I was asked how I plotted the recruitment curve from my last post. It's pretty easy to do in R.

[r]
Dn <- expression(r * (1 - N/K) * N) r <- 0.004 K <- 10^10 N <- seq(0,10^10,by=10^7) png(file="recruitment.png") ## sends output to .png file plot(N,eval(Dn),type="l", xlab="Human Population", ylab="Recruitment") dev.off() [/r]

Now we can generalize to the generalized $theta$ logistic model where

$$\frac{dN}{dt} = rN(1-N/K)^{\theta}.$$

This model changes the shape of the density dependence when $\theta<1$ the density dependence is felt only at higher densities, whereas when $\theta>1$ the density dependence kicks in at low population size.

Consider our problem of what the optimal harvest size of humans by Kanamits consumers would be if human population size was actually regulated according to the theta-logistic model. First, we assume that $\theta<1$.

[r]
Dn.gen <- expression((r * (1 - N/K))^theta * N) theta <- 0.1 png(file="recruitment-thless1.png") plot(N,eval(Dn.gen),type="l", xlab="Human Population", ylab="Recruitment") title("theta=0.1") dev.off() [/r]

Now, we assume that $\theta>1$.

[r]
theta <- 10 png(file="recruitment-thgr1.png") plot(N,eval(Dn.gen),type="l", xlab="Human Population", ylab="Recruitment") title("theta=10") dev.off() [/r]

I sent the output to a portable network graphics (png) device because I wanted to include the figures with this post. Normally, in interactive work, you would send this to the default on-screen device (e.g., x11 or Quartz). When making a figure for publication or to include with a $LaTeX$ document, one would send the output to a pdf device.

From this example, we can see that the shape of the density dependence can pretty drastically change the MSY, at least in the case where the discount rate is zero. When $\theta=10$, we can see that the Kanamits would only be able to sustainably harvest on the order of a billion people, whereas if $\theta=0.1$, they could take nearly 9 billion and still have a viable population. I'm guessing that the former is closer to the truth than the latter, but let's hope that the answer remains in the realm of absurdist speculation where it belongs.