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The Logistic Equation
Details of the Maths...
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Mathematical details of the logistic equation are explained, and applied to Australia's population projections....

The Logistic Equation

Let P(t) represent the population of a species at time t.

If growth is to end, the time rate of change of the population must fall to zero. The objective function is thus:

dP/dt = 0                                       ...(1)
We don't know what sort of function to use for dP/dt. Alfred Lotka postulated that it might be some function of the population size: 
dP/dt = f(P) = 0                                ...(2)
The Taylor series expansion of any function f(P) gives: 
dP/dt = bP + dP2 + gP3 + ....                         ...(3)
Using one term of the taylor series expansion for f(P) gives an exponentially increasing population - and can't satisfy the objective (1).

Taking the first two terms of the taylor series expansion is the simplest approximation to f(P) that will permit a non-trivial solution to (1).

dP/dt = bP + dP2                                     ...(4)
Solving (1) and (4) gives 
 0      = Plimit ( b + dPlimit )
hence 
b + dPlimit= 0
 
 Plimit = - b /d                     ...(4a)

( 
Since 

 Plimit ≥ 0
 
 we would expect that b  ≥ 0 and d ≤ 0  in equation (4).
 )

The growth rate of the population is defined as the growth in the population, divided by the size of the population. If the birth rate is 3.2 per 100 and the death rate is 1.8 per 100, then the growth rate is 3.2 - 1.8 = 1.4 per 100.

We then write dP/dt = 0.014P.

Suppose that in a given population the average birth rate (dP/dt divided by P) is a positive constant b. This is just as assumed in the exponential growth model - the birth rate does not vary with population size.

What is the death rate ?

Greater populations mean greater overcrowding and more competition for food and territory. In a market place, alternative technologies or modes of competing with your product arise. In the world at large, the growth of terrorism and outbreaks of disease act to limit the utility of international air travel.

Because of such pressures, the average death rate (dP/dt divided by P) is proportional to the size of the population. As population increases, so too does the death rate. So, it might start off at 0.8% in 1965 but rise to 6% by 1997 and 9.6% by 2025. The birth rate, however, remains constant with population size (say at 10% per annum) - just as is assumed in exponential growth models.

For the airlines business, if there was a "birth rate" of 10% for new passengers (every 10th passenger brings a new one along next year), by 1996 it would still be growing strong (at 10% - 6% = 4%). But by 2025, growth would have fallen to just 10% - 9.6% = 0.4%.

The differential equation (4) derived above from the Taylor Series expansion can be rearranged and solved:

eqn

The logistic equation produces an S shaped curve as shown below. For early years, the death rate is negligible (set d to zero in equation), and the curve is indistinguishable from an exponential growth curve. But eventually as the population increase, the death rate begins to have an effect, flattening the growth curve...

logistic example

A logistic curve can be fitted to Australia's population data based on Source: Australian Demographic Statistics (3101.0); Australian Demographic Trends (3102.0) - see www.abs.gov.au). This exercise gave a mean-square error of 0.31 million, b = 0.0192 (ie. 1.9%), and d = 0.000231 and P0 = 3.63 million (ie at 1900). The ratio b/d has the final (asymptotic) value of 83 million.

Least-square error fitting an exponential growth curve to the same data gives a growth rate of 1.68% per annum, P0 = 3.77 million and a mean-square error of 0.35 million. So the logistic equation is a better fit.

australian population

The goodness of fit can be seen in the graph below of Australia's Population 1900 to 2000.

australian population projection

Here is the ABS's population projection, which is considerably lower than the logistic estimate. The ABS is clearly expecting future growth rates unlike the past.

ABS Population Projection

ABS Population Projection

If the underlying data is truly exponential, fitting a logistic curve to it will give a d of 0.

Using It

Data can be fit to a logistic curve by using MS Excel's Solver tool, which uses a Generalized Reduced Gradient Algorithm.

  1. Construct a spreadsheet with the actual data points.
  2. Set up cells for b , d, & P0 and a column for the P(t) expression.
  3. Add a column showing the square error between P(t) and the actual data points, and
  4. put a cell at the bottom of it giving the mean (average) square error.
  5. Tick Assume non-negative in the solver options (disallows negative d etc.,. - but you might want to try some scenarios with this relaxed).
  6. Use Solver to minimize the mean square error cell by adjusting b & d and P0 cells.
  7. For some problems, formulating the logistic equation with P0 as 1 and using a time offset variable t0 instead may give better performance.
  8. Check solution is a global minima by choosing different initial values (change singly and in pairs, by small and large amounts). If you are really cute with Excel VBA, you could write a program to randomly vary the initial values over a likely range.

 

Other Logistic Curves

See the following pages for applications of the logistic curve to other situations:

 

First published 12th November 2000. Last 24th Jan 2017