Chapter 4

Density-dependent Population Growth Smith \((1963)\) proposed a model for the growth of a population of microorganisms whose reproductive rate decreases as the number of microorganisms increases. According to Smith's model, if \(N(t)\) is the number of \(\mathrm{mi}-\) croorganisms, then \(d N / d t=R(N) N\) where $$ R(N)=\frac{r(1-N / K)}{1+N / a} $$ where \(a, K\), and \(r\) are all positive constants. (a) Show that \(d N / d t \geq 0\) if \(NK\). (a) and (b) imply that the population grows if \(NK\) (c) Show that \(\frac{d}{d N}(R(N) \cdot N)>0\) at \(N=0\) and \(\frac{d}{d N}(R(N) \cdot N)<0\) if \(N=K .\) We will use these results in Chapter 8 to predict the population size \(N(t)\) as \(t \rightarrow \infty\) (d) Show that \(R^{\prime}(N)<0\) for all \(N \geq 0 .\) This result means that reproductive rate decreases as the number of microorganisms increases.

In this problem we will prove the quotient rule using an argument similar to the one used to prove the product rule in Section 4.4.1. Let \(u(x)\) and \(v(x)\) be differentiable functions, and define a quotient function \(f(x)=\frac{u(x)}{v(x)} .\) The derivative \(f^{\prime}(x)\), if it exists, is equal to: $$ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $$ (a) Assuming \(v(x) \neq 0\), show that the quotient on the right-hand side can be written as: \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(u(x+h) v(x)-u(x) v(x))-(u(x) v(x+h)-u(x) v(x))}{h v(x) v(x+h)}\) and then rearranged into: $$ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\left(\frac{u(x+h)-u(x)}{h}\right) v(x)-u(x)\left(\frac{u(x+h)-u(x)}{h}\right)}{v(x) v(x+h)} $$ (b) Using the limit laws, show that the equation from (a) can be rewritten as $$ f^{\prime}(x)=\frac{\left(\lim _{h \rightarrow 0} \frac{u(x+h)-u(x)}{h}\right) v(x)-u(x)\left(\lim _{h \rightarrow 0} \frac{v(x+h)-v(x)}{h}\right)}{v(x) \lim _{h \rightarrow 0}(v(x+h))} $$ provided all of the limits exist, and provided \(\lim _{h \rightarrow 0} v(x+h) \neq 0\). (c) Using the formal definition of a derivative write \(f^{\prime}(x)\) in terms of \(u(x), v(x), u^{\prime}(x)\), and \(v^{\prime}(x) .\) Hence prove the quotient law.