h a r a c t e r i s t i c s of g r ids A - J wi th pe rmeab i l i t y index expressed as per cent of sample passing through a l . l 9 n a m sieve. 38 1 G E N E R A L I N T R O D U C T I O N j The es tabl ishment of l eve ls of exploitat ion consistent with opt imal u t i l i za t ion of stocks of c o m m e r c i a l l y important species r emains as a p r i -m a r y focus of contemporary f i sher ies b iology. Th i s p rob lem of m a x i m u m sustained y i e l d i s , i n essence, a p r o b l e m of population ecology, and it i s not su rp r i s i ng therefore that f i sher ies and population theory have fol lowed s i m i l a r , though dis t inguishable, courses of evolut ion. A s i d e f r o m the fact that f i she r ies biology is concerned with fishes, i t is best cha rac t e r i zed by i t s pragmat ic , i f perhaps inelegant, nature. Thus, many management p r o -g rams are l a r g e l y concerned wi th the co l l ec t ion of catch and escapement s ta t i s t ics which, when added to previous records , should provide some insight into opt imal leve ls of escapement. Given enough of this type of informat ion , one has a reasonable expectation of successful management, p rovided that the va r i a t ion in r ec ru i tment of a stock of a g iven size i s not excess ive . Unfortunately, the s t o c k - r e c r u i t data which is now avai lable indicates that this v a r i a t i o n i s considerable and therefore h i s t o r i c a l approaches can only meet wi th l i m i t e d success . Thus, the development of better management p rograms would seem dependent on the incorpora t ion of r ec ru i tmen t va r i a t i on into management theory and a corresponding ana lys i s of the sources of this va r i a t ion . R i c k e r (1958) showed the i n i t i a l in teres t in this problem, and developed a conceptual f ramework and n u m e r i c a l model which s imula ted the behaviour of s t o c k - r e c r u i t systems when exposed to randomly-f luc tuat ing envi ronments . The bas ic model was de r ived e a r l i e r (1954) f r o m a cons idera t ion of the effects of predation and cann iba l i sm on numbers , the resu l t of which was the w e l l -. I i known s t o c k - r e c r u i t or reproduct ion cu rves . L a r k i n and R i c k e r (1964) con-f i r m e d e a r l i e r conclusions concerning the behavior of these systems by the use of computer techniques. The reproduc t ion curve i s dome-shaped which, i n theory, resu l t s f r o m an increase in the number of predators to a degree such that the abso-lute number of prey s u r v i v o r s i s decreased. Though one may jus t i f iably object to predat ion as the cont ro l l ing mechan i sm for some specif ic situations, the dome-shaped curve seems to be a reasonable fit for many f i she r ies (e.g, G a r r o d , 1966), and may be taken as one general f o r m of parent-progeny re la t ionships given density dependent mechanisms of regulat ion. The bedraggled debate oi density dependent versus density independent mor t a l i t y which occupied much of the ecologis t s ' t ime, never became a c r i t i c a l p r o b l e m in f i she r ies biology. It was c lear f r o m the outset that weather and c l imate played an important ro le i n influencing product ion of most s tocks. Consequently, the addi t ion of s tochast ic va r i a t i on to a sy s t em which was b a s i c a l l y density dependent i n form, was not considered as an outrageous d i s to r t ion of r e a l i t y . A t this l e v e l of invest igat ion one further attribute of the sys t em was i n need of study, namely, the effects of long t e r m trends in density independent mor t a l i t y fac tors . Many environmenta l var iab les ; e.g., cur rent patterns, tempera tures , etc., s eem to fol low a more regular pat tern of va r i a t i on than a r andom model would suggest. Thus, i t was of in teres t to determine i f the addit ion of these long t e r m phenomena would marked ly affect previous con-elusions concerning the gross behavior of these sys tems, Th is d i s cus s ion compr i s e s P a r t I of the fol lowing thesis . 1 While the above d iscuss ions provide a useful f r amework for the fo rmula t ion of genera l management pol icy , their value i s l i m i t e d for specif ic s i tuat ions. One cannot, for example, determine the c a r r y i n g capacity of a given s t r e a m and consequently a stock unit cannot be defined. Thus, a different l e v e l of inves t igat ion is r equ i red and a p r e l i m i n a r y study consistent wi th this approach i s desc r ibed i n P a r t II, which d iscusses some aspects of the spawning behavior of sockeye salmon, Oncorhynchus ne rka . A t the specif ic l eve l one must investigate those aspects of the population and environment relevant to the defini t ion of c a r r y i n g capaci ty. One need ask, for example, which areas of a s t r e a m are suitable for spawning, whether qual i ta t ive differences exis t i n the areas which are u t i l i zed , which factors influence neares t neighbor distances, and others . The pa r t i cu la r p r o b l e m invest igated in P a r t II i s , i n rea l i ty , pertinent to both leve ls of invest igat ion. A t the general l eve l i t cons iders the effect of env i ronmenta l v a r i a b i l i t y operating in a density dependent manner while, at the speci f ic l e v e l i t presents evidence for the mechan i sm of site se lect ion and the ab i l i ty of spawners to d i s t ingu ish habitats of different qual i ty. 1 P a r t I has been published as "Some effects of s imulated l o n g - t e r m e n v i r o n -mental fluctuations on m a x i m u m sustained y i e ld" , in the J . F i s h . R e s . B d . Canada, 26: Z71 5-Z726. P A R T I The B a s i c Sys tems The curves used in the present s imulat ions are iden t i ca l to those desc r ibed by R i c k e r (1958). Curves A , B and C are member s of the exponential f ami ly Z t = we a ( 1 ~ w ) where Z t = product ion i n year t w = spawning stock size ^ " a " i s a parameter de te rmin ing m a x i m u m production and, consequently, the shape of the reproduct ion curve . F o r curves A , B and C the p a r a -meter " a " assumes a value of 1.000,2.000, and 2.678 respec t ive ly (F ig . Curve A desc r ibes the situation -where m a x i m u m product ion occu r s at the e q u i l i b r i u m pos i t ion w = 1.000 = Z m a x . D i sp lacements to the. left of e q u i l i b r i u m (w < 1.0) resu l t i n a " c l i m b i n g " of the ascending l i m b of the curve . D i sp lacemen t s to the r ight of e q u i l i b r i u m (w > 1.0) resu l t in an immedia te compensation, the stock being depressed to a l e v e l below e q u i l i b r i u m density to which i t gradual ly re turns . Curve B . In this case the m a x i m u m product ion i s associa ted wi th a stock density of approx imate ly one- th i rd that of e q u i l i b r i u m densi ty . Compar ing Curve B with A , product ion per spawner at any g iven stock density i s greater in Curve B for 0<w<l, and sma l l e r for ^A.11 values are expressed i n stock units, where one stock unit i s the number of ind iv idua ls assoc ia ted with the unexploited e q u i l i b r i u m pos i t ion . i 1 1 r ]—r PARENT S T O C K F i g u r e 1. Stock r ec ru i tmen t curves A , B, C of R i c k e r (1958) in which z = w e a ( 1 " w ) for a = 1.000, 2.000, and 2.678 re spec t ive ly . w > 1. A r b i t r a r y d isp lacement f r o m e q u i l i b r i u m resu l t s in the p roduc-t ion of a damped o s c i l l a t i o n of abundance which re turns the stock to the e q u i l i b r i u m pos i t ion . Curve C i s a more extreme v e r s i o n of B i n that product ion rate are greater and compensat ion i s more severe . The sys tem i s unstable i n the sense that d isp lacements f r o m the e q u i l i b r i u m posi t ion resul t in the product ipn of permanent o sc i l l a t i ons and there i s no tendency for the stock to re turn to the e q u i l i b r i u m of w = 1. C u r v e s F , G and H ( F i g . 2) each belong to a different f ami ly of cu rves . Curve H represents the converted B ever ton-Holt re la t ionsh ip as desc r ibed i n R i c k e r (1958), whi le Curves F and G are cont r ived equations representat ive of the remain ing types of s t o c k - r e c r u i t sys tem Curve F i s cha rac t e r i zed by an ascending l i m b which conforms to Curve B for 0<Z<1, coupled wi th an asymptot ic par t which mainta ins the stock at the e q u i l i b r i u m density for a l l other values of w. Curve G may be cons idered as a representat ive of systems influenced by depensatory mor t a l i t y factors, i .e . , fac tors which are propor t ionate ly more severe at lower stock dens i t ies . The curve i s iden t i ca l to A except for w<0.4, i n which case product ion i s ca lcula ted , _ .433w by Z = we 7 I I 0 . 2 0 . 4 0 .6 0.8 1.0 1.2 P A R E N T S T O C K F i g u r e 2. Stock rec ru i tment curves F, G, H of R i c k e r (1958). See text for explanation. Simula t ion P r o c e d u r e The procedure of s imula t ion was as desc r ibed by L a r k i n and R i c k e r (1964) wi th modif ica t ions for the l o n g - t e r m effects. R e p r e -senting l o n g - t e r m fluctuations by a sin wave, each generat ion the value was assessed of where m i s the year in a cyc le and i i s the cyc le length. Thus, for . 2 the second year of a s i x -yea r cycle Q is the s in of — (2ir). T h i s i s — o scales by the mu l t i p l i e r , P, to define amplitude, augmented by one, and used as a mu l t i p l i e r or d i v i s o r depending on sign. Depending upon what i s envisaged, there are va r ious ways in which the s in function might be coupled with random n o r m a l deviates i n the s imula t ion of l o n g - t e r m fluctuations with super imposed random fluctuat ions. In addi t ion to using either effect separately, two a l t e rna -t ives were p rov ided for combined effects: r andom n o r m a l deviate, after each was scaled to a des i r ed o rder of magnitude. Then the absolute value was augmented by one and, depending on the sign, used as a m u l t i p l i e r or d i v i s o r . T h i s representat ion might be appropr ia te for a si tuat ion in which both effects were v i s u a l i z e d as occu r r i ng in the same environment . Q = P sin (1) S in plus random — the value of R was added to the value of a (2) S in t imes random — the value of R was scaled, augmented by one, used as a m u l t i p l i e r or d iv i so r , then was fol lowed by a r andom n o r m a l deviate used in the same way. T h i s type of s imula t ion might be taken to represent the situation i n which the effects were o c c u r r i n g i n different environments - - say, for example, l o n g - t e r m effects in the sea; s h o r t - t e r m random effects in f reshwater . A l g e b r a i c a l l y , the sin wave modif ied sys tem may be wr i t ten a( l - w t ) i / . ,m t 2TT . \ " l , H . / rru 2rh ^ n {1 + P s i n ( — Y ~ ) ) j ; m = 1, . . . - ; —J * 0 r a ( i - w j / i y-| L W t C V l+Pls inmTinT/J m = - , . where w t m P = product ion i n year t = spawning stock size i n year t = year i n sequence of cyc le ( i .e . , m v a r i e s f r o m 1 to 6 i n each 6-year cycle) = length of cyc le i n y e a r s = scal ing factor 10 F o r a f ixed rate of f i shing c t = F Z t w t + 1 = ( l - F ) Z t where = catch in year t F = f ixed rate of f ishing F o r complete s tab i l iza t ion of escapement: C t = Z t - w B ., Z t > w D = 0 , Z t < w B Zt+1 = W » e a ( 1 " W m ) , Z t - w B > 0 where wm = escapement assoc ia ted wi th m a x i m u m e q u i l i b r i u m catch (m. e. c. ) The computer s imulat ions conducted were a l l run for 200 s imulated y e a r s for a l l combinat ions of: (1) four l o n g - t e r m cyc le lengths: 6, 11, 12, 24 yea r s . (2) S ix stock r e c r u i t curves : A , B, C, F , G, H . (3) Two l eve l s of random effects. (4) Two l eve l s of l o n g - t e r m effects. (5) Three patterns of f i sh ing: f ixed percentage, pa r t i a l and complete s tab i l iza t ion of escapement ( R i c k e r 1958). (6) Three patterns of envi ronmenta l effects: s in effects only, s in plus random, s in t imes random. 11 D e t e r m i n i s t i c S in Effects In the s imula t ions using the _sin wave modif ier and a fixed percentage f i shery , the catch pattern was i n i t i a l l y one of e r r a t i c changes in abundance. Wi th in a few generations a repeatable pattern of catch and escapement was evident, the values o sc i l l a t i ng in phase wi th the sin modi f i e r but at lower ampl i tudes . (The i n i t i a l pe r iod of ins t ab i l i t y was a resu l t of s tar t ing the s imulat ions with the stock at i t s e q u i l i b r i u m density (w = 1). Two l e v e l s of ampli tude ( P = . 5, P = 2.0) were used i n conjunction with four cycle lengths and the resu l t ing mean catch and escapement values for two hundred s imulated y e a r s are s u m m a r i z e d in Table II, which can be compared with the s imple de t e rmin i s t i c case (Table I). F o r curves A , B and C, the tabulated escapement values differ only s l ight ly f r o m the de t e rmin i s t i c values assoc ia ted wi th m a x i m u m e q u i l i b r i u m catch (m. e. c . )* The differences a re at tr ibutable to the i n i t i a l pe r iod of ins tab i l i ty and the f ixed 200 year pe r iod of s imula t ion which did not a lways end on a year complet ing a cyc le . F o r curve F , the mean catch i s s l ight ly l e s s than the de t e rmin i s t i c mode l and curve G goes to ext inct ion under conditions of f ixed percentage The de t e rmin i s t i c values of m . e. c. i n Table I are de te rmined by setting the f i r s t der iva t ive of w t e ^ a _ w t ) -w t equa l to zero and solving for w. 12 T A B L E T. M e a n catch and escapement in de t e rmin i s t i c models of six s t o c k - r e c r u i t re la t ionsh ips . C U R V E C A T C H E S C A P E M E N T A 0.330 0.433 B 0.935 0.361 C 1.656 0.314 F 0.760 0.210 G 0.330 0.433 H 0.520 0.240 13 T A B L E II. M e a n catch and escapement in 200 year s imula t ions of f ixed percentage exploi ta t ion for each of s ix reproduct ive curve systems, wi th l o n g - t e r m envi ronmenta l fluctuations, represented by sin modi f ie r s, at cyc le lengths of 6, 11, 12 and 24 year s, explanation. and at two l e v e l s of ampli tude. See text for > C Y C L E 6 yea r s 11 year s 12 year s 24 year s Curve L e v e l Ca tch Escap , Catch E scap Catch E scap Catch E scap. A 1 2 .3 33 .338 .436 .442 .333 .338 .436 .442 .336 .346 .440 .453 .340 .358 .445 .469 B 1 2 .937 .948 .360 .365 .936 .945 .360 .364 .940 .956 .362 .368 .949 .981 .365 .377 C 1 2 1.657 1.673 .313 .316 1.656 1.673 .313 .316 1.660 1.684 .314 .318 1.676 1.723 .316 .325 F ' 1 2 .739 .746 .204 .206 .728 .711 .201 .197 .731 .742 .202 .20 5 .724 .750 .200 .208 G * 1 2 .0 13 .014 .018 .099 .019 .026 .024 .034 .0 20 .028 .026 .037 .037 .057 .049 .075 H 1 2 .543 .686 .250 .317 .550 .752 .254 .347 .553 .766 .255 .3 54 .562 .823 .259 .380 Cu rve G goes to ext inct ion i n a few generat ions under condit ions of f ixed percentage exploi ta t ion. exploi ta t ion. Curve H i s the only sys tem which responds favorably to the combinat ion of sin modi f ica t ion and fixed percentage exploi ta t ion. Curve H i s s i m i l a r to A in that m a x i m u m product ion i s assoc ia ted wi th a stock size much l a rge r than the stock size which provides m a x i m u m catch, but differs in that there i s no compensation at the higher stock dens i t ies . Since the catch i s p ropor t iona l to product ion in the f ixed percentage f i shery , gains are to be expected. Table III s u m m a r i z e s the resu l t s of s imula t ions using the same amplitude and cyc le length modi f ica t ions but with a s tab i l ized f i shery . Compar ing these resu l t s with Table I, i t i s apparent that a s tab i l ized f i shery re su l t s in an inc reased y i e l d in a l l cases . The Effect of Cyc l e Length and Ampl i tude Modi f i ca t ions In the prev ious section it was shown that the mean catches for curves A , B and C are unchanged with the combinat ion of sin m o d i f i -cat ion and f ixed percentage exploi ta t ion. Never the less , the pattern of the catch responds to both changes in cycle length and ampli tude. F i g u r e 3 compares , for curve C, the pattern of a 12-year cyc le with that of a 24-year cycle , while F i g u r e 4 shows the effect of va ry ing the ampli tude of the s in modifier". F o r the fixed percentage exploi tat ion, va r i a t i on in cycle length has a lmost no effect on the standard deviat ion of catch ( c r 0 ), and serves only to increase the length of the se r i e s of 4 u u u 0 B y»or eye* [1 24 y w r eyelt I 2 3 4 9 6 7 8 9 0 II 12 13 Generation * B 6 17 « 19 20 * 22 23 24 F i g u r e 3 . C a t c h i n d e t e r m i n i s t i c s i m u l a t i o n o f a f i x e d p e r c e n t a g e f i s h e r y o n c u r v e C s y s t e m s w i t h 1 2 - a n d 2 4 - y e a r s i n w a v e c y c l e s f o r a m p l i t u d e s c a l i n g f a c t o r P = 2 . 0 . 16 i I T A B L E III. Mean catch and escapement i n 200 year s imula t ions of a f i she ry with complete ly s tab i l ized escapement for each of six reproduct ive curve systems, when l o n g - t e r m e n v i r o n -mental f luctuations a re represented by sin modi f i e r s , at cyc le lengths of 6, 11, 12 and 24 yea rs and at two l eve l s of ampli tude. See text for explanation. C Y C L E • 6 year s 11 yea r s 12 year s 24 yea r s Curve L e v e l Catch E scap. Ca tch E scap. Catch E scap. Catch E scap. 1 .368 .433 .367 .433 .369 .433 .375 .433 A 2 .624 .370 .643 .351 .649 .348 .686 .330 1 .994 .361 . .994 .361 .996 .361 1.007 .361 JD 2 1.422 .361 1.436 .361 1.442 .361 1.482 .361 1 1.744 .314 1.744 .314 1.748 .314 1.765 .314 2 2.392 .314 2. 414 .314 2.424 .314 2.487 .314 TP 1 .837 .210 .837 .210 .839 .210 .847 .210 Jc 2 1.169 .210 1.179 .210 1.184 .210 1.214 .210 1 .368 .433 .367 .433 .369 .433 .375 .433 C j 2 .394 .322 .312 .250 .311 .243 .236 .174 T T 1 .557 .240 .556 .240 .558 .240 .564 .240 r i 2 .811 .240 .817 .240 .821 .240 .843 .240 o o o L k i B Lw ompntuO* 0 Miflh ompNtudt I 2 3 4 1 6 7 8 9 O II 12 13 14 6 16 Generation 17 M 20 21 22 23 24 F i g u r e 4. Catch i n de te rmin i s t i c s imula t ion of a f ixed percentage f i shery on a curve C sys tem with 12-year s in cycle for ampli tude sca l ing factor at two leve ls , P = 0.5 and P = 2.0. good and poor catches. A s might be expected, o"c responds to ampli tude modif icat ions , i nc reas ing as the ampli tude becomes l a r g e r . F o r the s tab i l ized f i shery two s l ight ly different catch patterns were evident. In the systems which had a r e l a t ive ly high product ion per spawner at low stock dens i t ies (Curves B and C), the product ion never dropped below w, and consequently the stocks a lways reproduced at an op t imal rate (F ig . 5). In the l e s s responsive systems (Curves A and G) the catch pat tern is t yp i ca l ly a ser ies of r e l a t i ve ly good catches fol lowed by a se r i e s of zero catches, the length of the se r ies i nc reas ing as a function of the ampli tude of the mod i f i e r s . Stochast ic Modi f i ca t ions The de t e rmin i s t i c s in modif ica t ions desc r ibed above were s tochas t i s ized by either additive or mu l t ip l i ca t ive super impos i t ion of random n o r m a l deviates. A l g e b r a i c a l l y , the modif ica t ions (R) m a y b e represented as: Add i t i ve Effects R = (1 + Q + RND) for Q + R N D > 0 ° ( I + Q ' + M T O ) for Q + RND < 0 i Curve A Q Curve 8 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 Generation F i g u r e 5 . Catch i n de te rmin i s t i c s imula t ion of a completely s t ab i l i zed f i shery on curve A and curve B systems with a 1 2 - y e a r s in cycle and ampli tude sca l ing factor P = 2 . 0 . 20 M u l t i p l i c a t i v e Effects = (1 + Q) (1 + RND) = ( l +Q) ( — ) \ 1 + R N D / U 4 ( - i ) \1 +QJ \ 1 .+ R N D / for Q > 0, R N D > 0 for Q > 0, R N D < 0 for Q< 0, R N D < 0 for Q < 0, R N D < 0 where Q = scaled s in modi f ie r R N D = scaled random modi f i e r In general , the addit ion of the stochastic mod i f i e r s serves only to accentuate the bas ic patterns desc r ibed for the de t e rmin i s t i c s in effects, the frequency of ze ro catches becoming greater for the s tab i l ized f i shery , and cre becoming l a r g e r for f ixed percentage exploi ta t ion. The catches for the s tab i l ized f i shery are l a rge r as a function of o"0 whi le no changes are observed for f ixed percentage exploi ta t ion. The Nature of the Modi f i ca t ions F r o m the preceding resu l t s three genera l conclus ions per taining to curves A , B and C are evident. (1) F o r a s tab i l ized f ishery, an increase in y i e l d over the de t e rmin i s t i c m a x i m u m i s apparent for a l l systems subjected to stochastic modi f ica t ions , and this inc rease i s commensurate wi th the va r i ance of the m o d i f i e r s . (2) The addi t ion of stochastic mod i f i e r s to a sys tem undergoing f ixed percentage exploi tat ion has no effect on the mean catch, i t s value remain ing the same as the de te rmin i s t i c m a x i m u m . (3) The pattern of modi f ica t ion ( i . e . , s inoidal , random, etc.) i s unimportant r e la t ive to the var iance of the modif ica t ions . The inc reased y i e ld s for the s tabi l ized f i shery can, i n a sense, be at t r ibuted to the technique of stochastic modi f ica t ion . Augment ing by one the absolute value of the random var ia te and subsequently using i t as a m u l t i p l i e r or d i v i s o r , has the effect of mul t ip ly ing product ion by an average modi f i e r which i s greater than 1. A l g e b r a i c a l l y , i f X i s a s y m m e t r i c a l l y d is t r ibu ted var iab le with mean 0, the frequency of a pa r t i cu l a r pos i t ive var ia te Xl i s the same as the frequency of the cor responding negative value. The appropriate mod i f i e r s would be 1 + X . for the fo rmer and ^r——i for the la t te r . ' 1 1 It i s ea s i ly demonstrated that the mean of a pa i r of these m o d i f i e r s i s greater than 1, f r o m which i t fo l lows that the mean of a se r i e s of these mod i f i e r s (R) i s a lso grea ter than 1. F o r example, i f X t = + 1, the cor responding m u l t i p l i e r = 2, and for X t = -1 , the appropr ia te m u l t i p l i e r = \ , F u r t h e r m o r e , since (1 +X) + ^ ^ \YL \} i s 2 an inc reas ing function of X , and since inc reas ing the standard deviat ion (<rx) r e su l t s in the more frequent occur rence of la rge values, i t fol lows that the mean modi f i e r R w i l l a lso inc rease as a function of tr 22 Thus, for a s tab i l i zed f i shery c" - [ w m e a ( 1 W m ) ] R - wm f o r Z t ^ w m where "c = mean catch w B = escapement assoc ia ted with m.e .c . c", then, i s a l i n e a r inc reas ing function of R . The case of the f ixed percentage f i shery i s somewhat l e s s s t ra ight forward . F o r curves A , B and C, average production per spawner i s higher than the de t e rmin i s t i c case, for the same reason given i n the preceding paragraph. However , the mean catch i s v i r t u a l l y iden t ica l to the de t e rmin i s t i c m a x i m u m . The sufficient conditions necessa ry for this resul t a re demonstrable a l g e b r a i c a l l y . F o r a fixed percentage f i shery r a ( l - w t ) n r „ -, r _-, ct = [ w t e W ] [R] [F] where c t catch i n generat ion t, wt spawning stock i n generat ion t, R s tochast ic modif ier , and F exploi ta t ion rate appropria te for m.e .c . ( (1 - e a ^ W m ^ ) . Substituting the express ion for F we have r a ( l - w t ) a ( w , - w t ) n r „ i c t = [ w t e t ; - w t e K *'] [R] . Harves t ing at the e q u i l i b r i u m rate i m p l i e s that w t = w, . Substituting wm for w t i n the express ion for catch, the expected catch i n a t ime per iod becomes a ( l - w m ) c t = R t w , e - R t w B . Because R t w m i s the escapement, the expected catch i n pe r iod (t + 1) i s a ( l - R t w B ) Ct+1 = R t + l R * w ° e - R t + l W n and i n (t + 2) i s ^ ( l - R t + i R t W . ) The success ive product of R t + ^ (i = 1» n) i s 1, and ignor ing the vaLue of th las t modi f i e r R , i t i s apparent that t - n r - a(l - w n ) C = Wm e -w m which i s the d e t e r m i n i s t i c m a x i m u m . B i o l o g i c a l l y , the ma themat i c s i m p l y that the product ion f r o m any g iven number of spawners exposed to a va ry ing environment w i l l , on average, be l a r g e r than i f condi t ions were constant at a mean value . In the s t ab i l i zed f i she ry this gain i n product ion i s ref lec ted by i n c r e a s e d y i e l d s , whereas i n the f ixed percentage f i she ry any gain i s e l imina ted by the subsequent compensa tory response . Because the s tandard devia t ion of catch (a 0 ) and mean ca tch (c) a re both i n c r e a s i n g functions of R, c: and crc are c o r r e l a t e d . T h i s r e l a t ionsh ip provides a means of de te rmin ing , in the absence of in fo rmat ion concern ing the exact shape of a reproduc t ion curve , whether a p a r t i c u l a r escapement i s appropr ia te for m . e . c . B y exp re s s ing "c i n stock units for a se r i e s of escapements of approx ima te ly the same s ize , i t m a y be assumed that va r i a t i ons in the resul tant ca tch may be a t t r ibuted to envi ronmenta l fac tors act ing independently of the densi ty of -the stock. Since crc and cV are unique for a g iven op t ima l escapement •ajid curve type, ( F i g . 6), i t i s poss ib le to obtain a d e t e r m i n i s t i c value • of " a " a s soc ia t ed wi th the p a r t i c u l a r "c and <re cons ide red . Hav ing an ' es t imate of " a " , soljving for w , the expres s ion - a ( l - w B ) a ( l - w j - a w „ e +e - 1 = 0 -.and compar ing th is value to the actual escapement, p rov ides the check. . 11 they are iden t ica l , i t may be concluded that m . e . c . i s being obtained. H a r v e s t i n g at l e v e l s other than m . e . c . r e su l t s i n a decrease of c --which w i l l cause the t heo re t i c a l and actual escapement va lues to di f fer . D i s c u s s i o n The supe r impos i t i on of s imula ted l o n g - t e r m envi ronmenta l effects on stock r e c r u i t systems does not a l t e r the genera l conclusions of R i c k e r ( 1 9 5 8 ) . E n v i r o n m e n t a l v a r i a b i l i t y , whether of a r andom or r egu la r nature, r e su l t s i n benefits f r o m complete s t ab i l i za t ion of escapement and, the greater the • - v a r i a b i l i t y , the greater the benefits . Benefi ts i n y i e l d a re obtained f r o m the combinat ion of s tochast ic v a r i a b i l i t y wi th complete ly s t ab i l i zed escapements, the s ize of the benefit being an inc reas ing function of the va r i ance of the m o d i f i e r s . ^ Th i s conc lus ion i n l a rge par t hinges on the va l i d i t y of the assumptions i m p l i c i t in the procedure of s imula t ing envi ronmenta l effects by using m u l t i -p l i e r s or d i v i s o r s of product ion. It might be more reasonable to suppose that a stock at high density would be more susceptible to harmful envi ronmenta l effects (such as low water l eve l s for salmon). In such a s i tuat ion the sca l ing 3.23 3 0 0 2.75 I 50 • .25 Curve C Curve B Curve A F i g u r e 6. Rela t ionship between mean catch (c) and standard deviat ion of catch ( 0 " c ) i n comple te ly s t ab i l i zed f i sher ies for 200 year s imulat ions for curves A , B , and C for which e n v i r o n -mental effects at various leve ls were s imulated by sca led r a n d o m n o r m a l deviates used i n the method of R i c k e r (1958). 27 f a c t o r f o r e n v i r o n m e n t a l e f f e c t s m i g h t b e m a d e l a r g e r a t h i g h e r d e n s i t i e s . A g r e a t m a n y t e c h n i q u e s m i g h t b e c o n t r i v e d t o a p p r o x i m a t e w h a t e v e r w a s e n -v i s a g e d . H o w e v e r , t h e r e i s r e l a t i v e l y l i t t l e i n f o r m a t i o n o n t h e m e c h a n i s m s by w h i c h e n v i r o n m e n t a l f a c t o r s i n f l u e n c e p r o d u c t i o n a t v a r i o u s s t o c k l e v e l s . I t i s a l s o n o t k n o w n t h a t t h e v a l u e s o f r e l e v a n t e n v i r o n m e n t a l v a r i a b l e s a r e d i s t r i b u t e d i n t h e s p e c i a l p a t t e r n o f R i c k e r ' s s y s t e m ( i t i s t o b e n o t e d t h a t t h e d i s t r i b u t i o n o f r a n d o m n o r m a l d e v i a t e s a u g m e n t e d b y o n e a s m u l t i p l i e r s a n d d i v i s o r s i s n o t l o g n o r m a l ) . F u r t h e r u n d e r s t a n d i n g o f s t o c k r e c r u i t s y s t e m s , a s i n f l u e n c e d b y e n v i r o n m e n t a l f l u c t u a t i o n s , p r o b a b l y d e p e n d s p r i m a r i l y o n m o r e d e t a i l e d k n o w l e d g e o f t h e m e c h a n i s m s b y w h i c h e n v i r o n -m e n t a l f a c t o r s i n f l u e n c e p r o d u c t i o n a t v a r i o u s s t o c k s i z e s . - P A R T II •Introduction In a s e r i e s of papers concerned wi th the techniques of harves t ing c o m -o m e r c i a l stocks subjected to va r ious fo rms and l eve l s of envi ronmenta l v a r i a -b i l i t y , R i c k e r (1958), L a r k i n and R i c k e r (1964) and Tautz, L a r k i n and R i c k e r '3(1969) have demonstra ted mathemat ica l ly the gains i n y ie ld assoc ia ted wi th ariaintaining a constant escapement. Tautz, et a l . , showed that the method used -to introduce s tochast ic v a r i a b i l i t y had the effect of i nc rea s ing average p roduc-A.tion per spawner. Th i s , combined wi th the l ack of severe compensatory m o r -t a l i t y (because s tocks were not a l lowed to r each high density conditions) - resul ted i n the inc reased y ie lds observed i n s imula t ion s tudies . Thus, for a g iven de t e rmin i s t i c re la t ionship , the increase in y i e ld was found to be some i n c r e a s i n g function of the var iance of the s tochast ic modif ica t ions . These conc lus ions a re n e c e s s a r i l y the product of a mode l i n which spec i f i c assumpt ions have been made wi th respect to the represen ta t ion of dens i ty dependent p rocesses and the ex te rna l s tochast ic v a r i a b i l i t y a s soc ia t ed w i t h them. Because they a re of some impor tance in the development of management po l i cy , i t i s useful to consider whether they are genera l for mechan i sms , other than those envisaged by R i c k e r i n h i s development of the m o d e l . Da ta wh ich have recen t ly been co l lec ted for the Skeena R i v e r sockeye suggest that highest r ec ru i tmen t ra tes a re assoc ia ted wi th in te rmedia te l eve l s ? o f escapement (I.S. Todd, pers.comm..). Thus, the dome-shaped reproduction curve would seem to be a better average relationship for salmon than the -asymptotic form suggested by Beverton and Holt. , ° / I n the existing model, however, the meaning of the deterministic - T elation ship is not explicitly stated. It is generally assumed that the curve(s) 'represent the relationship of stock to recruitment in the absence of environ-smental variability, and this in turn provides the basis for the concepts of "equilibrium density" and "stock unit." However, by assuming that stocks =are buffered against detrimental environmental change, the average recruit -ment curve in the stochastic version is different from the deterministic form (Fig. 7), with the result that average equilibrium density (i.e., the intersection o f the recruitment curve and the replacement line) is greater than one stock unit. In other words, although the deterministic curve may describe the behavior of a stock recruitment system for a particular set of environmental conditions, (i.e., average conditions) it does not necessarily follow that the same curve will be evident when the environment is allowed to vary about that average condition. Another feature of the existing model which may require modification, is the complete separation of density dependent and density independent mortality. At present, the degree of compensatory mortality is assumed to be only a function of the difference between the stock size and equilibrium density, a constant. In other words, the term e&^V W ^ in Ricker (1958), L a r k i n and Ricker (1964) and Tautz, et al., (1969)> i s independent of any 30 I STOCK F i g u r e 7. Stock rec ru i tment curve A and average curve when the s y s t e m is exposed to stochastic va r ia t ion A 1 . 31 var i ab le envi ronmenta l influence, and consequently no p r o v i s i o n would seem to be made for environmental changes to operate i n a density dependent manner . Thus, the sys tem now envisaged is one in which the c a r r y i n g capaci ty i s cons idered to be a r andom var iab le ; that i s , e q u i l i b r i u m density changes f r o m year to year and the compensatory response of the stock i s in r e l a t i on to that pa r t i cu la r condit ion ra ther than to an average value. Since , for most c o m m e r c i a l l y impor tant species, compensat ion is assumed to occur i n the reproduct ive and ea r ly stages of the l i fe h i s to ry , i t was felt that the above condition could be best demonstrated by examining the mechan i sms assoc ia ted with spawning and s u r v i v a l of f ry . Th i s , then, would a l l ow one to hypothesize the nature of the curve re la t ing s u r v i v a l of f ry (production) to stock size, and by an understanding of the re levant mechanisms, to suggest the nature of the v a r i a b i l i t y about that curve . F u r t h e r m o r e , a stock r e c r u i t curve would be suggested f r o m this re la t ionship , i f one assumes that the number of r e c r u i t s is some constant propor t ion of the number of fry, i .e . , that the density dependent effects are confined to a l i fe h i s to ry stage for wh ich the c a r r y i n g capacity of the assoc ia ted environment i s s m a l l , r e la t ive to the lake and oceanic life h i s to ry stages. T h i s i s , to some degree, erroneous and it would be su rp r i s i ng indeed if r ec ru i tment was found to be a constant p ropor t ion of the number of f ry produced. However , for some situations (e.g., Skeena R ive r ) i t serves as a reasonable approximat ion . E v e n i f i t were shown that l i m i t a t i o n was a proper ty of the lake and/or ocean environments , the above re la t ionsh ip would r e m a i n as an important management tool since fry product ion must s t i l l be de termined so as to opt imize the number of ind iv idua ls 32 entering the l i m i t i n g l i fe h i s t o r y stage. Spec i f i ca l l y , then, the purpose of this paper i s : (1) to descr ibe some of the behav io ra l mechan isms operating during spawning; (2) to develop the concept of p r e f e r r ed and m a r g i n a l habitats for spawning f ish; and (3) to con-s ider how this m e c h a n i s m would influence the exis t ing s t o c k - r e c r u i t model . Site Se lec t ion and Ext rapensa to ry M o r t a l i t y D u r i n g spawning aggress ive behavior i s m a x i m a l , there is obvious c o m -pet i t ion for space, and the eggs, when deposited i n the g rave l , are sensi t ive to environmenta l change. F lood ing , e ros ion , and dry ing up of ce r ta in areas often resul t i n severe egg mor ta l i ty ( Neave, 1953 ) and, consequently, the mechan i sms responsib le for the pattern of nest locat ion are of considerable in teres t . It i s evident that wi th respect to a given source of mor ta l i ty , ce r ta in a reas of the s t r e a m constitute m a r g i n a l habitates; e.g., shal lower areas are m o r e susceptible to dry ing up than deeper ones. It is a lso poss ib le that under different sets of c i r cums tances shal lower areas may be considered as opt imal habitats and deeper areas m a r g i n a l . Thus, over a la rge number of generations, the p robab i l i t i e s of s u r v i v a l assoc ia ted wi th phys ica l ly different areas of the s t r e a m may be s i m i l a r to one another, and therefore the mor ta l i ty rate assoc ia ted with var ious environmenta l changes may be density independent in the manner desc r ibed by R i c k e r (1958). T h i s i s not the only way that the con-di t ions of the R i c k e r formula t ion are fu l f i l led , however, since i t i s poss ible 33 that ce r ta in areas of the s t r eam are consis tent ly better than others, yet the adult spawners may not have the capacity to differentiate between these a reas . T h i s , too, would resul t i n a si tuation where mor ta l i ty rate was, on the average, independent of the density of the stock. Therefore , given either of these two situations, one might anticipate that the d i s t r ibu t ion of eggs would be random wi th respect to any measurable phys i ca l var iab le , since there would be no select ive p r e s su re which would influence the choice of a s i te . A n al ternat ive hypothesis would be one which would predic t that the " o p t i m a l " a reas of a s t r e a m would be colonized i n i t i a l l y and marg ina l a reas would only be used in the event of h igh stock dens i t ies . Th i s would suggest that the order of c o l o n i -zat ion would cor respond to some measurable phys ica l va r i ab l e . Therefore , f ie ld studies were conducted in an attempt to determine which patterns cha rac te r i zed the behavior of na tura l populations. The Study A r e a s I Fou r M i l e Creek The f i r s t study was conducted at F o u r - M i l e Creek, an inlet s t r eam of the Babine Lake system, which has been desc r ibed in deta i l by Hanson (1964). A se r i e s of pools along the length of the creek (depths of up to 1.5 metres) serve as res t ing areas for the f i sh during thei r mig ra t ion ups t ream. These a re i n t e r spe r sed wi th shal lower areas suitable for spawning. A wate r fa l l 1.8km ups t ream f r o m the lake confines the run to the lower area of the s t r eam and, under conditions of low discharge, mig ra t ion is further r e s t r i c t e d at 550in ups t r eam f r o m the lake due to a t r i fu rca t ion in the creek and resu l t ing shallowness 34 of the f low. The creek is heav i ly shaded throughout i ts length and contains a good representa t ion of the range of conditions i n 'wh ich sockeye are l i k e l y to spawn. The run in F o u r - M i l e Creek appears to be a d is t inc t stock in the sense that i t i s c l e a r l y separated in both space and t ime f r o m other segments of the Babine run. It i s one of the ea r l i e s t runs to a r r i v e and averages 2300 i n d i v i -duals per year (Hanson, 1964). Methods The 550m section of the creek below the t r i fu rca t ion was enclosed using two b r o o m st ick fences, the lower fence being equipped wi th a V t rap and a holding f ac i l i t y to aid i n counting, measur ing and tagging the sa lmon. Th i s pa r t i cu l a r a rea was chosen because most of the good spawning grounds are found below the t r i fu rca t ion and, i f the run were sma l l , i t could eas i ly be observed . It was also des i rab le because a natura l b a r r i e r to mig ra t ion per iod i c a l l y occurs at the t r i fu rca t ion . The conditions, however, were not idea l s ince i t was necessa ry to have enough ind iv idua l s in the study a rea to ensure co loniza t ion of the m a r g i n a l areas , yet not severe ly damage product ion by overc rowding the spawners . Th i s proved to be a dif f icul t s i tuation since neither the total s ize of the run, nor the t empora l d i s t r ibu t ion , could be e s t i -mated in advance. The run was l a rge r than ant icipated and, consequently, the upper fence had to be removed p e r i o d i c a l l y when la rge numbers of spawners accumulated at this loca t ion . With in the lower section ten f a i r l y un i fo rm areas, which differed f r o m one another, were selected for intensive observat ion ( F i g . 8). E a c h a rea was approx imate ly 7 m in length and was divided into l m s t r ips ac ross the width of the s t ream; e.g., i f the s t r eam were 5m wide, five l x 7 m g r i d s would be fo rmed . The contours, depths and gradients were determined using o r d i -na ry surveying techniques and ve loc i t i e s were taken using a Gur l ey flow mete r . The ve loc i t i e s were measured a standard distance f r o m the bot tom (10cm) in order to determine the range of ve loc i t i es a spawner would be sub-jected to, rather than de termining the average ve loc i ty of the s t r eam sect ion. F o r each of the 1x7m sections, nine measurements of depth and ve loc i ty were made, ' three measurements ac ros s the top of the section, three ac ross the middle , and three ac ros s the bot tom. G r a v e l samples in each of the sections were a lso taken using a core sampler and analyzed using the v o l u m e t r i c technique desc r ibed by M c N e i l (1964). Samples could not be obtained in a l l a reas since the grade was too coarse , but photographs of a l l g r id s were taken wi th the a id of a p lex ig lass -bo t tomed box and a wide angle lens . Staff gauges were ins ta l l ed i n each of the ten sections and an addi t ional staff gauge was set up as indicated i n F i g u r e 8, which served as a standard meter ing si te. These measurements were taken in the three-week per iod p r i o r to the onset of the run, the ve loc i t i e s being taken las t (July 22-23). It i s therefore neces sa ry to assume that the re la t ive differences in depth and ve loc i ty r emained constant during the spawning per iod, which is probably not quite true for a l l of the cases cons idered . However , this method was more des i rab le than the a l ternat ive of making da i ly measurements during the spawning pe r iod 36 F i g u r e 8 . F o u r M i l e C r e e k s h o w i n g l o c a t i o n of g r i d s A - J a n d the m e t e r i n g s i t e . (*)• as d a i l y d i s t u r b a n c e s w o u l d p r o b a b l y i n f l u e n c e t h e n o r m a l p a t t e r n o f c o l o n i z a -t i o n a n d e g g s u r v i v a l . E a c h d a y d u r i n g t h e s p a w n i n g r u n a l l o f t h e f i s h t r a p p e d d u r i n g t h e p r e v i o u s 2 4 - h o u r p e r i o d w e r e c o u n t e d a n d a r e p r e s e n t a t i v e s a m p l e o f t h e f e m a l e s p a w n e r s w a s t a g g e d , m e a s u r e d a n d c h e c k e d f o r s t a t e o f m a t u r i t y . T h e l a t t e r p r o c e d u r e i n v o l v e d g e n t l y s q u e e z i n g t h e b e l l y o f t h e f e m a l e s a n d , i f e g g s w e r e e x p e l l e d , t h e f i s h e s w e r e c o n s i d e r e d t o b e m a t u r e . P r i o r t o t h i s , e a c h d a y , t h e s p a w n e r s i n e a c h o f t h e g r i d s w e r e c o u n t e d a n d t h e i r p o s i t i o n r e c o r d e d o n m a p s o f t h e a r e a s . T h u s , e a c h d a y ' s e s c a p e m e n t w a s g i v e n a 1 5 - 2 0 h o u r p e r i o d t o d i s t r i b u t e i n t h e s t r e a m w i t h o u t b e i n g d i s t u r b e d b y o b s e r v a t i o n s o r b y n e w f i s h e n t e r i n g t h e s t r e a m . R e s u l t s P h y s i c a l c h a r a c t e r i s t i c s o f t h e s e c t i o n s . T h e d e p t h s , v e l o c i t i e s , a n d a n i n d e x o f s u b s t r a t e p e r m e a b i l i t y f o r e a c h o f t h e t e n s e c t i o n s d e s c r i b e d p r e v i o u s l y , a r e p r e s e n t e d i n T a b l e I Y • T h e p e r -m e a b i l i t y i n d e x i s e x p r e s s s e d a s a p e r c e n t a g e o f t h e c o r e s a m p l e p a s s i n g t h r o u g h a 1 . 1 9 m m s e i v e . T h i s w a s s l i g h t l y d i f f e r e n t f r o m t h e t e c h n i q u e d e s c r i b e d b y M c N e i l ( 1 9 6 4 ) i n t h a t h i s c a l c u l a t i o n s a r e e x p r e s s e d a s a p e r -c e n t a g e p a s s i n g t h r o u g h a 0 . 8 8 3 m m s e i v e . T h u s , t h e s e c a l c u l a t i o n s p r o b a b l y u n d e r e s t i m a t e a c t u a l p e r m e a b i l i t y , b u t e v e n i g n o r i n g t h i s d i s c r e p a n c y , a l m o s t all o f t h e a r e a s w o u l d b e c l a s s i f i e d a s m e d i u m t o h i g h i n p e r m e a b i l i t y ( c o e f f i c i e n t o f p e r m e a b i l i t y a s s o c i a t e d w i t h 10 p e r c e n t o f f i n e m a t e r i a l i s 38 <TABL.E I V . P h y s i c a l c h a r a c t e r i s t i c s of g r i d s A - J wi th p e r m e a b i l i t y ?index expressed as per cent of sample pass ing through a 1 . 1 9 m m s i eve . Depth (cm) , P e r m e a b i l i t y Index (%) V e l o c i ty (cfs) Grid (max) 4 # samples max m i n max m i n A 43.5 (15) 17.5 3.0 3.20 .63 B 69 ( 6) 0 0 4.16 .91 C 43 ( 9) 13.6 .0 2 3.73 .50 D 39 ( 6) 9.7 1.4 3.36 1.37 E 47 (9) 9.0 1.8 5.68 1.37 F 37 (6) 10.9 1.4 3.36 1.04 G 41 3.89 .59 H 48 2.93 .08 I 32 4.07 1.49 J 37 3.58 .78 39 approx imate ly 200cm/min) ( M c N e i l and AhneLl, 1964). These samples were taken shor t ly after freshet conditions and therefore represent m a x i m u m "na tu ra l " pe rmeab i l i t y . A r e a s of lower pe rmeab i l i t y were undoubtedly present in the s t r e a m along the sides and in ve ry shal low areas, but these were not used as spawning s i tes . It was a lso evident that the spawning m a t e r i a l was not a homogeneous mix tu re of the g r ave l s izes ; i .e . , the beds were s trat i f ied, wi th the finer m a t e r i a l being present in greater concentrat ions at lower depths. The spawning run . The run commenced on 29 Ju ly and a few f i sh were s t i l l entering the s t r e a m when the fence p r o g r a m was terminated on 1 5 August 1968. Because of two breaks in the fence, the numbers for August 3 and 4 are est imates, but even a l lowing for l a rge e r r o r s (in these estimates) it was apparent that the run had a d i s t inc t ly b imoda l d i s t r ibu t ion ( F i g . 9). E a c h day the number of females on each of the 1x7m gr ids was counted and the mean number of sightings per day (over a per iod of 23 days) was c a l -culated. These values were then used as the dependent var iab le in a mul t ip le r e g r e s s i o n analys is , i .e . , mean number of sightings i n each g r i d was r eg re s sed against the es t imated depth and ve loc i ty of that g r i d . The independent va r iab les , depth and veloc i ty , were not co r re l a t ed (r = .01) and produced a s ignif icant r e g r e s s i o n (p = .0 1) on the mean number of sightings ( F i g . 10). 40 d i s c h a r g e e s c a p e m e n t 3 0 a u g - 3 % D A T E F i g u r e 9. T e m p o r a l d i s t r i b u t i o n of spawning run and a r e l a t i v e measure of changes in d i s c h a r g e . 41 F i g u r e 10. R e g r e s s i o n l ines fitted to mean number of sightings per g r i d plotted against the depth and veloci ty of the g r i d . 42 The fit, though significant, s t i l l leaves over half of the v a r i a b i l i t y 2 unexplained (r = .36), and the analysis, confounds two effects, t ime of o c c u -pancy and density; e.g., a sect ion wi th four f i sh present for one day would receive, the same weight as a sect ion wi th one- f i sh present over a per iod of four days. In order to separate these effects, ind iv idua l g r ids were examined to determine whether the p r e f e r r ed areas were a lso the f i r s t co lon ized . F i g u r e 11 shows, for sect ion A , (which is wide and contains a la rge range of depths) mean depth of the spawners for any given day plotted against day and a l so percentage of m a x i m u m density (17 females) . It can be seen that the deep areas were colonized i n i t i a l l y and, as the density increased , shal lower a reas were u t i l i z e d as evidenced by a s m a l l e r mean depth. S t r e a m m i g r a t o r y behavior . The previous conclusions are further confounded by the m i g r a t o r y behavior of spawning sa lmon upon entering the s t ream. The best spawning g r a v e l i s situated i n the lower sect ion of the s t ream, but the typ ica l pattern of m ig ra t i on seemed to be one i n which the f i sh moved past these lower areas un t i l they were unable to migra te fur ther . In the study a rea the b a r r i e r was the upper fence, but those f i sh which were a l lowed to move past the fence continued their m ig ra t i on ups t r eam to na tura l b a r r i e r s . No quantitative data were col lec ted on this aspect of their behavior , but days on which la rge numbers of f i sh were counted pass ing through the lower fence, la rge schools of adults were subsequently observed at the upper b a r r i e r . Thus, i t would F i g u r e 11. M e a n depth of the redds i n g r i d A plotted against per cent of m a x i m u m spawner density (18 redds) . 44 seem that most of the f i sh moved ups t r eam to the fence and la ter t r ave l l ed downs t ream to select a si te. T h i s behavior would have some effect on the technique used for measur ing d e s i r a b i l i t y since the p robabi l i ty of phys i ca l ly iden t ica l areas being colonized i s a function of the re la t ive posi t ion of the si tes and the number of spawners a l ready present in the s t r eam. Thus, s ignif icant differences might be detected sole ly as a function of the posi t ion i n the s t r e a m of different areas of depth and veloci ty , without any behav io ra l m e c h a n i s m of se lec t ion operat ing. However , were this to be the case, one would expect some order ing of se lec t ion f r o m the top of the s t r e a m to the bot tom. No such posi t ion effects are revealed by the da t a - - f i sh were observed i n a l l of the sections (i .e. , A , B . C) on the second day of the run and la ter f i sh tended to f i l l in the remain ing s i tes . It is suggested, therefore, that the p h y s i c a l differences between areas were sufficiently great as to obscure any effects of pos i t ion . E g g retention. On the bas i s of the preceding observat ions, i t may be concluded that ce r t a in combinations of depth and ve loc i ty are subject to more intensive use by spawning fish, and that these aspects of the environment are of greater impor tance than the re la t ive posi t ions of the areas i n the s t r eam. Th i s may be in te rpre ted as evidence for the site se lec t ion and, consequently, op t imal and m a r g i n a l habitats . However , site se lec t ion per se, does not account for the descending l i m b of the reproduct ion curve; i .e . , at high densi t ies total product ion is l e s s than at lower dens i t ies . Thus, in order to conform to the 45 e m p i r i c a l curve, i t i s necessary to hypothesize an interference component such that deposi t ion success and/or s u r v i v a l rate of the eggs is reduced to a la rge degree at h igh density. S u r v i v a l of eggs a l ready deposited in the g r a v e l could be affected by super impos i t ion or by mechanica l dis turbances resu l t ing f r o m spawners i n close p rox imi ty , whereas deposi t ion success could be influenced by f i sh not being able to obtain t e r r i t o r i e s and a lso frequently in ter rupted spawning behavior . It was felt that the degree of egg retent ion might ref lec t deposi t ion success and/or the inab i l i t y to obtain a t e r r i t o r y and so dead females for which the date of entry, state of matur i ty on entering, and durat ion of stay, were known were examined for re ta ined eggs. A total of 231 spawners was col lec ted throughout the run, but no c lea r re la t ionsh ip of egg retention to date of entry, durat ion of stay, or state of maturi ty , was revea led , (2% reta ined over 2000 eggs; 10% retained over 500). However , l a rge numbers of dead eggs were observed in pools below areas of intensive spawning, i n d i -cating that super impos i t ion was of r e l a t ive ly common occur rence in these a reas . II P inkut and Weaver Creeks The F o u r - M i l e C reek study may be cons idered as an example of the co loniza t ion behavior of stocks at h igh densi t ies in a f a i r l y heterogenous environment . F o r the purpose of compar i son , two other situations were observed which provided informat ion for conditions of: (1) low stock density, m e d i u m envi ronmenta l heterogeneity (Pinkut Creek) ; and (2) low stock density, u n i f o r m environment (Weaver Creek spawning channel). P inkut Creek i s an in le t of the Babine sys t em and i s situated eleven m i l e s west of F o u r - M i l e Creek . The major spawning area is much l a r g e r than any found i n F o u r - M i l e and the creek, being fed f r o m a lake, i s l e ss subject to r ap id changes in d i scharge . The creek i s f a i r l y un i fo rm along the length of the spawning a rea and, in this sense, i t i s l e s s d iverse , although a greater range of depth and substrate type than was observed at F o u r - M i l e Creek was apparent i n c ros s sect ion. The run at P inku t i s often b imoda l wi th an ea r ly peak o c c u r r i n g i n mid -Augus t and a l a te r peak occu r r i ng in September (Department of F i s h e r i e s , Canada). Two g r ids were placed a c r o s s the width of the creek (as i n F o u r - M i l e ) , one near the ups t r eam end of the area , the other 4 0 - 5 0 m downst ream. Depths and ve loc i t i e s were recorded and da i ly counts of ind iv idua l f i sh in each of the areas were again made. The ea r ly peak, however, was ve ry s m a l l wi th a l a rge percentage of this ea r ly run spawning in the a rea around the ups t ream g r i d . D a i l y counts for this g r i d are presented i n F i g u r e 12, and it was apparent that a r e l a t i ve ly wide range of depth and ve loc i ty (17 .3cm-36 .8cm depth, 0.8 - - 2.3ft/sec veloci ty) was u t i l i zed , even though areas of depth and ve loc i ty s i m i l a r to the most p re fe r r ed area, were read i ly avai lable i n l o c a -tions further downstream. Al though this observat ion may be construed as being con t ra ry to the si tuation desc r ibed for F o u r - M i l e Creek, i t i s worth noting that the spawning ac t iv i ty was s t i l l confined to a s m a l l range of the 47 F i g u r e 12. D a i l y n u m b e r o f f i s h p l o t t e d a g a i n s t a c r o s s s e c t i o n o f t h e u p p e r g r i d a t P i n k u t C r e e k . 48 depths and ve loc i t i es ava i l ab le . The fact that the ups t ream g r i d was used exc lus ive ly tends to re inforce the idea that the p rocess of site se lect ion does not begin immed ia t e ly after the samon enter the s t ream. Thus, only after they reach some state of phys io lo -g i c a l matur i ty do they appear to begin searching and, at this point, two explana-tions are p laus ib le for their subsequent behaviour; 1) that the p rocess of site se lect ion i s influenced only by the presence or absence of ce r ta in environmenta l cues and that the range of op t imal cues i s re la ted to the total range avai lable in the s t ream; 2) that the presence of other ind iv idua ls in a spawning a rea tends to take precedence over s m a l l re la t ive differences in the environmenta l s t i m u l i (i .e. , s o c i a l fac i l i ta t ion) . Thus, i n order to obtain more in format ion on the ro le of soc ia l fac i l i ta t ion , further observat ions were made on the sockeye i n Weaver Creek (Lower Main land , B r i t i s h Columbia) . Because the major part of this run is d iver ted into a l a rge spawning channel, an opportunity was avai lable for the examinat ion of co loniza t ion behavior i n a completely un i fo rm environment . Since the channel was designed so as to provide "good" spawning conditions, in t e rms of depth, ve loc i ty , and substrate type, the previous speculations concerning the in i t i a t ion of spawning behavior in r e l a t ion to the presence of appropr ia te phys ica l and s o c i a l s t i m u l i could be tested i n a more r igorous manner. T h i s in format ion was obtained for a s m a l l number of females (42) by capturing f i sh as they entered the fishway, d iv id ing the f i sh into two groups, tagging the members of each group wi th Pe t e r s en d iscs , r e l ea s ing one of the groups into the channel, • 49 and holding the other group i n pens near the f ishway. When members of the re leased group were a l l seen to be defending t e r r i t o r i e s , the penned f i sh were al lowed to enter the spawning channel and compar i son of the d i s t r ibu t ion of groups was made. A s was anticipated, a l l of the ind iv idua ls which were re leased i m m e -diate ly into the spawning channel migra ted ups t ream as far as poss ib le and v i r t u a l l y a l l of the spawning ac t iv i ty was confined to the upper five legs of the 26 legs i n the channel. Because the f i sh in the pens should be at the same average state of matur i ty as those released, i t was felt that, i f the only factors involved in the in i t i a t ion of spawning ac t iv i ty were appropriate phys io log ica l state and envi ronmenta l cues, penned fish, when re leased, would occupy areas i n lower legs of the spawning a rea . T h i s did not occur and, without exception, these f i sh moved ups t ream to the areas of intensive ac t iv i ty . Thus, the specu-lat ions concerning the behavior of unripe f i sh and the ro le of soc ia l fac i l i ta t ion would seem to be substantiated. The element of c i r c u l a r i t y involved in defining behav iora l responses without having an independent measure of phys io log ica l state i s recognized and the same c r i t i c i s m may be made for the definit ion of op t imal and m a r g i n a l habitats . However , in the absence of long t e r m studies of mean s u r v i v a l and va r iance of s u r v i v a l for a reas which provide detectably different s t i m u l i for the spawner, the site se lect ion would seem to be sufficiently plausible as to war ran t considerat ion in the development of stock r e c r u i t theory. 50 I M P L I C A T I O N S O F T H E S T U D I E S | Wi th in the context of the o r i g i n a l intention of the study, namely, that of es tabl ishing the mechanisms of density dependent mor ta l i ty in the s t r eam environment and their r e la t ion to random environmenta l va r i ab i l i t y , the fo l lowing observat ions are of impor tance: 1) the rate of co loniza t ion of spawning areas in a s t r eam is re la ted to the depths and ve loc i t i e s of water i n those areas for conditions of h igh spawner density; 2) that the site se lect ion process does not immedia te ly commence on entering the str earn--ra ther , some phys io log ica l state of matur i ty would seem to be necessa ry to ini t ia te spawning ac t iv i ty ; 3) soc ia l fac i l i t a t ion i s l i k e l y to be of some consequence in the se l ec -t ion of a site; • 4) that f i sh spawning ea r ly in the run are susceptible to having their nests destroyed by the ac t iv i ty of la ter a r r i v a l s ; 5) egg retention does not serve as a good indicator of deposi t ion sue-ces, and the number of eggs retained does not re la te to t ime of entry, state of matur i ty (at least for the crude measure of matur i ty used here) or t ime of res idence . Before dealing wi th these observations in a specif ic manner, it w i l l be useful to examine one way in which the exis t ing model may b,e modif ied so as to include the effect of density dependent mor t a l i t y due to r andom envi ronmenta l changes. The model , as desc r ibed by R i c k e r (1958), L a r k i n and R i c k e r (1964), Tautz, et al., (1969), is of the form e a(l-w).. . z - w v '[R] -where R is a scaled random environmental variable, the value of which is ^-selected at random arid subsequently scaled for use as a multiplier or divisor in the manner of Ricker (1958). This variable is density independent in that , iithe proportional effect of the modifier is the same, irrespective of the size of rthe stock, w. R is distributed over the range (0 to oo ) with a mean greater „than 1 as a function of the variance (Tautz, et al., 1969). In this system, the degree of compensatory mortality is only a function of the term ea^~w^, i.e., the difference between size of stock and equilibrium density or carrying capacity. Therefore, if the changes in the environment "are to act on or change the carrying capacity, it seems reasonable to suggest "that the stochastic variable (RND) be added or subtracted to the 1 in the exponent, that is, a(l_+^^~ -w) where the RND is normally distributed with c L mean 0 and variance 0 " 2 . This formulation would allow for a stock of a given size to show different degrees of response, depending upon the difference between the size of the stock and the carrying capacity of the environment for that particular year. The complete formulation may then be written as , , , RND . a(l+ -w Z = w e — a However, while this may.at first seem to be more appealing, it is only trivially different from the existing model. This is apparent if the equation is re-written Z = w e a ( 1 - w ) [ e ( R N D ) ] where the b racke ted exp re s s ion is a l o g - n o r m a l d i s t r i bu t ion . T h i s d i s t r i bu t ion has s i m i l a r p rope r t i e s of a s y m m e t r y to the d i s t r ibu t ion of [R] but has the s l ight advantage of being soluble ana ly t i ca l l y for a mean, v i z : m m , - x 2 / 2 a 2 R N D = 1 e ' / 2 n ° " " 7 i_. R N D . a 3 / 2 . ' * \ •." E(e ) = e . ; •• -It can be seen that the expected value of the modi f ie r i s an i nc r ea s ing a 2 / 2 function of the va r iance , spec i f i ca l ly , e , and a lso that e q u i l i b r i u m • • • , , • i / - i 2 / - , • a/l+b 2 / 2 -w) dens i ty i s obtained for w = 1 + 0 12, s ince Z - w e subst i tut ing the expectation) reduces to Z = w for w = l + a 2 / 2 . V i e w e d i n this way, the d i f f i cu l ty i n defining a stock unit and e q u i l i b r i u m densi ty becomes apparent, s ince one stock unit cannot be re la ted to average or expected c a r r y i n g capacity, g iven m u l t i p l i c a t i v e sys tems of s tochast ic v a r i a t i o n . One might envisage different ways i n wh ich the pa ramete r a and the va r i ance of the environment could be a l t e red to c i r cumven t this d i f f icul ty , but a mode l which i s more reduc t ion i s t i n f o r m could perhaps more eas i ly o v e r -come th is d i f f i cu l ty . • A l though a model of the r educ t ion i s t type i s beyond the scope of this paper, i t i s useful to cons ider the way i n which the f i e ld observat ions d e s c r i b e d p r e v i o u s l y , may be of some consequence i n the cons t ruc t ion of such a mode l , and also to indicate how the phenomenon of site se lec t ion viola tes the a s sump-tions of the R i c k e r formula t ion . A s imp l i f i ed example of such a sys tem would be a s t r e a m i n which the m o r t a l i t y of eggs and alevins i s due so le ly to n o r m a l l y d is t r ibu ted random changes i n water l e v e l during the t ime for which the eggs are in the g r a v e l . In this situation, deep areas bf the s t r eam would be assoc ia ted consis tent ly wi th h igh s u r v i v a l rates, and shal low areas would be more " sens i t i ve" to changes in d i scharge . Thus, the process of site select ion would a l low for deeper areas to be colonized i n i t i a l l y and shal lower areas to be used only under conditions of h igh densi ty. F o r a given m i n i m u m discharge, the number of sites destroyed would be a density dependent phenomenon. Th i s i s d is t inc t f r o m R i c k e r ' s fo rmula t ion i n that, in the R i c k e r model , the number of sites des t royed would be some constant p ropor t ion of the total number of s i tes . In this sys tem a wide range of s t r e a m l eve l s may be complete ly equivalent i n t e r m s of their effect on a s m a l l number of w e l l p laced nests, a feature which i s not a proper ty of the exis t ing mode l . T h i s manner of argument suggests the use of threshold phenomenon as opposed to continuous functions in the de sc r ip t i on of some density dependent p rocesses , and a more reduct ionis t mode l would be better suited to this approach. In such a model, a s t r e a m may be considered as a m a t r i x of si tes and each site (over a per iod of years) i s assoc ia ted with a different p robabi l i ty density function of s u r v i v a l rate, wi th a different mean and poss ib ly a different va r i ance . The site se lect ion process enables the spawners to detect these 54 a reas of high p robab i l i ty s u r v i v a l . F u r t h e r m o r e , i n the same way as one may associa te different p robab i l i t i e s of s u r v i v a l to different locat ions i n the s t ream, one may also consider this m a t r i x of elements to change as some function of t ime, and there are many ways in which this t empora l va r ia t ion may be en-visaged as being important . F o r example, ea r ly f i sh entering the s t r eam are more l i k e l y to obtain better areas for spawning, but, at the same t ime, the eggs are more susceptible to being d is turbed by super impos i t ion , p a r t i c u l a r l y i f the site se lec t ion process i s ve ry accura te . A l s o , the accu racy of the site se lec t ion p rocess i s a var iab le which should be cons idered and is a lso density dependent since, at low densi t ies , the effect of making an " e r r o r " i n the se lec t ion of a site is r e l a t i ve ly important i n t e rms of inf luencing total p roduc-tion, while at higher densi t ies these e r r o r s are " c o r r e c t e d " by the " f i l l i n g i n " of la ter spawners . S e v e r a l other " c o m p r o m i s e " situations may be envisaged as being i m p o r -tant for desc r ib ing " o p t i m a l " escapements into different s t r e a m environments , and these considerat ions can only be viewed i n proper perspec t ive by the use of a theore t i ca l sys tem which i s more specif ic than present models . C O N C L U S I O N S The preceding paper has presented c i r cums tan t i a l evidence for the proces of site se lect ion and s o c i a l f ac i l i t a t ion in the spawning behavior of sockeye sa lmon . The existence of p re fe r r ed and marg ina l habitats and the ab i l i ty of the spawners to differentiate between these types of areas , suggest that R i c k e r ' s formula t ion of stock and rec ru i tment is not a genera l desc r ip t ion of this phenomenon. Al though the shape of the curve i s substantiated by f i e ld data, the assumptions concerning the var iance about the curve are probably not co r r ec t for si tuations i n which site select ion i s operating in conjunction wi th r andom environmenta l changes. The exact nature of the var iance i s a proper ty of the si tuat ion i n question, and is probably defined by the re la t ive abundance of m a r g i n a l and opt imal habitats, the degree of t empora l v a r i a b i l i t y i n a r r i v a l of the spawners, the accu racy of the se lect ion process , and, the absolute s ize of the spawning run, among others . A genera l model incorpora t ing these va r iab les i s now being considered, but for speci f ic management situations i t i s important to bear in mind that opt imal escapements must be re la ted not only to total escapement and total spawning a rea avai lable , but a lso to the degree of t empora l va r i a t i on in a r r i v a l t imes and spat ia l heterogeneity. 56 A C K N O W L E D G M E N T S I w i sh to thank M e s s r s . J . F e l l , I . S. Todd, W. L . Steigenberger, the Department of F i s h e r i e s and F o r e s t r y of Canada, and the International P a c i f i c Sa lmon F i s h e r i e s C o m m i s s i o n for their cooperat ion and ass is tance dur ing the var ious phases of this study. In pa r t i cu la r , I am grateful to D r . P . A . L a r k i n , who acted as super -v i s o r of this study and who was a continual source of encouragement and enthusiasm. This study was funded in part by the F i s h and Wild l i fe B r a n c h of the Department of Rec rea t i on and Conservat ion, and i n part by the B . C. Salmon Derby Schola r sh ip . BIBLIOGRAPHY • • Larkin, P .A . , and W. E . Ricker. 1964. Further information on sustained -yields from fluctuating environments. J . Fish. Res. Bd. Canada, - 2 i ( i ) : 1-7. • „ :"-V;'; McNeil , W.J., and W. H . Ahnell. 1964. Sue cess of pink salmon spawning relative to size of spawning bed materials. U.S. Fish Wildl. Serv. ;Spec. Sci . Rept.--Fisheries, No. 469. Neave, F e r r i s . 1953. Factors affecting the freshwater development of ^Pacific salmon in British Columbia. Proc. 7th Pac. Sci. Cong. (1949), Vol.4, Auckland, New Zealand, pp. 548-555. ^Ricker, W. E . 1954. Stock and recruitment. J. Fish. Res. Bd. Canada, _U (5): 559-623. ' ; . 1958. Maximum yields from fluctuating environments and mixed stocks. J . Fish . Res. Bd. Canada, 15(5):991-1006. Tautz, A . , P. A . Larkin and W. E . Ricker. 1969. Some effects of simulated long-term environmental fluctuations on maximum sustained yield. J . Fish. Res. Bd. Canada, 26:2715-2726. 

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