Exponential growth
Exponential growth is the simplest continuous model of population change. It describes a population whose per-capita growth rate remains constant.
Why this model makes biological sense
Suppose every individual contributes, on average, the same net amount to population growth per unit time. If the population doubles, there are twice as many individuals contributing to future growth, so the total rate of increase also doubles.
This leads directly to
\[\boxed{\frac{dN}{dt}=rN}.\]The equation says:
\[\text{rate of population change}=\text{per-capita growth rate}\times\text{population size}.\]What the symbols mean
| Symbol | Meaning |
|---|---|
| \(N(t)\) | population size at time \(t\) |
| \(N_0\) | initial population \(N(0)\) |
| \(r\) | net per-capita growth rate |
| \(dN/dt\) | instantaneous rate at which the total population changes |
Where does \(r\) come from?
A common interpretation is
\[\boxed{r=b-d},\]where \(b\) is the per-capita birth rate and \(d\) is the per-capita death rate.
Then
\[\frac{dN}{dt}=(b-d)N.\]If births exceed deaths, \(r>0\). If deaths exceed births, \(r<0\).
From per-capita growth to total growth
Dividing the model by \(N\) gives
\[\boxed{\frac{1}{N}\frac{dN}{dt}=r}.\]The left-hand side is the growth rate per individual. Exponential growth assumes this quantity is constant.
Deriving the solution
Starting from
\[\frac{dN}{dt}=rN,\]separate variables:
\[\frac{1}{N}\,dN=r\,dt.\]Integrating gives
\[\ln N=rt+C.\]Exponentiating,
\[N=Ae^{rt}.\]Using \(N(0)=N_0\) gives \(A=N_0\), so
\[\boxed{N(t)=N_0e^{rt}}.\]What the solution tells us
The sign of \(r\) determines the long-term direction:
| Case | Behaviour |
|---|---|
| \(r>0\) | population increases exponentially |
| \(r=0\) | population remains constant |
| \(r<0\) | population declines exponentially toward zero |
Why the curve becomes steeper
If \(r>0\),
\[\frac{dN}{dt}=rN.\]As \(N\) becomes larger, \(rN\) also becomes larger. So the absolute number added per unit time increases even though the per-capita rate \(r\) has not changed.
Doubling time
For \(r>0\), the doubling time \(T_d\) satisfies
\[2N_0=N_0e^{rT_d}.\]Therefore
\[2=e^{rT_d},\]so
\[\boxed{T_d=\frac{\ln 2}{r}}.\]A larger positive \(r\) gives a shorter doubling time.
Equal time intervals multiply the population by the same factor
For any fixed interval \(h\),
\[\frac{N(t+h)}{N(t)}=e^{rh}.\]So exponential growth is multiplicative rather than additive.
A numerical example
Suppose
\[N_0=100,\qquad r=0.2\text{ per year}.\]After 5 years,
\[N(5)=100e^{0.2(5)}=100e\approx271.8.\]The model therefore predicts about 272 individuals.
When is exponential growth a reasonable approximation?
The model can be useful when population density is low and resources, space or susceptible hosts are not yet strongly limiting growth.
Examples can include the early phase of microbial growth, an introduced population before strong density dependence develops, or the early phase of an epidemic when the susceptible population is still nearly unchanged.
When is it not appropriate?
Exponential growth assumes that \(r\) remains constant even as the population grows. Real biological populations usually encounter competition, resource limitation, disease, predation or environmental change.
Exponential growth versus logistic growth
The exponential model is
\[\frac{dN}{dt}=rN.\]The logistic model modifies the per-capita growth rate:
\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right).\]When \(N\ll K\),
\[1-\frac{N}{K}\approx1,\]so logistic growth is approximately exponential. As \(N\) approaches the carrying capacity \(K\), growth slows.
Why exponential growth appears in epidemic models
Near the start of a simple epidemic, the susceptible population may be almost unchanged. Under that approximation, the infectious population can satisfy an equation of the form
\[\frac{dI}{dt}\approx rI.\]This is why early epidemic growth is often approximately exponential before susceptible depletion and interventions become important.
Historical note
The idea of unrestricted population growth is strongly associated with Thomas Malthus's 1798 discussion of population increasing geometrically when unchecked. Modern continuous population models express this idea using the differential equation \(dN/dt=rN\).
What the model does not include
The simple model has no carrying capacity, age structure, migration, seasonal forcing, stochasticity or interaction with other species. Those features require extensions.