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Model of dispersal of fry of Sockeye Salmon (Oncorhynchus nerka) in Babine Lake Simms, Steven Eric 1974

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A MODEL OF DISPERSAL OF FRY OF SOCKEYE SALMON ( Oncorhynchus nerka ) IN EAEINE LAKE by Steven E r i c Simms B.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1971 A t h e s i s submitted i n p a r t i a l f u l f i l m e n t c f the requirements f o r the degree of Master of Science i n the Department of Zoology He accept t h i s t h e s i s as conforming t c the r e q u i r e d standard THE U N I V E R S I T Y OF B R I T I S H COLOMBIA August 1974 In p r e s e n t i n g 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 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 Co lumb ia , I a g ree 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 r e f e r e n c e and s tudy . I f u r t h e r agree 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 purposes 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 . It i s u n d e r s t o o d that 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 thout my w r i t t e n p e r m i s s i o n . Department o f Zoology The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada D a t e Seot. I6,1p7ll ABSTRACT A computer s i m u l a t i o n model was w r i t t e n to mimic the n a t u r a l movement of salmon f r y i n Babine Lake, B.C.. Simulated d i s t r i b u t i o n s of f r y were compared with f i e l d o b s e r v a t i o n s taken i n 3 sampling p e r i o d s during the summer and f a l l i n 1967, 1968, 1971, and 1972, i n order to evaluate the model's v a l i d i t y . Simulated d i s t r i b u t i o n s of f r y , when random and h e a v i l y - b i a s e d movements were combined, were i n reasonable accord with n a t u r a l l y observed d i s t r i b u t i o n s of f r y i n p e r i o d s 1 and 2. In p e r i o d 3 the model s u c c e s s f u l l y produced a d i s t r i b u t i o n s i m i l a r to t hat n a t u r a l l y observed when the f r y were programmed to undergo only random movement. F a c t o r s which might account f o r the v a r i o u s d i s t r i b u t i o n s of f r y i n d i f f e r e n t p e r i o d s i n c l u d e the e f f e c t s of c u r r e n t and innate b e h a v i o r a l responses of the f r y to l i m n o l o g i c a l c o n d i t i o n s . In c o n s t r u c t i n g my model, I assumed that f r y t r a v e l l e d at speeds observed i n the l a b o r a t o r y in s t i l l water. The model of f r y d i s p e r s a l i n Babine Lake could be improved as more i n f o r m a t i o n i s c o l l e c t e d on the limnology of the lake and on f r y behavior. In a d d i t i o n , the model has much g e n e r a l i t y and the techniques used may be a p p l i e d to the d i s p e r s a l of other organisms and to other l a k e s . i i i TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES FOR MAIN TEXT v LIST OF FIGURES FOR MAIN TEXT v i i i LIST OF TABLES AND FIGURES FOR APPENDICES x i ACKNOWLEDGEMENT X V INTRODUCTION 1 I n i t i a l Planning of the Model 3 METHODS 8 The Workings of the Model 8 Fi e l d Data and Goodness of F i t procedures 17 Swimming Speed of Fry 34 RESULTS 37 Random Dispersal - Period 1 37 Directionally-Biased Dispersal - Period 1 39 Dispersal in Periods 2 and 3 57 DISCUSSION 74 LITERATURE CITED 81 i v PAGE APPENDICES 86 APPENDIX 1 87 APPENDIX 2 99 APPENDIX 3 100 APPENDIX 4 102 APPENDIX 5 103 APPENDIX 6 107 APPENDIX 7a 113 APPENDIX 7b 128 APPENDIX 8 139 APPENDIX 9 146 V LIST OF TABLES OF MAIN TEXT TABLE PAGE Data for A (F - F) vs. number of step i n t e r v a l s . F i s the area under the- step function approximation of the normal d i s t r i b u t i o n function (Figure 2) and F i s the area under the normal d i s t r i b u t i o n function (Figure 2) ...14 The numbers and proportions of marked Fulton River f r y caught i n each of the 8 lake areas in period 1 in 1967, 1968, 1971, ana 1972 20 The numbers and proportions of marked Fulton River f r y caught in each cf the 8 lake areas in period 2 i n 1967, 1968, 1971, and 1972. 21 The numbers and proportions of marked Fulton River f r y caught i n each of the 8 lake areas in period 3 in 1967, 1968, 1971, and 1972 22 5 Mean proportions with 95% confidence -limits of Fulton River f r y i n each of the 8 lake areas for each of the 3 annual sampling periods i n 1967, 1968, 1971, and 1972. 23 6 The numbers and proportions of marked Pinkut Creek f r y caught i n each of the 8 lake areas i n period 1 in 1971 and 1972 .25 7 The numbers and proportions of marked Pinkut Creek f r y caught in each of the 8 lake areas in period 2 in 1971 and 1972 26 8 The numbers and proportions of marked Pinkut Creek fry caught i n each of the 8 lake areas in period 3 in 1971 and 1972. .........27 v i TABLE 9 10 11 12 13 14 15 16 17 18 19 P A G E Mean proportions of Pinkut Creek f r y in each of the 8 lake areas for each of the 3 annual sampling periods in 1971 and 1972 28 The numbers and proportions of t o t a l lake salmon f r y caught i n each of the 8 lake areas in period 1 i n 1967, 1968, 1971, and 1972. 29 The numbers and proportions of t o t a l lake salmon fry caught in each cf the 8 lake areas in period 2 in 1967, 1968, 1971, and 1972 ....30 The numbers and proportions of t o t a l lake salmon f r y caught in each of the 8 lake areas in period 3 i n 1967, 1968, 1971, and 1972. 31 Mean proportions of t o t a l lake salmon fry in each of the 8 lake areas for each of the 3 annual sampling periods i n 1967, 1968, 1971, and 1972 32 Random movement of f r y - no bias, period 1 38 Summary of results for period 1 - random movement. .....40 S t a t i s t i c s of linear regressions of Figures 6, 7, and 8 46 Biased movement of Fulton River f r y 'down the lake* - period 1. 49 Summary of res u l t s for period 1 - 30:70 biased movement ....50 Summary of results for period 1 - 20:80 biased movement. 51 v i i TABLE PAGE 20 Summary of results for period 1 - 10:90 biased movement. ....53 21 Data for combined simulated dispersal d i s t r i b u t i o n s of f r y in period 1 together with expected d i s t r i b u t i o n of f r y and calculated chi-square value 55 22 Movement of Fulton River f r y , ncn-biased and biased, »up the la k e 1 - period 2. 60 23 Summary of re s u l t s for period 2 - random movement 62 24 Summary of res u l t s for period 2 - 30:70 biased movement. 63 25 Summary of re s u l t s for period 2 - 10:90 biased movement. ....64 26 Data for combined simulated dispersal d i s t r i b u t i o n s of fry in period 2 together with expected d i s t r i b u t i o n of fry and calculated chi-square value 68 27 The chi-square test applied to the expected values of period 3 against the expected values of period 2 .......70 28 Random movement of f r y - no bias, period 3. 72 29 Summary of results for period 3 - random movement. 73 v i i i LIST OF FIGURES OF MAIN TEXT FIGURE PAGE 1 5x5 dispersal grid. Fish move ' from the center of the dispersal sguare to the center of each of the 25 squares 11 2 Step function approximations to the normal d i s t r i b u t i o n function using 3, 5, and 7 step i n t e r v a l s 1 3 3 Difference in 'area under the curve' between the normal d i s t r i b u t i o n function and the step function approximating i t , (F - F), plotted against the number cf step i n t e r v a l s 15 4 Babine Lake showing the Fulton River and fi s h i n g areas. 19 5 Computer simulation (observed) numbers of fry (N = 1000) at the best chi-square value (15436) for period 1 - random movement, and naturally observed (expected) numbers of f r y (N=1000) for period 1 in each of the 8 lake areas. Data are taken from Table 15 4 1 6 Linear regressions of numbers of recaptured tagged f r y from Fulton River vs. distance from shore when caught f c r period 1, 1967 and 1968 43 7 Linear regressions of numbers of recaptured tagged f r y from Fulton River vs. distance from shore when caught for period 2, 1967 and 1968. 44 8 Linear regressions of numbers of recaptured " tagged fry from Fulton River vs. distance from shore when caught f c r period 3, 1967 and 1968 45 Computer simulation (observed) numbers of fry (N=1000) at the best chi-square value (436) for period 1 - 10:90 biased movement, and naturally observed (expected) numbers of fry (N=1000) for period 1 in each of the 8 lake areas. Data are taken from Table 20. Computer simulation (observed) numbers of f r y (N=1000) of the combined d i s t r i b u t i o n patterns for period 1, and naturally observed (expected) numbers of f r y (N=1000) fo r period 1. Data are taken from Table 21. Rate of increase in length of sockeye (and squawfish) i n 1936 and 1937. The sockeye are of the brood spawned in the autumn of 1935 and hatched in A p r i l and May of 1936. (Taken from publication by Hoar (1938).) .. Computer simulation (observed) numbers of fry (N=1000) at the best chi-sguare value (1401) f o r period 2 - random movement, and naturally observed (expected) numbers of f r y (N=1000) for period 2 in each of the 8 lake areas. Data are taken from Table 23. . Computer simulation (observed) numbers of fry (N=1000) at the best chi-sguare value (1196) for period 2 - 10:90 biased movement, and naturally observed (expected) numbers of fry (N=1000) for period 2 in each of the 8 lake areas. Data are taken from Table 25 Computer simulation (observed) numbers of f r y (N=1000) of the combined d i s t r i b u t i o n patterns for period 2, and naturally observed (expected) numbers of f r y (N=1000) f o r period 2. Data are taken from Table 26. Computer simulation (observed) numbers of f r y (N=1000) at the best chi-square value (290) for period 3 - random movement, and naturally observed (expected) numbers of fry (N=1000) for period 3 in each of the 8 lake areas. Data are taken from Table 29. . FIGURE PAGE 16 General d i s p e r s a l t r e n d of f r y over summer and f a l l . 76 xi LIST OF TABLES AND FIGURES OF APPENDICES Appendix 1_ FIGURE PAGE 1 Examples of grid patterns 88 2 Examples of the r e f l e c t i o n p r i n c i p l e . 93 AEEendix 6 TABLE PAGE 1 Biases f o r d i f f e r e n t values of the parameter u. .112 FIGURE PAGE 3 4-para meter lognormal density functions, f (u,1.0,-3.0,3.0), together with the normal density function, n{0.0,1.0) ..........110 A££endix 7a TABLE PAGE 2 Average distance tr a v e l l e d by f r y i n the bias direction per i t e r a t i o n for varying biases. ...............127 FIGOBE PAGE *t Probability values obtained using the bivariate density function, n (0.0, 1.0) x f(u, 1.0,-3.0,3.0): u = 1.2810, with no rotation. Bias i s 10:90 115 x i i F I G U R E PAGE 5 Probability values obtained using the bivariate density function, n (0.0, 1.0) x f(u,1.0,-3.0,3.0): u = 1.0360, with no rotation. Bias i s 15:85 116 6 Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0) : u = 0.8420, with no ro t a t i o n . Bias i s 20: 80. 1 17 7 Probability values obtained using the bivariate density function, n (0.0, 1.0) x f(u,1.0,-3.0,3.0): u = 0.6740, with no rotation. Bias i s 25:75. 118 8 Pro b a b i l i t y values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0) : u = 0. 5240, with no ro t a t i o n . Bias i s 30: 70 119 9 Probability values obtained using the bivariate density function, n (0.0, 1.0) x f(u, 1.0,-3.0,3.0): u = 0.3850, with no rotation. Bias i s 35:65 120 10 Probability values obtained using the bivariate density function, n(0.0,1.0) x . f (u,1.0,-3.0,3.0) : u = 0. 2530, with no rota t i o n . Bias i s 40: 60. 121 11 Probability values obtained using the bivariate density function, n (0.0, 1.0) x f(u,1.0,-3.0,3.0): u = 0.1260, with no rotation. Bias i s 45:55 122 12 Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0): u = 0.0000, with no rota t i o n . Bias i s 50: 50. . 123 x i i i A££endix 7b FIGURE PAGE 13 Probability values obtained using the bivariate density function, n(0.0,1.0) x f(u, 1.0,-3.0,3.0) : u = 1.2810, with the -30 degrees rotation. Bias i s 10:90. ..........130 14 Probability values obtained using the bivariate density function, n(0.0,1.0) x f(u,1.0,-3.0,3.0): u = 1.0360, with the -30 degrees rotation. Bias i s 15:85 131 15 Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0) : u = 0.8420, with the -30 degrees rotation. Bias i s 20:80. 132 16 Probability values obtained using the bivariate density function, n(0.0,1.0) x f(u,1.0,-3.0,3.0): u = 0.6740, with the -30 degrees rotation. Bias i s 25:75. ................. 133 17 Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u r1.0,-3.0,3.0): u = 0.5240, with the -30 degrees rotation. Bias i s 30:70. 134 18 Probability values obtained using the bivariate density function, n(0.0,1.0) x f(u , 1.0,-3.0,3.0): u = 0.3850, with the -30 degrees rotation. Bias i s 35:65 135 19 Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0) : u = 0.2530, with the -30 degrees rotation. Bias i s 40:60. ..... 136 xiv FIGURE PAGE 20 Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u, 1.0,-3.0,3.0): u = 0. 1260, with the -30 degrees rotation. Bias i s 45:55 137 21 Probability values obtained using the bivariate density function, n (0.0,1.0) x f (u,1.0,-3.0,3.0) : u = 0.0000, with the -30 degrees rotation. Bias i s 50:50 138 X V ACKNOWLEDGEMENT I am very g r a t e f u l to Dr. P. A. L a r k i n , my s u p e r v i s o r , f o r his encouragement and support throughout my s t u d i e s and f o r a c r i t i c a l review of the manuscript. I a l s o thank Dr. D. C h i t t y f o r a c r i t i c a l review of the manuscript. 1 INTRODUCTION ' The movements of f i s h have far-reaching conseguences, both e c o l o g i c a l l y and commercially. Fish movements, as well as other animal movements, can be considered to be of two main types, migration and d i s p e r s a l . The migration patterns cf f i s h populations have been the subject of numerous investigations and in a few cases they are known with some accuracy. Far less study has been given to dispersal. Beverton and Holt (1957) l a i d some groundwork on f i s h dispersal by developing a t h e o r e t i c a l mathematical treatment of oceanic l o c a l movement of f i s h and dispersion of f i s h populations, using transport eguations. They then proceeded to use their model to obtain what they considered working sclutions to a number of problems raised by s p a t i a l variations in f i s h density and f i s h i n g e f f o r t . To date no ether major studies seem to have been done in t h i s f i e l d . For t h i s thesis, I decided to develop a model of another aspect of f i s h d i s p e r s a l , the inland dispersal of f i s h in a lake. Such a study would be a start toward investigating the relationship between f i s h (salmon fry) d i s t r i b u t i o n and the u t i l i z a t i o n of lake resources. The ultimate goal of such studies would be to assess how to achieve greater f r y production by the st r a t e g i c location of a r t i f i c i a l spawning grounds or the development of existing natural spawning areas. In e f f o r t s to increase numbers of salmon f r y in a lake the development of readily accessible and naturally good spawning 2 streams seems to be the simplest and most economical course to follow. However, such developments may not harmonize with s p a t i a l patterns of lake productivity and may cause l o c a l o v e r - u t i l i z a t i o n . A concerted e f f o r t to increase productivity of apparently l e s s suitable but better located spawning areas might contribute more to t o t a l production. Johnson (1956) suggested that to f u l l y u t i l i z e lake nursery areas and attain maximum production, d i s t r i b u t i o n of spawners to a l l available spawning areas i s e s s e n t i a l . And yet, t h i s might net be the answer e i t h e r . The d i s t r i b u t i o n of f r y r e s u l t i n g from th e i r dispersal depends at least i n i t i a l l y on the d i s t r i b u t i o n of the spawning parent population and may subsequently depend upon such things as the morphometry of the lake basin and the behavior of the f i s h . Whenever high densities of f r y occur, either because many f ry are produced from a few large spawning areas, or because moderate numbers are produced by many spawning areas i n r e l a t i v e l y close proximity, there may be i n e f f i c i e n t u t i l i z a t i o n of lake resources, and density dependent e f f e c t s cn mortality and growth. Differences in rate of growth of fry in various lakes may result from differences in the physical environment, differences in food supply (which are independent of size of stock), or differences in stock density. Within a lake, differences in rate of growth could be attributed to differences i n stock density alone. Both Foerster (1944) and Johnson (1956) demonstrated an inverse r e l a t i o n between density of stock and size of young sockeye. Reduced growth of fry may in turn lead to reduction i n numbers of f r y surviving to adults. Foerster 3 (1954) and Ricker (1962) substantiate that there i s a positive relationship between the size of sockeye smolts and subsequent survival to maturity. In addition, further mortality could occur i f a •closeness factor* i s present such that aggressive behavior between fishes i s p o s i t i v e l y related tc the density of the f i s h . Both the conditions of reduced growth and lower survival would ultimately af f e c t the yield of the commercial fis h e r y . What I have s p e c i f i c a l l y attempted tc do i s to model, as simply as possible, the natural movement of f r y in a lake; and to evaluate the v a l i d i t y of the model by comparing the model results with f i e l d observations. As mere data are accumulated on factors affecting the d i s t r i b u t i o n of f i s h , this information can be incorporated into the model so that i t w i l l become increasingly helpful in analyzing lake situations and predicting the outcome of new management schemes. I n i t i a l Planning of the Kodel The f i r s t obvious problem i n developing a dispersal mcdel was to decide generally how to go about i t . There are many d i f f i c u l t i e s in applying d i f f e r e n t i a l equations, which have generally been used to describe dispersal phenomena (Skellam 1951, Pielou 1969, Gadgil 1971, Beverton and Holt 1957), to the description and analysis of complex problems i n ecology and f i s h d i s p e r s a l . One d i f f i c u l t y i s assigning a par t i c u l a r density of animals to an area in space when there may be several wavelike movements of organisms into the area from several different epicenters, occuring perhaps at d i f f e r e n t times. A second d i f f i c u l t y i s determining densities when the p a r t i c u l a r system, such as a lake, has many boundary conditions which i n t e r f e r e , in two dimensions, with the freedom of movement which i s implied i n simple t h e o r e t i c a l approaches. For such reasons, t r a d i t i o n a l mathematical methods were undesirable for what was required f c r t h i s study. The only plausible a l t e r n a t i v e was to turn to computer simulation. A computer model allows the description of f i s h dispersal in a lake by a flow of events i n time and space in terms of a system that positions a l l causes and effects r e l a t i v e to r e a l space. It also does bookkeeping e f f i c i e n t l y and easily on any physical, chemical, and b i o l o g i c a l data that are required and can be adapted i n the future to handling increasing amounts of information. Although computer simulation i n ecology has made considerable progress in the l a s t few years, there are s t i l l r e l a t i v e l y few simulation models available concerning animal movements. S i n i f f and Jensen (1969) constructed a simulation model based on the movements of the red fox and sncwshce hare within t h e i r home ranges. Their techniques depended on determining, from telemetry data obtained in associated f i e l d studies, the probability d i s t r i b u t i o n s involved as the animals travelled in t h e i r normal daily a c t i v i t i e s . Kitching (1970) , using existing information about arthropod movement patterns, described the construction, properties, and implications of a simulation model which pertained to organisms dispersing in a 5 grid system containing a number of discrete habitats distributed at random within the gridded area. ft simulation model considering the dispersal of desert rodents to home ranges was carried out by French (1971), who u t i l i z e d ideas o r i g i n a l l y proposed by Hurray (1967). Some work towards computer simulation of the movements of f i s h has been done by S a i l a and Shappy (1963), who were concerned with trying tc explain the observed homing phenomenon of P a c i f i c salmon by means of random searching combined with a low^ degree of orientation tc an outside stimulus. In approaching the problem of f i s h dispersal in a lake, I decided to concentrate on Babine Lake, B.C., which supports one of the largest sockeye salmon ( Oncorhynchus nerka ) runs in B.C. and produces the bulk of the Skeena River catch. S u f f i c i e n t study of the lake has been done to meet most of the needs for an i n i t i a l simulation model. The lake i s also the s i t e of an extensive development program which i s now being carried out by the Resource Development Branch, Fisheries and Marine Service, and the Fisheries Research Board of the Department of the Environment (Department of Fisheries cf Canada and Fisheries Research Board of Canada, MS, 1965 and MS,1968) . Studies undertaken by the Fisheries Research Board indicated that Babine Lake could sustain a fry input many times greater than that which now occurs. Consequently, i n 1965, the Resource Development Branch embarked cn a major program of a r t i f i c i a l spawning channel construction coupled with flow control to increase f r y production in spawning streams tributary 6 to the main basin. The t r i b u t a r i e s of the main lake include one large stream, Fulton River, three moderate sized streams, Pinkut Creek, Morrison River, and Pierre Creek, and nine small streams. Fulton River and Pinkut Creek were chosen f cr i n i t i a l development. In 1966 J. McDonald of the Fisheries Research Board (Nanaimo) found that there were differences in the mean densities of sockeye f r y , both in summer and f a l l , between f i v e areas of the lake. Differences in the area means observed during period 1 (June 25 - July 27) indicated a progressive increase in density from the north end of the main lake basin (area 1) to the south end (area 5). Ey late summer (period 2: August 16 - September 9) mean catches were greater i n areas 3 and 4 than elsewhere, revealing that the underyearlings were concentrated more c e n t r a l l y i n the main lake basin. A further s h i f t northward i n areas of concentration occurred between late summer and f a l l , so that in October (period 3: October 6 - 25) underyearlings were most dense in area 2 . This d i r e c t i o n a l i t y of movement, f i r s t a north-to-south movement in the early summer and second, a south-to-north movement as the summer progressed, was also a general trend i n 1967, 1968, 1971, and 1972, when other data were c o l l e c t e d . This movement determines a pattern of d i s t r i b u t i o n of f r y i n the lake. It was the aim of this study to simulate the movement, and to attempt to generate a simulated pattern of d i s t r i b u t i o n of f r y . The techniques in the model could easily be applied to 7 other lakes. Perhaps the greatest additional benefit would come from investigating the important salmon-producing lakes of Alaska, notably those of the Kvichak Fiver system (Iliamna Lake and Lake Clark) and those of the Wood River system (Lake Nerka, Lake Aleknagik, Lake Kulik, and Lake Beverley), on which much work i n recent years has been done (Kerns 1968, Pella 1968, Burgner 1962, Rogers 1972) and for which there appear to be some similar p o s s i b i l i t i e s for complex patterns of u t i l i z a t i o n of lake resources. 8 METHODS i I i i ® Workings of the Model The basic idea in constructing the model was tc represent the lake on a large rectangular coordinate system, and then tc disperse numbers of fry from any grid sguaras tc various squares throughout the simulated lake, by a series of small steps, each step involving one i t e r a t i o n of a time lcop. One of the f i r s t decisions to be made concerned the square s i z e . A square size which represented 1/16 square miles (1/4 mile x 1/4 mile) of the actual lake was decided upon for twc reasons. F i r s t , i t proved to be a good compromise in achieving s u f f i c i e n t l y detailed representation of the complexity of the shoreline of the lake, and economy in the amount of computer time required i n dispersal of f i s h from one square tc another. Second, the location of f i e l d c o l l e c t i o n s cannot te measured with more precision than about 1/4 mile cn a lake the s i 2 e of Babine (as indicated by the Fisheries Research Board in their studies). Thus, any more detailed representation of the lake could not be tested with data, and a mere gross representation could f a i l to take advantage of the data. The lake grid was an assemblage of 3031 squares (1/4 mile x 1/4 m i l e ) . 1 Construction of the model was such that f i s h could be dispersed from any one of these squares to a distance not 1 See Appendix 9. 9 more than 2 squares i n any combination of north, south, east, west (N, S, E, W) directions per i t e r a t i o n . Thus, i n any given i t e r a t i o n , f i s h would disperse from a central square into 25 squares i n a 5x5 g r i d , when the i r movement was not inhibited by boundaries (i.e. the shoreline). Dispersal of f i s h from a square was determined by continuous bivariate density functions, summed appropriately tc give 25 discrete components. By means cf an appropriate bivariate density function, any type of biased movement could be simulated. In addition, d i f f e r e n t functions could be used for d i f f e r e n t parts of the lake and at d i f f e r e n t times and could therefore describe f i s h movements re s u l t i n g from such things as d i f f e r e n t behavior or limnological conditions. In the 'open1 lake, where the movement cf f i s h from a square to i t s surrounding sguares was uninhibited, there was nc dispersal problem. The main d i f f i c u l t y was to determine a proper method of ' t e l l i n g 1 the f i s h where to go i f a 5x5 grid overlapped the shore of the lake, thus causing the f i s h tc ' h i t ' land. Assuming f i r s t of a l l that movement might be e n t i r e l y random, what one r e a l l y needed was a means whereby, after a long enough time, a given number of f i s h , dispersed from any point in the lake, would assume a uniform d i s t r i b u t i o n over the lake and maintain i t i n d e f i n i t e l y . Several successive programs, written in an attempt to achieve t h i s , f a i l e d , and were based cn invented r u l e s : 10 1. A l l f i s h that ' f a l l * i n grid squares which are actually land squares are placed in existing water sguares cf 5x5 grid proportionately to the number of f i s h which dispersed d i r e c t l y to those squares. or 2. A l l f i s h that ' f a l l * i n grid squares which are actually land squares are returned to the o r i g i n a l dispersal square of g r i d . Respectively, with these kinds of rules, f i s h accumulated off shore towards the middle of the lake or on shore. Obviously, nc simple rule could be found to give what was wanted, and thus a more sophisticated approach was needed. A r e f l e c t i o n p r i n c i p l e method was eventually found to be what was required, and could be applied d i r e c t l y and r e l a t i v e l y simply to the dispersal problem. Details of the method involved in the dispersal process are given in Appendix 1. E r i e f l y , the method assumes that when a f i s h h i t s a land sguare of the grid i t i s reflected from the shore according to the normal laws of r e f l e c t i o n as could be used, for example, i n analyzing the path of a b i l l i a r d b a l l r e f l e c t e d off the edge of a b i l l i a r d table. The maximum possible distance a f i s h could move in an i t e r a t i o n from a p r a c t i c a l viewpoint was 0.625 miles, the distance from the middle of the dispersal square to the edge of the g r i d , but for computational purposes f i s h were always considered to disperse from the center cf one square to the center of another (Figure 1). Although f i s h could have been distributed over more squares at a time, the amount cf extra work in such an undertaking would have been immense cwing to the method of dispersal developed for boundary conditions, and was not considered worthwhile in bettering the model. The question 11 Figure 1. 5x5 dispersal g r i d . Fish move from the center of the dispersal square to the center of each of the 25 squares. 12 arises though, "How e f f e c t i v e l y does the step function approximate the continuous bivariate density function?". When the difference in •area under the curve' between the normal d i s t r i b u t i o n function (1-dimensional) and the step function approximating i t (Figure 2) was plotted against the number of steps, i . e . (F - F) vs. number of steps, the resulting curve resembled a negative exponential (Table 1 and Figure 3). The curve started to l e v e l off at 5 steps shewing that the 'returns' from increasing the number of steps beyond 5 for each axis of the bivariate normal density function were not compensated by the additional e f f o r t required. In this analysis, only the odd numbered steps should be considered because of the necessity for a central dispersal square. Some idea of the amount of error involved in approximating the movement of f i s h by moving them, i n each time i n t e r v a l , d i s c r e t e l y 5 steps along each axis of a bivariate function, can be gained by comparing the standard deviation of a one-dimensional density function and i t s 5-step approximation. The standard deviation of the bivariate normal density function used for random movement was 0.2083 2 so that the normal density function also had a standard deviation of 0.2083 . The standard deviation of the 5rstep approximation tc the normal density function i s found to be 0.2170 . 3 Thus, there i s an error of a b o u t : 0-2l70 t- Q0.208? x 1 0 0 = \ % z % 2 See Appendix 2 for derivation. 3 See Appendix 3 for derivation. Figure 2 . Step function approximations to the normal d i s t r i b u t i o n function using 31 5t and 7 step i n t e r v a l s . 14 Table 1. Data for (F - F) vs. number of step i n t e r v a l s . F i s the area under the step function approximation of the normal d i s t r i b u t i o n function (Figure 2) and F i s the area under the normal d i s t r i b u t i o n function (Figure 2). Area (F) under normal d i s t r i b u t i o n ( l i m i t s (-3.0,3.0) ) i s 2.99 . A A number of area (F) under D = F - F steps step function (-3.0,3.0) 1 6.00 3.01 2 4.50 1.51 3 3.99 1.00 4 3.74 0.75 5 3.59 0.60 6 3.49 0.50 7 3.42 0.43 8 3.37 0.38 9 3.33 0.34 10 3.29 0.30 11 3.26 0.27 12 3.24 0.25 Figure 3. Difference i n 'area under the curve 1 between the normal d i s t r i b u t i o n function and the step function approximating i t , (F - F), plotted against the number of step i n t e r v a l s . 16 in the normal density function approximation. Presumably, the error involved in other density function approximations would be si m i l a r . The simulation model was run cn an I.B.M. 360/67 computer. Several input values are required for any run.* These include, in general, the bivariate dispersal density, function or functions, a time parameter allowing change cf the functions when desired, numbers of f r y dispersing from given lake sguares, a mortality f a c t o r , and the number of i t e r a t i o n s or steps. Areal densities of f r y for d i f f e r e n t sampling periods are also input values. Hence, the observed d i s t r i b u t i o n s of fry in the lake could be compared with the simulated d i s t r i b u t i o n s of f r y as the simulation was run. The number of i t e r a t i o n s was found by multiplying an assumed average swimming speed (cm./sec. converted into miles/day) by the number of days since day 0 . This value was then divided by the average distance/iteration (miles/iteraticn) a f i s h moved on the grid according tc the bivariate density function used. The number of i t e r a t i o n s required to move f i s h , of course, was proportional to the assumed swinging speed. Thus, a day (24 hours) could be represented by a varying number of i t e r a t i o n s . In th i s model a day was represented from about 2 to 11 i t e r a t i o n s , representing time i n t e r v a l s of about 12 hours to about 2 hours, respectively. Although f i s h in nature move continuously, and can change dir e c t i o n instantaneously, the * See Appendix 8. 17 model s t i l l gives similar r e s u l t i n g displacements, although these displacements are 'packaged out' d i s c r e t e l y . F i e l d Data and Goodness of F i t Procedures Observed d i s t r i b u t i o n s of fry in Babine Lake used i n this study for comparison with r e s u l t s from the simulation model were taken from manuscripts prepared by the Fisheries Research Board of Canada (Scarsbrook and McDonald 1970, 1972, 1973). In addition to determining the d i s t r i b u t i o n of underyearling fry, in general, in lake areas in the years 1966 - 68 and 1971 - 72, the Fisheries Research Board also investigated the d i s t r i b u t i o n of marked fry from Fulton River i n the same years. Investigations were carried cut on the d i s t r i b u t i o n of marked fry from Pinkut Creek in 1971 - 72 . In investigating dispersal phenomena in the lake, the best approach would be f i r s t to deal with the dispersal of f r y from the d i f f e r e n t production centers separately, and then to integrate a l l r e s u l t s to relate them to the general dispersion of f r y throughout the whole nursery area. Unfortunately, not enough data have been collected on a l l production centers cr on conditions of the lake and behavior of the f i s h f c r one to be able to do t h i s s a t i s f a c t o r i l y . For t h i s reason I locked c h i e f l y at the dispersal of f r y from Fulton River, hoping tc discover as much as possible about dispersal 'laws* that the f r y might, i n general, follow. Fulton River fry showed the same trend i n d i s t r i b u t i o n a l changes throughout the summer and f a l l as noted for the underyearling population, in general, as I 18 described by McDonald (1969). The number of Fulton Biver fry caught and proportions caught i n each of the 8 lake areas (numbered 1,2,3,4,5,8,9,10) (Figure 4) in 1967, 1968, 1971, and 1972 for 3 sampling periods each year' are shown in Tables 2, 3, and 4.s Mean proportions together with 95% confidence l i m i t s were also calculated for the f i s h densities in each of the 8 lake areas for each period in a l l years (Table 5). Proportions were used since the f r y production in each year d i f f e r e d as well as the number of f i s h marked. although the 3 periods did not correspond exactly each year they were considered close enough for the purpose cf this study. Data obtained in 1966 were not used, since i t was the f i r s t year of f i e l d work, and catch procedures were s t i l l being refined. In addition, sampling i n 1966 was confined to areas 1 5 and did not include area 8 (Morrison Arm), area 9 (North Arm), or area 10 (Hagan Arm). In equating the proportion of marked f i s h recaptured with the actual proportion of Fulton Biver f i s h present in a particular area, I am assuming that a v a i l a b i l i t y and gear e f f i c i e n c y remained constant during the sampling periods, and that catch/unit e f f o r t was proportional to Fulton River fry population density. Catches/unit e f f o r t i n areas 8 and 10 were s The number of s i g n i f i c a n t figures shown i n the proportions may seem unwarranted. However, such accuracy i s reguired to reduce round-off error i n the simulation when large numbers cf f i s h are being looked at. 9 - NORTH ARM/ 8-MORRISON ARM IO-HAGAN ARM FULTON R, BABINE LAKE (BY AREA) MILES IO Figure k. Bablne Lake showing the Fulton River and f i s h i n g areas. T a b l e 2 . The numbers and p r o p o r t i o n s of marked F u l t o n B i v e r f r y caught i n each of the 8 l a k e a r e a s i n p e r i o d 1 i n 1 9 6 7 , 1968 , 1 9 7 1 , and 1972 . l a k e 1967 a r e a J u l y 10 - J u l y 23 number of f i s h p r o p o r t i o n o f c a t c h 1 0 0 . 0 0 0 0 0 0 0 2 2 0 . 0 2 8 1 6 9 0 3 13 0 . 1830986 4 41 0 . 5 7 7 4 6 4 8 5 9 0 . 1267606 8 (1) 2 0 . 0 2 8 1 6 9 0 9 2 0 . 0 2 8 1 6 9 0 10 (1) 2 0 . 0 2 8 1 6 9 0 t o t a l 71 1968 19 J u l y 5 - J u l y 17 J u l y 3 number p r o p o r t i o n number o f f i s h o f c a t c h o f f i s h 1 0 .0076336 0 3 0 .0229008 2 40 0 . 3 0 5 3 4 3 5 6 60 0 .4580153 8 22 0 .1679389 13 (0) 0 0 .0000000 ( - )0 e s t . 3 0 .0229008 (0) 0 (1) 2 0 . 0 1 5 2 6 7 2 (0) 0 131 29 1972 J u l y 12 J u l y 7 - J u l y 20 p r o p o r t i o n of c a t c h number of f i s h p r o p o r t i o n of c a t c h o . o o o o c o o 1 0 . 0 0 5 0 7 6 1 0 . 0 6 8 9 6 5 5 21 0 . 1065990 0 .2068966 30 0 . 1522643 0 .2758621 42 0 . 2 1 3 1 9 8 0 0 . 4 4 8 2 7 5 9 102 0 . 5 1 7 7 6 6 5 0 . 0 0 0 0 0 0 0 (0) 0 0 . 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 1 0 . 0 0 5 0 7 6 1 0 . 0 0 0 0 0 0 0 (0) 0 0 . 0 0 0 0 0 0 0 197 Table 3. The numbers and proportions of narked Fulton River f r y caught i n each of the 8 lake areas i n period 2 i n 1967, 1968, 1971, and 1972. lake area 1 2 3 4 5 8 9 10 t o t a l 1967 Aug. 31 - Sept. 12 number of f i s h 129 38 10 35 3 (1) 2 3 (2) 4 224 proportion of catch 0.5758929 0. 1696429 0.0446429 6. 1562500 0.0133929 0.0089286 0.0133929 0.0178571 1968 Aug. 28 - Sept. 4 number proportion of f i s h of catch 14 0.1666667 20 0.2380952 6 0.0714286 33 0.3928571 8 0.0952381 (1) 2 0.0238095 1 0.0119048 (0) 0 0.0000000 84 1971 Aug. 16 - Aug. 27 number proportion of f i s h of catch 8 0.2424242 10 0.3030303 3 0.0909091 • 8 0.2424242 2 0.0606061 (0) 0 0.0000000 0 0.0000000 (1) 2 0.0606061 33 1972 Aug. 19 - Aug. 27 number proportion of f i s h of catch 34 0.2575758 26 0.1969697 10 0.0757576 25 0.1893939 20 0.1515152 (1) 2 0.0151515 9 0.0681818 (3) 6 0.0454545 132 Tabla 4. The numbers and proportions of marked Fulton Siver f r v caught i n each of the 8 lake areas i n period 3 i n 1967, 1968, 1971, and 1972. lake 1967 1968 1971 1972 area Oct. 4 - Oct. 24 Oct. 9 - Oct. 18 Oct. 7 - Oct. 15 Oct. 6 - Oct. 16 number of f i s h proportion of catch number of f i s h proportion of catch number of f i s h proportion of catch nu mber of f i s h prcpcrtion of catch 1 4 0.0833333 2 0. 1538462 1 0.0454545 29 0.5370370 2 8 0.1666667 2 0.1538462 3 0. 1363636 ,4 0.0740741 3 12 0.2500000 6 0.4615385 5 0.2272727 2 0.0370370 4 5 0.1041667 2 0. 1538462 4 0.1818 182 3 0. 0555556 5 8 0. 1666667 1 0.0769231 6 0.2727273 10 0. 1851852 8 (2) 4 0.0833333 (0) 0 0.0000000 (0) 0 0.0000000 (1) 2 0.037C370 9 3 0.0625000 0 0.0000000 1 0.0454545 0 O.OOOCOOO 10 (2) 4 0.0833333 (0) 0 0.0000000 (1) 2 0.0909091 (2) 4 0.0740741 t o t a l 1 8 13 22 54 23 Table 5. Hean p r o p o r t i o n s with 95% conf i d e n c e l i m i t s of F u l t o n River f r y i n each of the 8 lake areas f o r each of the 3 annual sampling p e r i o d s i n 1967, 1968, 1971, and 1972. Pe r i o d 1 ( e a r l y J u l y ) l a k e area mean p r o p o r t i o n of f r y with 95 55 conf i d e n c e l i m i t s 1 0.0031774 + 0.0060691 2 0.0566586+0.0622780 3 0.2119058 + 0.1052959 4 0.3811350 + 0.2658147 5 0.3151855 + 0.3127675 8 0.0070422 + 0.0224084 9 0.0140365 + 0.0216519 10 0.0108590+0.0216381 £§£i2^ 2 ( l a t e August - e a r l y September) l a k e area mean p r o p o r t i o n of f r y with 95% confidence l i m i t s 1 0.3106399 + 0.2883715 2 0.2269345 • 0.0922933 3 0.0706846 + 0.0306516 4 0.2452313 + 0.1664522 5 0.0801881 + 0.0925843 8 0.0119724 .+ 0.0159848 9 0.0233699+0.0484776 10 0.0309794 + 0.0432720 P e r i o d 3 ( e a r l y October) l a k e area mean p r o p o r t i o n of f r y with 95% conf i d e n c e l i m i t s 1 0.2049178 + 0.3594425 2 0.1327377 + 0.0652847 3 0.2439620 + 0.2 762 916 4 0.1238467 + 0.0886385 5 0.1753756 + 0.1277508 8 0.0300926 + 0.0629329 9 0.0269886 + 0.0508026 10 0.0620791 +0.0667500 tc<= .OS C3 Jer«es of ^dornl "3--1'82 { 2 " t d i l e d ) 24 s t a n d a r d i z e d with the other areas. (Only 5 samples were taken i n areas 8 and 10 compared with 10 samples i n a l l other areas.) In one i n s t a n c e (area 8, p e r i o d 1, 1971) the c a t c h was estimated from other years' data owing to missed s a m p l i n g . 6 For completeness, numbers and p r o p o r t i o n s of f r y i n each lake area f o r each p e r i o d i n a l l years are a l s o given f c r Pinkut Creek f r y and t o t a l lake f r y (Tables 6,7,8,9,10,11,12, and 13). The plan f o r comparing the observed d i s t r i b u t i o n c f f r y with the d i s t r i b u t i o n p r e d i c t e d by the model was based on the chi-sguare t e s t . Using p ; , p z , p^ , p^, p^, p 6 , p 7 , and p 5 as the mean p r o p o r t i o n s of f i s h i n areas 1, 2, 3, 4, 5, 10, 8, and 9, r e s p e c t i v e l y , f o r a given p e r i o d , N as the t o t a l number of marked recaptured f r y , and the maximum confidence l i m i t : p x + t ^ s - , I c a l c u l a t e d a c h i - s g u a r e (X ) as f c l l c w s : For a c e r t a i n l e v e l of c o n f i d e n c e , t h i s c h i -sguare value would be the g r e a t e s t a l l o w a b l e to s t a t e that the d i f f e r e n c e between the n a t u r a l d i s t r i b u t i o n and computer model d i s t r i b u t i o n c f f r y was not s i g n i f i c a n t . The value of N could be set a r b i t r a r i l y . However, although any number of f i s h c o u l d be d i s p e r s e d using the computer model 6 The nature of the data d i d not warrant that ether harsher measures be taken. x = 8 x -1 Table 6. The numbers and proportions of marked Pinkut Creek fry caught i n each of the 3 lake areas i n period 1 i n 1971 and 1972. lake 1967 1968 1971 1972 area July 3 - July 14 July 7 - Ju l y 19 •>•- number • proportion number proportion number proportion number proportion of f i s h of catch of f i s h of catch of f i s h of catch of f i s h of catch 1 —. • 0 0. 0000000 0 0.0000000 2 0 0.0000000 1; 0.0163934 3 -: 2 0. 111 1 1 11 2 0.0327869 4 • 0 O.000OCO0 13 0.2131147 5 16 0. 8888888 45 0.7377049 8 (0)0 est. 0.0000000 (0) 0 0.0000000 i 9 — 0 O.OOOOCOO 0 0.0000000 •10 (0) 0 0.0000000 (0) 0 0.0000000 t o t a l 18 61 Table 7. The numbers ana proportions of marked Pinkut Creek f r y caught i n each of the 8 lake areas i n period 2 i n 1971 and 1972. lake area 1 2 3 4 5 8 9 10 t o t a l number of f i s h 1967 proportion of catch number of f i s h 1968 proportion of catch 1971 1972 Aug. 16 - Aug. 27 Aug. 19 - Aug. 27 number proportion number proportion of f i s h of catch of f i s h of catch 2 0.0740741 7 0. 1320755 7 0.2592593 12. 0.2264151 8 0.2962963 5 0.0943396 9 0.3333333 16 0.3018863 1 0.0370370 11 0.2075472 (0) 0 0.0000000 (0) 0 0.0000000 0 . 0.0000000 2 0.0377358 (0) 0 0.0000000 (0) 0 0.0000000 27 53 Table 8. The numbers and proportions of marked Pinkut Creek f r y caught i n each of the 8 lake areas i n period 3 i n 1971 and 1972. lake area 1 2 3 4 5 8 9 10 t o t a l 1967 number of f i s h proportion of catch 1968 number proportion of f i s h of catch 1971 Oct. 7 - Oct. 15 1972 Oct. 6 - Oct. 14 number of f i s h 0 3 6 9 1 (0) 0 0 (0) 0 19 proportion of catch 0.0000000 0. 1578947 0.3157895 0.4736842 0.0526316 0.0000000 0.0000000 0.0000000 number of f i s h 12 1: 1 3 9 d) 2 4 (3) 6 44 proportion of catch 0.2727273 0. 1590909 0.0227273 0.0681818 0.2045454 0.0454545 0.0909091 0. 1363636 28 Table 9. Mean p r o p o r t i o n s of Pinkut Creek f r y i n each of the 8 lake areas f o r each of the 3 annual sampling p e r i o d s i n 1971 and 1972. P e r i o d 1 ( e a r l y J u l y ) l a k e area mean p r o p o r t i o n of f r y 1 0.0000000 2 0.0081967 3 0.0719490 4 0.1065574 5 0.8132968 8 0.0000000 9 0.0000000 10 0.0000000 P e r i o d 2 ( l a t e August) l a k e area mean p r o p o r t i o n of f r y 1 0.1030748 2 0.2428372 3 0.1953180 4 0.3176100 5 0.1222921 . 8 0.0000000 9 0.0188679 10 0.0000000 P e r i o d 3 ( e a r l y October) l a k e area mean p r o p o r t i o n of f r y 1 0.1363636 2 0.1584928 3 0.1692584 4 0.2709330 5 0.1285885 8 0.0227272 9 0.0454545 10 0.0681818 Table 10. The numbers and proportions of t o t a l lake salmon f r y caught i n each of the 8 lake areas i n period 1 i n 1967, 1968, 1971, and 1972. lake 1967 area July 10 - July 23 number of f i s h proportion of catch 1 154 0.0100568 2 374 0.0244237 3 950 0.0620388 ii 4235 0.2765624 5 2164 0. 1413178 8 (231) 462 ' 0.0301704 9 6826 0.4457650 10 (74) 148 0.0096650 t o t a l 15313 1968 July 5 - July 17 • July 3 number of f i s h proportion of catch number of f i s h 218 0.0060532 245 469 0.0130227 1443 2363 0.0656134 2138 4643 0.1289221 2826 3740 0.1038485 4910 (413) 826 0.0229355 (-) 449 est. 23621 0.6558838 (2589)8630 (67) 134 0.0037208 (27) 67 36014 20708 1972 July 14 July 7 - Ju l y 20 propcrticn of catch number of f i s h proportion of catch 0. 0118312 854 0. 0181297 0. 0696832 1862 0. 0395287 0. 1032451 4209 0. 0893536 0. 1364690 5635 0. 1 196264 0. 2371064 26896 0. 5709797 0. 0216824 (281) 562 0. 0119308 0. 4167472 4967 0. 1054453 0. 0032355 (1060)2120 47105 0. 0450058 Table 11. The numbers and proportions of t o t a l lake salmon f r y caught i n each of the 8 lake areas i n period 2 i n 1967, 1968, 1971, and 1972. lake area 1967 Aug. 31 - Sept. 12 1968 Aug. 28 - Sept. 4 1971 Aug. 16 - Aug. 27 1972 Aug. 19 - Aug. 27 number of f i s h proportion of catch number of f i s h proportion of catch number of f i s h proport ion of catch number of f i s h proportion of catch 1 10588 0.3656330 1801 6.0287145 2757 0.0744974 6553 0. 1985818 2 2527 0.0872643 2085 0.0332425 4410 0. 1191634 3618 0.1096397 3 1060 0.0366047 839 0.0133767 2494 0.0673908 1530 0.0463650 4 4392 0.1516679 3926 0.0625947 12974 0.3505728 6393 0. 1938847 5 624 0.0215484 1373 0.0218906 870 0.0235084 2835 0.08591 17 . 8 (214) 428 0.0147800 (133) 266 0.0042410 (59) 108 0.0029183 (276) 552 0.0167278 9 8855 0.3057877 52301 0.8338674 12883 0. 3481 139 10969 0.3324040 10 (242) 484 0.0167139 (65) 130 0.0020727 (256) 512 0.0138348 (272) 544 0.0164853 t o t a l 28958 62721 37008 32999 Table 12. The numbers and proportions of t o t a l lake salmon f r y caught i n each of the 8 lake areas i n period 3 i n 1967, 1968, 1971, and 1972. lake 1967 1968 area Oct. 4 - Oct. 24 Oct. 9 - Oct. 18 number proportion number proportion of f i s h of catch of f i s h of catch 1 374 0.0618182 326 0.0545607 2 698 0. 1153719 133 0.0306276 3 876 0. 1147934 421 0.0704603 4 56 8 0.0938843 465 0.0778243 5 706 0.1166942 286 0.0478661 8 (119) 238 0.0393388 (100) 200 0.0334728 'J9 2266 0.3745455 4004 0.6701255 10 (162) 324 0.0535537 (45) 90 0.0150628 t o t a l 6050 5975 1971 1972 Oct. 7 - Oct. 15 Oct. 6 - Oct. 16 number of f i s h proportion of catch number of f i s h proportion of catch 778 0.0602074 5234 0. 1983177 1319 0. 1020740 1483 0.0561913 2764 0.2138988 562 0 .0212943 2632 0.2075530 1530 0.0579721 1616 0. 1250580 2505 0.0949151 (30) 160 0.0123820 (723)1446 0.0547893 2887 0.2234174 12452 0.4718096 (358) 716 0.0554094 (590)1180 0.0447105 12922 26392 32 Table 13. Mean proportions of t o t a l lake salmon f r y i n each of the 8 lake areas for each of the 3 annual sampling periods i n 1967, 1968, 1971, and 1972. Period 1 (early July) lake area mean proportion of f r y 1 0.0115177 2 0.0366646 3 0.0800627 4 0.1653950 5 0.2633131 8 0.0216798 9 0.4059603 10 0.0154 068 Period 2 (late August - early September) lake area mean proportion of f r y 1 0.1668567 2 0.0873275 3 0.0409343 4 0.1896800 5 0.0382148 8 0.0096668 9 0.4550432 10 0.0122767 Period 3 (early October) lake area mean proportion of fry 1 0.1668567 2 0.0760662 3 0.1126117 4 0,1093084 5 0.0961333 8 0.0349957 9 0.4349745 10 0.0421841 33 to detarmins ths proportion of f i s h dispersing to any area in a given time, the absolute number of f i s h had to be reduced to N for comparing chi-square values. was set to 1000 for a l l comparisons throughout the study. The value of X 7 for periods 1, 2, and 3 are: Period 1 V 2 - (6.069)Z . (62. 27S)2- . (105. 296) 2 (265.815)* A ~ 3. 177 56.659 ^ 211 .906 ^ 381.135 (312.768)2 ^ (22.408)2 . (21 .652) Z (21. 638) 2 + 315.185 + 7.042 14.036 10.859 = 775.947 ^ 776 Period 2 ^ - (288. 372) 2 (92. 293) 2 (30. 652) 2 (166. 452) 2 310.640 226.934 70.685 245.231 (92.584)2 (15.985)2 (48.478)2 (43. 272) 2 80. 138 1 1.972 23. 370 30.979 = 720.764 ^ 721 Period 3 (359. 442) 2 (65. 285) 2 (276. 292) 2 (88. 639) 2 X - 204.918 132.738 243.962 123.847 (127.751)2 (62.933)2 (50.803 ) 2 (66.750)2 175.376 30.093 26.989 62.079 =1431.016 ^ 1431 7 ^ . O S f a J = 3 ' 1 8 2 (2-tailed) 34 Swimming Speed of Fry. For the purpose of creating a basis for analysis i t was necessary to f i r s t run the simulation model tc obtain a d i s t r i b u t i o n of f r y where i t was assumed that the f i s h moved en t i r e l y at random, being influenced by no behavioral, genetic, or environmental factors. To do t h i s , I had to obtain some estimate of the speeds at which fry might move under these circumstances. The only relevant estimate obtained for sockeye fry was from data giving the speed of 2-to-3-week old fry roaming in s t i l l water at d i f f e r e n t temperatures (varying from 10.0 degrees C. - 17.0 degrees C.) obtained from a study done by Hear (1954). Hoar noted that sockeye f r y i n 225 cm. long troughs swam back and forth i n a predictable manner f c r long periods. However, he stated that the r e l a t i o n between the rates he tabulated to the rates of swimming of f r y i n nature, assuming no influencing factors, i s not known. Rates over a l l temperatures varied between 1.30 cm./sec. to 2.55 cm./sec. and had a mean cf 2.175 cm./sec. 8 This compares with actual rates of fry in the lake between 0.67 cm./sec. and 2.67 cm./sec. based cn a dispersal rate of 0.4 miles/day (24 hours) and 1.6 miles/day (24 hours), respectively, with an average of 1.55 cm./sec. (0.93 miles/day) 9 that were measured at Babine Lake in 1966 and 8 Values were based on average swimming distance covered in a 15-minute i n t e r v a l . ' See Appendix 4 for d e t a i l s . 9 The average was taken over a l l rates of dispersal given by McDonald (1969, p. 253, top table). 35 published by McDonald (1969). The s i m i l a r i t y between the 2 sets of data i s encouraging. I had some doubt at f i r s t about using swimming speeds of fry from Hoar's data disregarding temperature. If fry t r a v e l at di f f e r e n t speeds at d i f f e r e n t temperatures, then the temperature of the lake for a pa r t i c u l a r period should determine what speed to use i n that period. In a number of laboratory experiments J.R. Brett (1967) found that the r e l a t i o n between the optimum sustained swimming speed of sockeye f i n g e r l i n g s and temperature was dome-shaped with the peak performance occurring at 15 degrees C. . However, there i s no evidence that increased temperature, which raises metabolic rate, causes f i s h to travel faster under natural conditions. In addition, d i e l v e r t i c a l d i s t r i b u t i o n of f r y and temperature studies (McDonald 1973) indicate f r y pass through temperature gradients where the difference between maximum and minimum temperatures in summer can be about 10 degrees C. . I f i t i s assumed that the lake has an 'average* isothermal temperature of 10 degrees C. during the summer the corresponding speeds of the f i s h in the f i r s t dispersal period f a l l into the range of Hoar's data. Thus, the o r i g i n a l calculated swimming speeds which gave egual weight to a l l speeds at a l l temperatures were l e f t as they were for lack of any better measurements. Although a mortality factor was b u i l t into the simulation, mortality was set to 0 for a l l areas. (The mortality factor was set to 1. so that: o r i g i n a l population x mortality factor (=1.) = o r i g i n a l population.) This was based on the assumption that 36 m o r t a l i t y i n a l l areas was the same so that p r o p o r t i o n s of f i s h i n a l l areas would remain the same whether m o r t a l i t y occurred or not. 37 RESULTS Random Dispersal - Period _1 The duration of the f i r s t part of the dispersal was approximated to be about 49 days (Hay 25 - the date at which 50% of the f r y had been marked from Fulton River averaged over 1971 and 1972, 1 ° to July 12 - the mid-date of the f i r s t sampling period averaged over the years 1967, 1968, 1971, and 1972) Leeway i n the estimation of the actual swimming speed of the fry was given by treating the minimum and maximum speeds travelled by the f r y , as taken from Hoar's data, as 'averages* as well as the actual average speed determined from the data. V a r i a b i l i t y about these speeds was generated i n d i r e c t l y by the bivariate normal density function used for random movement and tcck form in terms of the variable distances fry travelled in a given time. The f r y dispersing randomly were found tc have an average speed of 0.262 m i l e s / i t e r a t i o n . 1 1 From this information, the number of i t e r a t i o n s of the program loop required to give the fry the 'right amount of dispers a l ' was calculated. Pertinent data are given in Table 14 . 10 'Marking of f r y ' data for 1967 and 1968 was unavailable at the time. However, when an average date was calculated l a t e r over 1967, 1968, 1971 , and 1972 i t was found to be May 29,^  close enough to May 25 for the difference in time not to be s i g n i f i c a n t (Coburn and McDonald, 1972, 1973). 1 1 See Appendix 5 for d e t a i l s . 38 Table 14. Random movement o f f r y - no b i a s , p e r i o d 1, speed d i s t a n c e number of i t e r a t i o n s t r a v e l l e d of loop r e q u i r e d at i n 49 days an average of 0.262 m i l e s / i t e r a t i o n 1.30 cm./sec. 34.2 mi. 131 (0.698 mi./24 hr.) 2.175 cm./sec. 57.2 mi. 218 (1.17 mi./24 hr.) 2.55 cm./sec. (1.37 mi./24 hr.) 67.1 mi. 256 39 The computer model was run through 260 it e r a t i o n s using one mill i o n f r y located i n i t i a l l y at the mouth cf Fulton River. On every tenth i t e r a t i o n , as s e l l as on i t e r a t i o n s 131, 218, and 256, the expected values p»N of number of fry/area were compared with the observed computer model values by means of the chi-sguare test . Cn i t e r a t i o n s 131, 218, and 256, the lake was also 'printed' to see how the f i s h were d i s p e r s i n g . 1 2 It was found from t h i s run that there was a highly s i g n i f i c a n t difference between expected d i s t r i b u t i o n and computer model d i s t r i b u t i o n no matter what the speed of the fry. The lowest pertinent chi-sguare value, 15,436, which was far above the acceptable chi-sguare l i m i t of 776, occurred at i t e r a t i o n 256, corresponding to the maximum allowed speed cf the f i s h (Table 15 - Summary of r e s u l t s for simulation of period 1 -random movement). Most f i s h never l e f t area 2, the Fulton River area, from which they dispersed. A small proportion, however, did manage to disperse to areas 1, 3, and 10 (Table 15 and Figure 5). Although the random dispersal printouts did not point out any major ef f e c t s of land boundaries, they did show how f i s h movement i s reduced by peninsulas and islands, as wculd be expected i n the r e a l world. Direct ion ally-Biased Disjpersal - Period 1 The next step was to introduce a d i r e c t i o n a l north - south (N - S) bias, as naturally observed, into the model i n an 1 2 See Appendix 8. Table 15. Summary of r e s u l t s f o r period 1 - random movement. observed numbers of f r y in each area frcm simulation -area 1 area 2 area 3 area 4 area 5 area 10 area 8 area 9 X slow speed (1.30 cm./sec.) - i t e r a t i o n 131 N=1,000,000 N=1,000 384 0.3 84 993611 993.677 35 0.035 0 0.000 0 0.000 5902 5.903 0 0.000 0 0.000 16432 average speed (2.175 cm./sec.) - i t e r a t i o n 218 H=1 ,000,000 N=1,000 3330 3.330 973240 973.344 855 0.855 0 0.000 0 0.000 22469 22.471 0 0.000 0 0.000 15796 f a s t speed (2.55 era./sec.) -> i t e r a t i o n 256 N=1,000,000 5473 962097 1777 0 0 30529 0 0 11=1,000 5.474 962.216 1.777 0.000 0.000 30.533 0.000 0.000 15436 optimum speed f o r best chi-square value - same as for f a s t speed - i t e r a t i o n 256 N=1,000,000 N = 1,000 5473 5.474 962097 962.216 1777 1.777 0 0.000 .0 0.000 30529 30.533 0 0.000 0 0. 000 15436 expected numbers of f r y in each area frcm lake data area 1 area 2 area 3 area 4 area 5 area 10 area 8 N=1,000 3. 180 56.660 211.900 381.130 315.190 10.860 7.040 area 9 14.040 776 N.B. There i s round-off e r r o r i n the 'N = 1,000,000' values. O 0.000 (obs.) 14.04-0 (exp.) 5 . W (obs.) 3.180 (exp.) 8 0.000 (obs.) 7.04-0 (exp.) 10 30.533 (obs.) 10.860 ( e x p j F U L T O N R 962.216 (obs.) 56.660 (exp.) BABINE LAKE (BY AREA) WLt8 1.777 (obs.) 211.900 (exp.) 0.000 (obs.) 381.130 (exp.) 0.000 (obs.) 315.190 (exp.) Figure 5. Computer simulation (observed) numbers of f r y (N=1000) at the best chi-square value (154-36) fo r period 1 - random movement, and natu r a l l y observed (expected) numbers of f r y (N=1000) f o r period 1 i n each of the- 8 lake areas. Data are taken from Table 15. 42 a t t e m p t t o a c h i e v e a c o m p u t e r m o d e l d i s t r i b u t i o n c f f r y s i m i l a r t o t h e n a t u r a l d i s t r i b u t i o n . B e f o r e a t t e m p t i n g t h i s , I made a c h e c k t o s e e i f t h e r e was a n y p o s i t i o n a l b i a s o f f i s h i n t h e l a k e c o n c e r n i n g t h e i r d i s t a n c e f r o m s h o r e . L i n e a r r e g r e s s i o n s o f n u m b e r s o f r e c a p t u r e d t a g g e d f r y a g a i n s t d i s t a n c e f r o m s h o r e f o r p e r i o d s 1 , 2 , and 3 i n 1967 a n d 1 9 6 8 ( F i g u r e s 6 , 7 , a n d 8 , a n d T a b l e 16) s h o w e d t h a t t h e r e was no r e l a t i o n s h i p b e t w e e n w h e r e f i s h were c a u g h t a n d d i s t a n c e f r o m s h o r e . (The p r o b a b i l i t y o f t h e s l o p e b e i n g z e r o f o r a l l r e g r e s s i o n l i n e s was 1 . 0 0 0 0 ( T a b l e 1 6 ) . ) T h u s , i t was n o t n e c e s s a r y t o g u i d e t h e f i s h , f o r e x a m p l e , down t h e m i d d l e o f t h e l a k e . The f a c t t h a t t h e r e i s no p o s i t i o n a l b i a s o f f r y i n B a b i n e L a k e i s n o t s u r p r i s i n g b e c a u s e t h e l a k e i s i n a l o n g n a r r o w b a s i n . ( F o r l a r g e r , w i d e r l a k e s t h i s p h e n o m e n o n may o c c u r f o r v a r i o u s r e a s o n s a n d m i g h t be s i m u l a t e d i n a m o d e l s y s t e m . ) A b i a s i n movement was c r e a t e d by means c f a 4 - p a r a m e t e r l o g n o r m a l d e n s i t y f u n c t i o n w h i c h was c h o s e n o v e r a number o f o t h e r p o s s i b l e d e n s i t y f u n c t i o n s f o r s e v e r a l r e a s o n s . The 4 - p a r a m e t e r l o g n o r m a l d e n s i t y f u n c t i o n n o t c n l y h a s f i n i t e l i m i t s b u t , b y c h a n g i n g i t s p a r a m e t e r u ( e i t h e r + o r - ) , i t c a n be s k e w e d i n e i t h e r d i r e c t i o n a s much a s o n e d e s i r e s w h i l e k e e p i n g t h e a s s o c i a t e d n o r m a l s t a n d a r d d e v i a t i o n {(t) c o n s t a n t . F o r u=0 i t a c h i e v e s s y m m e t r y a n d t h u s a l l o w s f c r a s m o o t h t r a n s i t i o n b e t w e e n s k e w n e s s i n o n e d i r e c t i o n a n d t h e n t h e o t h e r ^3 PERIOD 1, 13G7 AND 19G8 Y = 0-7375 + -0.1904 "X N = 140 > * 0-4577 «X N = 70 °Tfc-»0<yQCa^gMt»« PCX I K-X3i-gg-4«-X-+» „ „ „ , ^ . . . C O 0 » 1 0 » 2 0 - 3 0~4 0-5 0 - S 0 . 7 0 - 8 0 - 9 1*0 1 - i f.a 11 . 3i-T L S DISTANCE FROM SHORE IN MILES Figure 6. Linear regressions of numbers of recaptured tagged f r y from Fulton River vs. distance from shore when caught f o r period 1, 1967 and 1968. PERIOD 2, ±357 AND ^R=? Y 15- Y 14 13 12. 11< 10. 9«.. 8... 7. . 6... A-S77 * -4-331 + 0.3B57 » X N = N = 70 70 ° M - ^ « j H a « » a t M « B « xai K btae K I K — i x 1- * i i> - r - — u — m 1 i CO C l 0.2 0.3 0.4 0-5 0-6 0-7 0-8 0-9 i-0 1-1 £.2 i-3 i - 4 1. DISTANCE FROM SHORE IN MILES Figure 7. Linear regressions of numbers of recaptured tagged f r y from Fulton River vs, distance from shore when caught f o r period 2, 1967 and 1968. *5 . PERIOD 3» 13E7 AND 1SGB Y • 0.4134 *• 0«5150 «X N = 70 is. y • 0'iiBS * o.isaa «x N = 70 14... 13... IH». . 11'.. 10... x a*., e... 7... e... x s... . . X 3'.. X • b S«.. Xb 1 # + X X X X X X X K \>Oi b X X X X /767 f *J 0*0 0*1 0 « S 0*3 0 - 4 0*5 0 » S 0 » 7 0 - 8 0*3 1*0 1*1 1*5 1*3 1*4 1-3 DISTANCE FROM SHORE IN MILES Figure 8, Linear regressions of numbers of recaptured tagged f r y from Fulton River vs. distance from shore when caught f o r period 3, 1967 and 1968. Table l6 . S t a t i s t i c s of l i n e a r r e g r e s s i o n s of Figures 6, 7» and 8. r«A*0(X-XUARJ_«_ -A»_0 .6 6 <.2£_0 CU_ n a r 4 r»H 1 10(^7 XBAR- 0.A056E 00* AVAR" 0.2063E-01. ptSX-XUU X, 17"/ Y-C+BX. C- 0.737<tE 00. _ JCVARJ I_0 .5 223 E.-0.U-B«-a.ieo<tE_oo_ BVAR- 0.1920E 00 8--0.180<.E 00 _HVAIl.-—0...19 2 0 E_ 0U_ JOURCE- _DF_ REGRESSION 1 REMAINDER 138 .TOXAI 139_ NU. UF RCP." 1*6 THE PKObAlI 1LITY OF THE SLOPE BEING ZtHO IS 1.0000 . SUM_.a_.. 0.*B97E 00 0.3987E 03 _0.39V2E-.03_ HEANSQ 0.<.897E 00 0.2B89E 01 Y-A+BfX-XBARI• A- 0.1857E 01. _ J 4 1 O/CQ _X8AR«-0.«.l<K>C 00.—AVAKa-XU1254E-0O*-penoa 1 , lyoo Y-C*BX. c- O.H67E 01. B- 0.*577E 00 _i!VAK*_U.13 20E_0X-D> 0.4577E 00 CVAR- 0.3522E 00. BVAR- 6.1320E 01 -MO. C F RtP-.J—70 THE—PKUUAailU-Y-OF SOURCE -REGRESSION REMAINDER 68 TOTAL 69 SUM SO -0.1393E 01.. 0.S971E 03 O.iVUliE <J> MEAIiSO .0.13V3E C l . 0.V762E 01 Y-A+BtX-XUARI I A* 0.3157E 01. B—0.4391E 01 SOURCE DF SUMSO MCANSO — - — I ni3 n 1 Q<7 -XBAR- O.AlttE O0...-AVAR«..0.3337E..01»_UVAR".0.3->20E..02 ; REGRESSION. X 0.1298E_03 0.1298E 03. I"5*1"" FC * J-S^l Y-C+BX. C- 0.<t977E 01. B--0.4391E 01 REMAINDER 68 0.1612E 05 0.2371E 03 CVAR- 0.9<t3<>E 0l» BVAR" 0.3520E 02 • • TOTAL 69 0.1625E 05 ' __NO._eF_REP,« 70.-.IHE_PRCl>ABlLlTY- OF-JHE-SLULPE-aElNG ZURO_lS_l.J0000 Y-A*BU-X0AR». A* 0.1185E 01. B- 0.3B66E 00 perlOd Zf 1968 .—XBAR«-0.«.l»2E-00i~AVAR.-_0..<.319E-01.. B.VAR-...Q.<t5<.lE_00 Y-C+BX. C- 0.1025E 01. B- 0.3866E 00 CVAR- 0.1211E 00. BVAR" O . ^ I E 00 UO. IIF REP,. 7Q T H F PHr .F .AH .1 I T Y f lF T H F n p p B F l N f . SOURCE .REGRESSION. REMAINDER TOTAL DF _ 1 _ 68 69 •D000_ SUKS0 .0.9?_><>E 00. OtiOUiE 03 O.206iE 03 MiANSO 0.9V54E 00. 0.3023E 01 period 3, 1967 Y-A+BIX-XBAR)• A" 0.6285E 00. B« O.5150E CO SOURCE —XSAR«-C.*-i7-6E--6U-.—AVARn—0»3 2 A.7E-0X. B-VAR-—0. 3439E-00 DEGRESSION Y-C*0X» C« U.<.1J3L 00. 0" O.iliOE 00 RtMAlNUtR CVAR- 0.9252E-01. BVAR- 0.3«39E 00 TOTAL _ : .NO. OF—KEP..• XQ--THE—P-flOBABILi-T-Y—OF—THE SLOP.E—BEING-ZER SUMSO 0 . 1 7 5 3 E 0 1 . O . I S V J L 0 3 0 . 1 5 6 3 E 0 3 MEANSO 0.1753E 01 U.2273E 01 period 3, 1968 Y-A*B(X-XBARI. XUARa C.*ifc<.L. 00... Y-C*BX» A- 0.1857E 00. B- 0.1582E 00 „AVAK-..0.386VE-02> B.VAR-—0.A072E-QX C« 0.1185E O O t . B- 0.1582E 00 CVAR- 0.1120E-01. BVAR- 0.*072E-01 N U . O F — K U U A SOURCE DEGRESSION-DF L. REMAINDER 68 TOTAL 69 yif TH*- °"""'H'I I T V " F T U F <mPP HPimr. 7Lan li. l.oouo SUMSO .0.16fei>L..00. 0.18ME 02 0.1U50C 02 MEANSO _0.10t.iE. 00.. 0.2706E 00 47 i f such a change i s d e s i r e d . 1 3 In a d d i t i o n , i t r e p r e s e n t s , b e t t e r than other skewed d e n s i t y f u n c t i o n s , how f i s h might a c t u a l l y move i n nature when d i s p e r s i n g with a b i a s tc go i n a p a r t i c u l a r d i r e c t i o n . T h i s i d e a i s f u r t h e r d i s c u s s e d below. The u n i v a r i a t e 4-parameter lognormal d e n s i t y f u n c t i o n was m u l t i p l i e d by a normal d e n s i t y f u n c t i o n to form a b i v a r i a t e d e n s i t y f u n c t i o n over the 5 square x 5 square g r i d . Thus, movement up and down the lake was lognormal, and movement across the lake was considered 'random' normal. The lognormal x normal (logN x S) d e n s i t y f u n c t i o n s were c r e a t e d with biases between 90% (10:90) and 50% (50:50) i n steps of 5%.ltt These were f i r s t d i r e c t e d southward 'down' the l a k e but c c u l d be turned around to give northward 'up' the lake b i a s i n s t e a d . In the lower p a r t of Babine Lake the logN x N d e n s i t y f u n c t i o n s were r o t a t e d by -30 degrees so that the b i a s e s would conform b e t t e r with the l a k e ' s geographic c r i e n t a t i c n . 1 5 1 6 Other r o t a t i o n s c o u l d be done i n other p a r t s o f the lake as we l l to get b e t t e r 'down the l a k e ' or 'up the l a k e ' b i a s and thus o b t a i n more accuracy i n s i m u l a t i n g f i s h movements. Ecwever, the r o t a t i o n t h a t was done seemed to be the c n l y one that might be 1 3 See Appendix 6 f o r d e t a i l s concerning the lognormal d e n s i t y f u n c t i o n . See Appendix 7a f o r d e t a i l s . l s Thus, on any given run, there were 2 d i f f e r e n t b i v a r i a t e f u n c t i o n s used (the n o n - r o t a t i o n and -30 degrees r o t a t i o n d e n s i t y f u n c t i o n s ) which each r e g u i r e d a p r o b a b i l i t y matrix (P and Q). See Appendix 3. 1 6 See Appendix 7b f o r d e t a i l s . 48 important as far as a f f e c t i n g the r e l a t i v e l y crude results that were expected. In future models t h i s problem should be looked at i n more d e t a i l . In the runs giving the f i s h a d i r e c t i o n a l bias, the same swimming speeds of the f r y were used as for the e n t i r e l y random run. Thus, for lack of more information, i t i s assumed that whatever i s a f f e c t i n g the movement of the f i s h i s influencing only their d i r e c t i o n and not their speed within the given l i m i t s . T r i a l biases of 30:70, 20:80, and 10:90 were included in some preliminary computer runs. Precalculated information for each run i s given i n Table 17. For a bias of 30:70, the lowest value of chi-sguare was 678, achieved at i t e r a t i o n 196 corresponding to the maximum allowed speed of the f i s h (Table 18). This chi-sguare value was within the acceptable l i m i t of 776 . However, the model predicted that a ne g l i g i b l e number cf tagged f r y would be recaptured i n area 5, contrary to what had been observed naturally. It therefore seemed that the 30:70 bias might not be moving the f i s h down the lake fas t enough. For a bias of 20:80 more f i s h eventually entered area 5 but the best chi-sguare value was worse, having a minimum value cf 715, after 130 i t e r a t i o n s (Table 19). After 181 i t e r a t i o n s the number remaining in area 4 s t i l l seemed overly high. The 10:90 bias produced the best r e s u l t s , with a minimum chi-sguare value of 436 afte r 150 i t e r a t i o n s , which corresponded 49 Table 17. Biased movement of Fulton River f r y 'down the lake' period 1. speed distance number of i t e r a t i o n s t r a v e l l e d of loop required for in 49 days an average speed of: (miles/iteration,m/i) 0.343 m/i 0.371 m/i 0.414 m/i bias 30:70 bias 20:80 bias 10:90 1.30 cm./sec. 34.2 mi. 100 93 83 (0. 698 mi./24 hr.) 2.175 cm./sec. 57.2 mi. 167 155 139 (1.168 mi./24 hr.) 2.55 cm./sec. 67.1 mi. 196 181 163 (1.369 mi./24 hr.) Table 18. Summary of r e s u l t s f or period 1 - 30:70 biased movement. area 1 observed numbers of fey i n each area from simulation area 2 area 3 area 4 area 5 area 10 area 8 slow speed (1.30 cm./sec.) - i t e r a t i o n 100 N=1,000,000 N=1,000 0 0.000 198836 198.849 801031 801.083 67 0.067 0 0.000 0 0.000 0 0.000 average speed (2.175 cm./sec.) - i t e r a t i o n 167 N=1,000,000 N=1,000 0 0.000 11318 11.320 735306 735.387 253265 253.293 0 0.000 0 0.000 0 0.000 f a s t speed (2.55 cm./sec.) - i t e r a t i o n 196 N=1,000,000 N=1,000 0 0.000 5832 5.833 394623 394.674 599346 599.424 69 0.069 0 0.000 0 0.000 optimum speed f o r best chi-square value - same as for f a s t speed - i t e r a t i o n 196 H=1,000,000 S=1,000 0 0. 000 5832 5.833 39 46 2 3 394.674 599346 599.424 69 0. 069 0 0.000 0 0.000 S=1,000 area 1 3. 180 expected numbers-of f r y in each area from lake data area 2 area 3 area 4 area 5 area 10 area 8 56.660 211.900 381.130 315.190 10.860 7.040 H.B. There i s round-off e r r o r i n the »N = 1,000,000' values. Table 19. Summary of r e s u l t s f o r period 1 - 20:80 biased movement. area 1 observed numbers of f r y in each area from simulation area 2 area 3 area 1 area 5 area 10 area 8 area 9 slow speed (1.30 cm./sec.) - i t e r a t i o n 93 H=1,000,000 N=1 ,,000 0 0.000 10023 10.024 955555 955.603 34371 34.373 0 0.000 0 0.000 0 0.000 0 0.000 3314 optimum speed f o r best chi-square value - i t e r a t i o n 130 N=1 ,000,000 N=1,000 0 0.000 1430 1.430 289514 289.534 708981 709.032 4 0.004 0 0.000 0 0.000 0 0.000 715 average speed (2.175 cm./secw) - i t e r a t i o n 155 N=1,000,000 0 747 47666 948470 3032 0 0 0 N=1,000 0.000 0.747 47.670 948.550 3.032 0.000 0.000 0.000 1371 f a s t speed (2.55 cm./sec.) - i t e r a t i o n 181 N=1,000,000 N=1,000 0 0.000 492 0.492 5906 5.907 903740 903.829 89764 89.773 0 0.000 0 0.000 0 0.000 1169 N=1,000 area 1 3.180 expected numbers of f r y in each area from lake data area 2 area 3 area 4 area 5 area 10 area 8 56.660 211.900 381.130 315.190 10.860 7.040 area 9 14.040 x 776 N.B. There i s round-off e r r o r i n the »N = 1,000,000* values. 52 to a speed about half-way between the average allowable speed (2.175 cm./sec.) and the maximum allowable speed (2.55 cm./sec.) as given by Hoar's (1954) data. Numbers cf fry in area 4 and area 5 were more l i k e those naturally occurring although i t was now readily apparent that there were too few f i s h remaining in areas 1, 2, and 3 (Table 20 and Figure 9). By this time i t was realized that further manipulations with f i n e r d i v i s i o n s of biases were not going to achieve a better simulation and the o r i g i n a l ideas concerning the use of biases in smaller steps were dropped. The s c a r c i t y of f r y i n areas 1,2, and 3 could be accounted for and corrected to a large extent i f i t was assumed that a small percentage of the f i s h had no d i r e c t i o n a l tendency but, instead, moved randomly. With th i s idea, 10% of the f i s h (100,000) were allowed to move randomly from where they were o r i g i n a l l y positioned in the lake at the mouth of Fulton River. The other 90% of the f i s h (900,000) followed the 10:90 bias pattern down the lake. Observed areal numbers cf f r y for the randomly moving f i s h were taken from the program of random dispersal of f r y at i t e r a t i o n 218 corresponding to an average swimming speed of 2.175 cm./sec. . Fcr the ether f i s h t r a v e l l i n g with the 10:90 bias the data with the lowest chi-sguare value (436 at i t e r a t i o n 150) were used. Combined information of the two d i s t r i b u t i o n patterns i s given i n Table 21 and Figure 10. Table 20. Summary of r e s u l t s f o r period 1 - 10:90 biased movement. observed numbers of f r y in each area from simulation area 1 area 2 area 3 area 4 area 5 area 10 area 8 area 9 X2" slow speed (1.30 cm./sec.) - i t a r a t i o n 83 N=1,000,000 N=1 ,000 0 ,000 992 0.992 534939 535.013 463974 463.995 0 0.000 0 0.000 0 0.000 0 0.000 916 average speed (2.175 cm./sec.) - i t e r a t i o n 139 B=1,000,000 B=1,000 0 0.000 78 0.078 1128 1. 128 791857 791.917 206861 206.877 0 0.000 0 0.000 0 0.000 781 optimum speed f o r best chi-square value - i t e r a t i o n 150 N=1,000,000 N=1,000 0 0.000 62 0.062 390 0.390 558463 558.509 441003 441.039 0 0.000 0 0.000 0 0.000 436 f a s t speed (2.55 cm./sec.) - i t e r a t i o n 163 N=1,000,000 N=1,000 0 0.000 49 0. 049 151 0.151 289049 289.075 710662 710.724 0 0.000 0 0.000 0 0. 000 822 N=1,000 area 1 3. 180 expected numbers of f r y in each area frcm lake data area 2 area 3 area 4 area 5 area 10 area 8 56.660 211.900 381.130 315.190 10.860 7.040 area 9 14.040 I'-l l b H.B. There i s round-off e r r o r i n the "N = 1,000,000' values. 5k 0.000 (obs.) lk.OkO (exp.) 8 0.000 (obs.) 7.04-0 (exp. ) 0.000 (obs.) 3 .180 (exp.) 10 0.000 (obs.) 10.860 (exp.) FULTON R 0.062 (obs.) 56.660 (exp.) BABINE LAKE (BY AREA) 0.390 (obs.) 211.900 (exp.) 558.509 (obs.) 381.130 (exp.) 4/4-1.039 (obs.) 315.190 (exp.) mm 10 Figure 9. Computer s imulat ion (observed) numbers of f r y (N=1000) at the best chi-square value (4^6) f o r period 1 - 10s90 biased movement, and n a t u r a l l y observed (expected) numbers of f r y (N=1000) f o r period 1 i n each of the 8 lake areas . Data are taken from Table 20. T a b l e 2 1 . Data f o r combined s i m u l a t e d d i s p e r s a l d i s t r i b u t i o n s o f f r y i n p e r i o d 1 t o g e t h e r wi th e x p e c t e d d i s t r i b u t i o n o ( f r y ana c a l c u l a t e d c h i - s g u a r e v a l u e . . 10 x o b s e r v e d d i s t r i b u t i o n o f f r y moving w i th no b i a s a t i t e r a t i o n 218 .90 x o b s e r v e d d i s t r i b u t i o n o f f r y moving w i th a 10:90 b i a s a t i t e r a t i o n 150 a r e a 1 a r e a 2 a r e a 3 a r e a 4 a r e a 5 a r e a 10 333 97320 56 86 351 502617 396903 2247 a r e a 8 a r e a 9 new d i s t r i b u t i o n o f f r y (N - 1,000,000) new d i s t r i b u t i o n o f f r y (H * 1,000) 333 97380 437 502617 396903 2247 0 0 0.333 97.380 0.437 502.617 396.903 2.247 0.000 0.000 e x p e c t e d d i s t r i b u t i o n o f f r y (8 - 1,000) 3.180 56.660 211.900 361.130 315.190 10.860 7.040 14.040 secved-expected) expected .2. ( 0 , 3 " : 3 : 1 8 0 ' + <97.380-56.660) z + (0.U37-21 1 .900 i 2 ^ (502.617-381. 1301 2 ,3.180 ^ 56.660 * 211.900 * 3B1. 130 (396.903-315.190V*- (2.247-10.860V 2 - , ( 0 . 0 0 0 - 7 . 0 4 0 r t (0.000-14.040)* 315.190 * 16.866 * 7.040 * 1 4 .040 =S 331 I . B . T h e r e i s s o m e r o a n d - o f f e r r o r i n t h e d a t a . Ui 0.000 (obs.) 14-. 04-0 (exp.) 0.333 (obs.) 3.180 (exp.) F U L T O N R. 97.380 (obs.) 56.660 (exp.) 0.4-37 (obs.) 211.900 (exp.) 8 0.000 (obs.) 7.04-0 (exp. ) 10 2.247 (obs.) 10.860 (exp.) BABINE LAKE (BY A R E A ) MILES 502.617 (obs.) 381.130 (exp.) 396.903 (obs.) 315.190 (exp.) Figure 10. Computer s imulat ion (observed) numbers of f r y (N=1000) of the combined d i s t r i b u t i o n patterns f o r period 1, and n a t u r a l l y observed (expected) numbers of f r y (N=1000) f o r period 1. Data are taken from Table 21. 57 Dispersal i n Periods 2 and 3 To prepare for the second dispersal period cf the f r y , during which a general northerly movement of fry had been noted, f i s h were placed in each area of the lake according tc the expected values of sampling period 1. They were put i n each square of an area proportionately to the number already present in that square as indicated from the combined d i s t r i b u t i o n pattern described in Table 21. Since the f r y grow during the f i r s t dispersal period they could presumably move faster than they could upon entry into the lake. Thus, i t was necessary to try to obtain estimates cf new speeds of movement of the fry over the rest cf the summer and f a l l . Hoar (1954) found, along with his experiments on sockeye fr y , that sockeye smolts i n early June (cf the i r second year) swam about twice as f a s t as sockeye f r y for a temperature range of 8.5 degrees C. to 12.5 degrees C. . A graph cf growth rates of sockeye salmon prepared by Ricker (1938) (Figure 11) shows that from June to October of t h e i r f i r s t year sockeye have a f a i r l y constant growth rate and grow r e l a t i v e l y l i t t l e during the winter. Thus, from such information, swimming speeds cf the sockeye i n the second dispersal period were crudely estimated tc be 1.5 times t h e i r swimming speeds i n the f i r s t period and i n the t h i r d dispersal period to be twice their swimming speeds in the f i r s t period. The duration of the second part of the dispersal was T i i i i r i i i — i — i — i — i — i — i — i — r - I93G 1937 IV I V I VI IV!) IV!!1. I IX I X I XI I XII I I I II [ ill I IV 1 V |VI I vu IVlJH ix Figure 11. Rate of increase i n length of sockeye (and squawfish) i n 1936 and 1937. The sockeye are of the brood spawned i n the autumn of 1935 and hatched i n A p r i l and May of 1936. (Taken from publication by Hoar (1938).) 59 approximated to be 47 days (July 13 - the mid-date of the f i r s t sampling period, to August 28 - the mid-date of the second sampling period averaged over 1967, 1968, 1971, and 1972). T r i a l runs using no bias (random normal) and biases cf 30:70 and 10:90, th i s time up the lake (northward), were given to the 1,000,000 f r y on di f f e r e n t computer runs. Precalculated information for these runs i s given i n Table 22. For random dispersal the d i s t r i b u t i o n of f i s h did net come close to approximating the expected d i s t r i b u t i o n (Figure 12). A l l chi-sguare values remained near 1400, far abeve the allowable l i m i t of 721 (Table 23 - Summary of results f o r period 2 - random movement). For biases up the lake of 30:70 and 10:90 the chi-square values o s c i l l a t e d greatly. Although chi-sguare values i n the early parts of both runs were lower than in the lat e r parts the observed values made more sense in the la t e r parts of the runs in being better oriented with respect to the expected values. Thus, only chi-squares associated with the la t e r parts of the runs (for i t e r a t i o n s corresponding to speeds of f r y equal to or greater than the minimum speed at which the fry were thought to move) were examined (Tables 24 and 25). This made one especially aware of the great amount of variation in the available data. The lowest chi-square value was 1196, occurring after 130 it e r a t i o n s i n the run that gave the f i s h a 10:90 bias up the lake (Table 25). This number of i t e r a t i o n s corresponded tc a swimming speed s l i g h t l y above the minimum speed the f i s h were estimated to have. However, th i s chi-sguare value was not 60 Table 22. Movement of Fulton fiiver f r y , non-biased and biased, •up the lake 1 - period 2. speed distance number of i t e r a t i o n s t r a v e l l e d of loop required for i n 47 days an average speed of: (miles/iteration,m/i) 0.262 m/i 0.343 m/i 0.414 m/i no bias bias 30:70 bias 10:90 1.95 cm./sec. 49.2 mi. 189 144 120 (1. 047 mi./24 hr.) 3.2625 cm./sec. 82.3 o i . 315 242 200 (1. 752 mi./24 hr. ) 3.825 cm./sec. 96.5 mi. 369 282 234 (2.053 mi./24 hr.) 6i Figure 12. Computer simulation (observed) numbers of f r y (N=1000) at the best chi-square value (l^Ol) f o r period 2 - random movement, and natur a l l y observed (expected) numbers of f r y (N=1000) for period 2 i n each of the 8 lake areas. Data are taken from Table 23. Table 23. Summary of r e s u l t s f o r period 2 - random movement. area 1 observed numbers of f r y in each area from simulation area 2 area 3 area 4 area 5 area 10 area 8 area 9 slow speed (1.95 cm./sec.) - i t e r a t i o n 189 N=1,000,000 N=1,000 1731 1 17.313 77607 77.615 154250 154.267 401932 401.975 332103 332.138 9551 9.552 3034 3.035 4104 4. 104 1403 average speed (3.2625 cm./sac.) - i t e r a t i o n 315 N=1,000,000 N=1,000 18016 18.0 19 79595 79.608 146391 146.415 403336 403.402 335308 335.363 10090 10.092 3224 3.224 3876 3.877 1403 f a s t speed (3.825 cm./sec.) + i t e r a t i o n 369 N=1,000,000 N=1,000 18242 18.245 80445 80.460 143765 143.793 403876 403.953 336067 336. 131 10311 10.313 3277 3.278 3826 3.827 1401 optimum speed f o r best chi-square value - same as for f a s t speed - i t e r a t i o n 369 N=1 ,000,000 N=1,000 18242 18.245 80445 80.460 143765 143.793 403876 403.953 336067 336.131 10311 10.313 3277 3.278 3826 3.827 1401 S=1,000 area 1 310.640 expected numbers of f r y in each area from lake data area 2 area 3 area 4 area 5 area 10 area 8 226.930 70.680 245.230 80.190 30.980 11.970 area 9 23.370 721 N.B. There i s round-off e r r o r i n the 'N = 1,000,000' values. ON Table 24. Summary of r e s u l t s f o r period 2 - 30:70 biased movement. observed numbers of f r y i n each area from simulation area 1 area 2 area 3 area 4 area 5 area 10 area 8 area 9 X slow speed (1.95 cm./sec.) - i t e r a t i o n 144 N=1,000,000 H=1,000 56832 56.839 199579 199.605 391208 391.260 297663 297.703 4719 4.720 14828 14.830 5352 5.353 29686 29.690 1761 average speed (3.2625 cm./sec.) - i t e r a t i o n 242 N=1,000,000 N=1 ,000 121149 121.175 414145 414.233 319433 319.500 32713 32.720 52 0.052 27433 27.439 10040 10.043 74823 74.839 1524 optimum speed f o r best chi-square value - i t e r a t i o n 280 N=1,000,000 N=1,000 142050 142.085 519813 519.941 179482 179.526 9896 9.898 7 0.007 32205 32.213 12951 12.954 103350 103.376 1217 f a s t speed (3.825 cm./sec.) - i t e r a t i o n 282 N=1,000,000 N=1,000 142945 142.980 523568 523.698 173189 173.232 9245 9. 248 7 0.007 32524 32.532 13132 13.135 105143 105.169 1221 »=1,000 area 1 310.640 expected numbers of f r y in each area from lake data area 2 area 3 area 4 area 5 area 10 area 8 226.930 70.680 245.230 80.190 30.980 11.970 area 9 23.370 721 a.B. There i s round-off e r r o r i n the »N = 1,000,000' values. ON Table 25. Summary of r e s u l t s f o r period 2 - 10:90 biased movement. observed numbers of f r y in each area from simulation area 1 area 2 area 3 area 4 area 5 area 10 area 8 area 9 slow speed (1.95 cm./sec.) - i t e r a t i o n 120 H=1,000,000 N=1,000 151326 151.342 447825 447.869 255093 255.118 37735 37.739 31 0.031 16220 16.222 5885 5.886 85785 85.794 1211 optimum speed f o r best chi-square value - i t e r a t i o n 130 N=1,000,000 H=1,000 157657 157.674 499281 499.334 194940 194.961 201 18 20.120 11 0.011 17410 17.412 6326 6.327 104151 104.162 1196 average speed (3.2625 cm./sec.) - i t e r a t i o n 200 N=1,000,000 H=1,000 393877 393.939 227815 227.851 5302 5.303 46 0.046 0 0.000 36283 36.289 12160 12.162 324359 324.410 4287 f a s t speed (3.825 cm./sec.) - i t e r a t i o n 234 N=1,000,000 11=1,000 317083 317.141 77835 77.850 316 0.316 1 0.001 0 0.000 41049 41.056 17812 17.816 545718 545.820 12179 N=1,000 area 1 310.640 expected numbers of f r y in each area frcm lake data area 2 area 3 area 4 area 5 area 10 area 8 226.930 70.680 245.230 80.190 30.980 11.970 area 9 23. 370 721 N.B. There i s round-off e r r o r i n the 'N = 1,000,000' values. ON 65 s a t i s f a c t o r y . In addition, the associated observed values did not show the trend c h a r a c t e r i s t i c of the expected values (Figure 13) . The most l o g i c a l thing to do to improve the f i t was tc look at the same technique regarding the combination of d i f f e r e n t movement patterns as used in the f i r s t d ispersal period. I f t h i s time i t was assumed that there were two unknown factors working equally upon the population of f i s h , which resulted i n one half of them moving randomly and the other half moving with a 10:90 bias up the lake, a f a i r l y good r e s u l t was achieved (Figure 14). Values from the 2 sets of dispersal data were taken so that randomly moving f i s h travelled at their average speed of 3.2625 cm./sec. (1.752 mi./24 hr.), while the biased f i s h t r a v e l l e d at a speed corresponding to the best resulting chi-sguare value (which was 1196). The chi-sguare value which resulted from combining the movement patterns was 473, well below the acceptable value of 721 (Table 26). To prepare for the t h i r d and l a s t dispersal period the same thing was done as for the f i r s t dispersal period. Fish were placed i n each area of the lake according tc the expected values at the end of sampling period 2. They were put i n each sguare of an area proportionately to the number already present in that sguare as r e s u l t i n g from the combined d i s t r i b u t i o n a l patterns (Table 26) . McDonald (1969) noted, as mentioned previously, that fcr the t h i r d dispersal period of 1966 the f i s h continued their northward movement. However, from the accumulated data cf 1967, 1968, 1971, and 1972 for period 3 (Table 3) i t seems that the 104-.162 (obs.) 23.370 (exp.) 157.674 (obs.) 310 .640 (exp.) F U L T O N R. 499.334 (obs.) 226.930 (exp.) 194.961 (obs.) 70 .680 (exp.) 8 6.327 (obs.) 11.970 (exp.) 10 17.^12 (obs.) 30.980- (exp. ) BABINE LAKE (BY AREA) ttiLga 20.120 245.230 (obs. ) (exp. ) 0.011 (obs.) 80.190 (exp.) Figure 13. Computer s imulat ion (observed) numbers of f r y (N=1000) at the best chi-square value (1196) f o r period 2 - 10_90 biased movement, and n a t u r a l l y observed (expected) numbers of f r y (N=1000) f o r period 2 i n each of the- 8 lake areas. Data are taken from Table 25. 5^.013 (obs.) 23.370 (exp.) .87.837 (obs.) 310.64-0 (exp. ) 8 4.775 (obs.) 11.970 (exp.) 10 54-.013 (obs. ) 23.370.(exp.) F U L T O N R. 289.437 (obs.) 226.930 (exp.) 170.666 (obs.) 70.680 (exp. ). 4 211.727 (obs.) 245.230 (exp.) BABINE LAKE (BY A R E A ) MILES W 167.659 (obs.) 80.190 (exp.) Figure 14-, Computer simulation (observed) numbers of f r y (N=1000) of the combined d i s t r i b u t i o n patterns for period 2, and naturally observed (expected) numbers of f r y (N=1000) for period 2. Data are taken from Table 26. I?bJe,v2?; D'5aJB £ o r . C 0 B b i n e d simulated d i s p e r s a l d i s t r i b u t i o n s of f r y i n period 2 together with expected d i s t r i b u t i o n or f r y and c a l c u l a t e d chi-sguare value. .SO x observed d i s t r i b u t i o n of f r y moving with no bias at i t e r a t i o n 315 .50 x observed d i s t r i b u t i o n of f r y moving with a 10:90 bias at i t e r a t i o n 150 area 1 area 2 area 3 area 0 area 5 area 10 area 6 area 9 9008 39797 73196 201668 167650 5005 1612 1938 78829 209600 97070 10059 8705 3163 52075 new d i s t r i b u t i o n of f r y (H * 1,000,000) new d i s t r i b u t i o n of f r y (H - 1,000) 87837 289037 170666 211727 167659 13750 87.837 289.037 170.666 211.727 167.659 13.750 0775 50013 0.775 50.013 e x p e c t e d d i s t r i b u t i o n o f f r y (H > 1,000) 310.600 226.930 70.680 205.230 80.190 (observed-expected) 3 expected 30.980 11.970 (87.837-310.600> Z (289.037-226.9301* (170.666-70.6801* ^ (211.727-205.230>' 3T076«ur * . 226 .930 + 76.680 * 215.230 + M67 .659 -80 .190 ) * - ( 13 .750 -30 .980 ) 2 . ( 0 . 7 7 5 - 1 1 . 9 7 0 \ Z . ( 50 .013 -23 .370 ) * 80-190 * 36.9*0 + 11.970 * r • 1 23 .3V0 23.370 = 073 I.B. T h e r e i s soae r o a n d - o f f e r r o r i n t h e d a t a . ON co 69 f i s h are mixing uniformly among areas rather than following any obvious d i r e c t i o n a l tendency. The maximum allowable chi-sguare value of 1431 for sampling period 3 i s much higher than for the other sampling periods and thus allows more leeway in observed values generated by the simulation. Even when the chi-sguare test i s applied to the expected values of sampling period 3 against the expected values of sampling period 2, the res u l t i n g chi-sguare value of 442 (Table 27) f a l l s well below the allowable l i m i t . Thus, for any runs in the t h i r d dispersal period to have true si g n i f i c a n c e i t i s r e a l l y necessary to obtain observed values which generate chi-squares less than 442 . On the basis of what i s known, random movement is the only movement that need be postulated, to give a d i s t r i b u t i o n approximating that observed naturally (Figure 15). The length of dispersal period 3 was estimated to be 45 days (August 29 the mid-date of sampling period 2, to October 12 - the mid-date of sampling period 3 averaged over years 1967, 1968, 1971, and 1972). Precalculated information for a 'no bias' run with the f i s h now t r a v e l l i n g at twice the speed they did at the beginning of the summer i s given in Table 28. The run successfully produced chi-sguare values less than 442 (Table 29 - Summary of r e s u l t s for period 3). With the f i s h t r a v e l l i n g at t h e i r average speed of 4.35 cm./sec. (2.335 mi./24 hr.) the res u l t i n g chi-sguare value after 45 days was 297 . The chi-sguare values continued tc decrease slowly i f higher speeds were assumed for the f i s h . T a b l e 2 7 . The c h i - s q u a r e t e s t a p p l i e d t o t h e expec ted v a l u e s o f p e r i o d 3 a g a i n s t the e x p e c t e d v a l u e s o f p e r i o d 2. a r e a 1 a r e a 2 a r e a 3 a r e a 1 a r e a 5 a r e a 10 a rea 8 a r e a 9 e x p e c t e d d i s t r i b u t i o n o f f r y f o r p e r i o d 2 310.600 226.930 70.680 245.230 80.190 30.980 11.970 23.370 (N • 1,000) e x p e c t e d d i s t r i b u t i o n of f r y f o r p e r i o d 3 204.920 132.740 243.960 123.850 17S.380 62.080 30.090 26.990 (B » 1,000) — ( 3 1 0 . 6 4 0 - 2 0 4 . 9 2 0 ) z (226. 930-132.740)2" (70 • 680-243. 960) Z . (245.230-123.850)Z / \ — • • 204.920 132.740 243.960 123.BS0 ^ ( 8 0 . 1 9 0 - 1 7 5 . 3 8 0 ) Z . ( 3 0 . 9 8 0 - 6 2 . 0 8 0 ) 2 . (11.970-30.090)* . (23.370-26.990) Z + 1 nS.iaO + 62.080 .* 30.090 t 267990 = 442 -N3 O 71 34.307 (obs.) 26.990 (exp.) 240.906 (obs.) 204.92 0 (exp.) 8 27.954 (obs.) 30.090 (exp.) 10 61.625 (obs.) 62 .080 (exp.) 237.134 (obs.) 132.740 (exp.) 79.356 (obs.) 243.960 (exp.) 211.572 (obs.) 123.850 (exp.) 107.147 (obs.) 175.380 (exp.) BABINE LAKE ( B Y A R E A ) Figure 15. Computer s imulat ion (observed) numbers of f r y (N=1000) at the best chi-square value (290) f o r period 3 - random movement, and n a t u r a l l y observed (expected) numbers of f r y (N=1000) f o r period 3 i n each of the 8 lake areas. Data are taken from Table 29. 72 Table 28. Random movement of f r y - no bias, period 3. speed distance number of i t e r a t i o n s t r a v e l l e d of loop; required for i n 45 days an average of 0.262 mile s / i t e r a t i o n 2.60 cm./sec. 62.81 mi. 240 (1.3958 mi./24 hr.) 4.35 cm./sec. 105.09 mi. 402 (2.335 mi./24 hr.) 5.10 cm./sec. (2.738 mi./24 hr.) 123,21 mi. 472 Table 29. Summary of r e s u l t s for period 3 - random movement. area 1 observed numbers of f r y in each area from simulation area 2 area 3 area 4 area 5 area 10 area 8 area 9 slow speed (2.60 cm./sec.) - i t e r a t i o n 240 N=1,000,000 N=1,000 265870 265.898 234767 234.793 77639 77.648 219831 219.855 100353 100.364 49067 49.072 21483 21.486 30881 30.884 332 average speed (4.35 cm./sec.) - i t e r a t i o n 402 K=1,000,000 N=1,000 247432 247.481 236389 236.435 78785 78.801 213744 213.787 105408 105.429 58392 58.404 26285 26.290 33367 33.373 297 f a s t speed (5.10 cm./sec.) - . i t e r a t i o n 472 H=1,000,000 H=1 ,000 240849 240.906 237077 237.134 79337 79.356 211522 211.572 107121 107. 147 61610 61.625 27947 27.954 34299 34.307 290 optimum speed for best chi-square value - same as f o r f a s t speed - i t e r a t i o n 472 N=1,000,000 N=1,000 240849 240.906 237077 237.134 79337 79.356 211522 211.572 107121 107. 147 61610 61.625 27947 27.954 34299 34.307 290 N=1,000 area 1 204.920 expected numbers of f r y i n each area from lake data area 2 area 3 area 4 area 5 area 10 area 8 132.740 243.960 123.850 175.380 62.080 30.090 area 9 26.990 1431 H.B. There i s round-off e r r o r i n the •H = 1,000,000' values. ^0 74 DISCUSSION The l i t e r a t u r e on animal dispersal i s li m i t e d , and provides no precedent for the study of f i s h movements in areas with complex boundary conditions as found in a lake. Studies of such movements may, in the future, pay high dividends by providing information useful in the management of inland f i s h e r i e s . For these studies simulation seems the best approach. Ccnseguently, a computer model of f i s h dispersal was developed using Babine Lake, B.C., an important sockeye salmon producing nursery. Although the data collected on f i s h dispersal i n the lake show wide v a r i a b i l i t y from year to year, i t i s clear that the movement i s not random. Natural d i s t r i b u t i o n patterns suggest a southward movement i n early summer, then a northward movement la t e r on. Possible explanations for the observed movement of fry are that i t i s caused by currents alone or some aspect Of f i s h behavior. Regardless of the reasons for the fry movement, i t can be simulated by introducing d i r e c t i o n a l biases. Biases were created using the 4-parameter lognormal density function. Many examples of the lognormal d i s t r i b u t i o n have been found i n nature from a variety of f i e l d s from sedimentary petrology to astronomy (Aitchison and Brown, 1957). Aitchison and Brown (1957, p. 1) state that i t i s "our b e l i e f that the lognormal i s as fundamental a d i s t r i b u t i o n in s t a t i s t i c s as i s the normal". The present study suggests the usefulness of the di s t r i b u t i o n in describing f i s h movement. When freguencies of a lognormal density function are 75 replotted from a graph having a horizontal standard scale to a graph having a horizontal logarithmic scale, the frequencies take the normal symmetrical form. In t h i s transformation the successive units of the horizontal scale are readjusted to distances having a constant r a t i o rather than a constant difference (Davies 1925). Thus, in the dispersal model, the pr o b a b i l i t i e s of f i s h moving a certain distance are distributed normally 1 7 with respect to successive unit distances having a constant r a t i o . The hypothesis that f i s h may actually move in this way does not seem unreasonable. Simulation runs using single 'down the lake' (southward) biases for period 1 or single 'up the lake' (northward) biases for period 2 are not enough to account f c r the observed patterns of f r y i n period 1 and period 2. Consequently, the easiest hypothesis to account for the observed pattern in period 1 i s that some f i s h (about 10% of the f i s h population) do not show bias. Perhaps they either do not get caught in the main physical flow or do not behave as the majority do. For period 2, with the bias reversed, only an increased proportion of those fry that do not show any bias (to 50% of the f i s h population) i s reguired. For period 3 the observed pattern i s ' f i t t e d ' when a l l f i s h are allowed to t r a v e l at random. Thus, the general trend over the summer and f a l l i s as shown in Figure 16. The 1 7 To normalize the 4-parameter lognormal density function, p r o b a b i l i t i e s ( i - a x i s values) are replotted from a graph having a horizontal standard scale with 'x' along the X-axis tc a graph having a horizontal logarithmic scale with 'log (x- T)/(0 -x) • along the X-axis. / 76 Figure 16. General dispersal trend of f r y over the summer and f a l l . down bias 90% -{ no bias 10% t i t up bias 50% no bias 50% no bias 100% t t June July &ug. Sept. Oct. 77 idea represented here i s simple and could (should) be fi e l d - t e s t e d . Much more research i s required in investiqating factors which might play a role in causing the observed d i s t r i b u t i o n patterns of f r y . Whatever the f a c t c f s are, i t appears that their influence on susceptible fry i s strong, since a 10:90 bias southward and then northward in periods 1 and 2, respectively, was required to simulate d i s t r i b u t i o n patterns similar tc those observed in the lake. Adult salmon u t i l i z e spawning areas which are located both in i n l e t and outlet streams. Fry from spawning areas cf cutlet streams must move upstream against the current tc reach the nursery lake while fry from spawning areas of i n l e t streams must eventually move with the current. Such differences i n the behavior of sockeye f r y with respect to current have been observed i n the f i e l d by McCart (1967), Brannon (1967), and fialeigh (1967). Studies by Brannon (1967), Baleigh (1967), and Calaprice (1972) indicated that the behavioral response of the fry to current was under genetic control. Calaprice (1972) asserts that innate variat i o n in the responses of Eabine Lake sockeye f r y to current i s controlled by additive genetic factors. Genetic factors may play a complex part in influencing the behavioral responses of fry to currents i n the lake as well as in the spawning streams. It i s possible that current response behavior of sockeye f r y may vary i n degree. McCart (1967) observed that, after reaching their nursery lake, downstream migrants (which were negatively rheotactic) 78 became p o s i t i v e l y rheotactic. Hoar (1958) has also shewn experimental evidence of t h i s change i n sockeye f r y behavior with an increase i n age. When a f i s h i s long exposed tc current, f a i l u r e of positive rheotaxis might be expected as a resu l t of fatigue or adaptation of sensory mechanisms (Keenleyside and Hoar, 1954). Although some current measurements in the lake have been taken (Er. D. Farmer (Water Management, Environment Canada), personal communication), they have not yet been analyzed to develop a picture of lake c i r c u l a t i o n . Temperature p r o f i l e s of Babine obtained i n 1967 have been used to show major changes i n the thermal structure of the lake throughout the summer and f a l l (McDonald and Scarsbrcck, 1967). Some of the r e s u l t s are p a r t i c u l a r l y i n t e r e s t i n g . In early July a prominent feature was the wedge of r e l a t i v e l y warm surface water extending from the southeast end of the lake into the northwest end. In late August the thermal structure of the lake was f a i r l y uniform. In late September warmer water was i n the northwest end of the main basin. The d i s t r i b u t i o n of highest areal densities cf f i s h seems to correspond to the d i s t r i b u t i o n of warmest water. Whether or not there i s any causal connection between the two i s open tc conjecture. Although reversed movements due to temperature variation were not observed for sockeye f r y by Keenleyside and Hoar (1954) or Brannon (1967), such reversed movements for sockeye fry were observed by Baleigh (1971). If sockeye fry re t a i n t h e i r 79 positive rheotaxis i n the lake. Hoar (1958) states that they may be stimulated by r i s i n g temperatures to swim more strongly against the current in the lake owing to direct acceleration of metabolic processes or association with rheotaxis. Although the model described in this thesis i s r e l a t i v e l y simple, hopefully i t i s a modest step towards understanding the movement of salmon (sockeye) f r y better and w i l l generate new thinking which w i l l ultimately lead to better management schemes for salmon resources. The model could be further developed for Babine Lake as f i e l d data are collected concerning possible relationships of f r y d i s t r i b u t i o n patterns with various features of the limnology of the lake or variation i n behavior among populations of sockeye f r y . The model also has much generality. The technigues used in t h i s model could be applied to the dispersal cf salmon f r y i n other lakes, perhaps, p a r t i c u l a r l y , the large salmcn-prcducing lakes of Alaska. Intensive studies have been done on some of these lakes, especially Lake Clark, Iliamna Lake, and Lake Aleknagik. The movement of f r y in these lakes tends to be f a i r l y complicated (Kerns 1968, P e l l a 1968, Burgner 1962, Rogers 1972). For t h i s reason, simulation i s probably the best approach in analysis. The same modelling ideas could be applied to the dispersal of other f i s h besides salmon f r y , as well as to other organisms. The movements of the predators of salmon fry and f i s h which compete with salmon f r y for lake resources might prove worthwhile to study. 80 Talbot (1974) has examined the movement of oyster larvae i n estuaries and plaice larvae i n the open sea using d i f f u s i o n equations but such movements may be more easily and tetter investigated using the simulation techniques here developed. In general, the transport and dispersal of plankton species caused by t i d a l and other water movements are quite s i g n i f i c a n t from the point of view of the ecology of a region (Talbot 1974). In summary, f i s h dispersal i s a phenomenon which occurs i n a l l waters of the world. I t i s a d i f f i c u l t f i e l d of study and much more work should be done on i t i n the next few years. Computers should play an important part i n such an undertaking with the tremendous information-handling c a p a b i l i t i e s they possess. 81 LITERATURE CITED A i t c h i s o n , J . , and J.A.C. Brown. 1957. The lognormal iili£±iSii2£ * U n i v e r s i t y Press, Cambridge. 176 pp. Beverton, R.J.H., and S.J. H o l t . 1957. On the dynamics of e x p l o i t e d f i s h p o p u l a t i o n s . Her Majesty's s t a t i o n a r y o f f l c e T London. 533 pp. Brannon, E.L. 1967. Genetic c o n t r o l of migrating behavior of newly emerged sockeye salmon f r y . Int. P a c i f i c Salmon F i s h . Comm., Prog. Rep. 16. 31 pp. B r e t t , J.B. 1967. Swimming performance of sockeye salmon ( 0H£2iliJB£]lus nerka ) i n r e l a t i o n t o f a t i g u e time and temperature7~ ?J.~Fish. Res. Bd. Canada, 24 (8) : 173 1-174 1. Burgner, R.L. 1962. Sampling red salmon f r y by lake t r a p i n the Hood River Lakes, Alaska. Univ. Washington Publ. i n F i s h . , New Ser., 1(7):315-348. C a l a p r i c e , J.R. 1972. H e r i t a b l e v a r i a t i o n amcng popul a t i o n s of sockeye salmon Oncorhynchus nerka . I I I . D i f f e r e n c e s i n the migratory behavior'of f r y i n a c u r r e n t . Unpublished. Coburn, A.S., and J . McDonald. MS, 1972. The t r a p p i n g and marking of sockeye salmon f r y ( Oncorhynchus nerka ) at F u l t o n R i v e r , Babine Lake, B.C. "("1966 - 68). F i s h . Res. Bd. Canada, Tech. Rep. 348. 54 pp. . MS, 1 973. The marking of sockeye salmon f r y ( Oncorhynchus nerka ) at F u l t o n R iver and Pinkut Creek, BabIne~Lake7~B. c7 (~T971 - 72). F i s h . Res. Bd. Canada, Tech. Bep. 372. 17 pp. Davies, G.R. 1925. The l o g a r i t h m i c curve of d i s t r i b u t i o n . J . Amer. S t a t . Assoc., New S e r i e s , No. 152 (Vol. XX): 467-480. Department of Fisheries of Canada and Fisheries Research Board of Canada. MS, 1965. Proposed sockeye salmon development program for Babine lake. Dept. Fish. Canada, Vancouver, B.C., 53 pp. + 14 figures. . MS, 1968. The Babine Lake salmon development program. Dept. Fish. Canada and Fish. Res. Ed. Canada, ' Prog. Rep. 75 pp. Foerster, R.E. 1944. The r e l a t i o n of lake population density to size of young sockeye salmon ( Oncorhynchus nerka ) . J . Fish. Res. Bd. Canada, 6 (3):267-280. . 1954. On the r e l a t i o n of adult sockeye salmon ( Oj2£2lin£chus nerka ) returns to known smolt seaward migrations. J . Fish. Res. Bd. Canada, 1 1 (4):339-350. French, N.R. 1971. Simulation of dispersal in desert rodents. In: S t a t i s t i c a l ecology.j, volume 3: Banj snecies £2£u2aii2Slji ecggystemSj and systems analysis . Ed. by G.P. P a t i l , E.C. Pielou, and W.E. Waters. Pennsylvania State Univ. Press. 462 pp. Gadgil, M. 1971. Dispersal: population ccnseguences and evolution. Ecology, 52 (2) : 254-261. Hoar, W.S. 1954. The behavior of juvenile P a c i f i c salmon, with p a r t i c u l a r reference to the sockeye ( Oncorhynchus nerka ). J . Fish. Res. Bd. Canada, 11 (17:69-97. . 1958. The evolution of migratory behavior among juvenile salmon of the genus Oncorhynchus . J . Fish. Res. Bd. Canada, 15 (3):391-428. Johnson, W.E. 1956. On the d i s t r i b u t i o n cf ycung sockeye salmon ( 2££orhynchus nerka ) i n Babine and Uilkitkwa Lakes, B.C." J T Fish. Res.~Bd. Canada, 13 (5):695-708. Keenleyside, M.H.A., and W.S. Hoar. 1954. Ef f e c t s of temperature on the responses of young salmon to water current. Behaviour, 7:77-87. .83 Kerns, O.E.Jr. 1968. abundance, d i s t r i b u t i o n , and si z e of juvenile sockeye salmon and major competitor species i n Iliamna Lake and Lake Clark, 1966 and 1967. Univ. Washington Fish. Hes. Inst., C i r c . 68-11. 35 pp. Kitching, B. 1971. A simple simulation mcdel of dispersal of animals among units of discrete habitats. Oecologia (Berl.), 7:95-116. McCart, P. 1967. Behavior and ecology of sockeye salnscn f r y i n the Babine Biver. J. Fish. Res. Ed. Canada, 24(2):375-428. McDonald, J.G. 1969. D i s t r i b u t i o n , growth, and su r v i v a l cf sockeye f r y ( Oncorhynchus nerka ) produced i n natural and a r t i f i c i a l stream environments. J. Fish. Res. Bd. Canada, 26:229-267. . MS, 1973. Diel v e r t i c a l movements and feeding habits of underyearling sockeye salmon ( Oncorhynchus nerka ) at Babine Lake, B.C.. Fish. Res. Ed. Canada,~Tech.~Rep. 378. 55 pp. McDonald, J.G., and J.R. Scarsbrook. MS, 1969. Thermal structure of Babine Lake (main basin) in 1967. Fish. Res. Bd. Canada, MS Rep. 1070. 30 pp. Mood, A., and F.A. G r a y b i l l . 1963. Introduction to the theory of s t a t i s t i c s . 2nd edition. McGraw" H i l l , New York. 443 pp. Murray, B.G. 1967. Dispersal in vertebrates. Ecology, 48(6):975-978. P e l l a , J.J. 1968. Di s t r i b u t i o n and growth of sockeye salmon fry in Lake Aleknagik, Alaska, during the summer cf 1962. In: Further studies of Alaska sockeye salmon . Ed. By R.L. Burgner. Univ. Washington, Publ. In Fish., New Ser., Vol. I I I . pp. 45-111. 84 Pielou, E.C. 1969. Patterns r e s u l t i n g from d i f f u s i o n . In: An introduction to mathematical ecology .Wiley - ~ Interscience, New York. 286 pp7 Raleigh, R.F. 1967, Genetic control in the lakeward migrations of sockeye salmon ( Oncorhynchus nerka ) f r y . J. Fish. Res. Bd. Canada, 24 (12772613-26227 . 1971. Innate control of migrations cf salmon and trout f r y from natal gravels to rearing areas. Ecology 52:291-297. Ricker, W.E. 1938. A comparison of the seasonal growth rates of young sockeye salmon and young sguawfish in Cultus Lake. Fish res. Bd. Canada, Pac. Prog. Rep. 36. pp. 3-5. , 1962. Comparison of ocean growth and mortality cf sockeye salmon during t h e i r l a s t two years. J. Fish. Res. Bd. Canada, 19 (4):531-560. Rogers, D.E. 1972. Estimates of abundance and growth i n the early summer from beach seine catches i n Lake Aleknagik. Univ. Washington Fish. Res. Inst., C i r c . 72-4. 89 pp. Sa i l a , S.B., and R.A. Shappy. 1963. Random movement and orientation in salmon migration. J. Cu Conseil, 28 (1) : 153-166. Scarsbrook, J.R., and J.G. McDonald. MS, 1970. Purse seine catches of sockeye salmon ( Oncorhynchus nerka ) and other species of f i s h at Babine~Lake, British~Columbia, 1966 to 1968. Fish. Res. Bd. Canada, MS Rep. 1075. 110 pp. . MS, 1972. Purse seine catches of sockeye salmon ( Oncorhynchus nerka ) and other species of f i s h at Babine Lake7~British Columbia, 1971. Fish. Res. Bd. Canada, MS Rep. 1189. 48 pp. 85 . MS, 1973. Purse seine catches of sockeye salmon ( 2.Q£2£hynchus nerka ) and other species of f i s h at Babine Lake, B r i t i s h Columbia, 1972. Fish. Res. Ed. Canada, Tech. Rep. 390. 46 pp. S i n i f f , D.B., and CR. Jessen. 1969. A simulation model of animal movement patterns. Advances i n Ecol. Res., 6:185-219. Skellam, J.G. 1951. Random dispersal in theoret i c a l populations, Biometrika, 38:196-218. Sokal, R.R., and F.J. Rohlf. 1969. Biometry._ The p r i n c i p l e s and practices of s t a t i s t i c s in biolocjica 1 research . W.H. Freeman and Company, San Franciscc. 776 pp. Talbot, J.W. 1974. Diffusion studies in f i s h e r i e s biology. In: Sea Fisheries Research . Ed. by F.R. Harden Jones. John Wiley and Sons, New York. 510 pp. \ A P P E N D I C E S 87 APPENDIX J SSiSlis of the Dispersal Process One of the main problems in developing the dispersal process was to look at the various types of shoreline grid patterns and program dispersal so that when the fry h i t a 'land sguare* they would bounce from the shore according tc laws of r e f l e c t i o n . A computer program was written to lock at each water sguare of the lake and i t s 24 surrounding sguares. Each sguare i n the rectangular coordinate system was given a code value of 1 or 0, denoting, respectively, either a water sguare or land sguare. If a sguare covered both water and land i t was considered a water square i f 50% or mere cf i t covered water. Each grid pattern could thus be given a 25 place binary code. In setting up a 1-to-1 correspondence between grid sguares and place values for the code, I looked at the 25 sguares from l e f t to right as numbered: 4 -I 4 10 r H — H 4 4 4 11 12 13 14 15 16 17 21 22 18 23 19 20 4 4 4 24 25 Examples of grid patterns and their binary codes are shown in Figure 1. Figure 1. Examples of grid patterns. (1.) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -Binary 25 place code (2.) 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 -Binary 25 place cede 89 For storage purposes each binary code was converted to a base 10 number code in the computer. By keeping track of the coordinates of each sguare in the lake and i t s d i g i t code, the computer could sort together a l l squares having a similar grid pattern. A t o t a l of 847 d i f f e r e n t grid patterns were found. Each sguare i n the lake was then assigned a number between 1 and 847 depending on i t s grid pattern. Land sguares cn the coordinate system were assigned the number 0 (zero). After a bivariate density function was determined to represent the d i s p e r s a l movement1 i t was placed over the 'open lake' grid pattern where a l l sguares were water sguares. The volume of the function was then integrated over each sguare. Thus, the p r o b a b i l i t i e s are: 1 Although dispersal of f i s h in the lake can be described by more than 1 biva r i a t e density function at a time i t i s assumed here that 1 b i v a r i a t e density function describes the movement of f i s h throughout the whole lake to keep the explanation cf the dispersal process as simple as possible. 90 IP (21) I P(16) IP (22) - + — -I P(17) I P(23) |P (24) I —I- 4-I P{18) | P (19) I P (25) P (20) |P(11) I P(6) j P (1) P(12) 4 +_ P (7) P(2) P(13) |P(14) P(8) |P (9) P(3) |P(4) P(15) P(10) P (5) When each of these p r o b a b i l i t i e s was multiplied by the number of f i s h i n the central square i t gave the r e s u l t i n g number of f i s h i n each of the 25 squares. For the other pattern types, which had land squares, i t was necessary to • r e f l e c t 0 the p r o b a b i l i t i e s i n those land squares to water squares and add them to the p r o b a b i l i t i e s already existing i n those water squares. The p r o b a b i l i t y i n each land square would then be set to zero. Thus, each grid pattern involved 25 p r o b a b i l i t i e s regardless of how many land sguares i t had. A matrix P 2 was set up to take care of a l l possible combinations of p r o b a b i l i t i e s r e s u l t i n g from the procedure: P(1) to P(25) are the o r i g i n a l p r o b a b i l i t i e s generated from 2 I f , i n the program, more than 1 biva r i a t e density function i s used at the same time there must be a d i f f e r e n t matrix for each set of p r o b a b i l i t i e s . 91 the bivariate density function over the gri d . - P (26) = 0 for a l l land sguares. - P(27) to P ( U 1 9 ) are combinations of P { 1 ) to P (25) to take care of r e f l e c t i o n s . The r e f l e c t i o n p r i n c i p l e involved i n r e f l e c t i n g the f r y off land squares was based on laws of r e f l e c t i o n : 1 . The angle of incidence i s egual to the angle of r e f l e c t i o n . 2. The distance an object appears 'behind' a mirror eguals the distance the object i s i n front of the mirror. These rules made sure that a f i s h always bounced off land boundaries in the same way and went the same distance i t would have gone i f no boundaries had deflected i t . In many instances where there was a complex boundary, especially when peninsulas and islands were involved, a f i s h destined towards a land square had to be reflected several times. The problem of r e f l e c t i o n on a corner was overcome by considering a corner to be a straight boundary of very short length. Fish in a l l instances travelled only in straight l i n e s . In cases where i t was seen that no f i s h could get to another water square in a pa r t i c u l a r pattern type by t r a v e l l i n g i n straight l i n e s , these sguares were treated as 'not available', and assigned P(26) = 0 . The r e s t r i c t i o n of f i s h moving in straight l i n e s did not seem unreasonable since each movement considered i n a grid was small when compared with movement throughout the whole lake. A l l d e t a i l s were worked out by hand. Although the rules 92 set down describing f i s h movement were discrete the number of possible combinations of land and water sguares for a grid made th i s ah impractical programming job. 3 Diagrams in Figure 2 describe various examples of the r e f l e c t i o n p r i n c i p l e . 3 Theoretically, the number of combinations possible i s 22*;2si6.8 m i l l i o n . The middle sguare always has to be a water sguare from which f i s h can disperse. However, from a p r a c t i c a l viewpoint the number of combinations would be considerably less than t h i s . 93 94 V P(21) |P(22) |P(26) |P(26) | P (26 ) I I I I P(16) |P(17) |P (208) |P (174) |P (26) I I I I I I P(49) | P (12) |P(50) |P(59) | P (26 ) I I I I P(26) |P (26) | P (8) | P (9) |P(10) P(26) |P(26) I I P(4 8) |P (4) |P (5) I I i J j j j i i 9 5 P ( 4 ) , P ( 5 ) , P ( 9 ) , P ( 1 0 ) , P ( 1 2 ) , P ( 1 6 ) , P ( 1 7 ) , P ( 2 1 ) , a n a P (22) a r e o r i g i n a l p r o b a b i l i t i e s . P ( 4 8 ) = + P ( 3 ) - o r i g i n a l p r o b a b i l i t y i n s q u a r e 3 P (2) - r e f l e c t e d p r o b a b i l i t y f r o m s g u a r e 2 P ( 4 9 ) = P ( 1 1 ) - o r i g i n a l p r o b a b i l i t y i n s q u a r e 11 P ( 6 ) - r e f l e c t e d p r o b a b i l i t y f r o m s q u a r e 6 . a l t h o u g h s g u a r e 6 i s n o t a l a n d s q u a r e no f i s h c a n r e a c h i t t r a v e l l i n g i n s t r a i g h t l i n e s . P ( 5 0 ) = P ( 1 3 ) - o r i g i n a l p r o b a b i l i t y i n s g u a r e 13 P ( 7 ) - r e f l e c t e d p r o b a b i l i t y f r o m s q u a r e 7 P (59) = P (14) + P (15) P (208) = P (18) + P (23) P ( 1 7 4 ) = P ( 1 9 ) + P ( 1 ) • P ( 2 0 ) + P ( 2 4 ) + P ( 2 5 ) S q u a r e s 2 , 7 , 1 5 , 2 0 , 2 3 , 2 4 , a n d 2 5 h a v e P ( 2 6 ) = 0 b e c a u s e t h e y a r e l a n d s q u a r e s . S q u a r e s 1 a n d 6 h a v e P ( 2 6 ) = 0 b e c a u s e t h e y a r e ' u n a v a i l a b l e ' . B e c a u s e o f t h i s t e c h n i q u e I c o u l d h a v e u s e d l e s s t h a n 847 p a t t e r n t y p e s . S i n c e ' u n a v a i l a b l e ' s q u a r e s w e r e c o n s i d e r e d l a n d s q u a r e s ( i . e . t h e y w e r e a s s i g n e d t h e p r o b a b i l i t y P (26) = 0) f o r a p a r t i c u l a r p a t t e r n t y p e , a p a t t e r n t y p e i n some i n s t a n c e s had t h e same s e q u e n c e o f p r o b a b i l i t y (P) v a l u e s a s a n o t h e r p a t t e r n t y p e . 96 Example: (1.) I i 21 I I I 22 | H +-23 | +-24 | +_ 25 16 I I I 17 | 18 | /> i 20 -.+ +-i i — + -1 1 4 12 L- 14 | 15 \///A/ ' - — + - +-6 8 I <r—+-9 I +-10 1 i i 1 2 | I I 5 L (2. ) 21 | 22 | 23 f" + + + + \ 14 | 15 /. + + ^ 9 | 10 *-77^77-A\--+ + ^ \///,\//'A \ I I l> 1/l> 2/| | 4 | 5 24 | 25 In (1.), squares 1, 2, and 6 are ' u n a v a i l a b l e ' because f i s h d i s p e r s i n g from the c e n t r a l sguare (13) cannot reach them i f thsy t r a v e l o n l y i n s t r a i g h t l i n e s . Squares 1, 2, and 6 (as well as 7) are assigned P(26) = 0 . But i n (2.), squares 1 , 2,6, and 7 are a l s o assigned P(26) = 0 because here they are a l l land squares. a l l other p r o b a b i l i t i e s f o r (1.) and (2.) are the same. Thus, when a c o r r e c t i o n was l a t e r made f o r a square near Fulton R i v e r , new p a t t e r n types d i d not have to be invented because o l d patte r n types c o u l d be used as s u b s t i t u t e s . No attempt was made to reduce the number of p a t t e r n types, i n general, because the advantages did not warrant the e x t r a e f f o r t . The d i f f e r e n t p r o b a b i l i t i e s i n the computer program were i d e n t i f i e d by t h e i r a r r a y number. As a r e s u l t each g r i d p a t t e r n #1 - #847 was a s s o c i a t e d with 25 numbers between 1 - 419. These 97 data were punched on computer c a r d s , and st o r e d i n the f i l e FOFDS. Examples of how the data appear are as f o l l o w s . (Actual data are underlined.) S# i s square number; P# i s P array number, a) s# 1 2 3 4 5 6 7 8 9 10 1 1 12 13 p# 1 2 3 4 5 6 7 8 9 JO 11 11 13 J. s# 14 15 16 17 18 19 20 21 22 23 24 25 p# 14 15 16 17 18 19 20 21 22 23 24 25 1 g r i d p a t t e r n number - rep r e s e n t s 'open la k e * g r i d p a t t e r n b) s# 1 2 3 4 5 6 7 8 9 10 1 1 12 13 p# 26 26 48 4 5 26 26 8 9 10 49 12 50 336 s# 14 15 16 17 1 8 19 20 21 22 23 24 25 p# 59 26 16 17 238 174 26 21 22 26 26 26 3 36 g r i d p a t t e r n number - r e p r e s e n t s p a t t e r n of diagram on page 94 During execution of the s i m u l a t i o n , as the e n t i r e l a k e d i s t r i c t was scanned, sguare by square, the number of the g r i d p a t t e r n f o r each square (stored i n the matrix ATEST) was ' c a l l e d * , together with i t s p r o b a b i l i t i e s . 4 A program block then c a l c u l a t e d d i s p e r s i n g numbers of f r y . Numbers of f r y were s t o r e d i n a matrix AREF . The s i m u l a t i o n was w r i t t e n such that when a land square was found (having an ATEST value of 0) i t was a u t o m a t i c a l l y skipped over. The same t h i n g happened i f the * In case t h e r e were more than 1 b i v a r i a t e d e n s i t y f u n c t i o n some squares would c a l l one matrix of p r o b a b i l i t i e s , other squares would c a l l another matrix of p r o b a b i l i t i e s . 98 water sguare being looked at had no f i s h in i t . To make sure that the f i s h , after assuming a uniform d i s t r i b u t i o n over the lake given long enough to disperse, would maintain t h i s uniform d i s t r i b u t i o n , I checked the mcdel before i t was used in the study. 10,000 f i s h were placed in each square of the lake and the simulation was then run through 3 it e r a t i o n s using the bivariate normal 'random' density function. Since there were s t i l l 10,000 f i s h in each lake sguare after 3 it e r a t i o n s t h i s proved that the model was working as desired. 99 APPENDIX 2 The Standard D e v i a t i o n of the Normal Density F u n c t i o n To achieve enough accuracy i n the model, I s e t the l i m i t s of the normal d e n s i t y f u n c t i o n to 3.0 standard d e v i a t i o n s from the mean. For the s t a n d a r d i z e d normal d e n s i t y f u n c t i o n (mean (u) = 0 , standard d e v i a t i o n (**) = 1.0) the p r o b a b i l i t y P(-3.0 < z < 3.0) = 0. 9974 . To achieve t h i s same p r o b a b i l i t y f o r l i m i t s (-0.625 , 0.625) one would have: P(-0.625 < x < 0.625) = 0.9974 p ('0.625- < z < 0.6ZS ) _ 0 > 9974 # Thus: 0^625 = 3 0 <T = °'6Q^ = 0.2083 100 APPENDIX 3 The Standard Deviation of the 5-Step A££I£xifflation to the Normal Density Function (Reference: Sokal and flohlf (1969)) Calculations for finding the standard deviation cf the 5-step approximation to the normal density function with bounds (-0.625 , 0.625) and standard deviation 0.2083 are as fellows: class i n t e r v a l = 0.25 -0.625 j -oT375 " -oTl25 " oTl25 > L o T 3 7 5 i V oT625 class c l a s s class class class mark i s mark i s mark i s mark i s mark i s -0.500 -0.250 0.000 0.250 0.500 class frequency coded f »u f • u 2 mark (x) (f) class mark (u) (class mark x4) -0.500 0.0346 -2 -0.0692 0. 1384 -0.250 0.2384 -2 -0.2384 0. 2384 0.000 0.4514 0 0.0000 0. 0000 0.250 0.2384 1 0.2384 0. 2384 0.500 0.0346 2 0.0692 0. 1384 0.9974 0.0000 0. 7536 (approx. 1.0) = N 2. where s u i s the variance of the ceded variate u. Thu s: 1.0 = 0.7536 - 0/1.0 = 0.7536 101 s u = J 0.7536 = 0.8681 To f i n d s x , the standard d e v i a t i o n of the grouped continuous v a r i a t e ( i . e . the standard d e v i a t i o n of the step approximation), decode: s = ^ , where c = 4, the coding constant. Thus, f o r the 5-step approximation: Since the standard d e v i a t i o n o f the normal density f u n c t i o n i s 0.2083 the e r r o r i n the step approximation i s : 0 . 2 1 7 0 - 0 . 2 0 8 3 X L 0 0 _ ^ 2 % 0 . 2 0 8 3 X 1 U U ~ 102 APPENDIX 4 Rates of Sjgeed of the Fry From the data given by Hoar (1954, F.78), the lowest number of 225 cm. excursions/15 minutes by sockeye fry i s 5.2 . Thus, in 15 minutes or 900 seconds, the fry swim 5.2 X 225 cm. = 1170.0 cm. . Therefore th e i r speed i s : The greatest number of 225 cm. excursions/15 minutes by sockeye f r y i s 10.2 . Thus, in 15 minutes or 900 seconds, the fry swim 10.2 X 225 cm. = 2295.0 cm. . Therefore th e i r rate of speed i s : c 900 sec. 2 2 ^ ' ° c m ' = 2.55 cm./sec. . In the same table. Hoar gives the mean ^ number of excursions/15 minutes for a l l observation periods as 8.7 . Thus, in 15 minutes or 900 seconds, the fry swim 8.7 X 225 cm. = 1957.5 cm. * Therefore their speed i s : 199ol'leT.' = 2 ' 1 7 5 C m ' / s e c - * 103 APPENDIX 5 The Calculation of the Average Speed Jmiles^iterationX of the Fry Dispersing Randomly The bivariate normal density function i s : 1 f(x,y) = ( s-2 TT <TX 0y<Jl-J> where x and y are normally distributed random variables with means u x and Uy , respectively, and standard deviations T"x and (Ty , respectively, and J> i s the cor r e l a t i o n c o e f f i c i e n t of x and y (Mood and G r a y b i l l 1963). For the purpose of this ,study the following values were set: u x = 0 Uy = 0 (Ty = 0.2083 <Ty = 0.2083 f - 0 Thus, the function reduces to: 104 f(x,y) = T 27T (0. 2083P 2 LlO.2083/ 10.2083/ This function was integrated over the 5x5 sguare grid and gave the following p r o b a b i l i t i e s over each square. 5 (-0.625, 0.625) (0.625, 0.625) T ' 1 0.0012901 f-0.0085603 r 0.0162171 0.0085603 0.0012901 0.0085603 0.0568001 0.1076071 0.0568001 0.0085603 0.0162171 0.1076071 +-0.2038600 0.1076071 0.0162171 0.0085603 0.0568001 0.1076071 0.0568001 0.0085603 0.0012901 0.0085603 4 0.0162171 0.0085603 0.0012901 (-0.625, 0.625) — j . ; i (0.625, -0.625) 5 The small probability values of the density function outside the grid were added to the outermost squares of the grid and a l l probability values were then adjusted s l i g h t l y so as to add up to exactly 1.0 . 105 Each probability was d i r e c t l y proportional to a mu l t i p l i c a t i v e coded number of f i s h . The distance from the dispersal center to the center of each square i s shown i n each square below, 0.7071 0.5590 0.5000 0.5590 0.7071 0.5590 0.3535 0.2500 0.3535 0.5590 0.5000 0.2500 0.0000 0.2500 0.5000 0.5590 0.3535 0.2500 0.3535 0.5590 0.7071 0.5590 0.5000 0.5590 0.7071 i When the corresponding pro b a b i l i t y and distance values of each square of the grid were multiplied together the following products resulted: 106 r T T T 1 : 1 | 0.0009 | 0.0048 | 0.0081 | 0.0048 | 0.0009 | I 4 4 H 4 - 4 | 0.0048 | 0.0201 | 0.0269 | 0.0201 | 0.0048 | | 0.0081 | 0.0269 | 0.0000 | 0.0269 | 0.0081 | r 4 4 -I 4 -i | 0.0048 | 0.0201 | 0.0269 | 0.0201 | 0.0048 | r 4 1- 4 4 4 | 0.0009 | 0.0048 | 0.0081 | 0.0048 | 0.0009 | i L 1 . 1 ; L : i These products added together to give the value 0.2624, which i s the average distance any coded number (derived from the probability values) of f r y would t r a v e l . 107 APPENDIX 6 The Lognorraal Dis t r i b u t i o n Applied to the Model (Ref erence: . Aitchison and Brown (1957) ) If a positive variate X (0 < x < co) i s considered such that Y = log X i s normally distributed with mean u and variance <T then X i s said to be lognormally d i s t r i b u t e d . The d i s t r i b u t i o n of X i s completely specified by the 2 parameters u and tr"2". A(x|u, <TZ ) and N(y|u, <TX ) can be used to denote the d i s t r i b u t i o n functions of X and Y, respectively, so that: When there i s no p o s s i b i l i t y of confusion the abbreviated forms A (x) and N (y) may be used for the d i s t r i b u t i o n functions. Since X and Y are connected by the re l a t i o n Y = leg X the d i s t r i b u t i o n functions of X and Y are related by: A(x|u, (T^ ) = P{ X < x } and N{y|u, (T 2 ) = P{ Y < y } A(x) = N (log x) (x > 0) hence, A(x) = 0 (x < 0) and -1 1 2^ - (log (x) - u) dA(x) = e dx (x>0) x • <T • which i s the lognormal density function. 108 An extension of the lognormal d i s t r i b u t i o n can be introduced to allow f o r both a lower and an upper bound to the possible values of the variate - thus, the creation of a 4-parameter lognormal density function. The transformation of the variate i s : X -7" X» = * _^ where T < x < Q Thus: = H» (log -jjp-J-) d log (-J^ l) 0-x 0-x Hence, the 4-parameter lognormal density function, f (u, <T , r , 0) , i s : f ( u , ( T , r , 6 7 ) = — ' — (x- r ) (0-x) (T'j2T For t h i s study the bounds of the lognormal density function were set at -3.0 and 3.0 for T and 9 , respectively, and the associated normal standard deviation ((T) was set at 1.0 fcr a l l values of the parameter u. These values were chosen to maintain consistency with the normal density function used in random dispersal of fry which had bounds of -0.625 and 0.625 (with respect to a dispersal center with coordinates (0,0) ) and a 109 standard deviation of 0.2083, or after standardization, bounds of -3.0 and 3.0 and a standard deviation of 1.0 . The lognormal density function was not 'reduced', merely to make calculations easier. For T = -3.0, G = 3.0, <T = 1.0 6 1 f( u,1.0,-3.0,3.0) = * -(x+3.0) (3.0-x) J27T • e The graphs of f for d i f f e r e n t values of u are shown in Figure 3 together with the normal density function for comparison. For negative values of u the lognormal density function i s skewed to the l e f t , for u = 0 i t i s symmetrical, and for positive values of u i t i s skewed to the r i g h t . Only density functions for u > 0 were considered for the study since the density functions for u < 0 are merely inverse r e f l e c t i o n s of those for u > 0 . To determine the amount of bias each skewed d i s t r i b u t i o n had, lognormal density functions with d i f f e r e n t values cf u were integrated over (-2.9999, 0.0). 6 From the area calculated under the curve between -2.9999 and 0.0 the r e l a t i v e bias of the 6 The i n t e r v a l should r e a l l y have been (-3.0, 0.0) but the function does not exist at -3.0 . The same problem arises when the function i s integrated over (0.0, 3.0) since the function does not exist at 3.0 either. f(-1.0,1.0,-3.0,3.0) — — f(1.0,1.0,-3.0,3.0) - - - - f(-0.4,1.0,-3.0,3.0) f(0.4,1.0,-3.0,3.0) n(0.0,1.0) Figure 3. 4—parameter lognormal density functions, f ( u , l . 0 , - 3 . 0 , 3 . 0 ) , together with the normal density function, n ( 0 . 0 , 1 . 0 ) . 11.1 density function in each half of the curve could be determined. Pertinent biases for d i f f e r e n t values of the parameter u are given i n Table 1. Table 1. Biases for d i f f e r e n t values of the parameter u. value of u area under curve between -2.9999 and 0.0 bias l e f t side right side of curve of curve 0.000 0. 126 0.253 0. 385 0.524 0.674 0.842 1.036 1 .281 0.500 0. 450 0.400 0. 350 0.300 0. 250 0.200 0. 150 0.100 50:50 45:55 40 :60 35:65 30:70 25:75 20:80 15:85 10:90 113 APPENDIX 7a BivafiSl® Log^orm^l-Norimal Density Functions To produce the bivar i a t e lognormal-normal density function used for direct 'up and down the lake' movement cf f r y , I multiplied the standard normal density function, n (0.0, 1.0), by various 4-parameter lognormal density functions which had dif f e r e n t values of the parameter u to create pre-determined amounts of b i a s . 7 A l l lognormal density functions had bounds of (-3.0,3.0) and an associated normal standard deviation of 1.0 to maintain consistency with the standard normal density function. The normal density function, n (0.0,1.0), was positioned along the Y-axis, and the lognormal density functions, f (u,1.0,-3.0,3.0), were positioned along the X-axis. Thus: n (0.0,1.0) = JTrr 1 6 1 f ( u,1.0,-3.0 ,3.0) = (x+3.0) (3.0-x) J 21T zX U o 9 ( | ^ # ) - « > l e 7 See Appendix 6. 114 n x f = e 21T (x + 3.0) (3.0-x) For various values of u the function, n x f, was integrated over a 5 sguare x 5 sguare gri d to f i n d the p r o b a b i l i t y in each square of the grid. Since the l i m i t s of the bivariate density function now were (-3.0,3.0) along both the X and Y axes, proportionately larger sguares had to be integrated over (1.2 x 1.2) so as to maintain consistency with the rest of the model i n which the dispersal grid assumed l i m i t s of (-0.625,0.625) on both axes. Probability values were adjusted s l i g h t l y to account for small probability values outside the grid (from the normal density function) and to ensure a l l p r o b a b i l i t i e s summed to 1.0 . The following figures (Figures 4 - 12) give the results obtained. 115 Figure 4. Probability values obtained using the bivar i a t e density function, n(0.0,1.0) x f (u, 1.0,-3.0,3.0): u = 1.2810 , with no rotation. Bias i s 10:90. 0.0001373510.00151000I0.00520183I0.01262086I0.01645741 0.0009112010.01001675I0.03450655I0.0837200010.10916980 y H  0.0017262010.0189763010.0653712010.1586046010.20681820 r +- -J 10.00091120|0.01001675i0.03450655|0.08372000|0.10916980 0.000137351 0.00151000 I 0.00520183I0.0126208610.01645741 L L. -J. J 116 Figure 5. Probabi l i t y values obtained using the bivariate density function, n(0.0,1.0) x f (u, 1.0,-3.0,3.0): u = 1.0360 , with no rotation. Bias i s 15:85. | 0.00027709|0.0024 0772|0.00680632|0.01339270|0.0 130 4392| 10.0018 378010.01597 130|0.04514935|0.08883955|0.08652605| |0.00348170|0.03025710|0.08553380|0.16830330|0.1639 2050| I 0.0018378010.01597130|0.04514935|0.08883955|0.08652605| |0.00027709 | 0.00240772|0.0068063210.01339270|0.01304392| 117 Figure 6. Probability values obtained using the bivariate density function, n(0. 0,1.0) x f (u, 1.0,-3. 0,3. 0) : u= 0.8420 , with no rotation. Bias i s 20:80. r + -0.00046451 0.00308160 0.00583800 0.00308160 0.00046451 0.00334786 0.02220775 0.04207170 0.02220775 0.00334786 0.00808770 0.05364915 0.10163640 0.05364915 0.00808770 0.01349647 0.08952770 0.16960700 0.08952770 0.01349647 0.01053121 0.06985790 j 0.13234320 0.06985790 0.01053121 118 Figure 7. Probability values obtained using the bivariate density function, n(0. 0,1.0) x f (u, 1.0,-3. 0,3.0) : u = 0.6740 , with no rotation. Bias i s 25:75. 0.00070721I0.00432945I0.0091239210.0132111410.00855593 h- | 4 | I I I 0.00469140|0.02871920I0.06052310 I 0.08763515I 0.05675530 .4 + a 0.0088878010.0544 074010. 11465880I0. 16602160|0. 1075 2080 0.0046914010.0287192010.0605231010.0876 351510.05675530 I I I I 0.00070721I0.00432945I0.0091239210.01321114! 0.00855593 1 j 119 Figure 8. Pro b a b i l i t y values obtained using the bivariate density function, n(0. 0,1.0) x f (u, 1. 0,-3. 0,3. 0) : u = 0.5240 , with no rotation. Bias i s 30:70. 0.00100768 0.00668440 0.01266340 0.00668440 r 0.00100768 0.00532722 0.03533775 0.06694600 0.03533775 0.00532722 0.00993392 0.06589595 0.12483750 0.06589595 0.00993392 0.01267946 + -j 0.08410805 0.15933970 0.08410805 4 4 4 0.01267946 0.00697952 0.04629790 0.08770970 __4 ^ 0.04629790 0.00697952 ~ L 1 I / 120 Figure 9. Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u, 1. 0,-3. 0,3. 0) : u = 0.3850 , with no rotation. Bias i s 35:65. 0.00137443 0.00911730 0.00633664 0.04203365 0.01054613 0.06995645 0.01198175 0.07947950 0.00568895 0.03773715 0.01727240 0.07963120 0.13253000 0.15057100 0.0091 1730 0.04203365 0.06995645 0.07947950 0.00137443 0.00633664 0.01054613 0.01198175 0.07149 150 4 0.03773715 0.00568895 .JL I 121 Figure 10. Probability values obtained using the bivariate density function, n(0. 0,1.0) x f (u, 1. 0,-3. 0,3. 0) : u= 0.2530 , with no rotation. Bias i s 40:60. 0.00181709 0.01205345 0.02283480 0.01205345 0.00181709 0.00734883 0.04874760 0.09235040 0.04874760 + ^ 0.00734883 0.01097496 0.07280070 0.01116878 + 0.07408645 0.13791840 0.07280070 0.01097496 0.14035400 0.07408645 0.01116878 0.00461844 0.03063575 4 0.05803830 0.03063575 - 4 -i 0.00461844 122 Figure 11. Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u, 1.0,-3.0,3.0) : u= 0. 1260 , with no rotation. Bias i s 45:55. j + _ +. 0.00234358 0.01554570 0.02945080 0.01554570 r +-0.00234358 0.00834674 0.05536700 0.10489080 0.05536700 0.00834674 0.01122778 4 4 4 0.07447800 0.14109580 0.07447800 0.01122778 0.01028123 0.06819930 0.12920110 0.06819930 _ 4 4 _ x 0.01028123 0.00372872 0.02473395 0.04685750 0.02473395 0.00372872 12J Figure 12. Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0): u= 0.0000 , with no rotation. Bias i s 50:50. I I 0.00297587|0.00933195|0.01131246 I I 0.01974010|0.06190205|0.07503960 I I 0.03739695|0.11727115|0.14215980 I I I I I I 0.019740 1010.06190205|0.07503960 r + 4-I 0.00297587|0.00933195|0.01131246 0.00933195 0.06190205 0. 117271 15 0.06190205 0.00933195 0.00297587 4 0.01974010 0.03739695 0.01974010 0.00297587 L L L. 124 To find the average distance travelled by fry in the bias dir e c t i o n per i t e r a t i o n of the simulation for each bivariate function, I f i r s t multiplied the p r o b a b i l i t i e s (propcrticnal tc a coded number of fish) of the sguares for each bivariate function on the bias side of the dispersal grid by the corresponding distance that the center of each sguare was away from the center of d i s p e r s a l . 8 Only half the p r o b a b i l i t i e s in the middle column of sguares was used in order to obtain, ultimately, a better approximation of the average distance t r a v e l l e d by fry only in the biased d i r e c t i o n . These products were then summed and divided by the proportion of f i s h t r a v e l l i n g i n the bias d i r e c t i o n for the given bivariate function. This r e s u l t thus gave the average distance per i t e r a t i o n t r a v e l l e d by f r y going in the bias d i r e c t i o n fcr each bivariate function. Example: For a biased d i s t r i b u t i o n of 10:90 there are the following p r o b a b i l i t i e s on the bias side: 8 See Appendix 5. 125 T — T T— |0.00260091|0.01262086|0. 01645741| J ! |0.01725327|0.08372000|0. 10916980| . , , . , , . , J 1, „ „_ ,, i |0.03268560|0.15860460|0. i 206818201 r L ! |0.01725327|0.08372000 J O . 10916980 | — - l 10.00 2600 91 |0.01262086|0. 01645741| The distance from the dispersal center to the center of each square i s shown in each square below: 126 0.5000 0.5590 0.7071 0.2500 0.3535 0.5590 + +  H 0.0000 0.2500 0.5000 4 -I 4 0.2500 0.3535 0.5590 I 4 -+-0.5000 0.5590 0.7071 When the corresponding pr o b a b i l i t y and distance values of each sguare were multiplied together, and the products then summed, the result was 0.373 . This value, 0.373, was then divided by 0.9 (since there was a 90% bias) to give an average distance t r a v e l l e d by fry in the bias d i r e c t i o n per i t e r a t i o n of 0,414 (miles). The average distance t r a v e l l e d by fry in the bias d i r e c t i o n per i t e r a t i o n for a l l biases i s given in Table 2. 127 Table 2. Average distance t r a v e l l e d by f r y in the bias d i r e c t i o n per i t e r a t i o n for varying biases. bias average distance travelled by fry in bias d i r e c t i o n per i t e r a t i o n (miles/iteration) 10:90 15:85 20: 80 25:75 30: 70 35:65 40:60 45:55 50:50 0. 414 0.391 0.371 0.356 0.343 0.332 0.322 0.315 0.306 128 APPENDIX 7b Bivariate Lognormal-Norma1 Density Functions J-30 degrees rotation^ To provide a rotation of the logncrmal-ncrmal bivariate density function, I used the equations for rotation cf axes. For new X'-Y' axes where <x i s the angle of rot a t i o n : y' = y coscx - x sin oc Since a rotation of -30 degrees of the lognormal-normal bivariate density function was desired this gave: x« = x cos{-30") + y sin (-30°) = 0.8660 x - 0.5000 y y« = y cos (-30°) - x sin (-30°) = 0.8660 y + 0.5000 x The biva r i a t e density function from Appendix 7a: x cos cx + y s i n (x 6 n x f e 2 7T (x+3.0) (3.0-x) in terms of the new coordinates i s : 129 (0.8660 y + 0.5000 x ) 2 1 2 i i x f = • e Jrir . 6, (3.0 - (0.8660x-0.5000y) ) ' ( (0. 8660x-0. 5000 y) +3.0) Z±T \ o a /(0. 8660X - 0.5000y + 3.0) \ _ ~r\Z 2 L °\ 3.0 - (0.8660x - 0.5000y)/ J e For various values of u the new 30 degrees rotation function, n x f, was integrated over a 5 square x 5 sguare grid to f i n d the probability in each sguare cf the g r i d . Treatment of the problem was similar to the treatment given the problem for the bivariate density function, n x f, in Appendix 7a. Figures 13 - 21 give the re s u l t s obtained. The average distance t r a v e l l e d by the fry in the bias direction per i t e r a t i o n for a 30 degrees rotation function was the same as for the non-rotation bivariate density function with the same bias. Thus, new calculations i n t h i s regard were not necessary. 130 Figure 13. Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u, 1. 0,-3. 0,3.0) : u= 1.2810 , with the -30 degrees rotation. Bias i s 10:90. 0.0000118 0.0006633 0.0022502 0.0013652 0.0010596 0.0095163 0.0206842 0.0139144 0.0028697 0.0243960 0.0683876 0.0677167 0.0022936 0.0293482 0.1272561 0.1960971 0.0004176 -I 0.0186530 0.1282100 0.1585389 0.0001 142 0.0030057 0.0240019 0.0820870 0.0171417 i 131 Figure 14. Probability values obtained using the bivariate density function, n(0. 0,1.0) x f (u, 1.0,-3.0,3.0) : u = 1.0360 , with the -30 degrees rotation. Bias i s 15:85. 0.0000302 0.0014090 0.0041948 I +-0.0022581 0.0001691 0.0020307 0.0162264 0.0316945 0.0193264 0.0037955 0.0045277 0.0347388 0.0883917 0.0793127 0.0252205 0.0030485 0.0353212 0.1376137 4 -0.1868886 0.0680412 0.0004824 0.0186702 0.1092355 j, 0.1165342 4 x 0.0108384 .x. 132 Figure 15. Probability values obtained using the bivariate density function, n(0. 0,1.0) x f (u, 1.0,-3. 0,3. 0) : u = 0.8420 , with the -30 degrees rotation. Bias i s 20:80. 0.0000613 r 0.0024694 I + 0.0066254 r 0.0032388 0.0002217 0.0032800 0.0238692 0.0427975 0.0241309 0.0043936 0.0062580 0.0442499 -4 + H 0.1042697 0.0865516 0.0252678 0.0036756 0.0393813 0.1410735 0.1736795 0.0567520 0.0005198 0.0180096 0.0932294 0.0887067 0.0072878 133 Figure 16. Probability values obtained using the bivariate density function, n(0.0,1.0) x f(u,1.0,-3.0,3.0): u = 0.6740 , with the -30 degrees rotation. Bias i s 25:75. 0.0001103 r 0.0039137 0.0095919 r 0.0043079 0.0002724 0.0048433 0.0324782 0.0540299 0.0284553 0.0048513 0.0080620 0.0530994 0.1170643 0.0908456 0.0246378 0.0042049 0.0421080 0.1403354 0.1589034 0.0473597 0.0005388 0.0169994 0.0794107 0.0685345 0.0050419 134 Figure 17. Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0): u = 0.5240 , with the -30 degrees rotation. Bias i s 30:70. 0.0001823 |0.0067203 10.0104337 10.0046373 10.0005437 I 1 0.0057852 10.0418614 10.0611248 |0.0437259 0.0158015 0.0130722 10.0650913 10.1269729 10.1366867 10.0675075 0.0054371 10.0322446 10.0927942 10.1437730 |0.0534561 0.C003200 J 0.0051830 10.0235701 10.0395195 |0.0035557 135 Figure 18. Probability values obtained using the bivariate density function, n(0. 0,1.0) x f (u, 1.0,-3. 0,3.0) : u= 0.3850 , with the -30 degrees rotation. Bias i s 35:65. 0.0002857 0.0089609 0.0117516 0.0049903 0.0005386 0.0081814 0.052101 1 0.0684585 0.0445102 0.0145219 0.0171401 0.0760542 0.1345718 0.1311650 0.0572175 0.0066326 0.0355867 0.0930321 0.1289583 0.0418663 4. 4 4 0.0003648 0.0054158 0.0222434 0.0329187 0.0025325 136 Figure 19. Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u, 1. 0,-3. 0,3.0) : u= 0. 2530 , with the -30 degrees rotation. Bias i s 40:60. 0.0004306 0.0111981 r 0.0218261 0.0078785 0.0004060 0.0115966 0.0631152 0.0867210 0.0384284 0.0055513 0.0136155 0.0749704 + 0.1398410 0.0917114 0.0207075 0.0052609 0.0445144 0.1240650 0.1144819 4 0.0272630 0.0005244 0.0131844 0.0481778 0.0327243 0.0018063 .JL J 137 Figure 20. Probability values obtained using the bivariate density function, n (0. 0,1.0) x f (u, 1. 0,-3. 0,3. 0) : u = 0. 1260 , with the -30 degrees rotation. Bias i s 45:55. 0.0006301 0.0146600 0.0154573 0.0054518 0.0005032 0.0149435 0.0748301 0.0806063 0.0438557 0.0118388 0.0271624 0.0969505 0.1429468 0.1158779 0.0402884 0.009161 1 r 0.0004430 0.0407597 0.0891173 0.1006590 4  0.0055998 0.0190468 0.0224356 0.0254881 4 0.0012868 138 Figure 21. Probability values obtained using the bivariate density function, n(0.0,1.0) x f (u,1.0,-3.0,3.0): u = 0.0000 , with the -30 degrees rotation. Bias i s 50:50. 1 • T - ! I o. 0009067 |0 1 .0182557 10.0172814 L |0. i 0055658 |0.0004758 ~~\ 10. 0196408 T |0 .0873859 T •0.0853692 T |0. , 1 0425872 |0.0104888 10. 0332989 |0 1 .1067508 |0.1439860 T |0. 1 1067508 |0.0332989 |0. 1 0104888 -+— 10 1 .0425872 |0.0853692 1 |0. 1 0873859 _ jj 10.0196408 J L _ 1— |0. 0004758 -+-|0 .0055658 | .in i , : I - I . _ ,_ _ - -|0.0172814 1 — 10. 0182557 |0.0G09067 L L j L _ J L _ _ 139 APPENDIX 8 Programs and Data Sets A l l programs and data sets were contained in f i l e s cn disc and are referred to by t h e i r f i l e names. A l l programs and data sets are presently saved on magnetic tape stcred at the Computer Center, University of B r i t i s h Columbia. The programs are: S A V E U AMFUPL SAVE5 AMFUPL2 S A V E 6 PROB QPRCB CALC QCALC CHI D U B P 1 DUMP2 The reguired data sets are: > CODES FOFDS NDISTN LOG NDIST N D1090 D2080 D3070 A l l programs include extensive comments f u l l y explaining the steps taken. A description of the data set, CODES, i s found i n the f i l e , SUMDATA; a description of the data set, FOFDS, i s found i n the f i l e , NOTE; a description of the data set, NDISTN, i s found in the f i l e , REMARK; and a decription cf the data set, LOGNDISTN, i s found in the f i l e , MESSAGE. SAVEU, SAVE5, AUD SAVE6 are the main programs generating the dispersal of f r y in periods 1,2, and 3, respectively. A l l 3 programs are actually s l i g h t l y d i f f e r e n t versicns of the same 140 one. They d i f f e r mainly i n how the i n i t i a l number of f i s h are placed i n the rectangular coordinate system of the lake f o r a given period and how the p r o b a b i l i t i e s are read for *up and down the lake* movement of f r y . PROB, QPROB, CALC, QCALC, CHI, DUMP 1, and DUMP2 are a l l subroutines c a l l e d by the main programs. AMFDPL and AMFUPL2 are self-contained programs, completely separate from the main programs and the subroutines. Part of AMFUPL2 i s found i n the f i l e AREA8AREA9 . AMFUPL and AMFUPL2 manipulate numbers of f i s h from period 1 and period 2, respectively, to generate better dispersal patterns. As mentioned in the main text, t h i s involved combining 90% cf the f i s h of the 10:90 biased dispersal run with 10% cf the f i s h cf the random dispersal run for period 1. It involved combining 50% of the f i s h of the 10:90 biased dispersal run with 50% of the f i s h of the random dispersal run for period 2. The data sets CODES, FOFDS, HDISTN, and LOGNDISTS are used by a l l 3 main programs. The data set, CODES, as well as being contained in i t s own f i l e , i s also generated by the •summary of lake data program* contained in the f i l e , SUMDATA. LOGNDISTM contains p r o b a b i l i t i e s and dispersal rate 'CL0CK1 • values for biases 10:90, 15:85, 20:80, 25:75, 30:70, 35:65, 40:60, 45:55, and 50:50, as well as the reverse biases 55:45, 60:40, 65:35, 70:30, 75:25, 80:20, 85:15, and 90:10. In the model, however, only the p r o b a b i l i t i e s for biases 10:90, 20:80, and 30:70 were actually ever used. Thus, tc make programming easier, p r o b a b i l i t i e s and dispersal rate 'CLOCK 1' values f o r biases 141 10:90, 20:80, and 30:70 were put i n f i l e s D1090, C2080, and D3070, r e s p e c t i v e l y . i A l l programs had to be compiled before they c o u l d be run. A l l compiled programs were loaded i n t o f i l e s of t h e i r own. Below i s a complete l i s t of the source f i l e names and the names of the o b j e c t f i l e s the programs were loaded i n t o a f t e r they were compiled. SOURCE FILE SAME SAVE4 SAVE5 SAVE6 PROB QPROB CALC QCALC CHI DUMP 1 DUMP2 AMFUPL AMFUPL2 OBJECT FILE NAME S AV E4 0 SAVE50 S AVE60 PROBQPROBO PBOBCPROEO CALCO QCALCO CHIO DUMP 10 DUMP20 AMFUPLO AMFUPL20 Each s i m u l a t i o n run r e q u i r e d a s e r i e s of cards, i n c l u d i n g data c a r d s , t o be read. F o l l o w i n g i s a l i s t i n g of the cards needed i n each run. A sample p r i n t o u t of the random run f o r period 1 i s found i n the f i l e PRINTOUT. CUTS X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X I X X X X X X X X X X X I X I X I X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X I X X X X X X R F S H O . 6 7 3 1 1 0 O N I V B R S I T T O P B C C O R P O T I E G C E R T R E ( I T S (DL181) , 1 7 : 1 1 : 5 6 P R I A O G 0 2 / 7 * S S I G S M I S P R I N T « T H F O R M » B L A N K • • L A S T S I G M O N H A S : 1 7 : 1 1 : 3 1 S S S S S S S S S S M M M M i n r i i i m S S S S S S S S S S S S S S S S S S S S S S H M R B M M I I I I I I I I I I S S S S S S S S S S S S ss ss M H M H M M M H I I S S S S ss nn H R M H B M I I ss sss tin N H H H M M I I sss sssssssss M M H R nn I I sssssssss sssssssss M M M M I I sssssssss sss M M R H I I sss ss M M M M I I ss ss ss R H M R I I ss ss S S S S S S S S S S S S M M R H I I I I I I I I I I S S S S S S S S S S S S S S S S S S S S S S M M H R I I I I I I I I I I S S S S S S S S S S U S E R " S H I S " S I G N E D O H AT 1 7 : 1 1 : 5 6 O R P R X A U G 0 2 / 7 4 S L I S T RUMS 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 21 22 21 2<t 25 26 27 28 29 30 31 32 33 31 35 36 37 38 39 U0 4 1 02 4 3 44 45 46 47 48 49 50 5 1 52 53 54 55 56 57 58 59 60 6 1 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 RONS FOR PERIOD 1 NOTE: SAYE4, AS COMPILED FOR THE RAN DOR DISPERSAL RUN, CALLS THE SUBROUTINE "DUHP2" FOR STORING RESULTS. IF RANDOR DISPERSAL RESULTS ARE TO BE SAVED ALONG WITH RESULTS FROM ONE OF THE BIASED DISPERSAL RUNS THEN THE SUBROUTINE "DUMP1" MUST BE CALLED TO SAVE THE RESULTS FROM ONE OF THESE BIASED DISPERSAL RUNS. CONSEQUENTLY, SAVE4 MUST BE RECOMPILED AFTER THIS CHANGE TO THE PROGRAM HAS BEEN MADE. FOR THIS PERIOD RANDOM DISPERSAL RESULTS WERE STORED TN THE FILE "DF2" AND DIASED DISPERSAL RESULTS WERE STORED IN THE PILE "DF1". RESULTS WERE SEVER ACTUALLY STORED FOR THE DISPERSAL RUNS WITn A 10:70 AND 20:R0 BIAS. DISPERSAL OF FRY NITn NO BIAS. $RUN SAVE40*CALCO«QCALCO*PHOBQPHOBO»CHIUBJ«DUMP20 4*C0DES 3=FOFCS 7-NDISTR 2-DF2 260 131218256218 01 6 171 1000. 1000. 1000. 1000. 3.18 1. SENDFILE 1000000. 1000. 1000. 1000. 1000. 56.66 ITMLP IRITES AND IDUMP ( 4 3-PLACE INTEGERS) N O D C DISPERSAL CENTER COORDINATES AND NUMBER OF FRY 1000. 1000. 1000. 211.90 381. 13 1000. 1000. 1000. 315.19 FMORT 1000. 1000. 1000. 10.86 TTH TYH TYM TYH 7.04 14.04 1000. DISPERSAL OP FRY 8ITH A 30:70 BIAS DOWN THE LAKE. •»*••**•***••••*••••••**• SRUN SAVE4O»CALC0»QCALCO*PROB0PR0B0«CHI0BJ»DUMP10 4'CODES 3*FOFCS 7»D3070 1-DF1 200 100167196000 01 ITMLP IRITES AND I DUMP ( 4 3-PLACE INTEGERS) NO D C 171 1000000. DISPERSAL CENTER COORDINATES AND NUMBER OF FBI 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. TYM 18 56.66 211.90 381. 13 315. 19 10.86 7.04 14.04 1000. 1. FMORT SENDFILE DISPERSAL OF FRY WITH A 20:80 BIAS DOWN THE LAKE. **•*•»*»*«»»•»»«•»»••«••• SRUN SAVE40»CALCO»QCALCO»PROBQPROBO*CHIOBJ+DUHP10 4 = CODES 3-F0FCS 7*02080 1-DF1 190 93155181000 01 ITMLP IRITES AND IDUMP ( 4 3-PLACE INTEGERS) NODC 171 1000000. DISPERSAL CENTER COORDINATES AND NUMBER OF FBI 1000. 1000. 1000. 1000. 1000. TIN 1000. 1000. 1000. 1000. 1000. TYH 1000. 1000. 1000. 1000. 1000. TYH 1000. 1000. TYH 18 56.66 211.90 381. 13 315. 19 10.86 7.04 14.04 1000. 1. FIIOBT SENDFILE 81 82 83 8<l 85 86 87 88 89 90 9 1 92 93 94 95 96 97 98 99 100 10 1 102 103 10i» 105 106 107 108 109 110 111 112 113 111 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 1*0 DISPERSAL OF FBI K I T H A 10:90 BIAS DOWN THE LAKE. *•*«»«*«*****«»•«•«»*•«•• JRUN SAVE40*CALC0»Q,C»LC0»PR0BQPR0B0*CHI0BJ + DUMP10 4=C0CIS 3=FOFDS 7=D1090 1«DF1 170 83139163150 01 6 171 1000. 1000. 1000. 1000. 3.18 1 . SENOF1LE 1000000. 1000. 1000. 1000. 1000. 56.66 ITMLP I RITES • AND IDUI1P ( 4 3-PLACE INTEGERS) NODC DISPERSAL CENTER COORDINATES AND NUMBER O F F B I 1000. 1000. 1000. TYH 1000. 1000. 1000. TYH 1000. 1000. 1000. T I N T T H 211.90 381. 13 315. 19 10.86 7.04 14.04 1000. FHOBT C R E A T I O N O F C O H B I N E D D I S T R I B U T I O N P A T T E R N F O R P E R I O D 1 . f R U N A H P U P L O 1 - D F 1 2 - D P 2 1 2 - C F 3 4 = C 0 D E S •***»»*»•«»**••»**»*«**• RUNS FOB PERIOD 2 •**•*»••**«**«••**«»•*«»»••••••• N0T2:SAVE5, AS COMPILED FOR A BIASED DISPEBSAL RUN, CALLS THE SUBROUTINE "DUHP1" FOR STORING RESULTS. IF RESULTS OF ONE OF THE BIASED DISPERSAL RUNS AR R TO BE SAVED ALONG WITH RESULTS FROM THE RANDOM DISI'EHSAL RUN THEN THE SUBROUTINE "DUMP2" MUST BE CALLED TO SAVE THE RESULTS FROM THE RANDOM DISPERSAL RUN. CONSEQUENTLY, SAVES MUST UE RECOMPILED AFTER THIS CHANGE TO THE PROGRAM HAS EEF.N HADE. FOR THIS PERIOD BIASED DISPERSAL RESULTS HERE STORED IN THE FILE "DF4" AND RANDOM DISPERSAL RESULTS HERE STORED IN THE FILE "DF5". RESULTS HERE NEVER ACTtlALLI STORED FOR THE DISPERSAL RUN WITH THE 30:70 BIAS. DISPERSAL OF FRT WITH NO BIAS. »«*•••**»****»***»»*»**•*»•*•*»»«**««»•»«»»• SRUN SAVESO*CALC0>QCALCO»PHOBQPROB0»CHI0BJ*DUHP20 4'CODES 3=FOFCS 7»NDISTN 12=DF3 2=DF5 375 ITMLP 189315369315 IRITES AND IDUMP ( 4 3-PLACE INTEGERS) 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 310.64 226.93 70.68 245.23 1. S E H D F I L E 1000. 1000. 1000. 1000. 1000. 1000. 80.19 30.98 F H O R T T Y H T I N T Y H T Y H 11.97 23.37 1 0 0 0 . D I S P E B S A L O F F B I W I T H ft 3 0 1 7 0 B I A S U P T H E L A K E . » » • • » . • • • • • » » • » » • • • • » • » • » • • SBOI S A V E 5 O * C A L C O « Q C A L C 0 » P B O B Q P R O B 0 » C H I 0 B J » D D M P 1 0 4 - C 0 D E S 3 - F 0 F D S 7*03070 mi 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 16U 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 1U0 181 182 183 1B4 185 186 187 188 189 190 19,1 192 193 194 195 196 197 198 199 END OF 12-DF3 290 144242282000 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 310.64 226.93 70.68 245.23 1. SENDFILE ITMLP IRITES AND IDUMP ( 4 3-PLACE INTEGERS) 1000. 1000. 1000. 80. 19 FMORT 1000. 1000. 1000. 30.98 TTH TTH TTH TYM 11.97 23.37 1000. DISPEUSAL OF FRY WITH A 10:90 DIAS UP THE LAKE. SHUN S*VESO«CALCO«QCALC0*PROBQPRO0O«CHI0BJ»DUHP1O 4«CODES 3»FOFDS 7-D1090 12 = DF3 1=0F4 240 ITMLP 120200234130 IRITES AND IDUMP ( 4 3-PLACE INTEGERS) 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. TYM 310.64 226.93 70.68 245.23 80.19 30.98 11.97 23.37 1000. 1. FMORT SENDFILE CREATION OF COMBINED DISTRIBOTION PATTERN FOB PERIOD 2. ***•»»»•«*»•••»•«*»• SRUN AHFUPL20 1=DF4 2=DP5 12»DF6 4=CODES RUNS FOB PERIOD 3 NOTE: SAVE6 HAS ONLY COMPILED FOR THE RANDOM DISPERSAL RON. HO RESULTS HERE STORED FOR THIS PERIOD. DISPEUSAL OF FRY WITH NO BIAS. *•»••»*•*••*«•«**•»•*»»•»•*••••»•••*»•»*•«»• SRUH SAVE60«CALCO*QCALCO+PROBQPROBO+CHIOBJ 4=CODES 3=FOFDS 7-NDISTH 12-DF6 475 ITMLP 240402472000 IRITES AND IDUMP ( 4 3-PLACE IHTEGERS) 1000. 1000. 1000. 1000. 1000. TYH 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. 1000. 1000. 1000. TYM 1000. 1000. TYH 204.92 132.74 243.96 123.85 175.38 62.08 30.09 26.99 1000. 1. FMORT SENDFILE »»•»»•»«»•«««*•«*•*«»•*»****•»«»«•»••*»•»*•««*»«»•««*»*•»•»»»*»»»*»»««»»«»»« FILE 146 APPENDIX 9 Printout of the Lake Coordinate System This appendix contains a printout of the lake rectangular coordinate system; each »*» represents a land square and each »1* represents a water square. Each row of the grid i s numbered on the l e f t . The column ranqe i s 1 - 70. C U T 3 X X X X B F S N O . 6 X X X X X X X X X X I X X I X X X X X X X X X X X X I X I X I X X X X X X X X X X X X X X X X X X X I X X X X I I X Z I X I X I X X X X X X X X X X X I X X X X X X Z X X X X X X X X X X X I X I I X X X X X I X X X X X I I I X X X X X X X X I X X 7 1 3 2 9 0 H I V E R S I T T OF B C CONFUTING C E H T B B N T S ( U L 1 B 4 ) 1 1 : 4 1 : 0 9 THO 106 0 8 / 7 4 S S I G s n i s P I ) I N T = T N F0RH = B L A N K • ' L A S T S I G N O N H A S : i i : 1 2 : 1 4 S S S S S S S S S S no nn I I I I I I I I I I S S S S S S S S S S s s s s s s s s s s s s nnn n n n I I I I I I I I I I S S S S S S S S S S S S s s s s H M H H H H H H I I s s s s s s M H HB H B nn I I s s s s s nn H H H H nn I I s s s s s s s s s s s s n a HB n n I I s s s s s s s s s s s s s s s s s s nn nn X I s s s s s s s s s s s s M B n n I I s s s s s n n n n I I s s s s s s n a nB I I s s s s s s s s s s s s s s s s n n B B I I I I I I I I I I s s s s s s s s s s s s s s s s s s s s s s HB BH I I I I I I I I I I s s s s s s s s s s OSBB " S H I S " S I G N E D ON AT 1 1 : 4 1 : 0 9 ON THO AUG 0 8 / 7 4 t B U N • F T N P A R = S O U B C E » S T S T E M B X ECOTIOB B E G I N S MICHIGAN TEHHINAL SYSTEM FORTRAN G (41336) MAIN 08-08-74 2 1 : 4 0 : 0 6 PAGE P001 0008 0009 0010 00 1 1 0012 00 13 0014 0015 0016 0017 0018 0019 0020 c c c c c c c c c c c THIS PROG RAH PRINTS THE LAKE COORDINATE SYSTEM. EVERY STAR OR NUMBER (1) REPRESENTS A COORDINATE PAIR. n * " «S ARE LAND SQUARES AND "1" 'S ARE HATER SQUARES. ALL ELEMENTS OF THE MATRIX NA ARE FIRST GIVEN A VALUE OF 1000000 , TOO LARGE A VALUE FOR THE FORMAT FOR PRINTING NA VALUES. THEN, COORDINATES (I,J) OF WATER SQUARES ARE READ FROM THE F I L E "CODES " AND ELEMENTS NA(I.J) ABE GIVEN A VALUE OF 1 . THUS,WHEN THE COORDINATE SYSTEM IS PBINTED , THE " 1 " «S H ILL BE PRINTED AND "*" «S HILL BE PR INTED EVERYHHEBE E L S E . 0001 DIMENSION NA (70,332) 0002 DO 23 J = 1 , 3 3 2 0003 DO 24 1=1,70 0004 NA (I,J) = 1000000 0005 24 CONTINUE 0006 23 CONTINUE 0007 DO 33 K=1,3031 C ON IT 4 = CODES READ ( 4 ,1114) I , J 1114 FORflAT (5X , I2 ,1X , I3 ) NA (1*2,J+2) = 1 CONTINUE . HRITE (6,3) F O R M A T ( M » , 1 5 X , ' B A B I N E LAKE RECTANGULAR COORDINATE SYSTEM ' , / / / ) DO 10 J = 1 , 3 3 2 J J = 3 3 3 - J HRITE (6,4) J J , (NA ( I , J J ) , 1=1,70) FORMAT ( • 9 1 5 X , I 3 , 5 X , 7 0 1 1 ) CONTINUE STOP EN D •OPTIONS IN EFFECT* ID ,EBCDIC,SOURCE,NOL1ST,NODECK,LOAD,HONA P •OPTIONS IN E F F E C T * NAME = MAIN 1 , L INECNT « • 57 • S T A T I S T I C S * SOURCE STATEMENTS = 20,PBOGRAM S I Z E » 93730 • S T A T I S T I C S * NO DIAGNOSTICS GENERATED NO ERRORS IN MAIN 33 4 10 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9. 000 10.000 11.000 12.000 13.000 14.000 15.000 16.000 17.000 18.000 19.000 20.000 21.000 22.000 23.000 24.000 25.000 26.000 27.000 28.000 29.000 30.000 31.000 32.000 NO STATEMENTS FLAGGED IN THE ABOVE COMPILAT IONS. 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