Alaskan Slimy Sculpin I

  1. The plot is shown below. The data show a fairly obvious curve, a fairly obvious asymptote, no fish less than age-1 are shown, and there is considerable variability in length at each age.
  2. The plot is shown below. The model fits moderately well, but there is considerable variability in length at each age.
  3. The typical von Bertalanffy equation is \(E(L|t)=77.8(1-e^{-0.46(t+0.61)})\).
  4. The asymptotic mean length of Slimy Sculpins is 77.8 mm. The Brody growth coefficent, or the rate at which the mean length approaches the asymptotic mean length, is 0.46 year-1. The age at which the mean length is zero (i.e,. the x-intercept) is -0.61 years.
  5. The 95% confidence interval for the asymptotic mean length is 69.3 to 103.7 mm, for the Brody growth coefficient is 0.22 to 0.73 year-1, and for the x-intercept is -1.37 to -0.17 years. The intervals for the asymptotic mean length and Brody growth coefficient are quite wide because there was considerable variability in length-at-age.
  6. The predicted mean length at age-3 is 63.1 (with 95% confidence interval from 61.6 to 64.4) mm. This interval is fairly narrow because it is a confidence (rather than prediction) interval and age-3 is a well-represented age in the data.
  7. Slimy Sculpins will have reached half of their asymptotic length at age 1.50.

R Appendix

library(FSA)
library(tidyverse)
library(nlstools)

sculp <- read.csv("https://raw.githubusercontent.com/droglenc/FSAdata/master/data-raw/SculpinALTER.csv")
vbTyp <- vbFuns("Typical")
ggplot(data=sculp,mapping=aes(x=age,y=tl)) +
  geom_point(color=col2rgbt("black",1/3)) +
  stat_function(fun=vbTyp,args=list(Linf=coef(fitTyp)),color="red",size=1) +
  scale_x_continuous(name="Otolith Age (yrs)",
                     limits=c(0,NA),expand=expansion(mult=c(0,0.02))) +
  scale_y_continuous(name="Total Length (mm)",
                     limits=c(0,NA),expand=expansion(mult=c(0,0.02))) +
  theme_bw()

Alaskan Slimy Sculpin II

{:start=“9”} 1. The Gallucci-Quinn von Bertalanffy equation is \(E(L|t)=\frac{35.86}{0.46}(1-e^{-0.46(t--0.61)})\). 1. The growth of Slimy Sculpins near t0 is 35.86 mm/yr. The Brody growth coefficent, or the rate at which the mean length approaches the asymptotic mean length, is 0.46 year-1. The age at which the mean length is zero (i.e,. the x-intercept) is -0.61 years. 1. The common parameter estimates (for K and t0) between the two parameterizations of the von Bertalanffy are the same. This is because a parameterization is the same model, just with different parameters. 1. 35.86=77.8×0.461.

R Appendix

vbGQ <- vbFuns("GallucciQuinn")
svGQ <- vbStarts(tl~age,data=sculp,type="GallucciQuinn")
fitGQ <- nls(tl~vbGQ(age,omega,K,t0),data=sculp,start=svGQ)
bootGQ <- nlsBoot(fitGQ)
estsGQ <- cbind(Est=coef(fitGQ),confint(bootGQ))