Exponential Growth

You insert the initial size, growth rate and time into the equation $N(t) = N_{0}\cdot{}e^{(r\cdot{}t)}$. N₀ is the number at the start, r the growth constant and t the elapsed time. It is important that the units of r and t match, so that the product $r\cdot{}t$ is dimensionless. Example: if a bacterial culture starts with $N_{0} = 1000$ and $r =\frac{ 0.5}{h}$, then after 10 hours $N = 1000\cdot{}e^{(0.5\cdot{}10)} = 1000\cdot{}e^{5}$. Since $e^{5}$ is about 148.4, the culture grows to roughly 148,400 individuals. You recognise the typical property: at first the number rises slowly, then ever faster, because the increase is always proportional to the number already present. This model only holds as long as resources are unlimited.

Exponential growth describes a population that grows unchecked at a constant rate r. There is no upper limit, the number would theoretically become infinitely large and the growth curve rises ever more steeply. Logistic growth, by contrast, accounts for limited resources through a carrying capacity K and the braking factor $(1 -\frac{ N}{K})$, which slows the growth as K is approached until it stops. For small populations, as long as N is much smaller than K, both models are nearly identical, because the braking factor is then close to one. The exponential model is thus the initial phase of the logistic one. In nature growth is always logistic in the long run, because food, space and other resources are limited. The exponential model realistically describes only the early, unchecked phase.

The doubling time is the time in which the population exactly doubles. You obtain it by inserting the value $N = 2\cdot{}N_{0}$ into $N(t) = N_{0}\cdot{}e^{(r\cdot{}t)}$. Then N₀ cancels and $2 = e^{(r\cdot{}t)}$ remains. Taking the logarithm gives $\ln{} 2 = r\cdot{}t$ and from it $t =\frac{ \ln{} 2 }{ r}$. The doubling time therefore depends only on the growth rate r, never on the initial size N₀. Example: at $r =\frac{ 0.5}{h}$, $t\frac{1}{2} =\frac{ \ln{} 2 }{ 0.5} =\frac{ 0.693 }{ 0.5} \approx{} 1.39$ hours. A twice as large rate halves the doubling time. This quantity is intuitive and is often stated instead of the abstract rate r, for example for bacterial cultures or in epidemiology, to make the speed of a growth tangible.

The growth constant r states how fast a population grows relative to its current size. It is the difference between birth rate and death rate per individual and time unit and has the unit of an inverse time, such as $\frac{1}{h}$ or 1/year. A positive r means growth, a negative r shrinkage, at $r = 0$ the population stays constant. Intuitively r says which fraction is added per time unit: $r =\frac{ 0.5}{h}$ means that currently about 50 percent are added per hour. However, r must not be equated directly with the percentage increase per time step, because the continuous model uses $e^{(r\cdot{}t)}$ and not the discrete factor (1 + r). From two measurements one determines r via $r = (\frac{1}{t})\cdot{}\ln{}(\frac{N}{N_{0}})$. The larger r, the steeper the growth curve and the shorter the doubling time.

The exponential model applies only as long as resources are practically unlimited and the growth rate stays constant. This is true for the early phase of many populations, such as a bacterial culture in fresh nutrient medium or the initial phase of an epidemic. As soon as food, space or other resources become scarce, however, the growth rate falls and the model clearly overestimates the real size. Then logistic growth with its carrying capacity K takes over. No real growth stays exponential permanently, because no habitat is infinite. The exponential model is therefore not a contradiction to the logistic one but its limiting case for N ≪ K. One uses it to make short-term forecasts, compare growth rates or show the unchecked potential of a population, always aware of its limited validity.