Chapter 28, Problem 23P

Although the model in Prob. 28.22 works adequately when population growth is unlimited, it breaks down when factors such as food shortages, pollution, and lack of space inhibit growth. In such cases, the growth rate itself can be thought of as being inversely proportional to population. One model of this relationship is

$G=G\text{'}\left(p_{\text{max}}-p\right)\left(\text{P28}\text{.23}\text{.1}\right)$

where $G\text{'}=$ a population-dependent growth rate (per people-year) and $p_{\text{max}}=$ the maximum sustainable population. Thus, when population is small $\left(p<<p_{\text{max}}\right)$, the growth rate will be at a high constant rate of $G\text{'}{p}_{\text{max}}$,. For such cases, growth is unlimited and Eq. (P28.23.1) isessentially identical to Eq.(P28.22.1). However, as population grows (that is, p approaches $p_{\text{max}}$), $G$ decreases until at $p=p_{\text{max }}$ it is zero. Thus, the model predicts that, when the population reaches the maximum sustainable level, growth is nonexistent, and the system is at a steady state. Substituting Eq. (P28.23.1) into Eq. (P28.22.1) yields

$\frac{dp}{dt}=G\text{'}\left(p_{\text{max}}=p\right)p$

For the same island studied in Prob. 28.22, employ Heun's method(without iteration) to predict the population at $t=20$ years, using a step size of 0.5 year. Employ values of $G\text{'}={10}^{-5}$ per people-year and $p_{\text{max}}=20,000$ people. At time $t=0$, the island has a population of 6000 people. Plot p versus t and interpret the shape of the curve.