Logistic Growth

The logistic model describes growth that is slowed by a limited resource. The growth rate is $\frac{dN}{dt} = r\cdot{}N\cdot{}(1 -\frac{ N}{K})$, with the population size N, the growth rate r and the carrying capacity K. As long as N is small compared with K, the population grows almost exponentially at rate r. As N approaches the capacity K, the braking factor $(1 -\frac{ N}{K})$ becomes ever smaller and the growth slows down, until it stops completely at $N = K$. Example bacteria with $r =\frac{ 0.5}{h}$, $K = 10^{6}$ and $N_{0} = 1000$: after 10 hours N is about 860 000, and in the long run N tends towards K. The typical S-shaped growth curve results.

The carrying capacity K is the maximum population size an environment can sustain permanently, because resources such as food, space or oxygen are limited. In the logistic model K is the value towards which the population tends in the long run. When N reaches the capacity, the braking factor $(1 -\frac{ N}{K})$ becomes zero and growth stops; the population stays constant. If N is above K, for example after overpopulation, the factor becomes negative and the population shrinks back towards K. K is therefore a stable equilibrium. It depends on the specific environmental conditions and can change if resources or habitat change. K is one of the two central quantities of the model besides the growth rate r.

Exponential growth describes a population that grows unchecked at a constant rate; the growth speed keeps rising and the population would theoretically become infinitely large. Logistic growth, by contrast, accounts for limited resources through the braking factor $(1 -\frac{ N}{K})$, which slows the growth as the capacity K is approached. For small populations, when N is much smaller than K, both models are nearly identical, because the braking factor is then close to one. Only at larger population do they differ clearly: logistic growth flattens and tends towards K, exponential growth keeps growing without bound. In nature the logistic model is more realistic, because resources are always limited.

The absolute growth rate $\frac{dN}{dt}$ is largest at half the capacity, that is at $N =\frac{ K}{2}$. There the product of the existing population N and the free room $(1 -\frac{ N}{K})$ is maximal. At smaller N there is much room but few individuals to reproduce; at larger N there are many individuals but hardly any room left. Exactly in the middle the balance is optimal. At this point the S-shaped growth curve has its inflection point, where the slope is steepest. After that the growth decreases again until it stops completely at $N = K$. This inflection point at $\frac{K}{2}$ is a characteristic feature of the logistic model.

The logistic model describes many saturation processes beyond biology. In population ecology it models animal and plant populations with limited habitat. In microbiology it describes the growth of bacterial cultures that reach their limits in a nutrient medium. In epidemiology the spread of a disease in a population resembles the logistic course, because the number of not yet infected people decreases. The spread of new technologies or products also often follows an S-curve, because the market becomes saturated. Wherever initially fast growth is slowed by a natural upper limit, the logistic model is a fitting and widely used descriptive approach.