03 Depletion

03 Depletion - Stock DEPLETION MODELS(recruitment assumed to be zero RELATIONSHIP BETWEEN SURVIVAL and the APPLICATION of FISHING EFFORT there are

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Unformatted text preview: Stock DEPLETION MODELS (recruitment assumed to be zero) RELATIONSHIP BETWEEN SURVIVAL and the APPLICATION of FISHING EFFORT there are two models, one linear and one exponential LESLIE'S METHOD Assume that we have a fishery which is CLOSED (no in or out) and that we are dealing in a short interval for which M ¸ 0 then logically if we define Ct = catch at time t at time 1: at time 2: at time 3: N" = N! C ! N# = N! C ! C " N$ = N! C! C" C # or, in general t-1 N> = N! D Ci i=0 then if we define Kt = the number of individuals removed from the fishery from time 0 to time t-1, so t-1 K> = D Ci i=0 and Nt = N! Kt but, “N" is not an observable variable in most studies, so after multiplying through by q (the catchability coefficient) t-1 qNt = qN - q( D Ci ) i=0 Ct ft t-1 = b! + b" ( D Ci ) = b! + b" (K> ) i=0 which can be fitted by a simple linear regression line EXST025 - Biological Population Statistics Leslie's Model qN 0 Catch per unit of effort Leslie model q Cumulative catch a) the slope of the line is an estimate of q b) the intercept is qN! = CPUE! c) the initial population can be estimated by b! b" = qN! ^ q ^ = N! Page 2 EXST025 - Biological Population Statistics Page 3 DELURY'S METHOD first note that we never get all of the population as implied by the LESLIE MODEL, the line will approach the X axis, but will level off and not reach it Delury defines Ct ft = qNt = qNt N! N! = qN! Nt N! then, with a multiplicative error term we can get N Log( Ctt ) = Log(qN! ) + Log( N!t ) + Log(e> ) f he then describes an index; when q is small ( 0.02) then the fraction can be used as an index of stock remaining after Et units of effort are expended ie. Nt N! = e-qE> or N log( N!t ) = -qEt this then describes a linear relationship between the Log(ratio of stock) and the effort this is similar to the Leslie method in concept Catch per unit of effort qN0 Delury model q Cumulative effort EXST025 - Biological Population Statistics DELURY proceeds N Log( Ctt ) = Log(qN! ) + Log( N!t ) f Log( Ctt ) = Log(qN! ) - qEt f then CPUE = Ct ft = qN! eqE> where qN! = U! which is fitted by the model Yi = "! e-"" Xi %i Yi = log("! ) "" Xi + %i these methods (LESLIE and DELURY) generally do not fit well unless a very sizable portion of the stock is removed in a fishing season not that this is an exponential decay model with “X" as some variable other than time Another derivation is given by N> = N! DC where; C = qNf DC> ¸ qN! Df then N> ¸ N! qN! Df N> ¸ N! (1 qDf) and since 1 + X is approximately eX for small negative X (q is ) N> ¸ N! eqDf Page 4 EXST025 - Biological Population Statistics Page 5 REVIEW OF FISHERIES MODELS AND RELATIONSHIPS COVERED 1) fisheries relationships CPUE> = Ut = q*Nt = Ct ft Catch = C> = qN> f> 2) Mortality and survival Nt N! = ert = S = eZt = eF+M N! e(F+M)t or A = 1 - ert = the non-surviving portion 3) Exponential Growth Nt = N! ert which can be obtained by integrating ˜N ˜t = + rN ˜t ˜N = 1 rN Ê t = 1 r ' ˜N N Summary of Depletion models Leslie model Nt = N! - Kt Ê t qNt = qN - q( D Ci ) i=1 Ut = U! - Kt Ê U> = b! + b" (K> ) Delury Model Ct ft = U> = qNt = qNt Ut = U! e-qE> N! N! = qN! t-1 where E> = D fi i=0 Nt N! = 1 r ln(N) + c Page 1 EXST7025 – Biological Population Statistics 1 ***********************************************; 2 *** Data from Ricker (1974) ***; 3 *** Catch of Small-mouth bass ***; 4 ***********************************************; 5 options ps=256 ls=99 nocenter nodate nonumber; 6 7 ODS HTML style=minimal rs=none 8 body='C:\Geaghan\EXST\Exst7025New\Spring2004\Bass\bass.html' ; NOTE: Writing HTML Body file: C:\Geaghan\EXST\Exst7025New\Spring2004\Bass\bass.html 9 10 11 data one; infile cards missover; 12 TITLE1 'Catch and effort of small-mouth bass'; 13 input Day Catch CumCatch CumEffrt; 14 cpue = catch / 10; 15 lcpue = log(cpue); 16 label Day = 'Day of the experiment' 17 Catch = 'Bass caught (number)' 18 CumCatch = 'Cumulative catch' 19 CumEffrt = 'Cumulative effort (10 per day)' 20 CPUE = 'Catch per unit of effort'; 21 Cards; NOTE: Missing values were generated as a result of performing an operation on missing values. Each place is given by: (Number of times) at (Line):(Column). 1 at 14:18 1 at 15:13 NOTE: The data set WORK.ONE has 11 observations and 6 variables. NOTE: DATA statement used (Total process time): real time 0.03 seconds cpu time 0.03 seconds 21 ! run; 33 ; 34 proc print data=one; TITLE2 'Raw data print'; run; NOTE: There were 11 observations read from the data set WORK.ONE. NOTE: The PROCEDURE PRINT printed page 1. NOTE: PROCEDURE PRINT used (Total process time): real time 0.04 seconds cpu time 0.01 seconds Catch and effort of small-mouth bass Raw data print Obs Day Catch Cum Catch Cum Effrt cpue lcpue 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 . 131 69 99 78 56 76 49 42 63 47 . 131 200 299 377 433 509 558 600 663 710 1000 10 20 30 40 50 60 70 80 90 100 . 13.1 6.9 9.9 7.8 5.6 7.6 4.9 4.2 6.3 4.7 . 2.57261 1.93152 2.29253 2.05412 1.72277 2.02815 1.58924 1.43508 1.84055 1.54756 . 36 options ps=45; 36 ! proc plot data=one; plot catch*Cumeffrt; 37 TITLE2 'Scatter plot'; run; 38 NOTE: There were 11 observations read from the data set WORK.ONE. NOTE: The PROCEDURE PLOT printed page 2. NOTE: PROCEDURE PLOT used (Total process time): real time 0.06 seconds cpu time 0.00 seconds Page 2 EXST7025 – Biological Population Statistics Catch and effort of small-mouth bass Scatter plot 140 B a 120 s s c a u 100 g h t ( n u m b e r ) 80 60 40 NOTE: 39 40 41 42 NOTE: NOTE: NOTE: 43 NOTE: NOTE: NOTE: Plot of Catch*CumEffrt. Legend: A = 1 obs, B = 2 obs, etc. | | + | | |A | | + | | | | | + A | | | | | + | A A | | A | | A + | A | | A | A | A + | -+---------+---------+---------+---------+---------+---------+---------+---------+---------+10 20 30 40 50 60 70 80 90 100 Cumulative effort (10 per day) 1 obs had missing values. proc reg data=one lineprinter; TITLE2 'Delury model'; model lcpue = CumEffrt; plot residual.*predicted.; output out=next1 p=yhat r=e; run; 11 observations read. 1 observations have missing values. 10 observations used in computations. The data set WORK.NEXT1 has 11 observations and 8 variables. The PROCEDURE REG printed pages 3-4. PROCEDURE REG used (Total process time): real time 0.09 seconds cpu time 0.04 seconds Catch and effort of small-mouth bass Delury model The REG Procedure Model: MODEL1 Dependent Variable: lcpue Source Model Error Corrected Total DF 1 8 9 Analysis of Variance Sum of Mean Squares Square 0.70370 0.70370 0.41582 0.05198 1.11952 F Value 13.54 Pr > F 0.0062 Page 3 EXST7025 – Biological Population Statistics Root MSE Dependent Mean Coeff Var 0.22799 1.90141 11.99034 R-Square Adj R-Sq 0.6286 0.5821 Parameter Estimates Variable Intercept CumEffrt DF 1 1 Parameter Estimate 2.40937 -0.00924 Standard Error 0.15574 0.00251 t Value 15.47 -3.68 Pr > |t| <.0001 0.0062 Dependent Variable: lcpue ---+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+---0.3 + + | | | 1 1 | | | | | 0.2 + + | 1 | | 1 | | | | | 0.1 + + R | | e RESIDUAL | 1 | s | | i | 1 | d 0.0 + + u | | a | | l | | | | -0.1 + + | | | | | | | 1 | -0.2 + + | 1 | | 1 | | | | | -0.3 + 1 + ---+-------+-------+-------+-------+-------+-------+-------+-------+-------+-------+---1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 Predicted Value of lcpue PRED 44 proc univariate data=next1 normal plot; var e; 45 TITLE3 'Residual analysis'; run; NOTE: The PROCEDURE UNIVARIATE printed pages 5-7. NOTE: PROCEDURE UNIVARIATE used (Total process time): real time 0.06 seconds cpu time 0.01 seconds Catch and effort of small-mouth bass Delury model Residual analysis The UNIVARIATE Procedure Variable: e (Residual) N Mean Std Deviation Skewness Uncorrected SS Coeff Variation Moments 10 Sum Weights 0 Sum Observations 0.21494731 Variance -0.1476426 Kurtosis 0.41582113 Corrected SS . Std Error Mean Basic Statistical Measures Location Variability Mean 0.000000 Std Deviation Median 0.037964 Variance Mode . Range Interquartile Range 10 0 0.04620235 -1.8347444 0.41582113 0.06797231 0.21495 0.04620 0.55552 0.39774 Page 4 EXST7025 – Biological Population Statistics Tests for Location: Mu0=0 Test -Statistic-----p Value-----Student's t t 0 Pr > |t| 1.0000 Sign M 1 Pr >= |M| 0.7539 Signed Rank S -0.5 Pr >= |S| 1.0000 Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling Tests for Normality --Statistic--W 0.890183 D 0.19041 W-Sq 0.071035 A-Sq 0.454665 -----p Value-----Pr < W 0.1704 Pr > D >0.1500 Pr > W-Sq 0.2465 Pr > A-Sq 0.2181 Extreme Observations -------Lowest-----------Highest-----Value Obs Value Obs -0.2931394 2 0.0617519 10 -0.2354387 8 0.1602302 3 -0.2248254 5 0.1729125 6 -0.1736443 7 0.2555951 1 0.0141754 4 0.2623827 9 Stem 2 1 0 -0 -1 -2 Leaf 66 67 16 # 2 2 2 7 1 942 3 ----+----+----+----+ Multiply Stem.Leaf by 10**-1 Boxplot | +-----+ *--+--* | | | | +-----+ Normal Probability Plot 0.25+ *+++ * | *++*++ | *+*+++ | ++++ | +++++* -0.25+ * +++* * +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 47 48 49 50 NOTE: NOTE: NOTE: 51 NOTE: NOTE: NOTE: proc reg data=one lineprinter; TITLE2 'Leslie model'; model Cpue = CumCatch; plot residual.*predicted.; output out=next2 p=yhat r=e; run; 11 observations read. 1 observations have missing values. 10 observations used in computations. The data set WORK.NEXT2 has 11 observations and 8 variables. The PROCEDURE REG printed pages 8-9. PROCEDURE REG used (Total process time): real time 0.07 seconds cpu time 0.03 seconds Page 5 EXST7025 – Biological Population Statistics Catch and effort of small-mouth bass Leslie model The REG Procedure Model: MODEL1 Dependent Variable: cpue Catch per unit of effort Source Pr > F Model 0.0072 Error Corrected Total Root MSE Dependent Mean Coeff Var DF Analysis of Variance Sum of Squares Mean Square F Value 12.82 1 40.96602 40.96602 8 9 25.55398 66.52000 3.19425 1.78725 7.10000 25.17248 R-Square Adj R-Sq 0.6158 0.5678 Parameter Estimates Variable Intercept CumCatch DF 1 1 Parameter Estimate 11.99420 -0.01092 Standard Error 1.47890 0.00305 t Value 8.11 -3.58 Pr > |t| <.0001 0.007 Dependent Variable: cpue Catch per unit of effort -+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-3 + + | | | 1 | | | | | 2 + + | | | 1 | | | | 1 1 | 1 + + R | | e RESIDUAL | | s | 1 | i | | d 0 + 1 + u | | a | | l | | | | -1 + 1 + | 1 | | | | 1 | | | -2 + + | | | | | | | | -3 + 1 + -+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 Predicted Value of cpue PRED 52 proc univariate data=next2 normal plot; var e; 53 TITLE3 'Residual analysis'; run; NOTE: The PROCEDURE UNIVARIATE printed pages 10-12. NOTE: PROCEDURE UNIVARIATE used (Total process time): real time 0.07 seconds cpu time 0.01 seconds 54 ods html close; Page 6 EXST7025 – Biological Population Statistics Catch and effort of small-mouth bass Leslie model Residual analysis The UNIVARIATE Procedure Variable: e (Residual) N Mean Std Deviation Skewness Uncorrected SS Coeff Variation Moments 10 Sum Weights 0 Sum Observations 1.68503151 Variance -0.2552028 Kurtosis 25.5539806 Corrected SS . Std Error Mean Basic Statistical Measures Location Variability Mean 0.000000 Std Deviation Median 0.193296 Variance Mode . Range Interquartile Range 10 0 2.83933118 -0.7219366 25.5539806 0.53285375 1.68503 2.83933 5.44621 2.41171 Tests for Location: Mu0=0 Test -Statistic-----p Value-----Student's t t 0 Pr > |t| 1.0000 Sign M 0 Pr >= |M| 1.0000 Signed Rank S -0.5 Pr >= |S| 1.0000 Test Shapiro-Wilk Kolmogorov-Smirnov Cramer-von Mises Anderson-Darling Tests for Normality --Statistic--W 0.972752 D 0.155598 W-Sq 0.030955 A-Sq 0.191153 -----p Value-----Pr < W 0.9151 Pr > D >0.1500 Pr > W-Sq >0.2500 Pr > A-Sq >0.2500 Extreme Observations ------Lowest-----------Highest----Value Obs Value Obs -2.909292 2 0.462236 10 -1.663868 5 1.166398 6 -1.239466 8 1.172240 3 -0.998298 7 1.548781 9 -0.075644 4 2.536913 1 Stem Leaf 2 5 1 225 0 5 -0 1 -1 720 -2 9 ----+----+----+----+ # Boxplot 1 | 3 +-----+ 1 *--+--* 1 | | 3 +-----+ 1 | Normal Probability Plot 2.5+ +++*++ | * +*++*+ | +*++++ | +*++* | +*++*+ -2.5+ ++*+++ +----+----+----+----+----+----+----+----+----+----+ -2 -1 0 +1 +2 Page 7 EXST7025 – Biological Population Statistics Population estimates Leslie estimates : CPUE = 11.99420 - 0.01092 (cumulative catch) t 1 cpue qN i qN 0 q Ci , where q = 0.01092 , then i 0 N0 = qN0 / q = 11.99420 / 0.01092 = 1098.369963 Delury estimates : log(CPUE) = 11.99420 exp(-0.00924 (cumulative effort)) t 1 q Ei i 0 , where q =0.00924 , then N0 = qN0 / q = exp(2.40937) / 0.00924 = 11.12694896 / 0.00924 = 1204.215255 log(cpue) log(qN 0 )e cpue 14 12 10 8 Leslie Model Delury Model 6 4 2 0 0 100 200 300 400 500 Cumulative Catch Note, for small negative r, “1-er ≈ r” r ­0.6932 ­0.5000 ­0.0500 ­0.0100 ­0.0010 ­0.0001 er 0.5000 0.6065 0.9512 0.9900 0.9990 0.9999 1­er 0.5000014097 0.3934693403 0.0487705755 0.0099501663 0.0009995002 0.0000999950 600 700 800 ...
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This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.

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