De Lury method

Reading the data

Data <- read.csv("DeLury_R.csv", header=T)
Data
##    Period Catch Effort  CPUE CumCatch CumEffort
## 1       1    60  170.6 0.352        0       0.0
## 2       2   274  453.8 0.604       60     170.6
## 3       3   240  513.4 0.467      334     624.4
## 4       4   244  714.4 0.342      574    1137.8
## 5       5   301  679.1 0.443      818    1852.2
## 6       6   151  419.9 0.360     1119    2531.3
## 7       7   127  470.3 0.270     1270    2951.2
## 8       8    90  318.4 0.283     1397    3421.5
## 9       9    31  136.8 0.227     1487    3739.9
## 10     10     7  177.6 0.039     1518    3876.7
CPUE <- Data$CPUE
CumCatch <- Data$CumCatch
CumEffort <- Data$CumEffort

plot(CumCatch, CPUE, pch=19, col="red",
     main="CPUE against cumulative catch")

Analyzing data by De Lury method 1

Res.DeLury1 <- lm(CPUE~CumCatch)
summary(Res.DeLury1)
## 
## Call:
## lm(formula = CPUE ~ CumCatch)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.16798 -0.03658  0.01883  0.06804  0.10617 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.098e-01  6.013e-02   8.478 2.87e-05 ***
## CumCatch    -1.995e-04  5.883e-05  -3.391  0.00949 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1034 on 8 degrees of freedom
## Multiple R-squared:  0.5897, Adjusted R-squared:  0.5384 
## F-statistic:  11.5 on 1 and 8 DF,  p-value: 0.00949
plot(CumCatch, CPUE, pch=19, col="red",
     main="CPUE against cumulative catch")
abline(Res.DeLury1, col="purple", lwd=2, lty=2)

Para1 <- coef(Res.DeLury1)
Para1
##  (Intercept)     CumCatch 
##  0.509796545 -0.000199483
q.est1 <- (-1)*Para1[2]
N.est1 <- Para1[1]/q.est1

data.frame(q.est1, N.est1)
##               q.est1   N.est1
## CumCatch 0.000199483 2555.589

Analyzing data by De Lury method 2

Ignoring M (M=0)

Res.DeLury2 <- lm(log(CPUE)~CumEffort)
summary(Res.DeLury2)
## 
## Call:
## lm(formula = log(CPUE) ~ CumEffort)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.36018 -0.09308  0.18410  0.37092  0.46580 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -0.5562541  0.3303927  -1.684   0.1308  
## CumEffort   -0.0003425  0.0001338  -2.560   0.0336 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5946 on 8 degrees of freedom
## Multiple R-squared:  0.4503, Adjusted R-squared:  0.3816 
## F-statistic: 6.553 on 1 and 8 DF,  p-value: 0.03365
plot(CumEffort, log(CPUE), pch=19, col="red",
     main="log CPUE against cumulative effort")
abline(Res.DeLury2, col="blue", lwd=2, lty=3)

Para2 <- coef(Res.DeLury2)
Para2
##   (Intercept)     CumEffort 
## -0.5562540837 -0.0003424973
q.est2 <- (-1)*Para2[2]
N.est2 <- exp(Para2[1])/q.est2

data.frame(q.est2, N.est2)
##                 q.est2   N.est2
## CumEffort 0.0003424973 1674.036

Try to estimate unknown M

Period1 <- Data$Period-1

Res.DeLury3 <- lm(log(CPUE) ~ CumEffort + Period1)
summary(Res.DeLury3)
## 
## Call:
## lm(formula = log(CPUE) ~ CumEffort + Period1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.82722 -0.05662  0.16564  0.29739  0.38859 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -0.2169073  0.3107284  -0.698   0.5077  
## CumEffort    0.0014243  0.0008032   1.773   0.1195  
## Period1     -0.8726329  0.3929920  -2.220   0.0618 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4869 on 7 degrees of freedom
## Multiple R-squared:  0.6775, Adjusted R-squared:  0.5853 
## F-statistic: 7.352 on 2 and 7 DF,  p-value: 0.01905
Para3 <- coef(Res.DeLury3)
Para3
##  (Intercept)    CumEffort      Period1 
## -0.216907302  0.001424257 -0.872632934
q.est3 <- (-1)*Para3[2]
N.est3 <- exp(Para3[1])/q.est3
M.est3 <- (-1)*Para3[3]

data.frame(q.est3, N.est3, M.est3)
##                 q.est3  N.est3    M.est3
## CumEffort -0.001424257 -565.21 0.8726329

With M (known, fixed at 0.1)

Res.DeLury4 <- 
  lm(log(CPUE)~CumEffort+offset(-Period1*0.1))

summary(Res.DeLury4)
## 
## Call:
## lm(formula = log(CPUE) ~ CumEffort + offset(-Period1 * 0.1))
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.28395 -0.06134  0.17919  0.35625  0.43419 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.5173664  0.3152973  -1.641    0.139
## CumEffort   -0.0001400  0.0001277  -1.097    0.305
## 
## Residual standard error: 0.5674 on 8 degrees of freedom
## Multiple R-squared:  0.4752, Adjusted R-squared:  0.4096 
## F-statistic: 7.244 on 1 and 8 DF,  p-value: 0.02744
Para4 <- coef(Res.DeLury4)
Para4
##   (Intercept)     CumEffort 
## -0.5173663947 -0.0001400347
q.est4 <- (-1)*Para4[2]
N.est4 <- exp(Para4[1])/q.est4

data.frame(q.est4, N.est4)
##                 q.est4   N.est4
## CumEffort 0.0001400347 4256.718

With M (known, fixed at 0.01)

Res.DeLury5 <- 
  lm(log(CPUE)~CumEffort+offset(-Period1*0.01))
summary(Res.DeLury5)
## 
## Call:
## lm(formula = log(CPUE) ~ CumEffort + offset(-Period1 * 0.01))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.3526 -0.0899  0.1836  0.3700  0.4626 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -0.5523653  0.3288333   -1.68   0.1315  
## CumEffort   -0.0003223  0.0001332   -2.42   0.0418 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5918 on 8 degrees of freedom
## Multiple R-squared:  0.4527, Adjusted R-squared:  0.3842 
## F-statistic: 6.616 on 1 and 8 DF,  p-value: 0.03302
Para5 <- coef(Res.DeLury5)
Para5
##  (Intercept)    CumEffort 
## -0.552365315 -0.000322251
q.est5 <- (-1)*Para5[2]
N.est5 <- exp(Para5[1])/q.est5
data.frame(q.est5, N.est5)
##                q.est5   N.est5
## CumEffort 0.000322251 1786.144
Method <- 1:5
M <- c("NA", 0, round(M.est3,4), 0.1, 0.01)
N.est <- c(N.est1,N.est2,N.est3,N.est4,N.est5)
data.frame(Method, M, N.est)
##   Method      M    N.est
## 1      1     NA 2555.589
## 2      2      0 1674.036
## 3      3 0.8726 -565.210
## 4      4    0.1 4256.718
## 5      5   0.01 1786.144