The factor \(g(N)=r(1-N / K)\) in equation (2a) is a per capita growth rate. Smith (1963) observed that in cultures of the unicellular alga Daphnia magna \(g\) decreases at a nonlinear rate as \(N\) increases. To account for this fact, Smith suggested that the growth rate depends on the rate at which food is utilized: $$ g(N)=r \frac{T-F}{T} $$ where \(F\) is the rate of utilization when the population size is \(N\), and \(T\) is the maximal rate, when the population has reached a saturated level. He further assumed that $$ F=c_{1} N+c_{2} \frac{d N}{d t}, \quad\left(c_{1}, c_{2}>0\right) $$ as long as \(d N / d t>0\). (a) Explain this assumption for \(F\). (b) Show that the modified logistic equation is then $$ \frac{d N}{d t}=r N\left[\frac{K-N}{K+(\gamma N)}\right] $$ where \(\gamma=r c_{2} / c_{1}\) and \(K=T / c_{1}\). (c) Sketch the expression in square brackets as a function of \(N\). (d) What would be the qualitative behavior of this population growth? (For a deeper analysis of this problem see Pielou, 1977.)

The assumption for the rate of utilization, given by the formula \(F = c_1 N + c_2 \frac{dN}{dt}\), suggests that the utilization rate depends on the population size \(N\) and its rate of change \(\frac{dN}{dt}\). The coefficients \(c_1\) and \(c_2\) are positive constants. This means that as the population size increases, or as the rate of change of the population increases, the food utilization rate also increases.

To show that the modified logistic equation is \(\frac{dN}{dt} = r N \left[ \frac{K - N}{K + (\gamma N)} \right]\), start by substituting the expression for \(F\) into the growth rate equation. Given \(g(N) = r \frac{T - F}{T}\) and \(F = c_1 N + c_2 \frac{dN}{dt}\), we have:\[ g(N) = r \frac{T - (c_1 N + c_2 \frac{dN}{dt})}{T} \] Rewriting the equation, we get:\[ g(N) = r \left(1 - \frac{c_1 N + c_2 \frac{dN}{dt}}{T}\right) \]Since \(g(N)\) is also equal to \(\frac{1}{N}\frac{dN}{dt}\), equate both expressions to obtain:\[ \frac{1}{N}\frac{dN}{dt} = r \left(1 - \frac{c_1 N + c_2 \frac{dN}{dt}}{T}\right) \]Simplifying:\[ \frac{1}{N}\frac{dN}{dt} = r - r \frac{c_1 N + c_2 \frac{dN}{dt}}{T} \]Multiplying both sides by \(N\), we obtain:\[ \frac{dN}{dt} = rN - rN \frac{c_1 N + c_2 \frac{dN}{dt}}{T} \]Rearrange this into:\[ \frac{dN}{dt} + \frac{r c_2 N^2}{T} \frac{dN}{dt} = rN - \frac{r c_1 N^2}{T} \]Thus, combining coefficients, the modified logistic equation can be written as:\[ \frac{dN}{dt} = \frac{r N (T - c_1 N)}{T + r c_2 N} \]Substitute \(K = \frac{T}{c_1}\) and \(\gamma = \frac{r c_2}{c_1}\) to get:\[ \frac{dN}{dt} = r N \left[\frac{K - N}{K + \gamma N}\right] \]

To sketch \(\left[\frac{K - N}{K + \gamma N}\right]\) as a function of \(N\), note that when \(N = 0\), the value is \(1\). As \(N\) increases, the term in the square brackets decreases and approaches \(\frac{1}{\gamma}\) as \(N\) approaches infinity. The curve is a hyperbola that starts at 1 and asymptotically approaches \(\frac{1}{\gamma}\).

The modified logistic equation suggests that the population will grow rapidly when \(N\) is small, slow down as \(N\) approaches \(K\), and further slow down asymptotically as \(N\) increases beyond \(K\). This represents a logistic growth with an upper asymptote determined by \(\gamma\). Hence, initially, the population experiences logistic growth, but for very large \(N\), the growth rate decreases more than in a simple logistic model.