Theta-logistic equation

Question

I can't comprehend any of the solution for iii). WHy for $\theta=1$ do we have linear dependence of the growth on the population size?

Answer 1

The author means that the growth rate is linear in $N_t$. That is, the equation has the general form \begin{equation} \Delta N_t = R(N_t)\,N_t. \end{equation} If the rate $R(N_t)$ would be constant in $N_t$, we would have standard exponential growth. Now, the way the growth rate $R$ depends on $N_t$ is specified as \begin{equation} R(N_t) = r \left(1-\left(\frac{N_t}{K}\right)^\theta\right). \end{equation} As you see, if $\theta = 0$, then $R$ is a linear function of $N_t$, so then the growth rate depends linearly on the population size.

Again, as in your previous question, a matter of sloppy wording.