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Characteristics of cyclic fluctuations generated by stock-recruit systems. Basasibwaki , Pereti 1971

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CHARACTERISTICS OF CYCLIC FLUCTUATIONS GENERATED STOCK-RECRUIT SYSTEMS by PERETI BASASIBWAKI B.Sc, University of East A f r i c a , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of ZOOLOGY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 19 71 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the 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 , I a g r e e t h a t 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 r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r 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 g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . 1 Pereti Basasibwaki Department o f Z o o l o g y The 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 V a n c o u v e r 8, Canada Date 10th June, 1971 ABSTRACT Ricker (1954) derived a stock recruitment r e l a t i o n s h i p from assumptions involving cannibalism or other compensatory density dependent mechanisms. His r e l a t i o n s h i p indicates a decline i n reproduction at high stock densities of spawners, the r e s u l t of which may give a population a tendency to o s c i l l a t e i n numbers. The object of this study was to examine, using Ricker 1s stock-recruitment model, the period and amplitude of c y c l i c a l f luctuations i n numbers of animals as they are related to i n t r i n s i c factors such as age of f i r s t maturity, number of generations i n the spawning stock and the shape of the reproduction curve, and e x t r i n s i c factors such as e x p l o i t a -t i o n and random f l u c t u a t i n g environment. In l i g h t l y exploited populations the period of o s c i l l a -t i o n i s dependent on age of f i r s t maturity and number of generations i n the spawning stock, the period being equal to twice the mean age of maturity. In heavily exploited populations, the period of o s c i l l a t i o n i s dependent on age of f i r s t maturity but independent of the number of genera-tions i n the spawning stock, the period being equal to approximately twice the age of f i r s t maturity. When random fluctuations are added to the system, c y c l i c changes are maintained at low e f f e c t s of random factors but they introduce i n s t a b i l i t y i n the o s c i l l a t i o n and v a r i a b i l i t y i n the period of o s c i l l a t i o n as they become more e f f e c t i v e . The shape of a reproduction curve does not influence the period of population o s c i l l a t i o n but the steeper the r i g h t hand limb of the curve the greater i s the amplitude of the fl u c t u a t i o n . The amplitude of o s c i l l a t i o n i n numbers i s b a s i c a l l y determined by the shape of a reproduction curve and i t increases with delayed maturity while i t decreases with increasing number of ages spawning and increasing e x p l o i t a -t i o n . I t was considered doubtful whether the observed high reduction i n the period of o s c i l l a t i o n would every be big enough to notice, i n natural populations, before e x p l o i t a t i o n removed o s c i l l a t i o n s completely and s t a b i l i s e d population abundances. i i i TABLE OF CONTENTS Page TITLE PAGE ABSTRACT i TABLE OF CONTENTS i i i LIST OF FIGURES - i v LIST OF TABLES v i ACKNOWLEDGEMENTS v i i i INTRODUCTION 1 RICKER'S REPRODUCTION CURVE 4 THEORY 4 FECUNDITY AND RECRUITMENT - UNFISHED STOCKS 8 FECUNDITY AND RECRUITMENT - FISHED STOCKS 11 <£- METHODS 12 <c- CYCLE LENGTH 14 1. Unexploited populations 14 2. Exploited populations 20 3. Mean and maximum ages 2 8 4^ NUMERICAL FLUCTUATIONS 33 1. Amplitude of o s c i l l a t i o n 33 2. Equilibrium abundance 38 3. Age of f i r s t maturity 41 4. Year-classes 43 5. Reproduction curve 4 5 6. E x p l o i t a t i o n 47 7. 'Environment' . 49 DISCUSSION 52 SUMMARY 65 BIBLIOGRAPHY 67 APPENDIX . 70 i v LIST OF FIGURES FIGURE ^ ~" . P a 9 e 1. Reproduction curve conforming to the T X. • U- r , T 7 A(l-W) _ r e l a t i o n s h i p Z = We 7 2. Three types of o s c i l l a t i o n s which are generated by Ricker's Reproduction curve of the type Z = W e A ( 1 - W ) 15 3. Cycle length and i t s r e l a t i o n s h i p with age of f i r s t maturity and number of year-classes breeding i n unexploited populations 19 4. Cycle length of a 6-year-class breeding population at 4 d i f f e r e n t ages of f i r s t maturity and 6 d i f f e r e n t f i s h i n g mortality rates 21 5. Cycle lengths of 4 populations with 4, 6, 8 and 10 year-classes at 6 d i f f e r e n t f i s h i n g mortality rates. Age of f i r s t maturity i s 4 years i n each of the 4 populations 24 6 . Cycle lengths of 4 populations with 4, 6, 8 and 10 year-classes under f i s h i n g mortality (a) of 0, 10, 30 and 60 percent. Age of f i r s t maturity i n each of the 4 populations i s 4 years 27 7. Maximum age (Y m) of animal and mean age (M-p) of maturity of a 10-year-class breeding popu-l a t i o n under 6 d i f f e r e n t f i s h i n g mortality rates. Age of f i r s t maturity i s 4 years 32 8. Relationship between Recruitment abundance (Z Q) above equilibrium l e v e l (k) and time (T) taken to reduce Z Q to Z D/2, i n the r e l a t i o n s h i p Z = Z e ' C tSIN o \ m y 9. Fluctuations i n recruitment abundance with time i n a hypothetical population 37 10. Population o s c i l l a t i o n s determined by the repro-duction curve of the type Z = We-Ml-W) when there are 6 ages i n the spawning stock and spawning f i r s t occurs at 2, 3 and 5 years, res p e c t i v e l y 40 V FIGURE Page 11. Population o s c i l l a t i o n s determined by the reproduction curve of the type Z = WeMl-W) when there are 4, 6, and 10 ages i n the spawning stock, and spawning f i r s t occurs at 3 years 44 12. Population o s c i l l a t i o n s determined by the reproduction curve of the type Z = We A(^~ w) when there are 4 ages i n the spawning stock and spawning f i r s t occurs at 3 years. Fishing mortality (a) i s 5, 20 and 40 per-cent, respectively 48 13. Population o s c i l l a t i o n s determined by the reproduction curve of the type Z = We^(^ -^ when there are 6 generations i n the spawning stock and spawning f i r s t occurs at 3 years. (SD = standard deviation from stock abundance, equivalent to environmental factors) 50 v i TABLE I. I I . I I I . IV. V. VI. VII. LIST OF TABLES Page Cycle length (years) and i t s r e l a t i o n s h i p with age of f i r s t maturity and number of year-classes breeding i n unexploited populations 18 Rate of reduction i n cycle length per 10% increase i n f i s h i n g mortality. Age of f i r s t maturity i s 4 years 22 Cycle lengths of 4 breeding populations. Age of f i r s t maturity varies from 4 to 8 years, and f i s h i n g mortality varies from 0% to 60% 23 Parameters and c o e f f i c i e n t describing the trend i n cycle length reduction with increasing f i s h i n g i n t e n s i t y (a) at each age of f i r s t sexual maturity. The 3 breeding populations comprise 6, 8 and 10 year-classes 26 Maximum age (Y m) of animal and mean age (Mrp) of a 10 year-class breeding population under 6 d i f f e r e n t f i s h i n g m o r t a l i t i e s (a). Age of f i r s t maturity i s 4 years 31 (a) Influence of age of f i r s t maturity on amplitude of o s c i l l a t i o n . There are 6 year-classes i n the spawning stock and spawning f i r s t occurs at 2, 3, 4 and 5 years, respectively (b) Influence of number of.year-classes i n a spawning stock on amplitude of o s c i l l a t i o n i n numbers when there are 4, 6, 8 and 10 year-classes spawning (c) Influence of the shape of reproduction curve on amplitude and equilibrium abundance of an o s c i l l a t i o n . There are 8 ages i n the spawning stock and spawning f i r s t occurs at 4 years 42 Cycle lengths of 5 populations spawning at 2, 3, 4 and 5 years of f i r s t maturity. There are 3 ages i n the spawning stock. Reproduction curve i s of the type Z = We A ( 1" W ) 46 v i i TABLE Page VIII. E f f e c t of f i s h i n g mortality on amplitude of o s c i l l a t i o n , equilibrium abundance and cycle length i n the r e l a t i o n s h i p Z = Zoe ~C t SIN|(^) ( t - t Q ) j + k. There are 3 year-classes i n the spawning stock and age of f i r s t maturity i s 3 years (T i s time i n years taken to reduce Z Q to Z Q / 2 49 IX. Order of c a l c u l a t i o n s i n the model for each year f r y production 72 ACKNOWLEDGEMENTS The author would l i k e to express his sincere gratitude to Dr. P. A. Larkin for his supervision, guidance, and c r i t i c i s m throughout this study. Many thanks also to Dr. N.J. Wilmovsky and Dr. C.J. Walters for t h e i r useful suggestions and comments. F i n a l l y , the writer wishes to express thanks to Mr. S. Borden for his assistance i n data analysis. INTRODUCTION Numerical fluctuations i n populations have often been ascribed to i n t e r a c t i o n of many factors such as predator-prey systems (Lack, 1954), food competition (Nicholson, 1950), and v a r i a t i o n i n the physical environ-ment. There i s a p a r t i c u l a r kind of f l u c t u a t i o n , however, which may r e l a t e primarily to the mechanisms of population regulation and t h e i r i n t e r a c t i o n with properties of the reproductive structure of a population. These kinds of fluctuations have been dealt with by Ricker (1954). Ricker suggests that most well-known cycles seem not be of a simple reproduction-curve type, but the shape of this curve must profoundly influence the course of population abundance i n any animal. I t i s accordingly useful to examine the c y c l i c generating properties of stock-recruitment systems. Three d i f f e r e n t approaches have been taken to develop stock-recruitment curves. One approach has developed stock-recruitment r e l a -tionships on assumptions concerning the possible way density dependent regulatory mechanisms operate i n populations (Ricker 1954; Beverton and Holt, 1957; Cushing, 1969; and S i l l i m a n , 1970). Here the l i f e span of a f i s h i s a r b i t r a r i l y divided into two phases and each phase i s given i t s c h a r a c t e r i s t i c density dependent ef f e c t s on population. One e f f e c t , operating through mortality, i s considered to be strongest i n early l i f e , p a r t i c u l a r l y the l a r v a l and immediate post-l a r v a l stages. This density dependent mortality may operate through i n t r a s p e c i f i c competition for a l i m i t e d resource or i t may even be an i n d i r e c t e f f e c t of competition for food operating by lengthening of maturation time (Backiel and Le Cren, 1966) . The other density e f f e c t , operates through growth and i t s influence i s considered to increase as the f i s h ages. The second approach has developed reproduction curves based on observed relationships between recruitment and stock abundance. In one notable instance, this approach has u t i l i s e d a combination of two formulations, the L o g i s t i c model (Schaefer, 1968) which combines recruitment, growth and natural mortality as a s i n g l e function of population biomass, and the so-called Beverton and Holt (1957) "Dynamic pool model" which assumes recruitment constant i n i t s s i m p l i -est form, or omits i t e n t i r e l y from consideration ( y i e l d per r e c r u i t model). F i n a l l y simulation models have been used to generate reproduction curves which are related to whatever regulatory mechanisms are considered e f f e c t i v e for the species. Some simulation models (eg. Larkin and Hourston, 1964) u t i l i s e modified o r i g i n a l l y proposed r e l a t i o n s h i p s , while others (eg. Walters, 1969) use any desired stock-recruitment r e l a t i o n -ship . Ricker (1958) discusses the possible causes of various shapes of reproduction curves and he considers a reproduction curve that indicates a decline i n reproduction at high stock densities to be most applicable to many f i s h populations. Development of population o s c i l l a t i o n i n numbers i s assumed to be caused by a decline i n reproduction beyond stock densities associated with maximum recruitment. RICKER'S REPRODUCTION CURVE THEORY In his formulation concerning "Stock and Recruit-ment" and theory of population regulation Ricker (1954) stresses a basic idea that a f i s h population, even when not fished, i s lim i t e d i n size; that i s , i t i s held at some more or less f l u c t u a t i n g l e v e l by natural controls. Nicholson (1933), who f i r s t c l a r i f i e d and systematized ideas concer-ning the nature of such controls, showed that-, while the general l e v e l of animal abundance can be affected by innumer-able factor^of the physical and b i o l o g i c a l environment, the immediate mechanism of control must always involve competi-t i o n — t h e term competition including any factor of mortality whose effectiveness increases with the stock density. Among the population c h a r a c t e r i s t i c s a f f e c t i n g reproduction and recruitment Ricker considers the abundance of mature spawners to be often s u f f i c i e n t l y outstanding i n importance or to be s u f f i c i e n t l y well correlated with other important factors to make i t of r e a l value for analysis and pre d i c t i o n . The e f f e c t of parental stock density upon recruitment i s alleged to be exerted v i a the density of the eggs or larvae they produce, the s u r v i v a l of the l a t t e r being affected by density-dependent competition for food or oxygen, compen-satory predation and other related f a c t o r s . At low level s of stock abundance the number of progeny produced i s taken to vary d i r e c t l y as the abundance of spawners while at high stock densities net reproduction f a l l s o f f due to density-dependent factors. Thus the general theory of re-production indicates that density dependent causes of mor-t a l i t y set a l i m i t to the siz e which a population a t t a i n s . Ricker used "reproduction curves" or "recruitment curves" to describe the average r e l a t i o n s h i p between the abundance of the mature stock of a population and the number of progeny of the spawners (recruits) that survive to maturity. His general mathematical expression which relates the abundance of spawners to r e c r u i t s exists i n two equi-valent forms: F Z where the symbols are as follows: F the siz e of the f i l i a l generation (recruitment), measured at some stage after density-dependent mortality ceases, P the siz e of the parental generation, P parental stock abundance that produces maximum m c c recruitment, P^ the 'replacement' si z e of the parental genera-tion , that i s - - t h a t which, on the average, (P - P)/P , P r " m and e = W A ( l - W) just replaces i t s own numbers, or number required to keep the population at a steady density under conditions of natural mortality only, Z number of r e c r u i t s divided by the replacement number of spawners, A a c o e f f i c i e n t defined by the r a t i o of the replacement number of spawners over the number of spawners that give maximum number of r e c r u i t s . The second form of the mathematical expression was used. A graph of a t y p i c a l curve of the family i s shown i n F i g . 1 where the symbols are defined as follows: Wr replacement number of spawners, Z r replacement number of r e c r u i t s , Wm number of spawners that give maximum r e c r u i t -ment, Z maximum recruitment, m P 1 equilibrium p o s i t i o n , that i s , the stock i s i n equilibrium at the density at which the repro-duction curve cuts the 4 5 ° l i n e ; the stock i s producing enough progeny, and only enough, to replace i t s current numbers. When the population i s i n equilibrium, the replacement l e v e l of spawners i s equal to that of the r e c r u i t s , that i s , 7. F i g . 1 Reproduction curve conforming to the r e l a t i o n s h i p Z = We A ( 1- W ) 8 . W = Z r r FECUNDITY AND RECRUITMENT—UNFISHED STOCKS When a f i s h population contains a number of spawning year-classes, then fecundity, number of f i s h at various ages and proportion of f i s h maturing at various ages a l t e r as the f i s h grow older. Generally i n d i v i d u a l fecundity of f i s h increases with age though at very advanced age fecun-8--d i t y may not continue to increase (Be'genal, 1966; N i k o l s k i i , 1969). I n i t i a l l y the proportion of f i s h spawning i s very small since a l l f i s h of the same year-class do not mature at the same time, but, gradually, this proportion increases u n t i l the f i s h at a ce r t a i n advanced age of maturity a l l spawn. As fecundity and proportion of f i s h maturing change (increasing with age of f i s h ) , the number of f i s h at age declines with increasing age due to either natural mortality or f i s h i n g mortality or both. If age differences and age s p e c i f i c fecundities are ignored and the t o t a l abundance of adult animals i s the sum of animals at each age of spawning, then the t o t a l egg production from a spawning stock i s given by the product of weighted mean i n d i v i d u a l fecundity and the t o t a l abundance of adults (N) i n a spawning season, that i s , E = EN where E i s the population fecundity, E weighted mean i n d i v i d u a l fecundity which, i f age s p e c i f i c fecundities were known, i s estimated as, Y S m E.N. 1 Y y 1 1 E = -y S m N. i=Y 1 y where Y i s the age of f i r s t maturity, Y m the oldest age of spawning, E^ fecundity of f i s h of age i , N^ number of f i s h of age i . (The estimated population fecundity ignores the influence of mortality on adults except as i t might be incorporated into the values of N^ .) Recruitment abundance r e s u l t i n g from the abundance of eggs w i l l be affected by compensatory mortality and i s given by the Ricker r e l a t i o n s h i p : Z = EN e A ( 1 " E N ) where Z i s the recruitment abundance expressed as a pro-portion of replacement l e v e l , A the compensation c o e f f i c i e n t . Where age d i s t r i b u t i o n and age s p e c i f i c fecundities are included, t o t a l egg production from one year-class i n a spawning stock composed of many year-classes, when there i s no mortality, i s given by, 10. E = p^E^lNL, i n a spawning season, where E c i s the egg production from the abundance of a year-class, p^ the proportion of the year-class spawning at age i expressed i n r e l a t i o n to number at age (for f i s h of age below that of maturity p^ and E^ are equal to zero). Total recruitment i n a year j from one year-class i i s given by the Ricker r e l a t i o n s h i p as Y A ( l - E m p.E.Z . .) Z. . = p.E.N.e l j ^ l i i where N. = Z . . l 3-1 Population fecundity (E) from a stock composed of many spawning year-classes i s given by Y E = E m p . E . Z . . i= l 1 1 ^ Total recruitment i n a year j for a l l ages i s given by the Ricker r e l a t i o n s h i p as Y A( l - ( E m p.E.Z. .)) Y . , * i l j - l Z . = ( E m p . E . Z . . ) e 1 1 1 . , ^ 1 1 1 - 1 1=1 J when the stock i s at replacement l e v e l , 11. Y Y A ( l - ( r " p.E.Z . . )) = 1 i= l FECUNDITY AND RECRUITMENT—FISHED STOCKS The e f f e c t s of f i s h i n g mortality on populations are e a s i l y observed through the recruitment the exploited stocks produce. Harvest usually takes place over a l l ages of spawning and the ultimate e f f e c t of f i s h i n g mortality (and also natural mortality) on populations i s to reduce the number of spawners as a cohort of animals advances i n age from the age of f i r s t maturity onward. Thus e x p l o i t a t i o n progressively decreases the reproductive contribution of older f i s h spawning, except for species of animals that breed once i n t h e i r l i f e time, such as some species of salmon. Ignoring natural mortality causes, the e f f e c t of f i s h i n g mortality i s to reduce the number of animals accor-ding to a r e l a t i o n of the form where F i s the instantaneous rate of f i s h i n g mortality. In these circumstances t o t a l egg production (E c) from one year-class i n a multiage spawning stock i n a year j i s given Z . 3 Z . -F. . e by 12, E = p . E . Z . c * i l j- x -Fd(i-Y ) y where d = 0 i f i < Y and - y d = 1 i f i > Y . y Total recruitment i n year j from one year-class i s r e l a t e d to egg abundance by the Ricker r e l a t i o n s h i p Z. . = J O I ) „ „ -Fd(i-Y ) p . E . Z . . e v y' 1 1 3 - 1 / A t Y \ 1- E m p.E.Z. , e - F d ( i - y i = l 1 1 D _ 1 Population fecundity i s determined as the sum of egg produc-t i o n at each age of sexual maturity, and i s given by, „ vm ' _ _ -Fd(i-Y ) E = Z p.E.Z.-e y 1=1 J T o t a l recruitment i n a year j from a l l ages together i s given by the Ricker r e l a t i o n s h i p ,m z- P . E . Z . . e - F d ( i - y i = l 1 1 3 " 1 \ A e / v m TP n -Fd(i-Y ^ 1 Z p.E.Z..e y 1- . * i i 3-1 * \ I METHODS Many investigations i n f i s h e r y biology are d i f f i -c u l t to document with r e a l data c o l l e c t e d from the f i e l d be-cause of p r a c t i c a l problems. Due to these d i f f i c u l t i e s usually 13. g e n e r a l formulations are developed w i t h u n d e r l y i n g assumptions and approximations which are a p p l i c a b l e t o a s i t u a t i o n . This approach has been f a c i l i t a t e d by use of computers which make e a s i e r the problem of complex computations. A simple program was w r i t t e n i n FORTRAN IV and f o r an IBM 1130 computer. The program c a l c u l a t e s t o t a l number of f i s h by summing the numbers at each age, from which i t computes t o t a l number of spawners, i n a simulated year, by summing the number of mature f i s h t h a t spawn at each age. P o p u l a t i o n f e c u n d i t y i s computed from the sum of the products of numbers of spawning a d u l t s at"each age and i n d i v i d u a l f e c u n d i t y . Fry p r o d u c t i o n , every simulated year, i s r e l a t e d to egg abundance v i a compensatory density-dependent mechanisms formulated by R i c k e r (1954). Recruitment (Z) and number of spawners (W) are expressed as p r o p o r t i o n s of r e p l a c e -ment l e v e l (W ). When there i s no f i s h e r y , the p o p u l a t i o n i s i n e q u i l i b r i u m when Z = W = 1. When a f i s h e r y begins, the stock i s d e f l e c t e d and may be s e t i n t o o s c i l l a t i o n s i n numbers. The d e t a i l of the program i s given i n Appendix I . CYCLE LENGTH 1. unexploited populations Four factors were varied i n investigating c y c l i c a l fluctuations i n numbers of animals using Ricker's repro-duction curve: (a) value of the parameter A, the compensation c o e f f i c i e n t , (b) age of f i r s t maturity, (c) number of generations comprising a spawning stock, and . (d) f i s h i n g m ortality. The four factors were combined i n a variety of ways so as to assess the influence of each factor, i n d i v i d u a l l y or combined with others, on (a) the type of o s c i l l a t i o n i n numbers exhibited (b) the amplitude of o s c i l l a t i o n , and (c) the period of c y c l i c a l f l u c t u a t i o n . The effects of environmental factors on period and amplitude of a c y c l i c a l f l u c t u a t i o n were investigated i n an example involving one population. Displacement of a population from i t s equilibrium l e v e l r e s u l t s i n one of the 3 types of o s c i l l a t i o n s (Fig. 2). TIME (YEARS) F i g . 2 Three types of o s c i l l a t i o n s which are generated by Ricker's Reproduction curve of the type Z = We A ( 1~ W ) Type A: This i s an o s c i l l a t i o n that exhibits an amplitude (difference i n population l e v e l between peak and trough years) that i s gradually decreasing and i s eventually damped, Type B: This i s i d e n t i f i e d by a constant amplitude a l l the way through i t s progression, and Type C: The o s c i l l a t i o n has an amplitude which i n i t i a l l y increases but l a t e r o s c i l l a t e s with a uniform large amplitude. The three o s c i l l a t o r y types, as w i l l be dealt with l a t e r , depend on a number of factors, namely: (a) the shape of the reproduction curve of a population which i s defined by the c o e f f i c i e n t "A" i n the 1 4 - - u • n T i A(l-W) re l a t i o n s h i p Z = We , (b) age of f i r s t sexual maturity of the animal, (c) number of year-classes breeding, and (d) mortality rate (natural or fishing) experienced by the population. Cycle length associated with the hypothetical population that i s exhibiting regular fluctuations (Fig. 2) i s defined as the time i n t e r v a l between two adjacent levels of maximum animal abundance, that i s , the distance i n time from peak to peak. 17. From the re s u l t s (Table I ) , .two important r e l a -tionships can e a s i l y be described: (1) when age of f i r s t maturity i s constant and the number of year-classes breeding varies, for every year-class added to the population the cycle length i s lengthened by one year, and (2) when number of year-classes breeding i s constant and age of f i r s t maturity varies, for every one year increase i n the age of f i r s t maturity, cycle length of the population increases by two years. Since cycle length varies simultaneously with age of f i r s t maturity and number of year-classes breeding, combining (1) and (2) gives a l i n e a r r e l a t i o n s h i p (Fig. 3) described by equation, L = X + 2Y - 1 y where L i s the cycle length i n years, X number of year-classes breeding and age of f i r s t maturity. Alternately, cycle length can be expressed i n terms of the youngest and oldest individuals breeding i n a population. Thus the number of year-classes breeding i s related to the youngest and oldest f i s h by the re l a t i o n s h i p , 18. TABLE I. Cycle Length (years) and i t s r e l a t i o n s h i p with age of f i r s t maturity and number of year classes breeding i n unexploited populations. Age of f i r s t maturity Year classes breeding 2 Cycle 3 4 length 5 (years) 6 7 8 9 10 2 5 6 7 8 9 10 11 12 13 3 7 8 9 10 11 12 13 14 15 4 9 10 11 12 13 14 15 16 17 5 11 12 13 14 15 16 17 18 19 6 13 14 15 16 17 18 19 20 21 7 15 16 17 18 19 20 21 22 23 8 17 18 19 20 21 22 23 24 25 X = Y - Y + 1 m y where Y m i s the maximum age of spawning. Therefore, substituting for X gives L = (Y - Y + 1) + 2Y - 1 m y Y L = Y + Y . m y The sum of Y m + Y i s obviously twice the mean age of spawning, and these r e s u l t s show that the age of f i r s t maturity has a more s i g n i f i c a n t influence on cycle length than the num-ber of year-classes i n a spawning stock. 6. 4/? e /o 2 0 . 2 . E x p l o i t e d p o p u l a t i o n s The r e s u l t s ( F i g . 4) i n d i c a t e t h a t an i n t e r - r e l a -t i o n s h i p e x i s t s among c y c l e l e n g t h , age o f f i r s t m a t u r i t y and f i s h i n g m o r t a l i t y . The e f f e c t o f i n c r e a s i n g t h e age o f f i r s t m a t u r i t y i s t o l e n g t h e n the c y c l e o f t h e p o p u l a t i o n by 2 y e a r s a t a l l r a t e s o f f i s h i n g m o r t a l i t y ; thus t h e dependence o f c y c l e l e n g t h on the age o f f i r s t m a t u r i t y i s not a l t e r e d by e x p l o i t a t i o n o f p o p u l a t i o n s . I n c r e a s i n g f i s h i n g m o r t a l i t y by 10% on a 6 y e a r -c l a s s b r e e d i n g p o p u l a t i o n r e s u l t s i n s h o r t e n i n g t h e c y c l e l e n g t h by about 2/3 o f a y e a r , on t h e average , w i t h i n t h e range o f f i s h i n g m o r t a l i t i e s examined (row 2 , T a b l e I I ) . The r e s u l t s show t h a t f i s h i n g m o r t a l i t y d e c r e a s e s t h e c y c l e l e n g t h of a p o p u l a t i o n a t a l l ages o f f i r s t m a t u r i t y (Table I I I ) and the r a t e a t wh i c h i t s h o r t e n s i s " h i g h e s t i n t h e p o p u l a t i o n w i t h the l a r g e s t number o f spawning ages ( F i g . 5 ) ; t h e r e d u c t i o n i n c y c l e l e n g t h t a k e s p l a c e a l m o s t l i n e a r l y i n t h e 4 y e a r - c l a s s s t o c k , the l i n e a r i t y b e i n g d e s c r i b e d by the e q u a t i o n , L = X - (ba) + 2 Y y - 1 (1) where b ( e q u a l t o 3) i s a l i n e a r r e g r e s s i o n c o e f f i c i e n t . . 21. % FISHING MORTALITY (a) F i g . 4 Cycle length of a 6 year class breeding population at 4 d i f f e r e n t ages of f i r s t maturity and 6 d i f f e r e n t f i s h i n g mortality rates. 22. TABLE I I . Rate of reduction i n cycle length per 10% increase i n f i s h i n g mortality. Age of f i r s t maturity i s 4 years. Number of Cycle Reduction i n cycle length Average year-classes length from that at 10% lower reduction  (years) f i s h i n g mortality  4 11 .3 .3 .3 .4 .3 .3 .3 6 13 .8 .8 .7 .7 .5 .5 . 67 8 15 1.4 1.2 1.1 .9 .8 .6 1.0 10 17 2.3 1.9 1.4 1.1 . 8 .6 1.35 23. TABLE I I I . Cycle lengths of 4 breeding populations. Age of f i r s t maturity varies from 4 to 8 years, and f i s h i n g mortality (a) varies from 0% to 60% Column 1 Column 2 4-Year-Classes 6-Year-Classes Age of F i r s t Maturity Fishing 4 5 6 7 Mortality Cycle Length (Years) Age of F i r s t Maturity 4 5 6 7 Cycle Length (Years) 0 10 20 30 40 50 60 11.0 13.0 15.0 17.0 19.0 13.0 15.0 17.0 19.0 21.0 10 10 10 9, 9 9 .7 4 1 14 14 14 12 12 12 11. 8 11.4 11.1 13 7 16 4 16 1 16 13.8 15, 13.5 15, 2 15, 7 18.7 12.3 14.3 16.2 18.4 20.2 4 18.4 11.4 13.5 15.5 17.5 19.6 2 18.2 10.6 12.6 14.6 16.5 18.6 8 17.8 10.0 12.1 14.0 16.1 18.1 4 17.5 9.5 11.6 13.6 15.6 17.6 1 17.1 9.0 11.0 13.1 15.1 17.0 Column 3 Column 4 8-Year-Classes 10-Year-Classes Age of F i r s t Maturity 4 5 6 7 8 Fishing Mortality Cycle Length (Years) 0 10 20 30 40 50 60 15.0 17.0 19.0 21.0 23 14.2 16.1 17.9 19.9 21 16.1 18.2 20 15.2 17.1 19 12.1 14, 11.2 13, 2 6 10 9 9 12 11 14.4 16.4 18 13.7 15.7 17 ,0 ,9 3 2 ,4 7 Age of F i r s t Maturity 4 5 6 7 8 Cycle Length (Years) 17.0 19.0 21.0 23.0 25.0 0 11.0 13.0 15.0 17.0 15, 12, 11, 10, 9, 9, 17, 14, 13, 12, 11, 0 11 19. 16, 15, 14, 13, 13, 21, 19, 17, 16, 15, 15, 23, 21. 19, 18, 17. 8 17.1 24. F i g . 5 Cycle lengths of 4 populations with 4, 6, 8 and 10 year classes at 6 d i f f e r e n t f i s h i n g mortality rates. Age of f i r s t maturity i s 4 years i n each of the 4 populations. Since X = Y - Y +1, su b s t i t u t i n g for X i n (1) gives m y ' y ' ^  L = (Y - Y + 1) - (ba) + 2Y - 1 m y y L = Y + Y - ba m y Cycle lengths of larger populations (6, 8, and 10) decrease exponentially, the general decline being defined by the equation, L = A e ~ B a + C (2) where A,B and C are constants. However, i t may be observed that we may write A = X + n, and C = 2Y - (n+1) Y where n i s a variable c o e f f i c i e n t (Table IV). Substituting for A and C i n (2) gives L = (X + n ) e " B a + 2Y y - (n+1) (3) Substituting f o r X = Y - Y + 1 i n (3) gives ^ m m ^ L = (Y - Y + n + l ) e ~ B a + 2Y - (n+1) m y y Figure 6 represents 4 longitudinal sections of Figure 5 taken at f i s h i n g m o r t a l i t i e s of 0, 10, 30 and 60 percent, respectively. When there i s no ex p l o i t a t i o n , cycle 26. TABLE IV. Parameters and c o e f f i c i e n t describing the trend i n cycle length reduction with increasing f i s h i n g i n t e n s i t y (a) at each age of f i r s t sexual maturity. The 3 breeding populations comprise 6, 8 and 10 year classes. Age of F i r s t Number of Maturity Year-Classes Parameters C o e f f i c i e n t < v (X) (A) (B) (C) (n) (n) 4 8.6 1.05 4.4 2.6 5 9.1 0,94 5.8 • 3.2 6 6 8.2 1.09 8.8 2.2 (2.68) 7 8.0 1.14 11.0 2.0 8 9.4 0 . 90 11.6 3.4 4 9.4 1.75 5.6 1.4 5 9.4 1.75 7.7 1.3 6 8 9.2 1.76 9 . 8 1.2 (1.44) 7 9.6 1.65 11.4 1.6 8 9.7 1.62 13.3 1.7 4 10.1 2.96 7.1 .1 5 10.1 2.81 9.1 .1 6 10 10. 2 2.54 10.8 .2 ( .34) 7 10.9 •2.22 12.1 .9 8 10.4 2.38 14.6 .4 6 8 10 YEAR CLASSES (X) F i g . 6 Cycle lengths of 4 populations with 4, 6, 8 and 10 year classes under f i s h i n g mortality of 0, 10, 30 and 60 percent. Age of f i r s t maturity i n each of the 4 populations i s 4 years. 28. length increases by 1 year for every one year increase i n the number of year-classes breeding, the period of the cycle being equal to twice the mean age of maturity. How-ever, as f i s h i n g mortality increases, the dependence of cycle length on number of year-classes progressively becomes less s i g n i f i c a n t u n t i l i t i s independent of year-classes (beyond about 40% e x p l o i t a t i o n ) , but very much dependent on age of f i r s t maturity, when the cycle length i s approxi-mately twice the age of f i r s t maturity. 3. Mean and maximum ages Mean age of maturity and maximum age of a f i s h under d i f f e r e n t f i s h i n g mortality rates can be estimated using the r e l a t i o n s h i p s . L = Y + Y or m y L = X + 2Y - 1 y For example i f a population comprises individuals which a t t a i n age of f i r s t maturity at 4 years and die, on the average, at 13 years due to natural and f i s h i n g m o r t a l i t i e s , cycle length of the population w i l l be related to age of f i r s t maturity ( Yy) an<3- maximum age (Y'm) according to the form, L = Y + Y . m y Therefore L = 1 3 + 4 = 17 years. 29 Mean age of maturity (MT) of the population i s given by, m y 13 + 4 „ r M T = ^ — = 2 = years The same population exhibits a cycle length i n numbers of 9 years at f i s h i n g mortality of 60% (Column 4, Table I I I ) . Maximum age of the majority of individ u a l s i n the population under 60% e x p l o i t a t i o n i s given by, Y = L - Y m y Y = 9 - 4 = 5 years. m J Mean age of maturity w i l l be given by, 5 + 4 MT = — ^ — = 4 * 5 years. Change i n maximum age (AYm) due to 60% f i s h i n g mortality i s given by the r e l a t i o n , AY = (Y at 0% f i s h i n g mortality) m m -(Y at 60% f i s h i n g mortality) AY = 13 - 5 = 8 years, m J Change i n mean age of maturity (AMT) i s given by, AMT = (MT at 0% f i s h i n g mortality) -(M T at 60% f i s h i n g mortality) AMT = 8.5-4.5 = 4 years 30. Using the same procedure as i n the above ca l c u l a t i o n s , maximum age, mean age of maturity, change i n maximum age, change i n mean age for every 10% change i n f i s h i n g mortality ( a%) were estimated (Table V and F i g . 7). Mean age of maturity and maximum age of an i n d i v i d u a l decreases with increase i n f i s h i n g mortality, maximum rate of decrease i n this population occurring at low levels of e x p l o i t a t i o n (below 30%). At high f i s h i n g mortality the rate of reduc-t i o n i n cycle length i s not highly marked. Thus the popu-l a t i o n under 60% f i s h i n g mortality, cycles i n numbers i n the same manner as one i n which in d i v i d u a l s a t t a i n age of f i r s t maturity at 4 years and die, on the average, at 5 years of age. In this case a 10 year-class population i s v i r t u a l l y reduced to a 2 year-class population by 60% f i s h i n g mortality. Thus i f the i n i t i a l (unharvested) age structure of a popu-l a t i o n i s known, change i n maximum age and mean age of maturity i s l i k e l y to give an estimate of the magnitude of exp l o i t a t i o n rate to which the population i s subjected. I t can generally, therefore, be stated that cycle lengths of populations under e x p l o i t a t i o n depend on both l i f e h i story features of animals (age of f i r s t maturity and siz e of population i n terms of age structure) and factors of the environment (exploitation rate i n this example). Ex p l o i t a -tion shortens the cycle length by lowering mean age of breeding, that i s , making populations younger. At high TABLE V. Maximum age (Y^) of animal nad mean age (M^) of a 10 year class breeding population under 6 d i f f e r e n t f i s h i n g m o r t a l i t i e s (a). Age of f i r s t maturity i s 4 years. Fishing M o r t a l i t y -(a) (%) Maximum Age(Y m)-(Years) Change i n Maximum Age-AY m Mean Age of Maturity-Mrp (Years) Change i n Mean Age of AY m Aa% Maturity 0 13 0 8.5 0 0 10 11.6 1.4 7.8 0.7 1.4 20 8.5 4.5 6.2 2.3 3.1 30 7.4 5.6 ' .5.7 2.8 1.1 40 6.2 6.8 5.1 3.4 1.2 50 5.4 7.6 4.7 3.8 0.8 60 5.0 8.0 4.5 4.0 0.4 i n t e n s i t i e s of ex p l o i t a t i o n l i f e history features ( l i k e age composition of the breeding stock) which normally determine c h a r a c t e r i s t i c behaviour of unexploited populations w i l l no longer play t h e i r s i g n i f i c a n t r o l e . The change i n age com-pos i t i o n which governs growth and decline i n numbers of animals leads to change i n some population phenomenon among which c y c l i c behaviour i n numbers i s included. 32. 1 ! 1 1 1 1 i I i i I I I 0 10 20 30 40 50 60 % FISHING MORTALITY (a) F i g . 7 Maximum age (Y m) of animal and mean age (M ) of maturity of a 10 year class breeding population under 6 d i f f e r e n t f i s h i n g mortality rates. Age of f i r s t maturity i s 4 years. NUMERICAL FLUCTUATIONS 1. Amplitude of o s c i l l a t i o n The shape of a reproduction curve determined by the c o e f f i c i e n t A i n the r e l a t i o n s h i p Z = WeA ^ W^ determines the recruitment that i s produced by a p a r t i c u l a r abundance of mature spawners. Maximum recruitment (Z m) produced by spawners (W ) i s obtained, a l g e b r a i c a l l y , by d i f f e r e n t i a -A(l-W) ting Z = We and s e t t i n g the deriv a t i v e as equal to zero. The deriv a t i v e i s equal to zero where the peak of the Ricker curve has maximum recruitment and slope of zero, that i s , dZ = eA(l-W)_ A W eA(l-W) = Q dZ dW 1 - AW = 0 W = 1/A m A(1-W) Substituting for W i n Z - We gives ^ m ^ A - l m A The shape of the reproduction curve refers largely to the slope of the r i g h t hand limb of the curve and for a population to exhi b i t c y c l i c a l behaviour i n numbers Ricker alleges that the reproduction curve must s t a r t sloping down-ward above the 45° l i n e (Fig. 1). A population that i s e x h i b i t i n g c y c l i c behaviour i n numbers whose amplitude of o s c i l l a t i o n varies with time fluctuates i n numbers about the equilibrium l e v e l according to a mathematical expression of the form: Z 0 e ± C t SIN r ( t ) . When the amplitude of o s c i l l a t i o n i s uniform, the expression takes the form of Z Q SIN r ( t ) where Z Q i s i n i t i a l amplitude of o s c i l l a t i o n above equilibrium l e v e l r angular motion per unit time c a constant t time. To t a l abundance of recruitment (Z) at time t i s given by, Z = + ct Z e SIN o 2TT Y +Y • m y ( t - t Q ) + k and Z = Z SIN o 2TT Y +Y l m y ( t - t 0 ) + k (4) ( 5 ) where k i s the equilibrium l e v e l of the population 2TT jy—~—J i s the angular motion and IY +Y m y (Y^+Yy) i s the cycle length of a population t a constant adjusting time scale to an o r i g i n , Ricker's reproduction curve of the form A(l-W) Z = We produces 3 types of o s c i l l a t i o n s : (a) an o s c i l l a t i o n that advances with a decreasing amplitude and i s described by equation (4) where the parameter c takes a negative value, (b) an o s c i l l a t i o n with a uniform amplitude described by equation (5) and f i n a l l y (c) an o s c i l l a t i o n that progresses with an increasing amplitude which reaches i t s maximum and s t a b i l i s e s This i s par t l y described, up to the i n i t i a l time of a t t a i n i n g s t a b i l i t y , by equation" (4) and para-meter c takes a p o s i t i v e value. The parameter c i n equation (4) describes the rate at which the amplitude of o s c i l l a t i o n varies with time and the ampli-tude i s reduced to a half i t s o r i g i n a l s i z e i n time T (Fig.8 Its s i z e a f t e r time T i s related to o r i g i n a l s i z e by, Z /2 = Z e " C T o o • in(1/2) = -cT £n2 = cT c = £n2/T The symbols used to describe v a r i a t i o n i n numbers are shown in general schematic representation (Fig. 9). Maximum amplitude above equilibrium l e v e l i s given by maximum re-cruitment less equilibrium abundance, that i s , 36. F i g . 8 Relationship between recruitment abundance (Z^) above equilibrium l e v e l (k) and time (T) to reduce Z Q to Z Q/2, i n the r e l a t i o n s h i p taken = Z e ~ C t SIN o 2 TT y +Y m y ) (t- + k 37. F i g . 9 Fluctuations i n recruitment abundance with time, i n a hypothetical population. 38. ( eA-D om A Substituting for Z i n (4) gives ^ om ^ A - l / -> Z = ( C6-^-) -k) i C t S I N (—y-) ( t - t ) + k (6) > m y 2. Equilibrium abundance When a stock i s i n equilibrium as i n a f i s h e r y the i=x number of spawners (W) i s given by, W = pk £ (1-a) i = l where p i s the proportion of animals mature •• k i s equilibrium l e v e l of population x number of year-classes breeding a i s the annual percentage mortality and (1-a) = e" F Recruitment from the population i s rel a t e d to the abundance of spawners by i=x A(l-(pk Z (1-a) 1) (7) JL X • * - i Z = pk E (1-a) 1 e 1  r i = l but p = 1/x and Z = k r Therefore s u b s t i t u t i n g for p and Z^ i n (7) gives i=x 3 9 A(l-(^k E d - a ) 1 ) ) , i=x . x ._, k = —k E (1-a) e 1 1 X 1 = 1 , ±=x . A ( 1 - ( ! X ( 1 _ a , i ) ) 1 = — E (1-a) 1 e 1 = 1  X i = l i=x . , i=x log x = l o g o E (1-a) 1 + A(l-(£ E (1-a) 1)) e e i = l x i = l , i=x . i=x A - ( ^ E (1-a) 1) = l o g o (x - E (1-a) 1) 1=1 1=1 i=x x(A - log (x - E (1-a) 1)) k = : . — i z l 1=X A E (1-a) 1 1 = 1 (7a) Substituting for k i n 6 gives i=x a-, x(A-log (x- E (1-a) 1)) A- ± e _. _  n _, / Z = (^-=r-)- • 1 1 — e ~ c t SIN A i=x A E (1-a) 1 i = l 2-IT ( Z 7 T } ( t - t ) Y +Y ; V o ; ' \ m y / i=x x(A-log (x- E (1-a) 1)) + - i=± 1=X A E (1-a) 1 i = l E x p l o i t a t i o n removes amplitude of o s c i l l a t i o n s i n numbers (Fig. 10) so that recruitment abundance w i l l be equal to the abundance of spawners when the population i s i n equilibrium, that i s , i = x X(A-log (x- E (1-a) 1)) Z = , iz± (8) i=x . v ' A E (1-a) 1 i = l I I I 1 1 1 1 1 10 20 30 40 50 T IME ( Y E A R S ) F i g . 10 Population o s c i l l a t i o n s determined by the reproduction curve of the type Z = We A^ - w) when there are 6 ages i n the spawning stock and spawning f i r s t occurs at 2, 3 and 5 years, respectively. Substituting for X = - + 1 i n 8 gives Y -Y +1 m y Z = 1 , i=Y -Y +1 . \ A- Log Y -Y +1- y (1-a) 1 • el m y i = l i=Y -Y +1 A (1-a) 1 i = l Thus, the average abundance of recruitment i s dependent on age structure (Y m and Y ), the shape of the reproduction curve ('A1 i n Z = We A^ W^ and the f i s h i n g mortality (a). 3. Age of f i r s t maturity Age of f i r s t maturity varies as a r e s u l t of f i n i t e environmental factors. E f f e c t of delayed maturity on magnitude of fluctuations i n numbers i s evident from the resu l t s (Fig. 10 and Table VI(a)). There are 6 generations i n the spawning stock and spawning f i r s t occurs at 2, 3, 4 and 5 years, respectively. The stock i s deflected from i t s equilibrium abundance at each age of f i r s t maturity with a f i s h i n g mortality of 5%. The exhibited amplitude of o s c i l -l a t i o n increases very rapidly as age of f i r s t maturity i s more delayed. At low age of f i r s t maturity the population tends to go back to equilibrium when deflected, t h i s being achieved through an o s c i l l a t i o n of a decreasing amplitude, but at very l a t e age of maturity the type of o s c i l l a t i o n exhibited tends to become unstable so that when the stock TABLE VI. (a) Influence of age of f i r s t maturity on amplitude of o s c i l l a t i o n . There are 6 year-classes i n the spawning stock and spawning f i r s t occurs at 2, 3, 4 and 5 years, respectively. (b) Influence of number of year-classes i n a spawning stock on amplitude of o s c i l l a t i o n i n numbers when there are 4, 6, 8 and 10 year-classes spawning. (c) Influence of the shape of reproduction curve on amplitude and equilibrium abundance of an o s c i l l a t i o n . There are 8 ages i n the spawning stock and spawning f i r s t occurs at 4 years. T = time taken to reduce maximum amplitude by a h a l f . Section Number of Year-Classes Age of F i r s t Amplitude Maturity (Numbers) (Years) Time Equilibrium (Years)(Numbers) Percent Fishing M o r t a l i t y (a) Cycle Length (Years) (a) 6 2 3 4 5 0.079 0.116 0.163 0.54 8 26 * 35 1.113 2.67 5 9 11 13' 15 (b) 4 6 8 • 10 3 0 . 501 0.118 0.116 0.113 48 26 12 . . .11 . 1.081 1.113 1.145 1.178 2. 67 5 9 11 13 15 (c) 8 4 0.012 0.038 0.122 0.142 2 6 12 32 1.064 1.095 1.128 1.140 1.5 1.8 2.3 2.6 5 15 * O s c i l l a t i o n of uniform amplitude. i s deflected, even s l i g h t l y , i t establishes permanent o s c i l -l ations whose maximum amplitude i s determined by the shape of the reproduction curve. The abundance at equilibrium i s not altered by change i n age of f i r s t maturity--only the s t a b i l i t y i s affected. The population exhibits a d i f f e r e n t period of c y c l i c a l f l u c t u a t i o n whenever age of f i r s t maturity changes--the change i n the period being of two years when age of f i r s t maturity a l t e r s by one. year. 4. Year-classes Amplitude of o s c i l l a t i o n i n numbers, equilibrium abundance and number of generations comprising a spawning stock seem to be related (Table VI(b) and F i g . 11). The 4 stocks which have the same shape of reproduction curve (A = 2.67 i n Z = WeA ^ W^ d i f f e r i n the number of genera-tions spawning as 4, 6, 8 and 10. In a l l the stocks spawning f i r s t occurs at 3 years. The stocks are set into o s c i l l a -tions by a 5% f i s h i n g mortality. The amplitude of o s c i l -l a t i o n exhibited decreases as the number of generations i n the spawning stock increases; that i s , large magnitudes of fluctuations are associated with a simple population age structure while minor fluctuations are more common i n popu-lations of multi-age structure. In this example, a population with 4 year-classes spawning takes about 4 times as much time 44. TIME (YEARS) F i g . 11 Population o s c i l l a t i o n s determined by the repro-A(1-W) duction curve of the type Z = We when there are 4, 6 and 10 ages i n the spawning stock, and spawning f i r s t occurs at 3 years. to reduce maximum amplitude of o s c i l l a t i o n by half i t s o r i g i n a l s i z e as i t takes a 10 year-class population. A large age structure establishes a large equilibrium abun-dance which increases l i n e a r l y with increase i n number of generations i n a spawning stock. The period of c y c l i c a l f l u c t u a t i o n i s d i f f e r e n t i n the 4 populations—being larger the larger the number of generations and i t increases by 1 year for every 1 added year-class to the stock. The period i s determined by both age of f i r s t maturity and number of year-classes i n the spawning stock when ex p l o i t a t i o n i s low, but at high rate of exp l o i t a t i o n the period i s determined by age of f i r s t maturity. 5. Reproduction curve The shape of reproduction curve defined by A i n A(1-W) the r e l a t i o n s h i p Z = We , amplitude of o s c i l l a t i o n i n numbers and the equilibrium abundance of a population are in t e r r e l a t e d (Table V I ( c ) ) . There are 8 generations i n each of the 4 breeding stocks and spawning f i r s t occurs at 4 years of age. The populations are set into o s c i l l a t i o n i n numbers by d e f l e c t i n g them from equilibrium abundance by a f i s h i n g mortality of 5%. As the value of the c o e f f i c i e n t A i n the A(1-W) rel a t i o n s h i p Z = We increases, amplitude of o s c i l l a t i o n increases very rapidly i n an exponential form. A change from a stable type of o s c i l l a t i o n to an unstable one usually results as the value of the c o e f f i c i e n t i s increased; time a 46. population takes before i t establishes an equilibrium of abundance increases with increasing value of the c o e f f i c i e n t , The 4 populations e s t a b l i s h d i f f e r e n t equilibrium abundances due to t h e i r d i f f e r e n t reproductive d i f f e r e n c e s — t h e equilib-rium abundance being higher the larger the c o e f f i c i e n t . Although the 4 stocks have d i f f e r e n t shapes of reproduction curve, they exh i b i t the same period of c y c l i c a l f l u c t u a t i o n of 15 years. More results (Table VII) show that the period of c y c l i c a l f l u c t u a t i o n i s independent of the shape of repro-duction curve, the change i n the cycle length taking place only when age of f i r s t maturity v a r i e s . TABLE VII. Cycle lengths of 5 populations spawning at 2, 3, 4 and 5 years of f i r s t maturity. There are 3 ages i n the spawning stock. Reproduc-tion curve i s of the type Z = We A^ 1 - W^ . Age of F i r s t = W eA(l-W) Maturity " e 1.8 2.0 2.3 2.6 3.0 Cycle Length (Years) 2 6 6 6 6 6 3 8 8 8 8 8 4 10 10 10 10 10 5 12 12 12 12 12 6 . E x p l o i t a t i o n The si z e of the amplitude of o s c i l l a t i o n i n numbers i s related to the rate of e x p l o i t a t i o n i n a population (Fig. 12 and Table VIII); there are 4 generations i n the spawning stock i n which age of f i r s t maturity i s 3 years. The stock i s set into o s c i l l a t i o n by d e f l e c t i n g i t from equilibrium abundance with various f i s h i n g mortality rates. The amplitude of o s c i l l a t i o n displayed decreases rapidly as f i s h i n g mortality increases and as a r e s u l t of decreasing amplitude a population establishes s t a b i l i t y very quickly. At low f i s h i n g mortality (5%) the population, when deflected from equilibrium abundance, exhibits an unstable o s c i l l a t i o n which gains s t a b i l i t y as the rate of e x p l o i t a t i o n increases, thus at 20 and 40 percent e x p l o i t a t i o n the population f l u c -tuates i n numbers with an amplitude of decreasing s i z e which leads the population to a stable equilibrium. Increased recruitment, within c e r t a i n l i m i t s of f i s h i n g mortality, i s r e a l i z e d as f i s h i n g rate i s increased, this being indicated by increased equilibrium recruitment abundance (k). The period of c y c l i c a l f l u c t u a t i o n i s progressively decreased as f i s h i n g mortality increases. Thus, at 5% exp l o i t a t i o n the population has a cycle length of 9 years, but at 40% ex p l o i t a t i o n the period of o s c i l l a t i o n i s reduced by 1 year. Due to decreased amplitude by high rates of exp l o i t a t i o n , recognition of cycles becomes increasingly d i f f i c u l t as e x p l o i t a t i o n increases. 0 10 20 30 40 TIME (YEARS) 12 Population o s c i l l a t i o n s determined by the repro-duction curve of the type Z = We A^ 1 _ W^when there are 4 ages i n the spawning stock and spawning f i r s t occurs at 3 years. Fishing mortality (a) i s 5, 20 and 40 percent,respectively. TABLE VIII. E f f e c t of f i s h i n g mortality on amplitude of o s c i l l a t i o n , equilibrium abundance and cycle length i n the r e l a t i o n s h i p Z e ~ C t SIN o + k, There are 3 year-classes i n the spawning stock and age of f i r s t maturity i s 3 years (T i s time i n years taken to reduce Z to Z /2). J o o Amplitude of O s c i l l a t i o n (Numbers) Time (T) (Years) Equilibrium Abundance (Numbers) Cycle Length (Years)_ Fishing Mortality(a) (%) 0.501 48' 1.115 9.0 5 0.362 12 1.355 8.6 20 0.360 5.5 1.568 8.3 30 0.209 4.0 1.778 8.0 40 7. 'Environment' An additive normal random va r i a b l e with a mean of zero and a constant standard deviation (SD) from a varying abundance of a stock with time was used to investigate the e f f e c t of a f l u c t u a t i n g environment on c y c l i c property of a population which comprised 6 ages of spawning with spawning f i r s t occuring at 3 years (Fig. 13). 50. 1.4-1.2 1.0 4 WW SD = 0.05 WWM 1 1 1 1 1 0 10 20 .30 40 50 TIME (YEARS) F i g . 13 Population o s c i l l a t i o n s determined by the repro-duction on curve of the type Z = We when there are 6 generations i n the spawning stock and spawning f i r s t occurs at 3 years. (SD = standard deviation from stock abundance, equivalent to environmental f a c t o r s ) . When the population i s deflected from equilibrium abundance under no environmental e f f e c t s , i t exhibits a regular o s c i l l a t i o n i n numbers whose amplitude decreases as the population establishes s t a b i l i t y again, the period of c y c l i c a l f l u c t u a t i o n being equal to 11 years at 5% f i s h i n g mortality (SD = 0, F i g . 13). For the same population, when deflected from equilibrium abundance with the same f i s h i n g mortality but under increasing effects of the environment (ecj. SD = .05, .2, e t c . ) , the c y c l i c behaviour gets m o d i f i e d — a smooth, regular o s c i l l a t i o n , t y p i c a l of absence of environmental e f f e c t s , developes into one with a variable period which eventually becomes unrecognisable when random factors a t t a i n larger magnitudes. Not only does the period become i r r e g u l a r with increasing environmental effects but also the amplitude of o s c i l l a t i o n increases i t s variance with the r e s u l t that a population tends to lose i t s s t a b i l i t y . Random ef f e c t s tend to make the period and major peaks of a c y c l i c a l f l u c t u a t i o n d i f f i c u l t to recognise e s p e c i a l l y where the o s c i l l a t i o n has a small amplitude, and smaller cycles tend to lose t h e i r r e g u l a r i t y more quickly than larger cycles DISCUSSION Marked r e g u l a r i t y i n numbers of animals would suggest that there were i n t r i n s i c causes which could be of fundamental importance for forecasting y i e l d s . C y c l i c a l fluctuations i n numbers of animals caused by weather, predator-prey i n t e r a c t i o n , food competition, and many other causes, have been observed and reported i n natural and laboratory animal populations (eg. Nicholson, 1950; Cole, 1951 and Lack, 1954). While the above cycles are caused by innumerable factors of the animate and inanimate environment, the cycles developed i n Ricker's s t o c k - r e c r u i t -ment re l a t i o n s h i p are cycles which are primarily due to reproductive features such as the age of f i r s t maturity and longevity of the animal. The development of such cycles i s caused by the declining reproduction of a stock at high densities of spawners. Ricker (1954) outlines the nature of population o s c i l l a t i o n s i n numbers and t h e i r r e l a t i o n s h i p with the l i f e h istory features of animals. I t can be stated, from the r e s u l t s , that the age at which reproduction begins i s one of the most s i g n i f i c a n t c h a r a c t e r i s t i c s a f f e c t i n g the p o t e n t i a l c y c l i c behaviour of a population. Age of f i r s t maturity i s related to longevity and natural mortality of an animal (Beverton and Holt, 1959) and,therefore, partly determines age structure of the spawning 53. stock. Low age of maturity i s usually associated with a population of a simple age structure while late age of maturity i s common i n multiage populations ( M i l l e r , 1956 and N i k o l s k i i , 1962) . As reported e a r l i e r (Cole, 1954) shortening the period when maturity begins would increase b i o t i c p o t e n t i a l of the species and the increase would be more marked i n short-l i v e d species than long-lived ones. Shortening of age of f i r s t maturity i n natural populations i s usually caused by many factors such as poor growth r e s u l t i n g from poor food supply ( M i l l e r , 1956, 1957), or may be caused by f i s h i n g mortality which a l t e r s growth rate ( N i k o l s k i i , 1961). How-ever, age of f i r s t maturity varies within c e r t a i n l i m i t s since i t can gen e t i c a l l y be fixed i n d i f f e r e n t species or even i n d i f f e r e n t forms of the same species (Aim, 1959). The shortening of maturation time does not only increase the b i o t i c p o t e n t i a l of the species by enabling recruitment to come at f a i r l y shorter i n t e r v a l s of time (shortening of per-iod of c y c l i c a l fluctuation) but also the fluctuations i n numbers about the mean stock-density get th e i r amplitude reduced, the r e s u l t of which leads a population to a tendency to remain numerically constant (Murdock, 1970). Such a population would stand a high rate of removal either by f i s h i n g or predation, as has been reported by Neave (1953) who showed that Oncorhynchus gorbuscha which matures at 1 to 2 years can stand losses of over 60% of the adult population by f i s h i n g without reducing the number of f r y , while 0.keta, which matures at 2 to 7 years can only stand about 50% f i s h i n g mortality. The decreased cycle length due to lowering of age of f i r s t maturity would be very much dependent on rate of e x p l o i t a t i o n i n the population. The cycle length would be equal to twice the mean age of maturity i f the population was not heavily exploited and equal to twice age of f i r s t maturity i f e x p l o i t a t i o n i s intense. However, with consi-derable decrease i n age of f i r s t maturity the o s c i l l a t i o n s i n numbers are l i k e l y to be associated with such small amplitudes that the cycles would be d i f f i c u l t to i d e n t i f y . With delayed age of f i r s t maturity a population acquires a d i f f e r e n t respose from that displayed at an early age of maturity: enlarged recruitment (due to delayed reproductive s e l f - i n h i b i t i o n ) does not appear as adults u n t i l several years have elapsed. Furthermore, p r e - r e c r u i t mortali which varies considerably i n this stage (Beverton and Holt, 1957) i s prolonged i n the population, the r e s u l t of which could possibly reduce recruitment. Age structure of spaw-ning stock i s l i k e l y to be shortened since maximum age i s a f a i r l y fixed c h a r a c t e r i s t i c feature i n a species. Populations simulated at a late age of maturity (Fig. 10) show not only an increased time i n t e r v a l between peaks of abundance but also the v a r i a b i l i t y i n numbers about mean-stock density gets much larger because of the time lag before reproduction. Unstable o s c i l l a t i o n s generated by Ricker-type reproduction curves increase i n amplitude u n t i l the dome of the curve i s reached and surpassed, af t e r which a stable o s c i l l a t i o n series i s established. The dome becomes the l i m i t (maximum recruitment) imposed by the shape of repro-duction curve. Hutchinson (1954) suggests that o s c i l l a t i o n s i n numbers of any great amplitude would be extremely dan-gerous to a species because at the crests when equilibrium saturation i s surpassed epidemics can spread through the population or cover maybe i r r e v e r s i b l y destroyed and the f i n a l crash might i n some instances bring the species far below the safe density l e v e l where random events might exterminate a species at least l o c a l l y . Such high densities of animals can, however, be reduced through emigration. Although late age of maturity i s l i k e l y to influence the development of large amplitude, the magnitude of such amplitude i s a func-t i o n of many other factors such as number of generations comprising the spawning stock, the shape of reproduction curve and f i s h i n g mortality i n the population (Tables VI and VIII). Age structure of a spawning stock i s a matter of considerable p r a c t i c a l concern, e s p e c i a l l y i n economically valuable species such as commercial fishes where ce r t a i n age classes are more valuable than others. I t i s a function of replacement, growth, natural and f i s h i n g m o r t a l i t i e s , i t determines population condition, that i s , tendency to grow or decline. The s i g n i f i c a n c e of age structure of a spawning stock has been allegedly thought to be exerted i n a popu-l a t i o n through egg s i z e and s i z e differences i n f i s h . Egg s i z e , embryo size and v i a b i l i t y have been observed to increase with increase i n the number of generations spawning ( N i k o l s k i i , 1969). Ponomarenko (MS.) demonstrated that changes i n abundance of cod year-classes were p a r t l y i n f l u -enced by d i f f e r e n t age composition of spawners: populations dominated by f i r s t time spawners generally gave a poor year-class while multi-age spawner composition usually was associated with a r i c h year-class. The period of c y c l i c a l f l u c t u a t i o n i n numbers i s determined by age of f i r s t maturity and number of year-classes i n the breeding stock, but the period of the cycle i s more influenced by age of f i r s t maturity than the number of generations which, when the rate of e x p l o i t a t i o n i s high enough, do not have any r e l a t i o n s h i p with the period of the cycle. The decreased e f f e c t of the number of generations on the period of the cycle i s due to a l t e r a t i o n (decrease) i n population age structure by a f i s h e r y . Ricker (1954) makes the same observation that the period of c y c l i c a l f l u c t u a t i o n i s independent of the number of generations i n the spawning 57. stock. His observations seem to f i t the condition where stocks are heavily exploited. When f i s h i n g mortality operates i n such a way that a constant percentage of animals i s taken from populations, reduction i n cycle length i s noted to be higher i n larger population age structure (Fig. 5). This would imply that multi-age structure populations lose r e l a t i v e l y more of the older in d i v i d u a l s than populations of simple age structure. The higher reduction i n cycle length i n larger number of age groups i s due to a higher rate of e x p l o i t a t i o n which, i n this example, i s proportional to the number of generations breeding, that i s , the longer the r e c r u i t s stay i n a fishery, the higher w i l l be the ultimate rate of e x p l o i t a t i o n . S t a b i l i t y of a population depends on many factors. Stocks which perform regular o s c i l l a t i o n s (Fig. 11) show marked age structure differences i n v a r i a b i l i t y i n numbers and equilibrium abundances. A large age structure of a spawning stock i s a stable structure associated with minor fluctuations i n numbers and a high equilibrium abundance which, i n addition to shape of reproduction curve, age of f i r s t maturity and f i s h i n g mortality, i s a function of number of generations breeding (Equation 8). A large age composi-ti o n of a stock i s therefore l i k e l y to provide more r e s i s -tance to environmental influence as compared to a small num-ber of age composition. However, such s t a b i l i t y would depend 58. on other factors. If the number of generations i n the spawning stock i s very large and age of maturity i s very l a t e , re-cruitment produced by such a population comprises only an inconsiderable part of the population with the r e s u l t that the numbers of generations i n the spawning stock fluctuate considerably and hence fluctuations i n the catch occur. The population structure may even be very unstable i f i n d i -viduals mature late and do not spawn every year, such as Sturgeon (Roussaw, 1957), the i n t e r v a l between spawning some-times being f a i r l y long. A population of such reproductive behaviour i s not l i k e l y to tolerate a high rate of loss from the adult population, and, usually such species compensate for low reproductive rate by experiencing a small and constant natural mortality rate (Beverton and Holt, 1959) ; y i e l d s from such populations would fluctuate enormously. N i k o l s k i i e_t a l (MS.) pointed out that i n species with limited fluctuations a d i r e c t r e l a t i o n s h i p i s usually observed between the fecundity of the parent stock and the abundance of the broods, but the r e l a t i o n s h i p tends to disappear when fluctuations a t t a i n large magnitude. The s i g n i f i c a n c e of age composition of a spawning stock suggest that i n exploited populations an age structure that permits maximum take and normal reproduction should be maintained. Probably the most c o n t r o l l i n g element i n the devel-opment of o s c i l l a t i o n s and the associated amplitude i s the shape of the reproduction curve. The larger the c o e f f i c i e n t A(l-W) A i n Z = We the steeper is the r i g h t hand limb of the reproduction curve, and the smaller the c o e f f i c i e n t the shallower w i l l be the shape. The shape of the limb may, instead of sloping, increase asymptotically so that an i n i t i a l d e f l e c t i o n of stock abundance i s compensated by a gradual, asymptotic return to equilibrium, leading to f a i l u r e i n the development of o s c i l l a t i o n s , or the limb may slope so abruptly that when a stock i s deflected from equilibrium to any p o s i t i o n along the descending limb, i t o s c i l l a t e s about the equilibrium p o s i t i o n with an amplitude that may be damped or undamped, depending on age of maturity,, age struc-ture of the breeding stock and the f i s h i n g mortality. A population with a low dome of the reproduction curve tends to e x h i b i t o s c i l l a t i o n s which eventually get damped; when the slope increases asymptotically a population may esta b l i s h equilibrium abundance i n a few generations. A high dome of the reproduction curve tends to be associated with unstable o s c i l l a t i o n s . The shape of the reproduction curve determines the maximum amplitude an o s c i l l a t i o n attains, and how quickly t h i s l i m i t i s approached depends on reproductive features of the animal, the stock recruitment r e l a t i o n s h i p and f i s h i n g mortality rate. A l l unstable o s c i l l a t i o n s reach the l i m i t imposed by the value of A i n the reproduction curve. Damped o s c i l l a t i o n s may also reach the dome of the curve i f the i n i t i a l d e f l e c t i o n of the stock i s big enough (Fig. 12) 60. while those o s c i l l a t i o n s that hardly reach the l i m i t are usually those whose amplitude i s so small and gets so quickly reduced i n siz e before i t can be allowed to reach the dome. The shape of the reproduction curve has no r e l a -tion with the period of c y c l i c a l f l u c t u a t i o n but the degree to which the cycle i s recognised depends, among other fac-tors, on the shape of reproduction curve. Populations exhibit a wide variety of reproductive features and, therefore, are expected to have_ d i f f e r e n t reac-tions to ex p l o i t a t i o n . Fishing mortality exerts i t s influence on a population v i a a l t e r a t i o n i n age structure which i n turn affects reproductive capacity of a population. Fishing mortality may eliminate population fecundity e s p e c i a l l y i f the age of recruitment i s very low, but i t has, within cer-t a i n l i m i t s , a n e g l i g i b l e and usually i n d i r e c t e f f e c t on the age structure of the spawning stock of fishes with a short l i f e cycle (Fig. 5) expecially i f age of maturity i s low. Its e f f e c ts range from no d i r e c t e f f e c t on population age structure of those species (eg. salmon) i n which the whole mature stock dies af t e r i t s f i r s t spawning, to the greatest e f f e c t i n a population of fishes with r e l a t i v e l y long l i f e cycle and small v a r i a b i l i t y from year to year i n the siz e of recruitment. In the l a t t e r case e x p l o i t a t i o n very quickly reduces population age structure, but with r e l i e f from f i s h i n g the stock regains i t s o r i g i n a l age structure. Stocks which perform regular o s c i l l a t i o n s i n the absence of a fishery are modified when ex p l o i t a t i o n begins (Table V I I I ) . Amplitude of o s c i l l a t i o n i s reduced due to increased recruitment rate and a population tends to esta b l i s h a stable equilibrium. Period of c y c l i c a l f l u c t u a t i o n i s also reduced with a corresponding reduction i n the mean age of population (Fig. 7). The period of the c y c l i c a l f l u c t u a t i o n i s not only reduced but i t i s d i f f i c u l t to recognise because of decreased amplitude at high f i s h i n g i n t e n s i t y (Fig. 10). Equilibrium abundance, which i s a function of many factors (Equation 7a), i s increased for an intensively exploited population but these increases have c e r t a i n l i m i t s , and the population i s considered over exploited when these are exceeded, and each population has i t s maximum rate of exp l o i t a t i o n which w i l l not i n t e r f e r e with i t s reproductive features. Populations which exh i b i t large amplitude of f l u c -tuations when age of maturity i s low tend to get s t a b i l i s e d at higher rates of expl o i t a t i o n than populations with minor fl u c t u a t i o n s . A high dome of reproduction curve and an early age of maturity are l i k e l y to lead to development of large fluctuations i n populations. The d i f f i c u l t y of deciding the exact shape of a graph that relates the number of spawners to that of r e c r u i t s produced has been attributed to the e f f e c t s , on populations, of environment which have no general trend i n t h e i r opera-t i o n . A random f l u c t u a t i n g environment may exert i t s influence i n natural populations v i a l a b i l e food supply or cli m a t i c conditions which appear through changes i n food supply and hence i n the s u r v i v a l of the year-classes. However, some of the environmental factors may be absorbed by the organism and may cause no e f f e c t i f they are not of an appreciable magnitude, but, due to d i f f e r e n t reproduc-ti v e capacities and d i f f e r e n t s u r v i v a l mechanisms, popula-tions may display d i f f e r e n t reactions to such environmental factors. Probably the most s i g n i f i c a n t e f f e c t of a random fl u c t u a t i n g environment on the c y c l i c a l f l u c t u a t i o n i n numbers i s the loss of s t a b i l i t y , or tendency to remain numerically constant, and reg u l a r i t y i n numerical f l u c t u a t i o n . S t a b i l i t y tends to be a common feature of simulated multi-age populations when age of f i r s t maturity i s low or when the dome of a reproduction curve i s low. S t a b i l i t y can also be established at high rates of exp l o i t a t i o n by a population that normally fluctuates i n numbers v i o l e n t l y i n absence of a fishery and random factors of the environment (Fig. 12). When factors of environment are taken into account as among those factors that influence the abundance of animals, the simulated populations show that s t a b i l i t y i n abundance i s l o s t and the loss appears to be proportional to the effectiveness of the environmental factors which, when they are of considerable influence, a population fluctuates i n numbers very v i o l e n t l y . I t remains doubtful whether such v i o l e n t fluctuations would drive a stock to e x t i n c t i o n as has been proposed by some (eg. Hutchinson, 1954). In the presence of a random f l u c t u a t i n g environment r e g u l a r i t y i n the period of o s c i l l a t i o n would be hard to recognise and short cycles would generally lose t h e i r regu-l a r i t y before large ones while cycles of larger amplitude would tend to show the general trend i n the v a r i a t i o n of numbers more e a s i l y than cycles of smaller amplitude, depen-ding on how e f f e c t i v e the environmental factors operate. Apart from o s c i l l a t i o n s i n numbers of Haddock and Daphnia which Ricker (1954) alleges to be primarily due to steep reproduction curves, r a r e l y w i l l a single factor be considered as the cause of cycles. Many factors, i n t r i n s i c and e x t r i n s i c , are l i k e l y to i n t e r a c t to determine the abun-dance of a population. Recognition of cycles i n this i n v e s t i g a t i o n has been due to treating a population as having a r e l a t i v e l y steep reproduction curve and late age of maturity--a combination of which allows cycles to be i d e n t i f i e d with certainty even with high f i s h i n g mortality. This study suggests that c y c l i c a l fluctuations i n numbers d i r e c t l y related to l i f e history features of animals determine the period of o s c i l l a t i o n as equal to twice the mean age of maturity i n l i g h t l y exploited stocks and equal to twice the age of f i r s t maturity i n intensely exploited stocks. Random fluctu a t i o n s , when added to the system, introduce i n s t a b i l i t y i n the o s c i l l a t i o n and v a r i a b i l i t y i n the period. The magnitude of fluctuations i n numbers i s i n i t i a l l y determin-d by the shape of reproduction curve and i t increases with delayed maturity while i t decreases with increasing number of ages spawning and increasing e x p l o i t a -t i o n . However, i t seems doubtful whether the observed high reduction i n the period of o s c i l l a t i o n would ever be big enough to notice, i n natural populations, before e x p l o i t a t i o n removed the o s c i l l a t i o n s completely and s t a b i l i s e d population abundances. SUMMARY 1. Population o s c i l l a t i o n i n numbers examined here are relat e d to l i f e h i s t o r y features of animal and caused by a stock recruitment r e l a t i o n s h i p which assumes a decline i n reproduction at high densities of spawners. 2. When a population i s l i g h t l y exploited, i t exhibits an o s c i l l a t i o n i n numbers whose period i s dependent on age of f i r s t maturity and number of generations i n the spawning stock; the period of o s c i l l a t i o n i s equal to twice the mean age of maturity i n the population. 3. When a population i s heavily exploited, i t exhibits an o s c i l l a t i o n i n numbers whose period i s dependent on age of f i r s t maturity but independent of number of generations i n the spawning stock; the period of o s c i l l a t i o n i s approximately twice the age of f i r s t maturity. E x p l o i t a t i o n was considered to have a feature of reducing number of generations i n the spawning stock and therefore to a l t e r the c y c l i c behaviour of a population. 4. When random fluctuations are added to the system, c y c l i c changes are maintained at low eff e c t s of random factors, but they introduce i n s t a b i l i t y i n the o s c i l l a t i o n and v a r i a b i l i t y i n the period of o s c i l l a t i o n as they become more e f f e c t i v e . 5. Period of c y c l i c a l f l u c t u a t i o n i s independent of the shape of reproduction curve but the steeper the r i g h t hand limb of the curve the greater i s the amplitude of the f l u c -tuation. 6 . Amplitude of o s c i l l a t i o n i n numbers increases rapidly with increase i n age of f i r s t maturity, up to a l i m i t which i s set by the shape of reproduction curve. 7. Amplitude of o s c i l l a t i o n i n numbers decreases with increase i n number of generations spawning. Multi-age stocks are considered to constitute a stable structure which s t a b i l i s e s population abundance v i a large repro-ductive output. 8. A fishery reduced period and amplitude of o s c i l l a t i o n and increases, within certain l i m i t s , average abundance of populations. 9. I t was suggested that the period of o s c i l l a t i o n was d i r e c t l y related to l i f e h i story features of animal but that the period would be very much influenced by e x t r i n -s i c f a c t o r s . The magnitude of fluctuations i n numbers seem to be primarily determined by the shape of repro-duction curve but are influenced by many factors of animate and inanimate environment. BIBLIOGRAPHY Aim, G. 1959. Connection between maturity, s i z e , and age i n f i s h e s . Rep. Inst. Fresw. Res. Drotting., 40:1-145. Backiel, T. and E.D. LeCren. 1966. Some density r e l a t i o n -ships for f i s h population parameters. Symposium on 'The B i o l o g i c a l Basis of Freshwater Fish Production, 1 S.D. Gerking (ed.), Blackwell S c i e n t i f i c Publications, Oxford and Edinburgh. Bagenal, T.B. 1957. The breeding and fecundity of the long rough dab Hippoglossoides platessoides(Fabr.) and the associated cycle i n condition. J. Mar. B i o l . Ass. U.K., 36:339-373. Beverton, N.J.H. and S.J. Holt. 1957. On' the dynamics of exploited f i s h populations. F i s . Invest., Lond., Ser. 2, 19. • " Beverton, N.J.H. and S.J. Holt. 1959. A review of lifespans and mortality rates of f i s h i n nature, and t h e i r r e l a -t i o n to growth and other physical c h a r a c t e r i s t i c s . In Wolstenholme, G.E.W. and M. O'Connor (eds.), The l i f e -span of animals. Ciba Foundation Colloq. on Ageing, 5:142-177. Cole, C.L. 1951. Population cycles and random o s c i l l a t i o n s . J. W i l d l i f e Manag., 15 (3) :233-252. Cole, C.L. 1954. The population consequences of l i f e h i story phenomena. The Quart. Rev. B i o l . , 29 (2):103-137. Cushing, D.H. 1969. The f l u c t u a t i o n of year-classes and the regulation of f i s h e r i e s . F i s h . D i r . Skr. Ser. Hav Unders., 15:368-379. Hutchinson, G.E. 1954. Theoretical notes on o s c i l l a t o r y population. J. W i l d l i f e Manag., 18 (1) :107-109. Lack, D. 1954. C y c l i c mortality. J . W i l d l i f e Manag., 18 (l):25-37. Larkin, P.A., and A.S. Hourston. 1964. A model for simula-t i o n of the population biology of P a c i f i c Salmon. J . F i s h . Res. Bd. Can., 21 (5) :1245-1265. M i l l e r , R.B. 1956. The collapse and recovery of a small white f i s h f i s h e r y . J.Fish. Res. Bd. Can., 13 (1):135-146. 68. M i l l e r , R.B. 1957. Have the genetic patterns of fishes been altered by introductions or by s e l e c t i v e fishing? J . F i s h . Res. Bd. Can., 14 (6) : 797-806 . Murdock, W.W. 1970. Population regulation and population i n e r t i a . Ecology, 51(2):497-502. Neave, F. 1953. P r i n c i p l e s a f f e c t i n g the s i z e of pink and chum salmon populations i n B r i t i s h Columbia. J . F i s h . Res. Bd. Can., 9 (9):450-491. Nicholson, A. J. • 1933 . The balance of animal populations. J . Anim. Ecology, 2:132-178. Nicholson, A.J. 1950. Population o s c i l l a t i o n s caused by competition for food. Nature, 165:476. N i k o l s k i i , G.V. 1961. E f f e c t s of f i s h i n g on population structure of a commercial f i s h . F i s h . Res. Bd. Can., Translation Ser. No. 280. N i k o l s k i i , G.V. 1962. A study of dynamics of f i s h popu- . l a t i o n . Dept. Agric. F i s h . Scotland Marine Lab. Aberdeen. Translation No. 744. N i k o l s k i i , G.V. 1969. Fish population dynamics. Olive r and Boyd. Edinburgh. N i k o l s k i i , G.V., A. Bogdanov, and J . Lapin (MS.). On fecundity as a regulatory mechanism of f i s h population dynamics. A symposium on Stock and Recruitment (In press). Ponomarenko, P.V. (MS.) On probable r e l a t i o n between age composition of spawning stock and abundance of cod year-classes i n the Barents Sea. A symposium on Stock and Recruitment (In press). Ricker, W.E. 1954. Stock and recruitment. J . Fi s h . Res. Bd. Can., 11(5) : 559-623 . Ricker, W.E. 1958. Handbyok of computations for b i o l o g i c a l s t a t i s t i c s of f i s h populations. B u l l . 119 Fi s h . Res. Bd. Can., Ottawa. Roussaw, G. 1957.. Some considerations concerning Sturgeon spawning p e r i o d i c i t y . J . f i s h . Res. Bd. Can., 14(4): 553-572. Schaefer, M.B. 1968. Methods of estimating effects of f i s h i n g on f i s h populations. Trans. Amer. Fi s h . Soc. 97 (3): 231-241. S i l l i m a n , R.P. 1970. B i r e c t i l i n e a r recruitment curves to assess influence of lake s i z e on s u r v i v a l of sockeye salmon (oncorhynchus nerka) to B r i s t o l Bay and forecast runs. Special S c i e n t i f i c r e p o r t — F i s h e r i e s No. 600, U.S. Dept. I n t e r i o r , Fish and W i l d l i f e Service, Bureau of Commercial F i s h e r i e s . Walters, C.J. 1969. A generalized computer simulation model for f i s h population studies. Trans. Amer. F i s h . S o c , 98(3):505-512. APPENDIX I DESCRIPTION OF THE MODEL The computer simulation was constructed i n the following manner: Escapement i n a simulated year includes a l l f i s h , immature and mature, of a l l ages. Thus, the t o t a l escapement i n the population i s given by, i=Y Esc(I) = E m N. (1) i= l 1 where Esc(I) i s the t o t a l number of f i s h i n the population the number of f i s h of age i .Ym the oldest age of spawning The proportion of f i s h at maturity i n a spawning stock i s calculated by summing a l l the proportions that occur at each age. The proportion of mature at age i s given by numbers mature expressed as a proportion of mature stock, that i s p.E.N. p m a t i = 1 1 1 (2) E m E.N. • i=Y 1 1 where p^ i s the proportion of the year class spawning at age i E^ the fecundity of f i s h of age i Y the age of f i r s t maturity. The sum of a l l the proportion of f i s h spawning i n a popu-l a t i o n i s obtained by summing up a l l the Pmat^, that i s Pmat(I) = \ t — = 1 O) where age i=j The sum of a l l the proportions (Pmat^) at each age add up to The abundance of spawners (W), determined as eggs i s calculated by summing the product of Pmat^ and Esc^ from age of f i r s t maturity ( Yy) t o oldest age of spawning: Y m W = >Pmat. o Esc. = E. / 1 i t i=Y y where E^ _ i s population fecundity i n a spawning season. p.E.N. Y W = 1 1 1 . E m N. (4) E m E.N. i = Y y 1 i=Y 1 1 y Catch from a population i s obtained before spawning and i s computed as the sum of numbers of f i s h at each age of maturity removed as a fixed annual percentage, TABLE IX. Order of calculations i n the model for each year f r y production. Stage Equation Equa No. Legend Escape-ment Esc(I) Y 1 Esc(I)= t o t a l number of f i s h of a l l ages i n a pop-u l a t i o n . N. = number of f i s h of I age l Y m = oldest age of spawning Proportion • of a year class at maturity Pmat. 1 P i E i N i lm E.N. i=y 1 1 y 2 Pmat.= proportion of a year 1 class at maturity expressed i n r e l a t i o n to stock abundance. p. = proportion of the year class spawning at age i expressed i n r e l a -t i o n to number at age. E. = fecundity of f i s h of age l Proportions of a l l year classes at maturity „ ±.,T\ J Pmat(I) j=Y j = l = age Yy = age of f i r s t maturity Pmat(I) = proportions of a l l the year classes at maturity (cont'd) TABLE IX. (Continued) Population fecundity W p . E .N. E m E.N. i=Y 1 1 y Y Z m N. i=Y 1 y W = population fecundity i n a spawning season ex-pressed as a proportion of replacement l e v e l . Catch Y C = E m N.a 5 c = annual catch i n numbers i=Y 1 y a = percentage rate of annual removal Population fecundity p.E.N. Y for equi- W = —^ • E N. = 1 librium r £ m E.N. i = Y y 1 6 W r = replacement number of spawners. i=Y 1 1 y Fry pro-duction p.E.N. Y E m E.N. i=Y 1 1 E m N, i=Y J A 1-p.E.N. E.N. i=Y X x - y Y E i=Y m N, recruitment abundance expressed as a propor-t i o n of replacement l e v e l . 74. Y E m N.a i=Y 1 (5) where c i s the annual catch i n numbers a the percentage rate of e x p l o i t a t i o n . The program i s designed to s t a r t from the condition when the population i s i n equilibrium and there i s no exploitation, In this state the spawner abundance expressed as a proportion of replacement l e v e l i s given by, W p . E . N . Y •E m E.N. i=y 1 1 Y E m N . i=Y 1 y (6) Fry production (Z) from egg abundance i s related v i a com-pensatory density dependent mechanism described by Ricker's (1954) r e l a t i o n s h i p , Z = p.E.N. Y m £ E.N. x=Y l l y A 1-E m N. i=Y 2 p.E.N. Y E m E.N. J.=Y 1 1 m L N. i=Y • (7) where Z i s recruitment expressed as a proportion of replace-ment l e v e l . 

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