1. Individual Growth Population Biomass Recruitment Natural Mortality
Fishing Mortality
Immigration
Emigration
Growth

2. Length-At-Age Data TL Age Species 24 6 Rainbow 26 8 Rainbow
data(TroutBR)
rbt <- TroutBR[TroutBR$Species=="Rainbow",]
attach(rbt)
plot(jitter(TL,1)~jitter(Age,0.5),xlab="Age [jittered]",ylab="Total Length (in) [jittered]")
Size At Age

3. Length-At-Age Models Purposes Main models
Summarize growth with a few parameters.
Compare growth parameters among populations.
Used in key fisheries models, such as Beverton-Holt yield models.
Main models
Von Bertalanffy
Gompertz
Schnute
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4. Von Bertalanffy Growth Model
By far the most prevalent growth model.
Developed by Ludwig von Bertalanffy in 1938.
Derived from …
basic physiological principles of anabolism and catabolism.
by assuming rate of growth declines linearly with increasing length.
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5. Von Bertalanffy Growth Model
Explore parameters with growthModelSim(“vbTypical”)
L∞ = asymptotic mean length
K = Brody “growth” coefficient
not a true measure of the “growth rate”
controls how fast L∞ is approached
Log(2)/K is time to get halfway between a length and L∞
to = time when mean length is 0 (artifact)
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6. Von Bertalanffy Growth Model
growmodel.sim("vb",Age,TL,max.len=35)
svb <- list(Linf=31.3,K=0.3,to=0.6)
vbl1 <- nls(TL~Linf*(1-exp(-K*(Age-to))),start=svb)
windows(4,4); par(mar=c(3.5,3.5,1,1),mgp=c(2,0.75,0))
ylmt <- c(-2,32); xlmt<-c(1.5,10)
plot(jitter(Age,0.5),jitter(TL,1),xlab="Age",ylab="Total Length (in)",ylim=ylmt,xlim=xlmt,pch=19)
x <- data.frame(Age=seq(1.5,10,by=0.1))
y <- predict(vbl1,x)
lines(x$Age,y,lwd=1,lty=1,col="red")
Linf <- coef(vbl1)[1]; to <- coef(vbl1)[3]
lines(c(to,to),c(-5,2),lwd=2,lty=3,col="blue")
lines(c(1,2),c(0,0),lwd=2,lty=3,col="blue")
points(to,0,col="blue",pch=19,cex=1.25)
text(to,-4.5,expression(t[o]),xpd=T,col="blue",cex=1.25)
abline(h=Linf,lwd=2,lty=3,col="blue")
text(0.8,Linf,expression(L[infinity]),xpd=T,col="blue",cex=1.25)
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7. Galucci & Quinn Parameterization
New parameter is w=KL∞
a measure of growth rate in the vicinity of t0
“New” model looks like
Explore model parameters with growthModelSim(“vbGalucciQuinn”)
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