# functional forms for non-linear modeling

# exponential functions
a <- 1
b <- 1/10
curve( a*exp(b*x) , from=0 , to=20 , xlab="x" , ylab="y" )
a <- 2
curve( a*exp(b*x) , from=0 , to=20 , xlab="x" , ylab="y" , add=TRUE , col="red" )
a <- 1
b <- 1/20
curve( a*exp(b*x) , from=0 , to=20 , xlab="x" , ylab="y" , add=TRUE , col="blue" )

# negative exponential
a <- 2
b <- 1/10
curve( a*(1-exp(-b*x)) , from=0 , to=20 , xlab="x" , ylab="y" )
a <- 1
curve( a*(1-exp(-b*x)) , from=0 , to=20 , xlab="x" , ylab="y" , add=TRUE , col="red" )
a <- 2
b <- 1/20
curve( a*(1-exp(-b*x)) , from=0 , to=20 , xlab="x" , ylab="y" , add=TRUE , col="blue" )

# grid of exponential functions
a <- 2
b <- 1/10
par(mfrow=c(2,2))
curve( a*exp(b*x) , from=0 , to=20 , xlab="x" , ylab="y" , main="a*exp(b*x)" )
curve( a*(1-exp(-b*x)) , from=0 , to=20 , xlab="x" , ylab="y" , main="a*(1-exp(-b*x))" )
curve( a*exp(-b*x) , from=0 , to=20 , xlab="x" , ylab="y" , main="a*exp(-b*x)" )
curve( a*(1-exp(b*x)) , from=0 , to=20 , xlab="x" , ylab="y" , main="a*(1-exp(b*x))" )

# michaelis menten
a <- 2
b <- 5
curve( a*x/(b+x) , from=0 , to=20 , xlab="x" , ylab="y" )
a <- 1
curve( a*x/(b+x) , from=0 , to=20 , xlab="x" , ylab="y" , add=TRUE , col="red" )
a <- 2
b <- 10
curve( a*x/(b+x) , from=0 , to=20 , xlab="x" , ylab="y" , add=TRUE , col="blue" )

# power law
a <- 1
b <- 2
curve( a*x^b , from=0 , to=20, xlab="x" , ylab="y" )
a <- 2
curve( a*x^b , from=0 , to=20, xlab="x" , ylab="y" , col="red" , add=TRUE )
a <- 1
b <- 3
curve( a*x^b , from=0 , to=20, xlab="x" , ylab="y" , col="blue" , add=TRUE )

a <- 1
b <- 2
curve( a*x^b , from=0.1 , to=20, xlab="x" , ylab="y" , log="xy"  )
a <- 2
curve( a*x^b , from=0.1 , to=20, xlab="x" , ylab="y" , col="red" , add=TRUE , log="xy" )
a <- 1
b <- 3
curve( a*x^b , from=0.1 , to=20, xlab="x" , ylab="y" , col="blue" , add=TRUE , log="xy" )

# ricker
a <- 1
b <- -1/3
curve( a*x*exp(b*x) , from=0 , to=20 , xlab="x" , ylab="y" , ylim=c(0,2.5) )
a <- 2
b <- -1/3
curve( a*x*exp(b*x) , from=0 , to=20 , xlab="x" , ylab="y" , col="red" , add=TRUE )
a <- 1
b <- -1/5
curve( a*x*exp(b*x) , from=0 , to=20 , xlab="x" , ylab="y" , col="blue" , add=TRUE )

# tadpole predation exercise
library(bbmle)
library(emdbook)
data(ReedfrogFuncresp)
d <- ReedfrogFuncresp

plot(d$Killed ~ d$Initial)

d$freqkilled <- d$Killed/d$Initial
plot(d$freqkilled ~ d$Initial)

init <- c(5,10,15,20,30,50,75,100)
mfk <- sapply( init , function(z) mean(d$freqkilled[d$Initial==z]) )
plot(mfk ~ init , ylim=c(0,1) , xlab="Initial number" , ylab="Average freq killed" )