Logistic Model

The logistic model is a model that can frequently describe population growth quite accurately.

Let $P$ be the population of a species. The logistic model assumes first that $P$ grows roughly exponentially if $P$ is small--that is, $\frac{dP}{dt}\approx rP$ for small $P,$ where $r$ is a constant that influences the rate of roughly exponential growth. The second assumption is a carrying capacity, which we denote $K.$ This is meant to represent the maximum population that can be sustained. A simple mathematical condition that combines both of these conditions is: $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right).$$

To solve this differential equation, we rearrange: $$\frac{dP}{P(K-P)} = \frac{r}{K}dt$$ We then use partial fraction decomposition and then integrate: $$\frac{1}{K}\left(\frac{1}{P} + \frac{1}{K-P}\right)dP = \frac{r}{K}dt$$ $$\ln\left|\frac{P}{K-P}\right| = rt + C$$ $$\left|\frac{P}{K-P}\right| = e^{rt+C}$$ Removing the absolute value signs and dividing into cases, we get our answer. $C$ is simply a constant that depends on the initial value of $P,$ which we have called $P_0$: $$P = \frac{K}{1-e^{rt+C}} (P_0 < 0)$$ $$P = 0 (P_0 = 0)$$ $$P = \frac{K}{1+e^{-(rt+C)}} (0 < P_0 < K)$$ $$P = K (P_0 = K)$$ $$P = \frac{K}{1-e^{-(rt+C)}} (0 < P_0 < K)$$

Of course, the most interesting and applicable cases are the third and fifth cases. In general, the population tends toward the carrying capacity $K$ as time progresses, which should be no surprise. If the population starts out between $0$ and $K,$ the population grows quickly at first. The rate of growth then slows down and approaches zero as $P$ asymptotically approaches $K.$ If the population starts out above $K,$ then the population decreases, quickly at first, and then slower as $P$ asymptotically approaches $K$ from above.