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###############################################
## Soetaert and Herman (2008) ##
## A practical guide to ecological modelling ##
## Chapter 4. ##
## Parameterisation ##
## Chapter 4.4.2. ##
## Linear and Nonlinear fit of Pb210-data ##
###############################################
# the data:
x <- 0.5:9.5
y <- c(3.9,1.7,1.1,0.5,0.3,0.2,0.1,0.05,0.03,0.02)
plot(y,x,pch=16,ylab="depth,cm", xlab="dpm/cm3",
main="Pb210", ylim=c(10,0))
# the decay rate
lam <- 0.031
# linear fit
LL <- lm(log(y)~x)
C0 <- exp (coef(LL)[1])
Db <- lam/(coef(LL)[2])^2
xx <-seq(0,10,0.1)
lines(C0*exp(-sqrt(lam/Db)*xx),xx)
# nonlinear fit
fit<-nls(y ~C0*exp(-sqrt(lam/Db)*x),
start=c(C0=5,Db=0.1))
C02 <- coef(fit)[1]
Db2 <- coef(fit)[2]
lines(C02*exp(-sqrt(lam/Db2)*xx),xx,lty=2)
legend("topleft",lty=c(1,2),
c("linear fit", "nonlinear fit"))
par(new=TRUE,fig=c(0.5,1,0.1,0.6))
plot(y,x,pch=16,ylab="",xlab="",ylim=c(10,0),log="x")
lines(C0*exp(-sqrt(lam/Db)*xx),xx)
lines(C02*exp(-sqrt(lam/Db2)*xx),xx,lty=2)
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