{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Zoology, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Ssentongo, George William","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2011-05-03T21:49:42Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"1971","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"Simple algebraic relationships and yield equations that require the minimum of data are developed so as to enable quick and reliable assessments of relative rate of harvesting tropical freshwater fish populations.\r\nThe age of a fish at the inflexion point is inversely related to the growth rate (K) and directly related to the natural logarithm of the weight length exponent (b).\r\nAlgebraic relationships between the exponent of anabolism (m) and the weight length exponent are developed.\r\nEquations for estimating total mortality from age and length distributions in catch samples are given. Total mortality for both continuous and discrete recruitment are considered. The probability density function and the discrete probability function for a negative exponential are given.\r\nThe effect of a number of variables on trawl catches is studied and some multiple regression equations which might be used to assess the relative degree of exploitation are presented.\r\nThe parameters which have been recognized as vital to yield prediction are: the growth rate (K), the weight length exponent (b), the maximum length to which a fish grows (L\u221e) and the natural and total mortality rates. It is shown that one can replace age with a length expression in yield models and still have reliable yield predictions. It is also shown that a model with a few very relevant parameters, has almost the same predictive power as a model requiring more parameters.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/34234?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"YIELD EQUATIONS AND INDICES FOR TROPICAL FRESHWATER FISH POPULATIONS by GEORGE WILLIAM SSENTONGO B. Sc. Univers ity of East A f r i c a , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Zoology We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1971 In presenting th i s thesis in p a r t i a l f u l f i lment of the requ i re-ments for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree l y ava i lab le for reference and study. I further agree that permission for extensive copying of th i s thesis fo r scho lar ly purposes may be granted by the Head of my Department or by his representative. It i s understood that copying or publ icat ion of th i s thesis fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permission., Department of Zoology The Univers i ty of B r i t i s h Columbia Vancouver 8, B r i t i s h Columbia Canada. ABSTRACT Simple algebraic re lat ionsh ips and y i e l d equations that require the minimum of data are developed so as to enable quick and r e l i a b l e assessments of r e l a t i v e rate of harvesting t rop ica l freshwater f i s h populations. The age of a f i s h at the i n f l e x i on point i s inversely re lated to the growth rate (K) and d i r e c t l y re lated to the natural logarithm of the weight length exponent (b). Algebraic re lat ionsh ips between the exponent of anabolism (m) and the weight length exponent are developed. Equations for estimating to ta l morta l i ty from age and length d i s t r i bu t i on s in catch samples are given. Total morta l i ty fo r both continuous and d i sc rete recruitment are considered. The p robab i l i t y density funct ion and the d i sc rete p robab i l i t y function for a negative exponential are given. The e f fec t of a number of var iables on trawl catches i s studied and some mul t ip le regression equations which might be used to assess the r e l a t i v e degree of exp lo i ta t i on are presented. The parameters which have been recognized as v i t a l to y i e l d predict ion are: the growth rate (K), the weight length exponent (b), the maximum length to which a f i s h grows (L<\u00bb) and the natural and to ta l mor ta l i t y rates. i i i i i I t i s shown that one can replace age with a length expression in y i e l d models and s t i l l have r e l i a b l e y i e l d predict ions. I t i s also shown that a model with a few very relevant parameters, has almost the same pred ic t i ve power as a model requir ing more parameters. TABLE OF CONTENTS Pafle LIST OF TABLES v i LIST OF FIGURES vi i i ACKNOWLEDGEMENTS x INTRODUCTION . . . . 1 ESTIMATION OF GROWTH CHARACTERISTICS WITH PARTICULAR REFERENCE TO TILAFIA IN EAST AFRICA 3 V a r i a b i l i t y of Growth Rate 3 The Point of Inf lexion on the Growth Curve 6 The Weight Length Exponent and the Exponents of Metabolism . . 14 V a r i a b i l i t y of Weight Length Exponent for Tilapia nilotioa . . 21 SPAWNING, SEXUAL MATURATION AND FECUNDITY 23 Spawning 23 Sexual Maturation 24 Fecundi ty 27 ESTIMATION OF MORTALITY RATES FOR TROPICAL FISH 34 Continuous Recruitment Model 35 Estimation of Total Morta l i ty Rates for Fish with Discrete Age Groups 43 Estimation of Total Mor ta l i t y Rates and the Ratio Z\/K from Length Data 47 Estimation of Total Morta l i ty Rates and the Ratio Z\/K using Extreme Values 52 iv V Page CATCHES AND FACTORS AFFECTING CAPTURE IN AFRICA 57 Variables Af fect ing Catches 57 Mu l t ip le Regression Equations 68 YIELD EQUATIONS 70 Beverton and Holt Y ie ld Model 70 Other Y ie ld Models 79 GENERAL DISCUSSION 91 RECOMMENDATIONS FOR FUTURE MANAGEMENT 95 LITERATURE CITED 97 APPENDIX 1 - Derivation of P robab i l i t y Density Function for a Negative Exponential 101 APPENDIX 2 - Derivation of Discrete P robab i l i t y Function for a Negative Exponential 104 LIST OF TABLES Table Page 1 Estimated age at the point of i n f l e x i on for various values of K and b assuming t i s zero 12 2 Parameters a and b of the model W = aL b for Tilapia nilotica in various l o c a l i t i e s in East A f r i c a 22 3 The ra t io s ^ \/ L \" and W \/W\u00b0\u00b0 for some Tilapia species in East A f r i c a . The numbers enclosed in brackets re fer to maximum length based on the largest f i s h in samples taken 26 4 Estimates of t o ta l morta l i ty for Tilapia esoulenta in Lake V i c t o r i a . I t i s assumed that recruitment i s continuous, t = 2.5 years and n i s the sample s ize 42 5 Estimates of to ta l mor ta l i t y for Tilapia esoulenta in Lake V i c t o r i a . I t i s assumed that recruitment i s d i s c re te , t = 2.5 and n i s the sample s i ze 46 6 Estimates of to ta l morta l i ty rates for Tilapia esoulenta i n Lake V i c t o r i a . Length of f i r s t capture L = 22 cm., K = 0.32, L\u00bb = 33.8 and n i s the sample s i z e c 50 7 Estimates of to ta l morta l i ty rates for Tilapia nilotica (normal population) of Lake A lber t , Uganda. Length of f i r s t capture 28 cm., K = 0.5, L\u00ab = 49.0 cm. and n i s the sample s i ze 51 8 Comparison of variance of exponential and extreme value functions for to ta l mor ta l i t y , Z = 0.5 55 9 Mean catch per e f f o r t (catch per net per set) fo r Tilapia species from 1959 to 1965 58 10 Mean catches in kilograms of f i s h caught per hour at various depths during exploratory bottom trawl ing in Lake V i c t o r i a 60 v i v i i Table Page 11 Correlat ion coe f f i c i en t s between catches and variables a f fect ing catches in Bugoma-Salisbury Channel to Rosebery Channel in Lake V i c t o r i a , Uganda 67 12 Comparison of y ie ld s estimated with equations (8.13) and (8.16) and which are based on age and length respec-t i v e l y . Both equations use the incomplete Beta funct ion. The parameters used are K = 0.5, F = 0.2, M = 0.3, Z\/K = 1, b = 3.0, tx = 6.39, L<=\u00b0 = 49 cm. and l x = 47 cm.. . . 83 LIST OF FIGURES Figure Page 1 Contours of age at the i n f l ex i on point for various values of K and b. Age i s given i n years 13 2 The ef fects of exponents of anabolism and catabolism on the weight length re la t ionsh ip of f i s h 20 3 Number of eggs produced by Tilapia at d i f f e ren t s i ze s . (1) Tilapia karome, (2) Tilapia esculenta3 Tilapia nilotica and Tilapia variabilis, (3) Tilapia galilaea, (4) Tilapia zilli. A f te r Lowe McConnell (1955). 4 Changes in mean catch per net (expressed as number of f i s h per 25 yd. set) of (A) Tilapia esculenta (B) Tilapia variabilis and (C) Tilapia zilli for sampled f i s h landings in Tanzania and Uganda. (D) i s Lake water leve l in meters at J i n j a 59 5 Mean catch per hour ( i n kilograms) at various depths of bottom trawl ing in Lake V i c t o r i a , East A f r i c a . A, HaplochromiSj B, Tilapia esculenta^ C, Tilapia (other species) and D-j Bagrus docmao. ! 62 6 A hypothetical curve of catches against time of day with nodes at 7 and 19 hours. b_ i s the amplitude of the sine wave (see equation 7.1; 63 7 Y i e l d i sopleth diagram for Tilapia esculenta (Lake V i c t o r i a ) 75 8 Y i e l d i sopleth diagram for Tilapia nilotica (with normal growth) in open water, Lake A lber t , Uganda . . . 78 9 Y i e l d i sopleth diagram for the stunted Tilapia nilotica in Lake A lber t , Uganda 78a 10 Y ie ld per 100 gm. r e c r u i t plotted against f i sh ing morta l i ty rate ( for Tilapia nilotica in Lake A lber t , Uganda). L i s length of capture in cm., M = 0.3, K = 0.5, and LS = 49.0 cm 87 v i i i i x Figure Page 11 Y ie ld per 100 gm. r e c r u i t p lotted against length of capture at d i f fe rent f i sh ing rates (F) ( for Tilapia nilotica in Lake A lber t , Uganda). M = 0.3, K = 0.5 and L\u00bb = 49.0 cm 88 12 Y ie ld per 100 gm. r e c r u i t plotted against f i sh ing morta l i ty rate ( for stunted Tilapia nilotica* Lake A lber t , Uganda). M = 3.37, K = 2.77, L\u00bb = 17.0 cm. and L c i s the length of capture 89 13 Y ie ld per 100 gm. r e c r u i t plotted against length of capture at d i f fe rent f i sh ing rates (F) ( for Tilapia nilotica in Lake A lber t , Uganda). M = 3.37, K = 2.77 and L\u00bb = 17 cm 90 ACKNOWLEDGEMENT I wish to express my thanks to my supervisor Dr. Peter A. Larkin who provided the i n i t i a l stimulus which resulted in th i s pa r t i cu la r study and who remained a source of advice, c r i t i c i s m and encouragement throughout th i s study. I am very grateful to Dr. N. J.Wilimovsky and Dr. T. G. North-cote, fo r c r i t i c a l l y reading the manuscript and o f fe r ing useful suggestions. My thanks also go to the fol lowing people: (1) Mr. Stephen Borden who did the step-wise mult ip le regression analysis of Lake V i c to r i a data and permitted me to use i t . (2) Mr. Robin A l len for g iv ing useful suggestions and for al lowing me to use his computer programme to construct y i e l d i sop leths. (3) Dr. D. L. Alverson (U.S.A. National Marine Fisher ies Service) with whom I have had in teres t ing and useful discussion on th i s work. F i n a l l y , I wish to extend my thanks to the Canadian International Development Agency, the East Afr ican Community and the F.A.O. (of the United Nations) Lake V i c t o r i a Fisheries Research project for contr ibut ing in various ways to th i s study. x INTRODUCTION The current trend in the study of f i s h population dynamics i s towards a deta i led analys is of var iables a f fec t i ng y i e l d and the construct ion of theoret ica l models which describe the i n te r re l a t i on s of these var iab les . Unfortunately, attempts to a t t a i n r e a l i s t i c models introduce complexities that require more and more basic data. The c o l l e c t i o n and analyses of these data take much time. Moreover, i t i s necessary to have many long sampling periods to reduce the e f fec t of annual v a r i a b i l i t y on est imation. In the circumstances of rapid develop-ment of some Afr ican freshwater f i s h e r i e s , i t may obviously be d i f f i c u l t to meet the data requirement of modern management models. Add i t i ona l l y , management of natural populations of f i s h i n t r op i ca l freshwater i s made d i f f i c u l t by: (a) lack of read i l y detectable growth rings on ske leta l s t ructures, (b) v a r i a b i l i t y in growth rates , (c) absence of d e f i n i t e spawning periods, (d) l imi ted catch s t a t i s t i c s . I t i s therefore des i rable to consider the functional r e l a t i o n -ships among the population var iables that inf luence y i e l d . This might enable development of some simple and useful approximations that give r e l i a b l e ind icat ions of the r e l a t i v e degree of exp l o i t a t i on . 1 2 The purpose of th i s study i s to focus on the most relevant population parameters and to develop simple ana l y t i ca l methods and theoret ica l models that require the minimum of data so as to enable quick and r e l i a b l e assessments of r e l a t i v e rates of exp lo i ta t i on of t r op i ca l freshwater f i s h populations. In th i s thes i s , consideration i s f i r s t given to the separate processes of growth, fecundity, morta l i ty and some simple a lgebraic re la t i ons ip s are developed which could be used in management. Throughout, Tilapia has been used as an example to te s t the v a l i d i t y and usefulness of the f ind ings. Add i t i ona l l y , a few references are made to Bagrus doamao and Haploohromis spp. A separate section i s devoted to the recent data on catches of Tilapia in Lake V i c t o r i a . The analys is of the catches by mul t ip le r e -gression methods provide useful information on the abundance and d i s -t r i bu t i on of Tilapia species and indicates the var iables re lated to high catches. This type of analys is could be used in managing other f i s h species in t r op i ca l freshwater. La s t l y , consideration is given to y i e l d equations and some s imp l i f i c a t i on s which might lead to quick, easy and yet r e l i a b l e e s t i -mation of f i s h y i e l d s . ESTIMATION OF GROWTH CHARACTERISTICS WITH PARTICULAR REFERENCE TO TILAPIA IN EAST AFRICA V a r i a b i l i t y of Growth Rate The estimation of growth rates of t r op i ca l f i s h poses several problems. The absence of seasonal environmental f luctuat ions means that growth at each age and the age of maturity cannot eas i l y be determined by reading annual rings on scales or other ske leta l s t ructures. This d i f f i c u l t y has to now l imi ted the use of y i e l d models which are based on age, in the management of t rop ica l freshwater f i s he r i e s . In some cases rings have been observed on bony parts of some t rop i ca l f i s h , some-times the rings are apparently a re su l t of spawning and other times as a re su l t of drought and starvat ion (Garrod 1959 and Lowe 1956). Species of Tilapia show considerable v a r i a b i l i t y in growth (Lowe 1956). In order to understand the growth processes, extensive and intens ive studies must be car r ied out to determine the growth ra te , the maximum length and weight attained by these f i s h in various waters and the length, weight and age of maturity. Even with in a s ing le lake, one observes differences in growth rate and s i ze of maturity and th i s i s exemplified by Tilapia esculenta in Lake V i c t o r i a . Garrod (1959) used the scale method to determine the age of Tilapia esculenta^ mouth brooder with two spawning seasons. 3 4 Related to the spawning behavior of mouth brooding, there i s a cessation of growth re su l t i ng in r ing formation. On the basis of th i s observation, Garrod (1959) determined 12\"ring years\" for Tilapia esoulenta. In terms of actual years Tilapia esoulenta l i v e s for 6 years. Other methods may be employed in growth studies. Rings on ske leta l structures are made more read i l y detectable by heating but th i s has not been done for Tilapia. Lowe-McConnell (1956) obtained some ind icat ions of growth rate by analysing length frequency d i s t r i bu t i on s . The main l im i t a t i o n of th i s method i s that a number of f i s h species spawn a l l the year round so that there i s considerable overlap of s i ze ranges of d i f f e ren t ages. The adult and young Tilapia l i v e in d i f f e r en t habitats and therefore sampling in one l o c a l i t y w i l l give length frequen-c ies that are truncated. Growth rates can also be determined by conducting mark and recapture experiments but th i s has not yet been done for Tilapia. Management of Tilapia and other t r op i ca l f i s h species, i s poss ible even without d i r e c t determination of age, provided we can estimate the von Berta lanffy growth equation parameters. The maximum length (Loo), maximum weight (W<=\u00b0), the rate at which a f i s h approaches i t s asymp-t o t i c s i ze (K) and the weight length exponent (b), have to be determined i f we are to understand the growth of f i s h species. With the above parameters plus a few s imp l i f y ing assumptions a number of useful r e l a t i o n -ships can be establ i shed. Ursin (1967) reports that Putter in 1920 f i r s t r ea l i zed the truism that food absorbed i s the d i f ference between food ingested and 5 that ejected and advanced a metabolic growth model. The rate of intake of food i s surface dependent whereas the rate of breakdown i s weight dependent because i t occurs in a l l parts of the body, v i z : dw = HW 2 \/ 3 - kw (1.1) dt where w i s the weight, t i s the time, H i s the coe f f i c i en t of anabolism and k i s the coe f f i c i en t of catabolism. Put ter ' s work was continued by von Berta lanffy (1934, 1938) who regarded an organism as a reacting chemical system. The processes of anabolism and catabolism control the weight of an organism. The rate of change of weight of an organism dw\/dt i s expressed in terms of exponents m and n of the body weight. dw = HWm - kw n (1.2) dt where m i s the exponent r e l a t i ng anabolism to weight and n i s the exponent re l a t i ng catabolism to weight. The equation given by von Berta lanffy (1934) describes the rate of change of length with time and i s the equivalent of equation (1.1) when expressed in terms of length and asymptotic length: -d l = K(L\u00bb - 1) (1.3) dt where 1 i s the length of a f i s h at time t and L\u00bb i s the asymptotic length and K i s the rate at which a f i s h approaches i t s maximum length. When integrated (1.3) gives the von Berta lanffy growth equation: l t = L\u00bb (1 - e \" K ( t \" V) (1.4) 6 where t i s the time at which the length of a f i sh is t heo re t i c a l l y zero. Other workers have investigated the von Berta lanffy growth equation and have made several developments (Beverton and Holt 1957, Taylor 1959 and 1962, Paloheimo and Dickie 1965 and 1966.) Ursin (1967) looked at the processes of anabolism and catabolism and also examined the exponents m and n in equation (1.2). He out l ined ways in which m and n could be measured. The Point of Inf lex ion on the Growth Curve Rational exp lo i ta t ion of a f i shery requires knowledge of the weight and age of a f i s h at the i n f l ex i on point. At the point of i n f l ex i on a f i s h has maximum change in weight dw\/dt. In some species of f i sh the maximum growth increment i s attained before sexual maturity. Exp lo i tat ion of such a f i s h species requires catching the f i s h at a s i ze or age beyond the point of i n f l e x i on so that there w i l l be s u f f i c i e n t ind iv iduals of s u f f i c i e n t age to reproduce. For those f i s h populations with a high natural morta l i ty r a te , th i s could mean loss of biomass. The stunted population of Tilapia nilotica i n Lake Albert exemplifies th i s case. This Tilapia population has a natural morta l i ty of 3.37 and sexual maturation i s attained at 10 to 12 cm. ( l i e s , MS.). The maximum biomass occurs at about 8 to 9 cm. In contrast, the population of Tilapia nilotica in Lake Albert (open water) and Tilapia esculenta in Lake V i c t o r i a a t ta i n maturity pr io r to the s i ze and age corresponding to the point of i n f l e x i on . Therefore maximum y i e l d can be obtained by catching the f i s h at the s i ze corresponding to the point of i n f l e x i on . I f we take equation (1.2) and take a second der i va t i ve der i va t i ve equals zero at the point of i n f l e x i on . dw = Hwm - kw11 dt d 2w = mHw\"1\"1 - nkw 1 1 - 1 = 0 so that 5 ? InHwn,-, = nkw\"\"1 which with rearrangement gives the equation w ( n - m ) = mH nk Let the weight at the i n f l ex i on be Wj. Then W T ( n \" m ) = mH 1 nk 1 When a f i s h atta ins the maximum weight (W\u00b0\u00b0)> dw\/dt = 0 Therefore dw = Hwm - kwn = 0 dt Hwm = kwn and H = v\/n-m> \u00bb = w where w i s the maximum weight (W\u00b0\u00b0). Therefore the weight of a f i s h at the point of i n f l ex i on i s given by 1 Wj = \/ m \\ ( n \" m ) . W=o (1.8) The parameters m and n are d i f f i c u l t to measure and therefore we cannot read i l y estimate the weight at the point of i n f l e x i on (1.8). A generalized growth equation in terms of weight i s Wt = W\u00bb (1 - e \" K ( t \" V) (1.9) where b i s the weight length exponent. I f we take W^ to be the weight of a f i s h at the i n f l ex i on point, the s i gn i f i cance of b in (1.9) i s very evident. The weight of a f i s h i s re lated to length by the equation W = ql_ b (1.10) where q i s the constant of p ropor t i ona l i t y . Many f i s h have b = 3.0 and therefore show isometric growth. However there are other f i s h for which b i s not 3.0 and which change in shape with increase in length. Paulik and Gales (1964) have discussed the consequences of assuming isometric growth, on the shape of y i e l d curves. We are aware that the value of b for known f i s h species ranges from 2.5 to 3.5. The f i r s t der i vat i ve of equation (1.9) i s dW t = b Woo dT~ (1 - e-\u00ab* \" V)b \" 1 . Ke-\u00ab* \" V and the second der ivat ive i s d\"W t = bWco dt ' (1 - e^ \" V)b \" 1 . (-K 2 e\"** \" V) + (b - 1) . (1 - e \" ^ \" V ) B \" 2 . ( K V 2 ^ \" V Factoring leads to bW\" e v o ( l-e-\u00ab* \" V) K 2 . (b - 1) . ( e \" K ( t \" V) b - 2 = 0 K 2 ( l - e\"^* \" V) + (1.11) The above equation has 3 square bracketed terms mul t ip ly ing each other and any of them being zero could make the whole equation zero. Taking the f i r s t square bracketed expression, the parameters b and W\u00b0\u00b0 could not be zero at the point of i n f l e x i o n . For the f i r s t square bracketed term to be zero, i t w i l l be necessary that e-K ) t = _ l n (| b j . \u00a3 + t Q (1.16) But both equations (1.13) and (1.16) define the age of a f i s h at i n -f l e x i o n . We can equate the r ight hand expressions of these equations 1 1 ln b + t = - 1 K 0 \u201e A . v b (n - m) \\ . 1 + t n (J ) t Subtract ( t Q ) from each s ide and mult ip ly by K 1 In b = - In ( l - m b ( n \" m ) ) ( 1 ' 1 7 ^ Rearranging equation (1.17) gives 1 - ln b . ln (l - m b ( n \" m ) j 1 1 = i - \/ m \\ b ( n \" m ) (1.18) b ( * ) von Berta lanffy (1957) dealt with a l l omet r i c re lat ionsh ips between an animal 's metabolic rate and i t s weight. He claimed that the slope m of the a l lometr i c l i n e is e i ther 2\/3 for species obeying the surface ru le of metabolism, unity for cases where oxygen consumption 16 i s proportional to weight instead of surface area, and that for other f i s h species in ranges between .66 to 1.0. Parker and Larkin (1959) and Ricker (1960) c r i t i c i z e the der ivat ion of the von Berta lanf fy growth equation because of the assumption of the surface law of metabolism. Taylor (1962) discusses the parameters of the von Berta lanf fy equation and points out factors l i m i t i n g metabolism, von Berta lanffy argues that the rate of metabolism i s proportional to the m power of the weight where m i s the exponent of metabolism. But the rate of catabolism i s proportional to weight i t s e l f , thus n = 1. Ursin (1967) has pointed out that the assumption of n being one does not hold for a l l animals. I f we assume that n = 1, we can establ i sh a re la t ionsh ip between m and b in equation (1.18), i . e . , 1 i = 1 . m b 11 \" m ) (1.19) b It fol lows from equation (1.19) that 1 1 _ i = m M n - m) ( 1 < 2 0 ) b Raising both sides of the above equation to the power b y ie ld s (\u2014]\u2014) b=m]~m (1.21) Lett ing 1 - 1_ = x b we have b = 1 17 We can replace b in equation (1.21) by the appropriate expression, to obtain 1 1 x ] \" x = m 1 \" \" 1 (1.22) Equation (1.22) holds i f x = m for a l l real numbers. But x = 1 - 1_ and also x = m b Therefore from equation (1.21) and 1.22) we can establ i sh the re lat ionsh ip 1 - 1 = m (1.23) b or 1 =1+ m b or b = _ J (1.24) 1 - m From the re la t ionsh ip above i t i s evident that f i s h with isometric growth (b = 3), have m = 0.67 as proposed by von Berta lanffy (1957). I f the exponent of catabolism (n) i s one, then i t can be deduced from equation (1.23) that f i sh with b greater than 3 w i l l have m greater than 0.67 and a f i s h with b less than 3 w i l l have m less than 0.67. Thus a f i s h with b = 2.5 has m = 0.60 and a f i s h with b = 3.5 has m = 0.72. The processes of anabolism change several times during the l i f e span of a f i s h . But a f i s h does not change i t s body shape during 18 i t s l i f e span (except for ear ly l i f e h i s tory stages and with maturity which are beyond the range of the growth period here considered). Therefore a re la t ionsh ip must ex i s t between b and m. This re la t ionsh ip i s expressed by equations (1.23) and (1.24). Changes in the weight length exponent r e f l e c t changes in the processes of anabolism. Hecht (1916) reports that f i s h and frogs have uniform but indeterminate growth. The body form of a f i s h i s l a i d down very ear ly in l i f e and th i s body form i s maintained with in narrow l im i t s throughout the period of growth. This i s in contrast to growth of higher ve r te -brates in which body form cont inua l ly changes during the period of growth. However, i t must be added that th i s conclusion applies only to external surfaces for K e l l i c o t t (1908) has shown that in a dogfish, the brain and v i scera d i f f e r in t he i r rates of growth in much the same way as in the higher vertebrates. When the exponent of catabolism (n) i s less than one, the weight length exponent b cannot be expressed e x p l i c i t l y in terms of m and n. Equation (1.18) i s a transcendental equation, i . e . 1 1 1 - 1. = b ( n \" m ) 1 1 . ! _ \/ m \\ b ~ m ) = o (1.25) 19 For given values of m and n, we can by i t e r a t i on processes f i nd values of b which make equation (1.25) zero. ' I f we use Newton's method of so lv ing transcendental equations, the i t e r a t i on process converges rap id l y . However, i t i s necessary to set the lower and upper l im i t s of b. I f b s a t i s f i e s the inequa l i t ie s 2.5 < b < 3.5, then the values of m range from 0.60 to 0.90 and n ranges from 0.8 to 1.0. For many f i s h species, so fa r studied the weight-length exponent l i e s wi th in the l im i t s 2.5 to 3.5. Carlander (1969) reports 3 populations of Coregonus artedi with b ranging from 3.62 to 3.69. But the values of b are based on samples in which length ranges from 200-230 mm. There are also f i v e populations of Coregonus artedi with values of b less than 2.5. But these values of b are based on samples with maximum length of 164-179 mm. Biased sampling may lead to estimates of b outside the range 2.5 to 3.5. The weight .length exponent outside the range 2.5 to 3.5 cannot apply over a wide range of length without causing profound changes in body form. There may be a few exceptional f i s h species with b greater than 3.5 but i t i s doubtful whether such f i s h species obey the law of uniform and indeterminate growth. The exponent of anabolism m cannot be 1 as th i s would make the value of b tend to i n f i n i t y (see equation (1.24)). The value of m most probably does not exceed 0.90 for 0.8 s n <; 1. However some evidence is needed to ve r i f y th i s propos i t ion. Figure 2 - The effects of exponents of anabolism and catabolism on the weight length re lat ionsh ip of f i s h . V a r i a b i l i t y of Weight Length Exponent b for Tilapia nilotica 21 Analysis of data co l lec ted by Lowe (1958) shows that the weight length exponent for Tilapia nilotica in various l o c a l i t i e s in East A f r i ca is va r iab le . Table 2 gives the parameters a and b for the r e l a t i o n : W = aL b Since the age of a f i s h at the i n f l ex i on point i s a function of the rec iprocal of K and b, i t follows that i f natural mor ta l i t i e s are the same, the age of maximum biomass in the d i f f e ren t l o c a l i t i e s i s d i f f e r en t . The v a r i a b i l i t y of b shown in Table 2 for Tilapia nilotica i s most probably true for other species of Tilapia in various waters in East A f r i c a . There i s need to determine the weight length exponent for other species of Tilapia . Piennar and Thomson (1969) have pointed out the importance of a l lometr ic weight length re lat ionsh ip and the s t a t i s t i c a l problems of such re la t ionsh ips . Under the assumptions of isometric growth, the von Berta lanffy growth equation has proved extremely a t t r a c t i v e to y i e l d model bu i ld ing , for example the Beverton and Holt (1957) y i e l d model. The importance of v a r i a b i l i t y of the weight length exponent has been given l i t t l e cons iderat ion. Because the parameters K and b control the i n f l ex i on point and because b is re lated to the processes of anabolism and catabolism, the f i r s t step in studying the dynamics of a f i s h population might be the estimation of K and b. TABLE 2 - Parameters a and b of the model W = aL for Tilapia nilotica in various l o c a l i t i e s in East A f r i ca LOCALITY a b pH CONDUCTIVITY LAKE ALBERT OPEN WATER 0.017 3.34 9.0 710 LAKE ALBERT BUHUKU LAGOON 0.028 3.33 9.2 7200 LAKE EDWARD 0.479 2.99 9.1 900 LAKE GEORGE 0.010 3.29 9.1 900 MALAGARASI SWAMPS 0.39 2.96 7.6 300 LAKE RUDOLF 0.927 3.19 9.7 2800 SPAWNING, SEXUAL MATURATION AND FECUNDITY Spawning The species of the genus Tilapia do not seem to have a c lear spawning season. In favourable and uniform environmental conditions Tilapia may spawn at frequent interva l s (Lowe 1955). In waters with marked seasonal changes Tilapia may have one or more well defined breeding seasons. The frequency of spawning and the mechanism under which i t works i s not understood. Lowe-McConnell (1955) reports some of the approaches that have been used to determine the frequency of spawning. A Tilapia esculenta marked on the 13th Ap r i l 1953 in Lake V i c t o r i a , had f r y in the mouth and when captured 9 1\/2 weeks l a t e r on the 20th June 1953, was found to have eggs in her mouth. Another Tilapia esculenta ( in Lake V i c t o r i a ) , having f r y in her mouth, was marked on the 4th March 1953 and when captured 7 weeks l a t e r , the ovary was found to be in a r ipening stage. Examination of the ovary of Tilapia species reveals dark yel lowish or brown specks which are signs of recent spawning. Many times, an ovary in a r ipening stage w i l l have small ova s ta r t ing to develop and these ova form the next batch of eggs to develop (Lowe 1955). On the evidence of ovary observations several species of Tilapia may have three or more batches of young in succession. I f we can determine the time taken by each 23 24 batch to develop, th i s would give some measure of the length of the breeding season. The absence of a wel l marked breeding season creates several problems in the management of Tilapia populations. With no d e f i n i t e breeding season i t i s d i f f i c u l t to determine annual recruitment and to re la te recru i t s to the many batches of young that occur in a year. Repeated spawning with in a year creates what may be termed \"sub-year c lasses \" in a year c lass . Because of d i f ferences in growth rates the length frequency d i s t r ibut ions show considerable overlap and i t becomes extremely hard to d issect them into age groups. Sexual Maturation Sexual maturation may be governed by attainment of a cer ta in s i ze rather than age. There are differences in growth rates and these differences mean that a year class or a batch of young hatching at the same time w i l l reach maturity at d i f f e r en t ages. This point i s emphasized by N i ko l s k i i (1969). There are very few species of f i s h in which maturity for a year class occurs at the same age, an exception being the viviparous Poec i l i i d ae . Even in th i s family var iat ions in food supply cause va r ia t ion in age of maturation. Size of maturation i s a v i t a l parameter in management of f i s h populations. Russell (1931) and Graham (1935) stress the importance of \"a l lowing f i sh to grow\" before catching them. Beverton (1963) has establ ished a re lat ionsh ip between length at maturation (lm) and the maximum length (L\u00b0\u00b0). The bigger the s i ze 25 to which a f i s h grows, the bigger i t i s on f i r s t reaching matur ity. This means that the r a t i o lm\/L\u00b0\u00b0 i s r e l a t i v e l y constant for a family of f i s h . Though th i s i s general ly true, exceptions do occur. Table 3 shows va r i a t i on in the r a t i o lm\/L\u00ab in the genus Tilapia i n East A f r i c a . Holt (1962) found corre lat ions between the r a t i o lm\/L\u00b0\u00b0 and K. Fish with high K have low lm\/L\u00b0\u00b0 and mature at a smaller s i ze whi le f i s h with a low K mature at a bigger s i ze . From the r a t i o lm\/L\u00b0\u00b0, the r a t i o Wm\/W\u00ab> can be establ ished i f we know the weight length exponent b. The r a t i o of weight at maturity to maximum weight (Wm\/W\u00b0\u00b0) i s about 0.3 for many f i s h species (Holt 1962). I t i s also known that the weight of a f i s h at the i n f l e x i on point i s 0.3 of maximum weight for f i s h species with b = 3.0 (see equation 1.12). For a l l the f i s h species which mature before a t ta in ing the s i ze of maximum dw\/dt, catching the f i s h at the s i ze corresponding to the point of i n f l ex i on would be the best way of gett ing maximum y i e l d . The ra t io s lm\/L\u00b0\u00b0 and Wm\/W\u00b0\u00b0 in Table 3 below are based on data in Lowe (1958), Garrod (1959, 1963) and l i e s (MS.). 26 TABLE 3 - The rat io s Lm\/L\u00b0\u00b0 and Wm\/W\u00bb for some Tilapia species in East A f r i c a . The numbers enclosed in brackets re fer to maximum length based on the largest f i s h in samples taken. LOCALITY Fish Species 1m Loo lm\/L\u00b0\u00b0 Wm\/W\u00b0\u00b0 LAKE ALBERT OPEN WATER Tilapia nilotioa 36 49 .73 0.35 LAKE ALBERT BUHUKU LAGOON Tilapia nilotioa 10 17 .58 0.16 LAKE EDWARD Tilapia nilotioa 25 (36) .69 0.33 LAKE GEORGE Tilapia nilotioa 28 (40) .70 0.31 LAKE RUDOLF Tilapia nilotioa 39 (63) .61 0.21 MAGALASI SWAMPS Tilapia nilotica 22 (30) 0.73 0.39 LAKE VICTORIA Tilapia esoulenta 22.8 34 0.67 0.30 LAKE VICTORIA JINJA REGION Tilapia variabilis 22 (30) 0.73 From the resu l t s of Table 3 above i t i s evident that mesh s i ze of g i l l n e t s or codend mesh s i ze w i l l be d i f f e ren t in the various l o c a l i t i e s . The s i ze of maturation and the weight at maturation must be considered ser ious ly when set t ing the mesh s i ze of the f i sh ing gear. 27 Fecundity One of the factors con t ro l l i ng the s i ze of a year c lass i s the number of eggs l a i d . The number of eggs l a i d i s governed by the fecundity of a species and the number of mature females. There i s no simple re la t ionsh ip ex i s t ing between number of eggs and the number of o f f spr ing that survive to sexual maturity; the main reason being var iable mor ta l i t y in the several stages of development between egg lay ing and sexual mor ta l i t y . Svardson (1949) gave several general izations about fecundity and egg production a l l of which are noted in various ways with in Tilapia populations of East A f r i c a . The general izat ions are: (1) There i s a negative co r re la t i on between number of eggs and ind iv idua l s i ze of the eggs. (2) The number of eggs produced i s po s i t i ve l y correlated with female s i z e . (3) The growth of a f i s h i s great ly dependent upon the amount of food ava i l ab le . Since growth and consequently s i ze i s modified by environment, egg number might be strongly influenced by environment. (4) Fish species with some parental care produce r e l a t i v e l y fewer eggs than f i s h with no parental care. (5) Closely re lated species may have egg number showing geographical c l i ne s . (6) Egg numbers may show i n t r a s pec i f i c va r ia t ion and th i s might correspond to geographical c l i n e s . (7) The largest larvae hatch from the largest eggs. Tilapia species are subdivided according to mode of reproduction into guarders and mouth brooders. Tilapia zilli i s a guarder and the eggs are guarded by both male and female parents. Character i s t ics of 28 the guarders i s a large number of eggs (see Figure 3). Tilapia nilotioa^ Tilapia esoulenta and Tilapia variabilis belong to the mouth brooders. For these species, development of f r y takes place in the mouth. In the case of Tilapia leuoostiota, also a mouth brooder, f r y are f i r s t r e -leased when about 8 mm. or with in 11 to 15 days of egg f e r t i l i z a t i o n (Welcomme 1966). The fecundity of Tilapia species increases with length fo l lowing an exponential curve. The model descr ibing the re la t ionsh ip of fecundity and length i s : -F = aL B (2.1) where F i s the fecundity at the length L and B i s the exponent r e l a t i ng fecundity to length. A logarithmic transformation of the above model leads to Log F = log a + B log L (2.2) In cases where the parameter B i s equal to the weight length exponent b, fecundity i s said to vary d i r e c t l y with weight. Note that , F = aL b and W = qL B Therefore W = s i * F a T If b = B, then 29 or F = a_ W q . Sett ing C = a_ q F = CW (2.3) where W i s the weight of a f i s h and C i s the c oe f f i c i en t of regression of F on weight. The estimated parameters of equations (2.1) and (2.2) for Tilapia leuoostiota and Tilapia nilotica are given below: Tilapia leucosticta F = 0 .131L 2 - 3 0 or log^F = -0.118 + 2.30 log L Tilapia nilotica F g ) 2 . 6 5 L 2 ' 9 6 or l o g j 7 = 0.423 + 2.96 log L Therefore the fecundity of Tilapia leucosticta increases with about the square of length while the fecundity of Tilapia nilotica increases with about the cube of length. This means that the fecundity of Tilapia nilotica increases l i n e a r l y with weight as shown in equation (2.3). Several factors including seasonal changes in weight of a f i s h and improper sampling ser ious ly a f fec t the values of the parameters in 30 equation (2.1). I f some length groups are not sampled, there w i l l be bias in the estimated parameters. In f i s he r i e s management, i t i s important to know how fecundity varies with age so as to assess the e f fec t of f i sh ing on to ta l egg produc-t i on and i t s consequences on recruitment. Though i t i s not easy to determine age of t r op i ca l f i s h and fecundity with age d i r e c t l y , we can determine i n d i r e c t l y the age of a f i s h of a given fecundity. The age of a f i s h of a given fecundity is determined using the von Berta lanf fy growth equation l t = L- (1 - e \" K ( t \" V) and the fecundity length model F - aL B Let be the fecundity of a f i s h of length L and age t . Then the fecun-d i t y at age t i s given by the re la t ionsh ip F t = a(b\u00bb (1 - e \" K ( t \" V))B (2.4) I t must be noted that th i s equation i s only true for ages that produce eggs. Equation (2.4) gives fecundity of a f i s h as a function of time. Thus fecundity increases with age to an asymptotic value. This i s to be expected since length of a f i s h reaches an asymptote with time. Knowing the parameters K and L\u00b0\u00b0 and knowing the fecundity weight re l a t i on sh ip , the age of a f i s h can a l gebra i ca l l y be expressed. F t = (L\u00ab (1 - e \" K ( t \" V))6 a 31 NUMBER OF RIPE OVA Figure 3 - Number of eggs produced by Tilapia at d i f f e ren t s i zes . (1) Tilapia karome, (2) Tilapia esoulenta, Tilapia nilotica, and Tilapia variabilis, (3) Tilapia galilaea, (4) Tilapia zilli. A f te r Lowe McConnell (1955). 32 F , j B = bo (1 - e - * * - V) F A * - L- - L- e \" K ^ \" V so that . ]_ e - K ( t - t Q ) =(?t\\ B - Lc -K ( t - t Q )= l n \/ \/ F 4 t - t Q = - In (2.5) The accuracy of the estimated age t in (2.5) depends on whether our estimates of K, L\u00ab>, B and t are r e l i a b l e . A l l the above parameters can be determined with reasonable accuracy provided sampling i s conducted in such a way that many length groups are covered. For many species of Tilapia the parameter t i s about zero so that equation (2.5) reduces to 1 (2.6) t = - In \/\/F. \\ B - L\u00bb \\ 33 Again i t must be underlined that th i s expression i s only v a l i d for estimating age of f i s h which are in mature age groups. Determinist ic expressions for age such as that in (2.5) and (2.6) could enable b io log i s t s working in the t rop ics to make approximations of r e l a t i v e indices of y i e l d from fecundity, growth rate and the weight length exponent. ESTIMATION OF MORTALITY RATES FOR TROPICAL FISH The theoret ica l foundation for so lv ing the problem of natural mor ta l i t y was given by Baranov (1918) when he said that the age l i m i t determines the c o e f f i c i e n t of natural mor ta l i t y . Beverton and Holt (1954, 1959), Taylor (1960), Beverton (1963) and several other f i shery b i o l og i s t s have pointed out that l i f e span i s dimensionally the same as the c o e f f i c i e n t of to ta l mor ta l i t y . The equations formulated here for estimating morta l i ty rates from the mean age of f i s h in the catch, are based on the usual assump-tions of negative exponential models of mor ta l i t y . Below are the symbols used in the equations: E = expected value K = the growth rate (von Berta lanffy growth parameter) 1 = length T = mean length l c = length of recruitment L~ = maximum length (a von Berta lanf fy growth equation parameter) M = instantaneous rate of natural morta l i ty n = sample s i ze t = time or age t = mean age 34 35 t = age of f i r s t capture t^ = age of oldest f i s h in the catch t^ = age of e x i t from a f i shery t = time at which the s i ze of a f i s h i s t heo re t i ca l l y zero Z = the instantaneous rate of to ta l morta l i ty In developing the fo l lowing models, i t i s assumed that r e c r u i t -ment i s constant and the instantaneous rate of to ta l morta l i ty Z i s constant. The recruitment can e i ther be of a d i screte or continuous form. The model based on continuous recruitment should be appl icable to the Tilapia species which breed several times in a year. The model based on d i sc rete recruitment i s very useful in temperate lat i tudes where f i s h spawn once a year. Continuous Recruitment Model In an unexploited f i s h population the number of f i s h at any age t i s given by where i s the number of f i s h at the age or time t and NQ i s the i n i t i a l number at time t . N ! t = N n e^ \" V t o (3.1) o We can express N. as a proportion of N (3.2) 36 In the case of exploited f i s h populations, the number of f i s h at any of the expoited ages i s given by N t = R e \" Z ( t \" t c ) (3.3) where R i s the number of f i s h recru i ted at age t c < The number of f i s h N^ . can be expressed as a proportion of R N t = e \" Z ( t \" t c ) (3.4) IT In an unexploited f i s h population the to ta l area under the negative exponential curve i s unity. S im i l a r l y for exploited f i s h populations, the to ta l area under the negative exponential between age t and i n f i n i t y i s unity. This property can be used to f i nd the prob-a b i l i t y of a f i s h a t ta in ing age t . The p robab i l i t y density function i s defined as F(x) = ( f (x ) dx = 1 This means that the sum of the p robab i l i t i e s of a l l ages in a population w i l l be equal to one. For the der ivat ion of a p robab i l i t y density function for the negative exponential, see Appendix 1. In an unexploited f i s h population, the p robab i l i t y of a f i s h a t ta in ing age t i s P(t) = P ( T = t ) = M e \" M ( t \" V P( t) = M e \" M ( t \" t o ) f o r * > *o ( 3 - 5 ) 37 In the case of an exploited f i s h population the r e l a t i v e prob-a b i l i t y of catching a f i s h of age t i s P(t) = Z e \" Z ( t \" t c ) f o r t > lc (3-6> An observation t has an expected value 1 + t and the variance 7 2 i s 1\/Z . The expected mean age i s given as E ( T ) = l + t (3.7) Z c The variance of the mean age t of the catch i s Var ( t ) = T o (3.8) The to ta l instantaneous morta l i t y i s a parameter but in pract ice i t i s estimated as a s t a t i s t i c . Let us suppose that the mean age t i s t = E(t) + e (3.9) where e i s a random error and the expected error i s zero E(e) = 0 Then the variance of the error i s Var (e) = U (3.10) From equations (3.7) and (3.9) the re la t ionsh ip i s establ ished:-t - t = 1 + e (3.11) c Z 38 Notice that the error term e can e i ther be negative or po s i t i ve . From equation (3.11) we can derive an expression for estimating to ta l morta l i ty from a catch sample drawn from a population. V = 1 (3.12) t - t c - (e) where e i s pos i t i ve or negative. When the error i s large and negative the to ta l mor ta l i t y i s under estimated. But when the error i s large and pos i t i ve V i s over estimated. Let T - t = U. I f we assume t i s a constant, a formal expression for the d i s t r i bu t i on of Z i s obtained by a binomial expansion of the equation: Z = _ J (3.13) U + e V = t f 1 - e U ' 2 + e 2 U \" 3 - e 3 U \" 4 + . . . . (3.14) The expected Z' i s E(Z ') - I f 1 + E(e 2 ) I f 3 Therefore E(Z ' ) = Z + Z 3 (3.15) nT E(Z ' ) = Z + Z (3.16) n Square V in equation (3.14) Z ' 2 = U\" 2 - e 2 U \" 4 + (3.17) 39 I f we t reat t o ta l morta l i ty V as a random var iable and i f we denote the expected to ta l morta l i ty as d, i t i s poss ible to measure the d i s -persion of the expected value of to ta l morta l i ty (Z 1 - d ) 2 The dispers ion of the expected value i s known as the variance of Z, denoted as Var (Z). The variance of a random var iable for example to ta l morta l i ty Z, i s Var (Z) = E(Z' - d ) 2 (3.18) As Hodges and Lehmann (1965) put i t , th i s variance i s the expectation of the squares of the d i f ference between Z and i t s expectation. From equation (3.18) the variance of Z can be expressed as Var (Z) = Z 2 - 2Zd + d 2 (3.19) From the law of expectation, the variance becomes Var (Z) = E(Z 2) - 2dE(Z) + d 2 (3.20) But expected Z equals d. Therefore E(Z) = d Var (Z) = E(Z 2) - d 2 (3.21) But from equation (3.16) E(Z 2) = (Z + Z ) 2 (3.22) 40 and 6 = 2+1+.... n Therefore d 2 = (Z + Z + . . . . ) 2 (3.23) Therefore the variance of to ta l morta l i ty estimated by equation (3.12) i s Var (Z) = (Z + Z ) 2 - (Z + Z + . . . . ) 2 Var (Z) = Z 2 (3.24) n From the above considerations E ( T ^ T C ) = ( J V 1 ) . Z (3.25) Therefore the t o t a l estimated morta l i t y based on a taken sample i s Z = 1 . n (3.26) t - t c This i s an unbiased estimator of t o ta l morta l i ty Z. The variance of Z i s a 2 (Z) =\/ n \\ 2 . Z 2 (3.27) (n + y n The variance of Z can be estimated from the sample as 41 Total mor ta l i t y estimates for Tilapia esculenta are made using equation (3.26) and given in Table 4. Note that Table 4 does not give to ta l morta l i ty at each age. But i f one sampled the catch and found the mean age t where t >_ t , to be 3 years, the to ta l morta l i ty of Tilapia esculenta which i s recru i ted at 2.5 years, would be 1.99 and the variance of th i s estimate would be 0.0079. I t i s expected that the mean age in the catches from various l o c a l i t i e s w i l l vary and each region w i l l be characterized by i t s own to ta l mor ta l i t y . In the case of maximum l i k e l i h o o d , the tota l morta l i ty i s estimated as Z = 1 (3.29) * \" *c Equation (3.29) gives a biased estimate of Z whose variance i s a 2 ( Z )= 1 (3.30) n ( t - t c ) 2 The variance of Z in the case of maximum 1 ikel ihood i s bigger than the variance determined from the expected value of Z. While estimates of t o ta l morta l i ty can be determined with equation (3.29), i t i s better to estimate to ta l morta l i ty with equation (3.26) so as to avoid bias in the estimate. 42 TABLE 4 - Estimates of t o ta l morta l i ty for Tilapia esculenta in Lake V i c t o r i a . It i s assumed that recruitment i s continuous, t = 2.5 years and n i s the sample s i ze . c n = 500 n = 1000 n = 1500 n = 2000 t Z o 2 ( Z ) Z o 2 ( Z ) Z o 2 ( Z ) Z o 2 ( Z ) 3.0 1.99 .00796 1.99 .00399 1.99 .00266 1.99 .00199 3.5 .99 .00199 .99 .00099 .99 .00066 .99 .00049 4.0 .66 .00088 .66 .00044 .66 .00029 .66 .00022 4.5 .49 .00049 .49 .00024 .49 .00016 .49 .00012 5.0 .39 .00031 .39 .00015 .39 .00010 .39 .00007 5.5 .33 .00022 .33 .00011 .33 .00007 .33 .00005 6.0 .28 .00016 .28 .00008 .28 .00005 .28 .00004 6.5 .24 .00012 .24 .00006 .24 .00004 .24 .00003 7.0 .22 .00009 .22 .00004 .22 .00003 .22 .00002 43 Estimation of Total Mor ta l i t y Rates for Fish with Discrete Age Groups I f a quantity X takes on the possible d i screte values x.-j, x,,, x k and i f x ] < x 2 < x k > then the p robab i l i t y that X takes a value x.. ( for f i n i t e ser ies) i s defined by px i = P(X = xn.) = f (x . ) ( i = 1,2 k) and the sum of f (x^) i s unity (Burington and May 1958) k I f ( x j = 1 i = 1 S im i l a r l y for an i n f i n i t e se r ie s , the p robab i l i t y that X takes the values x.j i s px i = P(X = x.) = f (x . ) ( i = 0, 1, 2...,\u00bb) CO P(X = x.) = f (x . ) = 1 0 From the properties of a d i screte p robab i l i t y d i s t r i b u t i o n , we can derive expressions for estimating t o ta l morta l i ty for f i s h with d i sc rete age groups. But two assumptions have to be made: (1) constant recruitment and (2) constant to ta l morta l i ty for a l l ages. If we draw a sample of s i ze n from an exponential d i s t r i b u t i o n , the p robab i l i t y of gett ing age t i s P(t) = P(T = t ) = (1 - e ' Z ) e ' Z t for t > tQ P(t) = (1 - e * Z ) e \" Z t fo r t > t (4.1) 44 The mean age t for a population with d i screte age groups i s t = e \" Z (1 - e \" Z ) (4.2) The der ivat ion of equations (4.1) and (4.2) i s given in Appendix 2. The surv iva l rate is given by S = e \" Z Therefore the mean age i s given by t = S (4.3) 1 - S From equation (4.2) and (4.3) i t i s obvious that i f t o ta l morta l i ty Z i s zero the mean age of a f i s h in a sample w i l l be i n f i n i t y . Re-arranging equation (4.3), we have t = _ J - . 1 (4.4) 1 - S so that t + 1 = 1 (4.5) 1 - S Taking the inverse of equation (4.5), we have 1 = 1 - S (4.6) t - 1 Therefore the surv iva l rate S i s given by S = 1 - 1 (4.7) t - 1 45 which i s rearranged to give S = t (4.8) 1 + T But S = e \" Z Therefore e~ Z = t (4.9) 1 + T - Z = In Vi + t' Z = In ^1 + t j (4.10) Equation (4.10) estimates to ta l morta l i ty Z i f the age t \u00a3 i s zero. The age of f i r s t capture t i s not zero and therefore i t must be sub-tracted from the denominator and numerator of equation (4.9), i e . , e \" Z = t - t c (4.11) 1 + t - t c The to ta l morta l i ty in the case of d i sc rete recruitment i s Z = In \/ t + 1 - t \\ (4.12) The above estimator of Z has s t a t i s t i c a l bias i f the mean age i s deter-mined from a small sample. A more r e l i a b l e estimate of Z i s given by 46 The variance of the to ta l morta l i ty Z i s a2 = \/ n \\ 2 . 1 . Z 2 (4. \\p + 1\/ n Total morta l i ty estimates fo r Tilapia esculenta are made using equation (4.13) and given in Table 5. The estimates given in Table 5 are very close to those in Table 4. Under d i sc rete recruitment a mean t of 5 years in the catch would mean that the t o t a l morta l i ty i s about 0.33. But under continuous recruitment a mean age of 5 years gives a to ta l mor ta l i t y estimate of 0.39. Note that the age of f i r s t capture i s 2.5 years. The variances of the estimates in Table 5 show that using large samples makes the estimated Z more r e l i a b l e . TABLE 5 - Estimates of t o t a l mor ta l i t y for Tilapia esculenta i n Lake V i c t o r i a . I t i s assumed that recruitment i s d i s c re te , t = 2.5 and n i s the sample s i ze . t n = 500 n = 1000 n = 1500 n = 2000 Z o 2 (Z ) Z a 2 (Z ) Z a 2 (Z ) Z a 2 (Z) 3.0 1.09 .00239 1.09 .00120 1.09 .00080 1 .09 .00060 3.5 .69 .00095 0.69 .00047 .69 .00031 .69 .00023 4.0 .50 .00051 .50 .00025 .51 .00017 .51 .00013 4.5 .40 .00032 .40 .00016 .40 .00010 .40 .00008 5.0 .33 .00022 .33 .00011 .33 .00007 .33 .00005 5.5 .28 .00016 .28 .00008 .28 .00005 .28 .00004 6.0 .25 .00012 .25 .00006 .25 .00004 .25 .00003 6.5 .22 .00009 .22 .00004 .22 .00003 .22 .00002 7.0 .19 .00007 .19 .00004 .20 .00002 .20 .00002 47 Estimation of Total Morta l i ty Rates and the Ratio Z\/K from Length Data Since i t i s d i f f i c u l t to age t rop ica l f i s h species, the use of age in estimating to ta l mor ta l i t y may not eas i l y apply to t rop i ca l species. Instead of age, one can use length to estimate the to ta l morta l i ty (Z) i f the parameter K i s known. Where K i s unknown, the r a t i o Z\/K i s estimated from the negative exponential curve. The r a t i o Z\/K i s important in determining y i e l d s , for f i s h with a l lometr ic growth, by means of the incomplete Beta funct ion. In an explo ited f i s h population, we can express the number of f i s h at any age t as I t i s assumed that the number of f i s h at age t i s constant and equal to un i ty. The von Berta lanf fy growth equation for length i s In the above equation, time t can be expressed as a function of length. Then t and t in (5.1) are given as = L~ (1 - e - K ( t - t 0 ) } t = 1 (- In (1 - 1.)) + t K j 1 Loo (5.2) (5.3) Let X ] = - In L\u00bb 48 and X c = - In ( l - i j Then t = 1 X, + t (5.4) K *c = 1 X c + *o ( 5 ' 5 ) Subtracting tQ from t m \\ ( X 1 - Xc\u00bb Replacing t - t in equation (5.1) N = e \" Z \/ K ( X l \" X c } ( 5 ' 6 ) The p robab i l i t y of X^ i s given by the p robab i l i t y density function below P(X,) = Z e \" Z \/ K ( X l \" X c ) fo r X, > Xr (5.7) K 1 c Note that equation (5.7) i s s im i l a r to equation (3.6). Therefore i f we know the length d i s t r i bu t i on in the catch and the length of f i r s t capture, the r a t i o Z\/K can be estimated. Z = n . _ J (5.8) K n + 1 Y Y x l \" *c 49 where Xj i s the mean of X-j from various samples. The variance of Z\/K i s o 2 Z = \/ _ n \\ 2 . Z 2\/K 2 (5.9) which can be estimated by 2 7 \/ \u201e \\2 1 (5.10) a Z = ' \" x n ( X r - X c ) 2 Note that to determine X^ we have to take several samples each of s i ze n. For each sample we determine X-j according to: x i - - L N ( ] - V) Then X-j i s given by m r, = X. (5.11) where m i s the number of x-| each determined from equation (5.8). Table 6 shows estimates of t o ta l morta l i ty based on the above method for Tilapia esoulenta in Lake V i c t o r i a in the North Buvuma area. The length of f i r s t capture 1 i s 22 cm. and corresponds to age t = 2.5 years. In Table 6, the mean length in the catch i s given instead of X^. The to ta l morta l i ty rates estimated with length are very close to the estimates determined from age data. For example, i f the mean age in the catch i s 3 years a population of Tilapia esoulenta would have a to ta l morta l i ty rate of 1.99. A 3 year old Tilapia esculenta i s about 24 cm. long. I f the mean length in the catch i s 24 cm., the r a t i o Z\/K and the to ta l morta l i ty (Z) would be 5.37 and 1.71 respect ive ly . Also note that a mean age of 5 years and a mean length of 29 cm give to ta l morta l i ty estimates of 0.39 and 0.35 respect ive ly (see Table 4 and Table 6). TABLE 6 - Estimates of to ta l morta l i ty rates for Tilapia esculenta in Lake V i c t o r i a . Length of f i r s t capture L = 22 cm., K = 0.32, Lo\u00b0 = 33.8 and n i s the sample s i z e : n = 500 n = 1000 n = 1500 n = 2000 Z o Z (Z) Z a 2 (Z) Z a 2 (Z) Z a2(z) 23 3.60 .02590 3.60 .01300 3.61 .00868 3.61 .00651 24 1.71 .00589 1.72 .00295 1.72 .00197 1.72 .00148 25 1.08 .00236 1.08 .00118 1.09 .00079 1.09 .00058 26 0.77 .00118 0.77 .00059 0.77 .00039 0.77 .00029 27 0.57 .00066 0.57 .00033 0.58 .00022 0.58 .00016 28 0.44 .00040 0.45 .00020 0.45. .00013 0.45 .00010 29 0.35 .00025 0.35 .00012 0.35 .00008 0.35 .00006 30 0.28 .00015 0.28 .00007 0.28 .00005 0.28 .00003 31 0.22 .00009 0.22 .00004 0.22 .00003 0.22 .00002 32 0.16 .00005 0.17 .00002 0.17 .00001 0.17 .00001 51 TABLE 7 - Estimates of to ta l morta l i ty rates for Tilapia nilotica (normal population) of Lake Albert Uganda. Length of f i r s t capture 28 cm., K = .50, Loo = 4 9 . 0 and n i s the sample s i z e . n = 500 n - 1000 n = 1500 n = 2000 l t cm Z o 2 (Z ) Z a 2 (Z ) Z a 2 ( Z ) Z a 2 (Z ) 29 10.22 .20837 10.23 .10460 10.24 .06982 10.24 .05240 30 4.98 .04951 4.99 .02485 4.99 .01659 4.99 .01245 31 3.23 .02087 3.24 .01047 3.24 .00699 3.24 .00524 32 2.36 .01110 2.36 .00557 2.36 .00372 2.36 .00279 33 1.83 .00670 1.83 .00336 1.83 .00224 1.83 .00168 34 1.48 .00438 1.48 .00219 1.48 .00146 1.48 .00110 35 1.23 .00301 1.23 .00151 1.23 .00101 1.23 .00075 36 1.04 .00215 1.04 .00108 1.04 .00072 1.04 .00054 37 .89 .00158 .89 .00079 .89 .00053 .89 .00039 38 .77 .00118 .77 .00059 .77 .00039 .77 .00029 39 .67 .00090 .67 .00045 .67 .00030 .67 .00022 40 .58 .00069 .58 .00034 .58 .00023 .58 .00017 41 .51 .00053 .51 .00026 .51 .00017 .51 .00013 42 .45 .00041 .45 .00020 .45 .00013 .45 .00010 43 .39 .00031 .39 .00015 .39 .00010 .39 .00007 44 .34 .00024 .34 .00012 .34 .00008 .34 .00006 45 .30 .00018 .30 .00009 .30 .00006 .30 .00004 46 .25 .00013 .25 .00006 .25 .00004 .25 .00003 47 :21 .00008 .21 .00004 .21 .00003 .21 .00002 48 .16 .00005 .16 .00002 .16 .00001 .16 .00001 52 Estimation of Total Mor ta l i t y Rates and the Ratio Z\/K Using Extreme Values The oldest age in a f i s h population has s t a t i s t i c a l properties of extreme values. The age of a f i s h at death i s a s t a t i s t i c a l v an ate and the negative exponential curve gives the p robab i l i t y of dying a f te r a certa in age. Fish populations with high to ta l morta l i ty have r e l a t i v e l y fewer age groups than populations with low to ta l mor ta l i t y . By reducing the f i sh ing i n tens i t y one expects more f i s h reach an older age. Several workers have invest igated the app l i cat ion of s t a t i s t i c s of extreme values in estimating the to ta l morta l i ty of f i s h (Gumbell 1954, Kendall 1955, Beverton 1963 and Holt 1965). Suppose we have n independent observations x-j, X2,..., x n with a common d i s t r i bu t i on Then i f y-|, y n are the same n observed numbers rearranged in des-cending order of magnitude, the largest value y-j and the smallest value y n and the range (y-j - y n ) are new random variables the j o i n t d i s t r i bu t i on of which depends on the d i s t r i bu t i on function F ( t ) . The negative exponential d i s t r i b u t i o n expressing morta l i ty with age i s F(X) is the p robab i l i t y that a given observation has a value equal to or less than x. I f y i s the largest value of x (age) in the sample of s i ze n, then F(t) = Prob F(X) = 1 - e \" x fo r X >_ 0 53 y = v + In (n) (6.1) As n tends to i n f i n i t y , v = 0.5772 (Euler ' s constant). By taking several samples of s i ze n, the mean largest y becomes y = .5772 + In (n) (6.2) Holt (1965) derived an equation for estimating the mean age of the oldest f i s h in a ser ies of samples of s i ze n t , = 0.5772 + In (n) + t (6.: Z where t ^ i s the mean age of the oldest f i s h in a ser ies of samples of s i ze n. Equation (6.3) can be wr i t ten as T, - t = 0.5772 + In (n) (6.< L C ^ The standard deviat ion of y in (6.1) i s ay = n From equation (6.4) an expression for ' estimating to ta l morta l i ty i s der i ved: -and the variance of y i s a y = n_ 6 The variance of t, - t in equation (6.4) i s 54 Z = 0.5772 + In (n) (6.5) But the expected to ta l morta l i ty E(Z) i s E(Z) = Z 1 4- n' 6 (.5772 4- ln(n)) (6.6) The variance of E(Z) i s Var (Z) = it 6 (.5772 4- in(n)) (6.7) I t i s important to note the differences between the negative exponential d i s t r i b u t i on and the extreme value d i s t r i bu t i on as estimators of t o ta l mor ta l i t y . (1) The mean age of a population estimated from a negative exponential i s smaller than the mean age estimated from the extreme values. (2) The variance of the mean age (of a negative exponential) i s bigger than the variance of the mean age determined from extreme values. (3) The coe f f i c i en t of va r ia t ion for the mean age estimated from a negative exponential i s unity because mean age i s equal to the standard dev iat ion. But the coe f f i c i en t of va r i a t i on of the mean age from the extreme value function i s less than one. (4) As estimators of to ta l morta l i ty Z, the negative exponential i s more r e l i a b l e than the extreme value funct ion. The variance of Z estimated from a negative exponential i s smaller than the variance of Z as estimated from the extreme value funct ion. 55 A comparison of variances from both estimators is given in Table 8. I t i s assumed that the to ta l morta l i ty Z = 0.5, and the v a r i -ances of Z from samples of various s i ze s , are ca lcu lated. TABLE 8 - Comparison of variance of exponential and extreme value functions for Z = 0.5 Sample S ize (n) Variance Exponential Var (Z) = Z 2\/n Extreme Value Var (Z) n 2 Z 2 6 (.5772 + In ( n ) r 10 0.025 0.049 100 0.0025 0.015 1000 0.00025 0.0073 10000 0.000025 0.0043 The extreme value function can be used to determine the r a t i o Z\/K from length data. I f we replace ages t and tj_ in equation (6.5), the longest f i s h in the catch can be used for estimating to ta l morta l i ty X 1 = - In (1 - l t \/L\u00ab) where 1 t i s the longest f i s h in the catch of sample s i ze n X c = - In (1 - l c \/L~) and t - t = 1 I + t - 1 X + t L c 1 o j\u00a3 c o 56 Therefore * L \" * c = 1 (X1 \" Xc> Then the r a t i o Z\/K i s given by Z = .5772 + ln(n) (6.8) I f the r a t i o Z\/K i s constant for a given f i s h population, then as we increase the s i ze of the sample, we would expect the parameter to increase. I f Zj does not increase with n then the r a t i o Z\/K estimated with large samples, w i l l be over estimated. Instead of taking one very large sample from a population, one could take small samples of s i ze n from the several s t ra ta and reduce the variance of Z j . For each stratum the r a t i o Z\/K would be estimated and the mean of the various ra t io s would be the parameter for the population. Extensive sampling i s required to show that the extreme age and length in a population have properties of the extreme value funct ion, which in t h i s case i s a double exponential. CATCHES AND FACTORS AFFECTING CAPTURE IN AFRICA Variables A f fect ing Catches The most serious problem connected with determining y ie ld s from t rop i ca l lakes i s the estimation of annual recruitment. Many f i s h species, especial l y those of the genus Tilapia,have, no de f i n i t e breeding season and i t i s extremely d i f f i c u l t to re la te the notions of r e c r u i t -ment to several batches of young that appear in a year. For Tilapia, which spawns in the inshore waters, f luc tuat ion of water level i s an important environmental factor inf luencing the success of spawning. Welcomme (1966) reports that Lake V i c t o r i a levels show seasonal o s c i l l a t i o n with a maximum in May-June and a minimum in October to November. Long-term f luctuat ions of water level also occur. P r io r to 1927, Lake V i c t o r i a had a 10 or 11 year cycle of water level maxima. From 1927 to about 1961 the pattern of f luctuat ions changed markedly and the water leve l rose considerably. In 1964, the water level was 1.4 meters above previous records. The r i s e in water level was accompanied by changes in catch per unit e f f o r t for Tilapia esculenta (see Figure 4). Mean catches of three species of Tilapia from 4 and 4.5 inch g i l l nets are given in Table 9. These mean catches are based on catch e f f o r t data from several f i s h landings in Tanzania and Uganda. 57 58 TABLE 9 - Mean catch per e f f o r t (catch per net per set) for Tilapia species from 1959 to 1965. Fish species MEAN CATCH PER NET PER SET 1959 1960 1961 1962 1963 1964 1965 Tilapia esoulenta 1.09 0.84 0.92 i 1.11 1.59 5.85 3.80 Tilapia variabilis 0.72 0.81 0.99 1.16 1.03 0.93 0.26 Tilapia zilli 0.03 0.06 0.08 0.17 0.15 0.19 0.11 The spawning grounds of Tilapia esoulenta are swampy sheltered margins and these areas were increased considerably by f looding in 1961 to 1962 (Welcomme 1964). Lowe (1956) reports that breeding a c t i v i t y of Tilapia esoulenta increase with heavy r a i n f a l l . The heavy r a i n f a l l of 1961 and 1962 seem to have induced a high response in breeding a c t i v i t y of Tilapia esoulenta. The year classes of 1961 and 1962 resulted in high catches in 1964 and 1965. Note that Tilapia esoulenta takes two to three years to a t t a i n maturity (22 cm. to 24 cm.) and i t i s at th i s s i ze that a f i s h is caught in 4 and 4.5 inch g i l l nets. Tilapia zilli and Tilapia variabilis spawn on harder bottomed exposed beaches (Fryer 1961 and Welcomme 1964). The r a i n f a l l of 1961 and 1962 did not s i g n i f i c a n t l y a f f ec t the catches of Tilapia zilli and Tilapia variabilis (see Table 9 and Figure 4). 59 600 I960 1961 1962 1963 1964 1965 YEAR Figure 4 - Changes in mean catch per net (expressed as number of f i s h per 25 yd. set) of (A) Tilapia esculenta, (B) Tilapia variabilis and (C) Tilapia zilli fo r sampled f i s h landings in Tanzania and Uganda. (D) i s Lake water leve l in meters at J i n j a . 60 Information on other var iables a f fect ing catches i s furnished by F.A.O. exploratory bottom trawl ing in Lake V i c t o r i a . These variables include depth of bottom, time of day of f i sh ing and mesh s i ze of codend. For some f i s h species, e.g., Tilapia esculenta and other Tilapia species, the catches decl ine with increasing depth of the lake. But catches of Haplochromis increase with increasing depth and the maximum catch occurs at about 44.5 metres. Beyond a depth of 44.5 metres, the catches dec l ine . One of the important cat - f i shes {Bagrus docmac) gives low catches at a mean depth of 6.5 metres. The catches increase with depth to about 24.5 metres beyond which the catches de l ine. A comparison of the e f fec t of depth on catches of some f i s h species i s given in Table 10 and Figure 5. TABLE 10 - Mean catches in Kilograms of f i s h caught per hour at various depths during exploratory bottom trawl ing in Lake V i c t o r i a . MEAN DEPTH IN METERS 6.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 Kg. Kg. Kg. Kg. Kg. Kg. Kg. Kg. Haplochromis 320.4 524.8 462.8 524.0 465.9 496.7 185.2 28.8 Tilapia esculenta 52.6 31.7 3.5 0.3 0.1 0.0 0.0 0.0 Other Tilapia 15.0 0.9 0.0 0.0 0.0 0.0 0.0 0.0 Bagrus docmac 24.6 42.3 45.1 35.5 31.3 38.6 21.9 0.3 61 The time of day when f i sh ing i s conducted i s another var iab le inf luencing catches. Regier (1970) assumes that the d i e l v e r t i c a l move-ment of some f i s h species affects trawl catches fo l lowing a sine curve with nodes at dawn 07:00 hours and at dusk 19:00 hours. The nodes are the periods of minimum catches and the antinodes are periods of maximum catches. The antinodes occur at 13:00 hours and at 01:00 hours. There is therefore a period of 12 hours between nodes as shown in Figure 6. The catch i s expressed as a function of time of day of f i s h i ng as y = b 3 s in n (T - 7.00) 12 (7.1) where T i s the time of day of f i sh ing and ranges from 1 to 24 hours, b^ i s the amplitude of the sine wave and y = bg at the antinodes. A ser ies of prel iminary analyses support the assumption of catches fol lowing a sine curve. The mult ip le regression analyses were done using the sine curve in the form of (7.1). However through personal interviews with loca l fishermen in the northern end of Lake V i c t o r i a , I learnt that a number of fishermen u t i l i z e the 01:00 hour antinode. Most fishermen set the i r g i l l n e t s between 17:00 and 19:00 hours and pick up the i r g i l l nets between 02:00 and 05:00 hours. This implies that some fishermen are aware that a f te r 02:00 hours, the catch declines. Local fishermen in Lake V i c t o r i a do not conduct day time f i sh ing and information on the catches at 13:00 hour antinode i s from trawl catches, Since there are two antinodes (one at 01:00 hour and another at 13:00 hours i t might be des irable to take the absolute values of (7.1) y = b^ s in n (T - 7.00) 12 62 MEAN DEPTH IN METRES Figure 5 - Mean catch per hour ( in kilograms) at various depths of bottom trawl ing in Lake V i c t o r i a , East A f r i c a . A, Haplochromis, B, Tilapia esoulenta, Cj Tilapia (other species) and D1 Bagrus doemao. 63 Figure 6 - A hypothetical curve of catches against time of day with nodes at 7 and 19 hours, b~ i s the amplitude of the sine wave (see equation 7.1). 64 I t i s necessary to estimate the e f fect of a number of var iables on the catch of d i f f e ren t species. Regier (1970) suggested a step-wise mult ip le regression analys is on the catches from bottom t rawl ing. This analys is gives useful information on the d i s t r i b u t i on and standing crop of cer ta in f i s h species. An out l ine of a step-wise mul t ip le r e -gression analys is i s given below. The re la t ionsh ip between y i e l d and variables l i k e depth of a lake and mesh s ize of codend of a trawl may not be simply l i nea r . Therefore a simple l i nea r re la t ionsh ip i s commonly modified by use of a polynomial regress ion, v i z . y = b Q + b 1 X + b 2 X 2 + . . . . + b n X n (7.2) The e f fec t of type of bottom, time of f i s h i n g , depth of lake and mesh s i z e , on y i e l d i s investigated for the genus Haplochromis and for Tilapia esoulenta and Tilapia nilotioa. The area studied i s that between Bugoma-Salisbury channel and Rosebery channel fo r depth less than 50 metres. Below i s a symbolic notation of the independent var iab les . = type of bottom h = so f t mud bottom = mud bottom *h = hard bottom h = depth of bottom h = time of day of f i sh ing h = mesh s i ze of codend 65 The re la t ionsh ip between the y i e l d y and the independent v a r i -ables i s given by a mul t ip le regression model below. y = B 0 + B 1 X 1 + B 2 X 2 + B 3 X3 + p 4 X 4 + U (7.3) The parameters PQ, B 1 , B^, P3 and B^ are unknown population c o e f f i c i e n t s . U i s an unknown random var iable measuring the departure of observed y from the predicted y. The above parameters are estimated from samples taken from a population: y = b Q + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X 4 + e (7.4) where e is a random error term and the coe f f i c i en t s b^, b 2 , b 3 and b 4 are coe f f i c i en t s giving the slope of y on the variables X-j, X 2 , X 3 and X 4 respect ive ly . Because the regression of y i e l d on each of the var iab les , except X 3 , i s of a polynomial form, the catches have been subjected to a logarithmic transformation. This transformation helps to reduce the polynomial terms, s t a b i l i z e s the variance of the mean and make the regression model more e f f i c i e n t . I f we wr i te a function for each of the independent var iab les , the model (7.4) becomes: y = log (Z) = U + f ^ X , ) + f 2 ( X 2 ) + f 3 ( X 3 ) + f 4 ( X 4 ) (7.5) where Z i s a discr iminant function and a l i nea r function of the independent var iab les . Each of the independent var iables contributes an e f fec t independent of the other var iables to the logarithm of the catch. In the case of a mul t ip le regression where a dependent var iab le (e.g. catch) i s af fected by several var iab les , i t i s necessary to d iscr iminate 66 among the independent var iab les , and leave only those variables which c o n t r i -bute to the regression sum of squares. In an exploratory manner various combinations of the var iables , X 2 > X^ and X^ are chosen in such a way as to minimize the unexplained res idual va r i a t i on . Any var iab le which does not s i g n i f i c a n t l y contr ibute to the regression sum of squares i s dropped. We use the co r re la t i on coe f f i c i en t s between y i e l d and the other var iables as a c r i t e r i o n for entering variables in equation (7.4). Correlat ion coe f f i c i en t s for the genus Haplochromis and for Tilapia esoulenta and Tilapia nilotica are given in Table 11. 67 TABLE 11 - Corre lat ion coe f f i c i en t s between catches and var iables a f fec t ing catches in Bugoma-Salisbury Channel to Rosebery Channel in Lake V i c to r i a Uganda . FISH SPECIES Independent Variables Haplochromis log y Tilapia esoulenta log y Tilapia nilotica log y X s 0.0160 - 0.1179 - 0.1247 Xm 0.1796 - 0.5325 - 0.0760 X h - 0.1866 0.5784 0.1279 x 2 0.4424 - 0.7505 - 0.3413 X 3 0.4634 - 0.3136 - 0.2184 X4 - 0.7371 0.2196 0.1243 X 2 *2 0.4459 - 0.6581 - 0.2867 X 2 - 0.0200 0.0833 0.1231 X 2 - 0.7852 0.2057 0.1351 X 3 0.4369 - 0.5606 - 0.2358 X 3 3 0.3913 - 0.2546 - 0.1990 X 3 *4 - 0.8169 0.1886 0.1367 = so f t mud bottom = mud bottom = hard bottom = depth of bottom = time of day = mesh s i ze of codend 68 Mu l t ip le Regression Equations By using the simple cor re la t ion coef f i c ient s in Table 11 as a c r i t e r i o n for entering variables in a mul t ip le regression model, the equations below were establ i shed. Because the regression model i s of a polynomial form, the mul t ip le regression equation contain some variables raised to cer ta in powers. The equation to describe catches for Haplochromis species i s log y = 1.9869 - 0.104 X 3 + 0.0608 X 2 + 0.2267 X 3 + 0.0003 x\\ (7.6) Equation (6.7) shows that the catch of Haplochromis depends on mesh s i z e , time of day of f i sh ing and bottom depth. I t i s also evident that the smaller the mesh s i ze X^, the bigger the catch. The time of day of f i s h i n g X^ w i l l contr ibute to the catches depending on the time function (7.1). From above i t i s evident that high catches of Haplochromis w i l l occur at a greater depth X 2 -For Tilapia esculenta, the mul t ip le regression equation i s : log y = 2.0407 - 0.0996 X 2 + 0.00118 X 2 + 0.2936 X ] (7.7) Equation (7.7) shows that catches for Tilapia esculenta are more influenced by depth and type of lake bottom than any of the other var iab les . Also note that the co r re la t i on c o e f f i c i e n t between log y and depth is - 0.7505 meaning that catches decl ine with depth. From the cor re la t ion coe f f i c i en t s 69 in Table 11, i t appears high catches of Tilapia esculenta occur in hard bottom l o c a l i t i e s . The equation for Tilapia nilotica i s log y = 0.3213 - 0.02 X \u00a3 + 0.00028 X2, (7.8) Equation (7.8) shows that the catch of Tilapia nilotica i s very much influenced by the depth of lake. I f a l l the important parameters are included, the mult ip le regression equations and the cor re la t ion coe f f i c i en t s provide a short -cut method of assessing the state of explo ited f i s h stocks. Under steady state condit ions, the catches of each year should be close to what i s predicted. I f there i s overf i sh ing and the stocks are dec l in ing , the catches w i l l be less than what the mult ip le regression equations p red ic t . Analyses of catch data of two or more periods w i l l give r e l i a b l e ind icat ions of the r e l a t i v e degree of exp lo i t a t i on . The mul t ip le re -gression equations could also be used in improving f i sh ing success, since they provide information on the d i s t r i bu t i on of f i s h , mesh s i ze of codend of trawl and time of day of f i s h i n g , l i k e l y to give high catch. YIELD EQUATIONS Beverton and Holt Y i e l d Model In explo ited f i s h populations, f i s h are recru i ted to the f i shery at age t (the age of recruitment), but are not caught un t i l the age of f i r s t capture ( t c ) . The only exception to th i s i s the case of knife edge recruitment where t = t r > The change in numbers with time in exploited f i s h populations i s given by dN t = - (F + M) N t (8.1) d t ~ Integrating the above der ivat ive with the lower l i m i t of the integra l equal to t and the upper l i m i t t x , the \"age of e x i t \" from a f i she ry , gives N = R e \" Z ( t \" V (8.2) where R i s the number of rec ru i t s at the age t and Z i s the to ta l mor ta l i t y . Equation (8.2) describes change in number of recru i t s with age, in a f i shery with knife edge recruitment. Normally between age t and age t ,natural morta l i ty reduces the rec ru i t s R. Therefore the recru i t s reaching age t are R' = R e ' M ( t c \"V (8.3) 70 71 The number of f i s h at each age for the exploited age groups i s given by N t = R' e \" Z ( t ' V (8.4) Fishing morta l i ty i s responsible for a proportion of the numbers dying and the catch C i s given by the integral tx C = j F R' e \" Z ( t \" V . dt which leads to an equation descr ibing catch C = R' F . (1 - e \" Z ( t A \" lch (8.5) For f i s h species with a large t x , the expression in brackets approaches one and the catch i s approximated by C = FR' (8.6) Z In terms of weight, the y i e l d at any time i s given by f t = F N t W t dt The to ta l y i e l d in weight from a year class i s given by the integra l tx Y = \\ F N t Wt . dt (8.7) 72 A major problem facing f i shery b io log i s t s i s one of f ind ing an unbiased expression for weight in equation (8.7). The von Berta lanf fy growth equation for length i s l t = L - (1 - e \" K ( t \" V) Beverton and Holt (1957) assumed isometric growth and expressed weight in terms of a cubic expression of length W t = \u2022 W- (1 - e \" K ( t \" V)3 Replacing weight i n (8.7) by a cubic expression is a convenient method for evaluating the integra l in (8.7). Though i t i s well accepted now that growth of many f i s h species i s not i sometr ic, the eas iest approach to evaluating the y i e l d integra l (8.7) i s the assumption of isometric growth. However, numerical evaluation of (8.7) for f i s h with a l lometr i c growth can be done using an incomplete Beta funct ion. The above cubic equation when expanded and rearranged can be wr i t ten as a summation. 3 W. = Y U e - n K ^ ~ t o ) (8.8) * j p r o n where U =1.0, - 3.0, 3.0, - 1.0, for n = 0, 1, 2, 3 respect ive ly . I f we replace in (8.7) by the expression in (8.8) and replace N t by the expression for R' in (8.3), the integral (8.7) leads to the Beverton and Holt y i e l d equation. Y = FW~ e-M<*c \" V^T U n e - ^ c \" V . (1 - e ^ Z + \"M* \" V) (8. 0 F + M + nK 73 where Y i s the y i e l d per r e c r u i t . For f i s h species with large tx , the l a s t expression in brackets i n (8.9) could be eliminated without a f fec t ing s i g n i f i c a n t l y the value of y i e l d . But most t rop ica l f i s h species have a short l i f e span and therefore a small t x . Without the expression with in the brackets, equation (8.9) would give biased estimates of y i e l d for f i s h species in the t rop i c s . Using equation (8.9) y i e ld s have been calculated and y i e l d isopleths constructed for Tilapia esculenta in Lake V i c t o r i a and Tilapia nilotica in Lake Albert Uganda. The ages for Tilapia nilotica have been estimated from length by the equation t = 1 (- l n (1 - l t \/ L - ) ) + t Q expressed in the same conventional form of the von Berta lanffy equation: l t = f ( t ) = L- (1 - e \" K ( t \" V) where fo r Tilapia nilotica L\u00b0\u00b0 = 49.0 cm., K = 0.5 and t = 0. Tilapia esculenta i n Lake V i c t o r i a l i ve s for 12 \" r i n g \" years, equivalent to s i x calendar years (Garrod 1963). Tilapia esculenta i n the J i n j a region of Lake V i c t o r i a have the fol lowing population para-meters: L\u00ab> = 33.8 cm., K = 0.32, t = - 0.8. The natural morta l i ty estimated by Garrod (1963) i s 0.17 which is close to the natural morta l i ty rate of 0.16 estimated on the assumption that the mean natural morta l i ty i s a rec iprocal of l i f e span. With th i s natural mor ta l i t y , there i s a p robab i l i t y of 0.07 that a f i s h w i l l reach a maximum age of 6 years. 74 In the ca l cu la t i on of y i e l d , W\u00b0\u00b0 was taken as 730 gm. It i s poss ible that Tilapia esoulenta now being caught in bottom trawling may exceed that weight. Whether the actual maximum weight i s less or greater than 730 gm., does not a f fec t the shape of the y i e l d i sop leths. The age of r e c r u i t -ment t i s taken as zero but th i s does not mean that the young and adult Tilapia esoulenta l i v e in the same habitat. Actua l l y , Tilapia esoulenta i s recru i ted at a length of about 20 cm. corresponding to about 2 years of age. Sett ing t as zero in model (8.9) i s a matter of computational convenience. But th i s i s based on p r io r information that t does not r inf luence the shape of y i e l d isopleths but only reduces the value of y i e l d . Observations on y ie ld s of Tilapia esoulenta as revealed by the y i e l d isopleths in Figure 7, are given below. The greatest y i e l d can be obtained by catching Tilapia esoulenta at a s i ze of 28 cm. but th i s would require a f i sh ing morta l i ty of 2.55. The r e l a t i v e y i e l d obtained under such conditions i s 186.66 gm. per r e c r u i t . I f we catch Tilapia esoulenta at the same s i ze 28 cm. but with a f i s h i ng morta l i t y of 0.9, the y i e l d i s - 182.51 gm\/recruit. This means that i f we increase the f i s h i ng morta l i ty by 183 per cent, the y i e l d increases only by 2.1 percent. I f these f i s h are caught at 26 cm., a f i s h i n g morta l i ty of 0.5 would be necessary to obtain maximum y i e l d . From the to ta l morta l i ty estimates of Garrod (1963) for the years 1958 to 1960, the mean to ta l morta l i ty for that period was 0.3. Since natural mor ta l i t y fo r Tilapia esoulenta i s about 0.17, the f i sh ing morta l i ty for that period was about 0.13. Doubling or t reb l i ng the f i sh ing morta l i ty 36.01 76 would have no adverse ef fects on the f i shery i f the length of capture of 26 cm. and mesh s i ze 4.5 inches were maintained and provided there was no f i s h i n g in the inshore waters where the f i s h spawn. Catching Tilapia esculenta at any length less than 17 cm. would mean catching a l o t of immature f i s h , and th i s w i l l have adverse ef fects on spawning and recruitment. Tilapia nilotica in Lake Albert forms two subpopulations, a stunted population in the Buhuku lagoon and a normal population in the open lake. The normal population has the fo l lowing estimated para-meters: L\u00bb =49 cm., K = 0.50, length of maturity Lm = 36 cm. and natural morta l i ty M = 0.30. The p robab i l i t y density function for the negative exponential i s P ( t ) = M e \" M ( t \" V Tilapia nilotica could l i v e up to 8 years and with a natural morta l i ty of 0.3 about 3 f i s h out of 100 would a t ta i n an age of eight i f there was no f i s h i n g . The maximum weight W\u00ab for the normal Tilapia nilotica in Lake A lbert i s unknown. For the purpose of ca l cu la t i ng r e l a t i v e y ie ld s and constructing y i e l d i sop leths , W\u00bb i s taken as 1000 gm. It i s accepted that absolute y i e l d values are not of primary importance to f i she r ie s management. But what i s of prime importance i s y i e l d response to f i sh ing i n tens i t y and mesh s i ze . At a low f i s h i ng in tens i ty F = 0.1 the best s i ze to catch Tilapia nilotica i s 24 cm. However at 24 cm., the f i s h i s s t i l l immature and the r e l a t i v e y i e l d i s small (76.37 gm\/recruit). I f the s i ze of capture i s increased to 34 cm., the f i sh ing morta l i ty 77 required to give a maximum y i e l d would be 0.4 (see eumetric f i sh ing curve in Figure 8). With a f i sh ing morta l i ty of 0.4 and length of capture of 34 cm. the r e l a t i v e y i e l d would be 159.12 gm\/recruit. I f we ra i se the length of capture to 36 cm., the f i s h i ng morta l i ty required for maximum y i e l d is 0.7. The highest y i e l d for Tilapia nilotica i s obtained at 39 cm., but th i s length of capture requires a f i sh ing morta l i ty rate exceeding 2.1. Note that i f we catch these f i s h at 39 cm. and with a f i sh ing morta l i ty rate of 0.5, we obtain y i e l d of 159.2 gm\/recruit. Increasing the f i s h i ng morta l i t y four times increases y i e l d by a factor of only 1.2. I t i s possible to obtain sustained y i e l d s , i f Tilapia nilotica i s caught at 35 cm. and above with a f i sh ing morta l i ty of 0.5 to 0.6. I t i s a lso of in teres t to note that the highest y i e l d i s obtained a f te r the length of maturity 36 cm(see Figure 8) . l i e s (MS.) reports a natural mor ta l i t y rate of 3.37 for the stunted population of Tilapia nilotica. But th i s population has a high growth rate (K = 2.77) and maximum length i s 17 cm. The l i f e span i s for about one year and sexual maturity i s attained at 10 - 12 cm. corres-ponding to an age of 4 months. With a natural mor ta l i t y rate as high as 3.37 about one f i s h out of a hundred would survive to an age of one year. Is i t poss ible to manage r a t i o n a l l y a f i shery of th i s nature? Because of a high natural mor ta l i t y , the maximum biomass occurs at a length before sexual maturity. The highest y i e l d would be obtained by catching the f i s h at 9 cm. but a high f i s h i ng morta l i ty exceeding 1.8 would be needed (see Figure 9). I f we fo l low the eumetric f i s h i ng curve we should catch the stunted Tilapia nilotica at 8 cm. at about 2.5 months of age. But catching 52.0-48.0-44.0-LU 40.0-f - 36.0-=> Q; 32-0-O LU Q: 26.0-u. o 24.0-X h- 20.0-<\u00a3) ~Z. LU 16.0-_ J 12.0-8.0-4.0-Figure 8 0.8 1.0 1.2 1.4 1.6 IB 2D z\\z 2A 2JS FISHING MORTALITY CM - 0.30, W\u00bb = 1000.0, tx = 8.00, K = 0.50, ^ = 0, t = 0] 195.80 194.29 2j8 3D 00 I r\u2014 2.0 2 2 ~2A i ' i \" 2.8 i 1 \u2014 i 1 1 1\u2014 0.2 0.4 0.6 0.8 -i r 1.0 1.2 1.4 1.6 1.8 FISHING MORTALITY 3.0 Figure 9 - Y ie ld i sopleth diagram for the stunted Tilapia nilotioa in Lake A lbert , Uganda. [M = 3.37, Woo = 100.0, t\\ = 1.10, K = 2.77, t = 0, t =0 ] oo 79 these f i s h at 8 cm. would require a f i sh ing morta l i ty of 1.9 so as to obtain maximum y i e l d . The to ta l morta l i ty (F + M) in such circumstances would be 5.27. With a to ta l morta l i ty of th i s magnitude, one f i s h out of 1000 reaching the age of 2.5 months, would survive to the age of maturity of 4 months. This deduction i s made from the p robab i l i t y density function P(t) = Z e ' Z ( t \" V where t = 4 months, t^ = 2.5 months and Z = 5.27. In these circumstances, i t might be better to allow f i s h to a t t a i n maturity at 10 - 12 cm. and have at least one spawning. Then a high f i sh ing morta l i t y rate can be applied to the f i s h of a s i ze greater than 12 cm. For f i s he r i e s of th i s nature, the eumetric f i s h i ng curve i s not he l p f u l . Populations such as the stunted Tilapia nilotioa, have l i t t l e commercial value. Other Y ie ld Models Because age of t r op i ca l f i s h species cannot eas i l y be determined, there i s a need for use of y i e l d models that are based on the length of a f i s h . The y i e l d model presented by Thompson and Be l l (1934) uses age as well as length. Thompson and Be l l assume that weight increase by some constant percentage in each year of l i f e . Ricker (1944) expresses growth as a simple exponential funct ion. Under the assumption of exponen-t i a l growth and i f we suppose l i f e span to be of i n f i n i t e durat ion, y i e l d i s given by Y = F W (8.10) F + M - g 80 where Y is the y i e l d , F the f i sh ing mor ta l i t y , M the natural morta l i ty and g the instantaneous rate of growth and W i s the to ta l weight of each year ' s brood of r e c r u i t s . I f natural morta l i ty M i s greater than the growth rate g, then fo r a l l values of F, there i s a pos i t i ve y i e l d which approaches an asymptotic value as F approaches i n f i n i t y . When M = g, the y i e l d is simply the i n i t i a l weight of a l l the r ec ru i t s . F i n a l l y , when M < g, the y i e l d i s i n f i n i t e l y large when F <. (g - M). A number of f i s h species show a l lometr ic growth and the y i e l d for these f i s h species can be determined by means of an incomplete Beta funct ion. Jones (1957) and Paul ik and Gales (1964) discuss the usefulness of the incomplete Beta function in determining y i e l d s . The function denoted by B (p, q) i s defined by the integra l B x (p , q) = j XP\" 1 (1 - X ) * \" 1 dx (8.11) where p > 0 and q > 0. In the above i n t e g r a l , the parameters X, P and Q are defined as: Y _ -K(t - t ) x = e c o P = Z\/K Q = 1 + b where b i s the weight length exponent. It i s obvious from the parameters above that y i e ld s can be determined with j u s t a few parameters. The integral above leads to the equation given by Wilimovsky and Wicklund (1963). Y = F W~ e \" Z ( t c \" V ^ B (X, P, Q) - B ( X r P , Q)| (8.12) 81 where X ] = e \" K ^ t x \" V and tx i s the age of e x i t from a f i shery . The y i e l d per gram r e c r u i t i s then given by Y = F e Z ( t c \" V l p (X, P, Q) - p (X,, P, Q) } (8.13) The point of i n f l e x i on on the growth curve, i s the point of maximum biomass. For many f i s h species the greatest y i e l d i s obtained by catching them at an age or s i ze corresponding to the point of i n f l e x i o n . At the point of i n f l e x i on the re lat ionsh ips below hold. b Therefore by knowing the weight length exponent b, the growth rate K and to ta l mor ta l i t y Z, one can predict the maximum y i e l d expected for various f i s h i ng morta l i ty rates. The y i e l d when t i s the age of a f i s h at the i n f l e x i on point i s Y = F e Z ( t c \" V { B ( 1 , P , Q ) - p (X,, P, Q) } (8.14) R W~ K I ' v b 1 J For f i s h species with a large age of e x i t from a f i shery , t x , equation (8.14) reduces to Z ( t , \" 0 { P ( \u00a3 . P > Q ) j ( 8 J 5 ) = F e v c o R W\u00b0\u00b0 K I f we replace age in (8.13) by a length expression an incomplete Beta function that can be used for t r op i ca l f i s h species i s obtained. 82 From equation (5.3) the age of capture t i s 0 t = - In 1 - 1 1 + t 1 0 where 1 i s length of capture corresponding to age t and L\u00b0\u00b0 = maximum C c length. Let X 1 = - In 1 - l c Therefore t = X, 1 + t C 1 T7 o S im i l a r l y tx =- In K \/ 1 - l x 1 + t \" U K 0 tx = xx . l + t K 0 where l x i s the length of e x i t from a f i shery corresponding to the age tx and where XX = - In (1 - 1X\/L\u00bb) Then the parameters for the incomplete Beta function (B) are X, P, Q and X^. Note that 1 xi = V \" *n K 1 c 0 and 1 Xx = tx - t K 0 Therefore X = e 1 P = Z\/K 83 Q = 1 + b If we use length, the y i e l d per gram r ec ru i t i s given by Y = F e ( Z \/ K ) X l \/B (X, P, Q) - B (X. , P, Q) \\ (8.16) R W\u00bb K t \/ Tables of the incomplete Beta function for ca lcu la t ion of f i s h population y i e ld s are given by Wilimovsky and Wicklund (1963). According to equation (8.16), one can determine y ie ld s and construct y i e l d curves with 4 parameters namely (1) maximum length (Loo), (2) the growth rate (K), (3) the weight length exponent (b) and (4) the to ta l morta l i ty (Z). The incomplete Beta function gives unbiased y i e l d estimates for f i s h with a l lometr i c growth. TABLE 12 - Comparison of y i e ld s estimated with equations (8.13) and (8.16) and which are based on age and length respect ive ly . Both equations use the incomplete Beta funct ion. The parameters used are K = 0.5, F = 0.2, M = 0.3, Z\/K = 1, b = 3, tx = 6.39 years, L\u00b0\u00b0 = 49 cm., and l x = 47 cm. L o m c . Age Yrs. Equation(8.13) Equation (8.16) Y ie ld in gn\/recruit Y ie ld in gm\/recruit 28 1.69 0.171 0.172 32 2.11 0.189 0.191 36 2.65 0.207 0.210 40 3.38 0.214 0.220 44 4.56 0.187 0.203 46 5.58 0.104 0.142 84 There i s l i t t l e d i f ference between y ie ld s estimated with equations (8.13) and (8.16). Table 12 shows y ie ld s ca lculated with both equations. I f f i s h are caught at a length close to the maximum length (L\u00b0\u00b0), the y ie ld s estimated with (8.16) d i f f e r s i g n i f i c a n t l y from y ie ld s estimated with equation (8.13). This i s caused by logarithmic transformations made when replacing age by length in equation (8.13). Note that in equation (8.16) is given by X ] = - In (1 - 1 C \/ H As 1 approaches L\u00ab>, the expression (1 - 1 \/L<=\u00b0) approaches zero and therefore X^ tends to i n f i n i t y . In pract ice very few f i s h are caught at a length close to the maximum length (L\u00ab) and therefore the above observation does not a f fec t y i e l d predict ions made from normally exploited length groups. I f one plotted y i e l d against length of capture, the extreme lengths would r e -present the descending limb of the y i e l d curve as shown in Figure 11. Equation (8.16) requires very few parameters and provides a quick way of determining y ie ld s for those f i s h which are d i f f i c u l t to age. Another y i e l d model based on length i s given by Beverton and Holt (1964). The model was derived from the von Berta lanffy growth equation and has the inherent assumption of isometric growth. The para-meters required are M, K, the r a t i o L \/L\u00b0\u00b0 and the f i sh ing morta l i ty (F). With these parameters, the y i e l d and the eumetric f i sh ing curve can be d i r e c t l y read from the tables of y i e l d functions given by Beverton and Holt (1964, 1966). 85 From the von Berta lanffy growth equation we obtain the r a t i o c c -c The r a t i o c represents the to ta l growth in length which is made up before f i s h enter the exploited phase. n \u201e _ a-K(t - t ) I - c = e o The exponential term within the summation in equation (8.9) can be wr i t ten as n (1 - c ) n where the above expression s a t i s f i e s the equal i ty (1 - C ) n = e \" n l < ( t ' V (8.17) The rate of exp lo i ta t ion E i s given by E = F = F F + M Z The f i s h i ng morta l i t y F i s expressed a l gebra i ca l l y in terms of E and M F = M__E (8.18) 1 - E and the rec iprocal of f i sh ing morta l i ty i s 1 = 1 - E (8.19) F M E Af ter transforming the age variables to the length expressions and i f we replace F and 1\/F by the appropriate expressions, the y i e l d model (8.9) 86 becomes iM\/K M = E (1 - c ) M \/ K V~Un (1 - c ) n (8.20) R M \" \/ 1 + nK (1 - E) r\\=o Using equation (8.20), y i e ld s have been calculated for the normal and stunted population of Tilapia nilotica in Lake A lbert Uganda. For the normal Tilapia nilotica the highest y i e l d i s obtained by catching the f i s h at a length of 36 cm. (see Figure 10). I f we catch a f i s h at a small s i z e , we require a low f i sh ing morta l i ty rate to obtain maximum y i e l d (see Figure 11). For the stunted Tilapia nilotica with a natural mor ta l i t y of 3.37, y i e l d increases at a l l rates of f i s h i ng morta l i ty up to 1.05 (see Figure 12). The stunted Tilapia nilotica mature at 10 to 12 cm. but maximum y i e l d i s obtained between 7 cm. and 10 cm. (see Figure 13). For t rop i ca l f i s h species whose age can be d i r e c t l y or i n -d i r e c t l y determined the y i e l d can be estimated with equation (8.9). But th i s y i e l d model assumes isometric growth which i s not true for a l l f i s h species. Ricker (1958) gives a y i e l d model which does not require age and th i s could be used in the tropics to make y i e l d predict ions. The incomplete Beta function (8.13) and (8.16) are unbiased estimators of y i e l d for f i s h species with a l lometr ic growth. Equation (8.16) requires four parameters namely L\u00b0\u00b0, K, Z and the weight length exponent b. Though equation (8.20) has the assumption of isometric growth, i t i s s t i l l very valuable in estimating y i e l d for f i s h that are d i f f i c u l t to age. I f the parameters L\u00b0\u00b0, K, M and F are known, the y i e l d for the r a t i o Lc\/L\u00b0\u00b0 i s read from the tables of y i e l d functions given by Beverton and Holt (1964, 1966). 87 Figure TO - Y ie ld per 100 gm. r e c r u i t plotted against f i sh ing morta l i ty rate ( for Tilapia nilotica in Lake A lbert , Uganda). L i s length of capture in cm., M = 0.3, K = 0.5, and L\u00b0\u00b0 = 4\u00a7.0 cm. 88 19,0 30 35 40 45 50 LENGTH OF CAPTURE IN CM. Figure H - Y ie ld per 100 gm. r e c r u i t p lotted against length of capture at d i f f e ren t f i s h i ng rates (F) ( for Tilapia nilotica in Lake A lber t , Uganda). M = 0.3, K = 0.5 and L~ = 49.0 cm. o o ZD or o UJ or < o: o a: L U o_ o _j UJ FISHING MORTALITY RATE F Figure 12 - Y ie ld per 100 gm. r e c r u i t p lotted against length of capture at d i f f e ren t f i sh ing rates (F) ( for stunted Tilapia nilotica Lake A lber t , Uganda). M = 3.37, K = 2.77, L~ = 17.0 cm. and L \u00a3 i s the length of capture. 4.0 90 ~ \u2014 I I 1 r- 1 1 1 1 1 j 1 4 5 6 7 8 9 10 II 12 13 14 15 16 LENGTH OF CAPTURE F l ' q u r e 13 - Y i e l d per 100 gm. r e c r u i t plotted against length of capture at d i f f e ren t f i sh ing rates (F) ( for Tilapia nilotica in Lake A lbe r t , Uganda). M = 3.37, K = 2.77 and L~ = 17 cm. GENERAL DISCUSSION The alegebraic re la t ionsh ip s , ana l y t i ca l methods and the models here developed should provide short-cut methods that require a minimum of factual data to manage a f i shery . Some general izat ions can be made about the growth processes. The growth rate K and the weight length exponent b determine the age and s i ze of a f i s h at the i n f l ex i on point on the growth curve. The age of a f i s h at i n f l ex i on i s inversely re lated to K and d i r e c t l y re lated to the natural logarithm of b. The s i gn i f i cance of th i s to f i s he r i e s management i s evident when sett ing the mesh s i ze of g i l l n e t s and codend of t raw l . Since maximum biomass occurs at the i n f l ex i on point, the mesh s i ze should be chosen so as to catch f i s h at the i n f l e x i on . For Tilapia species, i t i s shown that the age and s i ze at the i n f l ex i on correspond to the age and s i ze of maturity. Since i t i s always easier to determine the s i ze at matur ity, th i s could serve as a measure of the s i ze of maximum biomass. The rat io s lm\/L\u00b0= and Wm\/W\u00b0\u00b0 are useful constraints in safeguarding against exp lo i ta t ion of immature f i s h . With the parameters K and b and the ra t io s lm\/L\u00b0\u00b0 and Wm\/W\u00b0\u00b0 one w i l l have some rough appreciation of r e l a t i v e rate of exp lo i t a t i on ,o f a f i s h population. A f i s h species inhabit ing d i f f e ren t l o c a l i t i e s may have d i f f e ren t weight length exponents as exemplified by Tilapia nilotica in East A f r i c a . 91 The di f ferences in b for a f i s h species are due to the differences in metabolic processes of the f i s h in the various l o c a l i t i e s . Assuming the exponent of catabolism i s unity as proposed by von Berta lanffy (1957), f i s h with b = 3.0 have m = 0.67. Fish with b > 3.0, have m > 0.67 while f i s h species with b < 3.0, have m < 0.67. These re lat ionsh ips may par t l y explain why some f i s h species may not obey the von Berta lanffy growth equation. In order to appreciate some of the causes of population f l u c -tuations and the decl ine of catches and catch per e f f o r t , one should make estimates of natural and to ta l morta l i ty rates. The quickest and eas iest way of descr ibing morta l i ty rates i s with a negative exponential. One should determine the mean age or mean length in the exploited popu-l a t i on by analysing catch samples. I f the age or length of f i r s t capture i s known, to ta l morta l i ty rates can be estimated with the equations given. These estimators are based on the assumption that the age d i s t r i -bution of explo ited populations conforms to the expectations of the p robab i l i t y density function or the d i screte p robab i l i t y function of the negative exponential. It i s also possible to use the extreme age or length to estimate to ta l morta l i ty and th i s has previously been suggested by Holt (1965). But i t has been noted that the extreme value estimator i s less r e l i a b l e than the negative exponential estimator. The to ta l morta l i ty rate estimated with the extreme value has a larger variance. Therefore extensive sampling is required to show that the extreme age and length in a population have the properties of the extreme value funct ion. Some of the d i f f i c u l t i e s met in estimating growth and morta l i ty rates are due to population f luctuat ions and seasonal changes in d i s t r i bu t i on 93 of f i s h species. Analyses of catch s t a t i s t i c s by mult ip le regression methods provide useful information on abundance and d i s t r i bu t i on of f i s h . A number of var iables a f f ec t catches and one should know which var iables are of greatest importance. Such information may be given by the step-wise mul t ip le regression analysis of trawl catches. Some of th i s information i s useful in advis ing fishermen where and when to f i s h . I f th i s analys is i s car r ied out at spaced periods, one can eas i l y assess the r e l a t i v e degree of exp lo i t a t i on . For example, the analys is of recent catches of Tilapia in Lake V i c t o r i a shows that depth of the bottom, mesh s i ze of codend and time of day of f i sh ing are the most important var iables determining catches. Assessing the state of the f i s h stocks poses special complexit ies. I t i s very un l i ke l y that i t w i l l be possible i n pract ice to qu ick ly solve the problems of taxonomy and to c o l l e c t , a l l the l imno log i ca l , b i o l og i ca l and s t a t i s t i c a l data i dea l l y des i rable for a deta i led evaluation of t r op i ca l f i s h stocks and y i e l d s . Instead we must use simple mathematical models that require a minimum of parameters for making predict ions. The parameters that are v i t a l to y i e l d predict ion are: K, b, L<=\u00b0, M and Z. These parameters can be used to determine y i e l d and construct y i e l d curves as has been shown for Tilapia species. Without age, the same parameters can be used to determine y i e ld s for f i sh with a l lometr i c growth by means of the incomplete Beta funct ion. For f i s h species with enough estimated parameters, the y i e l d model of Beverton and Holt (1957) may be used to determine y ie ld s as shown for Tilapia nilotica and Tilapia esoulenta. I t i s important to note that a model with a few very relevant 94 parameters, has almost the same pred ict ive power as a model requir ing more parameters. Rea l iz ing that the resources for sampling and analysis are often l i m i t e d , simple y i e l d models should be used to speed up assessment. RECOMMENDATIONS FOR FUTURE MANAGEMENT In order to assess qu ick ly the f i s h stocks in t rop i ca l fresh water, research should fo l low the l i nes of attack given below: (a) Representative samples should be co l lec ted from the population to estimate the weight length exponent b. In large lakes l i k e V i c t o r i a , s t r a t i f i e d sampling w i l l give more r e l i a b l e estimates of b than unrestr icted random sampling. (b) Large random samples should be taken in order to determine the largest s i ze to which cer ta in f i s h species grow. The mean maximum length in the d i f f e ren t samples could serve as a rough measure of L\u00b0\u00b0. (c) At present the eas iest way of estimating K for t r op i ca l f i s h i s to carry out tagging experiments and make Walford graphs. The present methods of estimating K needs refinement to avoid the shortcomings i n t r o -duced by age. The p o s s i b i l i t y of estimating K by means of maximum l i ke l i hood should be invest igated. (d) The length of f i r s t capture should be set using information on maturity and s i ze of maximum biomass. (e) The catch samples at various f i s h landings should be sampled to give estimates of mean length in the catch. By means of the equations given the r a t i o Z\/K and to ta l morta l i ty Z, can eas i l y be estimated. ( f ) I t i s r ea l i zed that several var iables a f fec t the catch. The abundance and d i s t r i b u t i on of f i s h species should be investigated by 95 96 mult ip le regression methods. I f th i s analys is i s repeated at certa in in terva l s one can appreciate the r e l a t i v e degree of exp lo i t a t i on . Information on f i s h d i s t r i b u t i on i s necessary for e f f i c i e n t f i s h i n g . (g) A f ter determining the parameters K, b, L\u00b0\u00b0, M and Z some y i e l d predict ions should be made and y i e l d curves constructed. No ca lcu lat ions are needed where the length y i e l d model of Beverton and Holt i s used. The y i e l d values can be read from the tables given by Beverton and Holt (1966). I f the incomplete Beta function is used very few simple ca l cu -la t ions are needed and the y i e l d values can be read from the tables of the incomplete Beta function given by Wilimovsky and Wicklund (1963). LITERATURE CITED Baranov, F. I. (1918). K vopresy biologicheskogo osnovaniya rybnogo khozyaistva. [On the question of the b io log i ca l basis of f i s h e r i e s . ] Nauch. i s s l edova te l s k i i i k h t i o l og i che sk i i Inst. Izv., 1 (1): 81-128. Ber ta lan f fy , L. von (1934). Untersuchungen uber die Gesetz l i chke i t des Wachstums I. Allgemeine Grundlagen der Theorie. Arch.Entw Mech. Org., 131:613-653. (1938). A quant i tat ive theory of organic growth. Hum. B i o l . , 10(2): 181-243. (1957). Quantitat ive laws in metabolism and growth. Quart. Rev. B i o l . , 32:217-231. Beverton, R.J.H. (1963). Maturation growth and morta l i ty of c lupeid and engraulid stocks in r e l a t i on to f i s h i n g . Rapp. Proces-Verb. Cons. Intern. Explor. Mer. 154:44-67. Beverton, R.J.H. and Hol t , S. J . (1954). A review of methods for estimating morta l i t y rates in exploited f i s h populations with special r e f e r -ence to sources of bias in catch sampling. Rapp. Proces-Verb. Cons. Intern. Explor. Mer., 140: 67-83. (1957). On the dynamics of explo ited f i s h populations. F i sh. Invest. London (2) 19, 153 p. (1959). A review of the l i fespan and morta l i ty rates of f i s h in nature and the r e l a t i on to growth and other phys io logical charac-t e r i s t i c s . In: Wolstenholme, G. E. and O'Connor, M. (ed.). The l i fespan of animals. Ciba Foundation, Colloquia on ageing 5:142-177. Churchi l l London. (1964). Tables of y i e l d functions for f i shery assessment F.A.O. Fish Tech. Paper (38) 49 p. (1966). Manual of methods fo r f i s h stock assessment. Part II. Tables of y i e l d functions F.A.O. Fish Tech. Paper (38) Rev. 1. 97 98 Burington, R. S. and May, D. C. (1958). Handbook of P robab i l i t y and S t a t i s t i c s with Tables. Handbook Publishers Inc., Sandusky, Ohio. Carlander, K. D. (1969). Handbook of Freshwater Fishery Biology Vo. 1. Iowa State Univers i ty Press. Fryer, G. (1961). Observations on the biology of the c i c h l i d f i s h Tilapia variabilis Boulenger in the northern waters of Lake V i c to r i a (East A f r i ca ) Rev. Zool., Bot. A f r . 64: 1-33. Garrod, D. J . (1959). The growth of Tilapia esculenta Graham in Lake V i c t o r i a . Hydrobiologia 12: 268-298. (1963). An estimation of the morta l i ty rates in a population of Tilapia esculenta Graham (Pisces c ich l idae) in Lake V i c t o r i a , East A f r i c a . J . Fish. Res. Bd. Can., 20(1):195-227. Graham, M. (1953). Modern theory of exp lo i t ing a f i shery and appl icat ion to North Sea t rawl ing. J . Cons. perm. i n t . Explor. Mer. 10(2): 264-274. Gulland, J . A. (1969). Manual of methods for f i s h stock assessment. Part 1. Fish population analys i s . F.A.0. Manuals in Fisheries Science No. 4. Gumbell, A. J . and L i eb l e i n , J . (1954). S t a t i s t i c a l theory of extreme values and some p rac t i ca l app l i cat ions . Appl. Math. Ser. U.S. Nat. Bur. Stand., (33). Hecht, S. (1916). Form and growth in f i shes. J . Morph., 27:379-400. Hodges, J . L. and Lehmann, E. L. (1965). Elements of f i n i t e p robab i l i t y . Hoi den-Day Inc. San Francisco, London. Hol t , S. J . (1962). A prel iminary comparative study of the growth maturity and morta l i ty of Sardines. Proc. World S c i . Meet. B i o l . Sardines r e l a t . Spec. 2:553-561. F.A.0. Rome. (1965). A note on the r e l a t i on between the morta l i ty rate and the duration of l i f e in an exploited f i s h population. Jo int S c i . Meet. ICNAF, ICES and F.A.O., Lisbon. ICNAF Res. B u l l . , (2). l i e s , T. D. (MS.). Dwarfing or stunting in the genus Tilapia ( c ich l idae) a possibly unique recruitment mechanism. International Council for the Exploration of the Sea, F.A.0. and ICNAF. A symposium on Stock and Recruitment [ in press]. Jones, R. (1957). A much s imp l i f i ed version of the f i s h y i e l d equation. Document No. P 21 presented at the Lisbon j o i n t meeting of ICNAF\/ ICES\/FA0, 8 pp. 99 K e l l i c o t t , W. E. (1908). The growth of the brain and v i scera in the smooth dogfish (Mustelus oanis). Amer. J . Anat., 8:319-354. Kendal l, D. G. (1955). S t a t i s t i c s of w indfa l l s and d i sasters . Nature, Lond., 175 (4450):290. Lowe, McConnell R. H. (1955). The fecundity of Tilapia species. East A f r . Agr ic . J . , 21(l):45-52. (1956). 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A l lometr ic weight-length regression model. J . Fish. Res. Bd. Can., 26(1): 123-131. Regier, H. A. (1970). Fish stock assessment on Afr ican inland waters. Working paper No. 1. Current problems in assessing Lake V i c t o r i a stocks, F.A.O. 9 p. Ricker, W. E. (1944). Further notes on f i s h i ng morta l i ty and e f f o r t . Copeia, 1944 (1): 23-44. (1958). Handbook of computations for b io log i ca l s t a t i s t i c s of f i s h populations. F i sh. Res. Bd. Can., B u l l . No. 119. 300 p. (1960). Ciba Foundation Col loquia on ageing Vol. 5. The l i f e -span of animals. Review in J . Cons. perm. i n t . Explor. Mer., 26:125-129. 100 Russe l l , E. S. (1931). Some theoret ica l considerations on the over-f i s h i ng problem. J . Cons. perm. i n t . Explor. Mer, 6(1):3-27. Svardson, G. (1949). Natural se lect ion and egg number in f i s h . Rept. Inst. Freshwater Research, Drottningholm 29:115-122. Taylor, C. C. (1959). Temperature an,d growth - the P a c i f i c razor clam. J . Cons. perm. i n t . Explor. Mer., 25:93-101. (1960). Temperature, growth and morta l i ty - the P a c i f i c cockle. J . Cons. perm. i n t . Explor. Mer., 26:117-124. (1962). Growth equations with metabolic parameters. J . Cons. perm. i n t . Explor. Mer., 27:270-286. Thompson, W. F. and B e l l , F. H. (1934). B io log ica l s t a t i s t i c s of the P a c i f i c Hal ibut f i shery . (2) Ef fect of changes in i n tens i t y upon to ta l y i e l d and y i e l d per unit of gear. Rept. Int. Fish [ P a c i f i c Hal ibut] Comm. No. 8, 49 pp. Seat t le , Washington. Urs in , E. (1967). A mathematical model of some aspects of f i s h growth, re sp i ra t i on and morta l i t y . J . F ish. Res. Bd. Can., 24(11): 2355-2453. Welcomme, R. L. (1964). The habitats and habitat preferences of the young of the Lake V i c t o r i a Tilapia (Pisces: C i ch l idae ) . Rev. Zool. Bot. A f r . 70:1-28. (1966). The e f fec t of rap id ly changing water level in Lake V i c t o r i a upon the commercial catches of Tilapia (Pisces: C ich l idae) . In: Obeng (ed.) 1969; Man-Made Lakes. The Accra Symposium,Ghana Univers i ty Press, Accra. Wilimovsky,N. J . and Wicklund, E. C. (1963). Tables of the incomplete Beta function for the ca l cu la t i on of f i s h population y i e l d . Vancouver, Univers i ty of B r i t i s h Columbia, I n s t i tu te of F i sher ies , 291 p. APPENDIX 1 DERIVATION OF PROBABILITY DENSITY FUNCTION FOR A NEGATIVE EXPONENTIAL For an unexploited f i s h population the number of f i s h at age t i s given by N t = NQ e \" M ( t \" V (9.1) The proportion of number of f i s h at 'age t to the i n i t i a l numbers at age N, - e \" ^ \" V (9.2) S i n c e t h e a r e a u n d e r a n e g a t i v e e x p o n e n t i a l c u r v e i s u n i t y , summing up t h e p r o p o r t i o n s d e s c r i b e d by e q u a t i o n (9.2) s h o u l d y i e l d 1. The p r o b a b i l i t y d e n s i t y f u n c t i o n i s d e f i n e d a s F (X = x ) = Jf(x) dx = 1 CL I n t e g r a t i n g t h e r i g h t hand s i d e o f (9.2) b e t w e e n t = o a n d t = <\u00bb g i v e s oo e 4 1** - V d t = 1 (9.3) J M o The r i gh t hand side of (9.2) i s normalized to a p robab i l i t y density function by d iv id ing by i t s i n t e g r a l . 101 102 Therefore e-\"'* - V = H e-M<' \" V (9.4) f e - ^ - V dt The p robab i l i t y of age t i s given by P(T = t ) = M e \" M ( t \" t o ) for t > t Q (9.5) F ( T ) = ( M e ^ - V ,\u00ab3 and ( ) = 1 M e \" n v u \" V dt = 1 where F(T) i s the integra l of the der i vat i ve below: M e 4 1** - V For a f i shed population we can think of the proportions of f i s h beyond age t as adding to un i ty. The numbers at each age t can be expressed as a proportion of the r e c r u i t s , so that N t = e \" Z ( t \" V (9.6) The r i g h t hand side of equation (9.6) i s normalized to a p robab i l i t y density function by d i v id ing by i t s i n t e g r a l , e - ^ - *c> = Z e - \u00ab * - * c > (9.7) e-Z(t - t c ) dt and the p robab i l i t y of obtaining age t in a sample catch i s P ( t ) = Z e \" Z { t ' t c ) f o r t i *c (9.8) 103 Equation (9.8) integrated gives 1 and hence i t i s a p robab i l i t y density funct ion. F (T) - I Z e \" 2 ^ \" t c ) d t = 1 APPENDIX 2 DERIVATION OF DISCRETE PROBABILITY FUNCTION FOR A NEGATIVE EXPONENTIAL In an explo ited f i s h population the number of f i s h at age t i s given by N t = N o e \" Z ( t \" ^ ( 1 0 J ) and the number of f i s h age t + 1 i s given by N t + 1 = NQ e \" Z ( t + 1 \" t c ) (10.2) The proportion of N^ . + 1 over N^ . i s N t + , = ,-\u00abt \u2022 1 - t c - (t - t \u00a3 \u00bb ( 1 0 i 3 ) N t With a d i screte time model with an i n f i n i t e number of time periods, the to ta l explo ited population i s given by N = N + N r t e \" Z + N e \" 2 Z + . . . . + N e \" 2 \" (10.4) o o o o where N i s the to ta l number of f i s h from a l l exploited age groups N , here refers to the number of f i s h recru i ted at age t . Because the area 3 c under the exponential curve between t and \u00b0\u00b0 i s unity, we can re fer to t as zero time. 104 105 Then N = NQ (1 + e \" Z + e \" 2 Z + . . . . + e\"Z\u00b0\u00b0) (10.5) for ( t = 0, 1, 2, oo) Under the assumption of constant recruitment NQ i s taken as unity and equation (10.5) reduces to N = 1 + e~ Z + e \" 2 Z + . . . . + e\"Z\u00b0\u00b0 (10.6) Note that equation (10.6) i s an i n f i n i t e se r ie s . The sum of th i s ser ies i s given by N = 1 (1 - e\"Z\u00b0\u00b0) (10.7) 1 - e \" Z when t = \u00b0\u00b0 , the numerator of the r i gh t hand expression of (10.7) becomes unity. Then N i s estimated by N = _ J (10.8) 1 - e \" Z But for f i n i t e age groups N i s given by N = 1 - e \" Z t (10.9) 1 - e \" Z The expressions in (10.9) can be evaluated for a given t and Z. Note that equation (10.9) can be used to estimate the adult exploited population provided we know the rec ru i t s or i f we can estimate the r e c r u i t s . The exploited population i s given by N = R (1 - e ~ Z t ) (10.10) (1 - e \" Z ) 106 where R are the rec ru i t s and t the oldest age in a population. From the der ivat ion of the geometric series in (10.5) and (10.6), the p robab i l i t y of any age t i s given by e- 2 t By the de f i n i t i o n of a d i sc rete p robab i l i t y d i s t r i bu t i on CO 0 where px = P (X = x.) = f (x.) i 1 1 Note that f (x^) above represents e ~ Z t but oo 0 The term e ~ Z t can be normalized to a d i screte p robab i l i t y function by d i v id ing by o Therefore the p robab i l i t y of age t in the case of d i screte recruitment is P (T = t) = e \" Z t (10.11) 0 The denominator of the r i gh t hand side of (10.11) i s an i n f i n i t e ser ies and fo r e~ Z < 1, the expression i s equal to 107 e \" Z t = 1 1 - e \" Z Therefore for a d i sc rete time model the p robab i l i t y of age t i s P (t) = (1 - e \" Z ) e \" Z t for t > t c (10.12) The mean age U of the exploited age groups i s given by the equation below (Burington 1958). U = > t . f ( X l ) Note that f ( x . ) = e \" Z t (1 - e \" Z ) \u2022It -Z, Therefore U = t . e \" . (1 - e~L) (10.13) 0 U can be estimated by mean age t\" of a sample. CO t 1 \u2022 e \" Z t \u2022 ( 1 - e _ Z ) (10.14) Evaluate expression X~ t . e \" Z t (1 - e \" Z ) factor ing 1 - e \" Z . > t e \" Z t 108 This summation above gives the i n f i n i t e ser ies : S 0 Let CO ] T t e \" Z t S = 0 + e ' Z + 2 e \" 2 Z + 3 e ' 3 Z +.... + oo e ~ Z \" S . i z '=\u2022\u2022 + e \" 2 Z + 2 e _ 3 Z + . . . . +(\u00bb - l ) e \" Z \u00b0 Subtracting S e \" Z from S S - s e \" Z = 0 + e ' Z + e ' 2 Z + e \" 3 Z + . . . . + e ' Z a The above i s a geometric progression which can be wr i t ten as S (1 - e _ Z ) = e _ Z (1 - e~Z\u00b0\u00b0) 1 - e \" Z S (1 - e \" Z ) =e~Z . 1 l - e \" Z Therefore the summation gives S = e \" Z (1 - e ' Z ) (1 - e _ Z ) S = e \" Z (1 - e - \u00a5 The mean age t i s therefore given by t = (1 - e ' Z ) e \" Z (1 - e \" Z ) 2 t = e \" Z (10.15) 1 - e \" Z ","attrs":{"lang":"en","ns":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","classmap":"oc:AnnotationContainer"},"iri":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","explain":"Simple Knowledge Organisation System; Notes are used to provide information relating to SKOS concepts. 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