A Tale of two cheeses - Adrian Oldknow a_oldknow@compuserve.com

Chapter 4: Limits to growth: the Listeria
Chapter 2 used a model of population growth often attributed to the Rev. Thomas Malthus, an eighteenth century English cleric. This does not take into account the finite size of the Camembert in question and its incapacity to sustain a population of bacteria greater than 10 000. The modified model (due to Pierre Verhulst) assumes that the rate of growth decreases (or is damped) as the count c gets closer to the maximum m = 10 000 by a factor (l - c/m). If the rate of growth without resource constraints is r (as a decimal) then the formula for the rate of increase at a time t when the count is c is given by: r.c.(1- c/m). If we assume, as before, that the increase is roughly constant for our little intervals of 1 second then we can build a modified table:
t 0 1 2 . 3 4 5
c 1000 1180
How many bacteria will there now be at the time the Cheddar breaks the sound barrier? How long does it take the count to reach 99% of its maximum? What does the graph of count c against time t now look like? This is known as a logistic curve.

----------------------------------------------------------------------------------------------------------------

Now we have the differential equation: and initial condition c= c₀ at time t= 0. Otherwise we have the difference equation: . This time there is no analytic solution to the quadratic (non-linear) difference equation so we cannot predict results other than by computing all the in-betweens. This is the simplest form of difference equation which can exhibit chaotic behaviour.

We will work with lists to hold the bacteria count and time. First let's have some constants:

Now we'll build a multistatement definition (a simple program) using a For..Next loop to define the c list:

Now we can graph a scatterplot of the data from t and c,
If the cheese could reach the speed of sound it would be after a time t secs given by:

Here is the definition of the logistic function which we'll derive shortly!

So here the scatterplot of the finite approximation is closely fitted by the theoretical model. We'll leave you to explore the convergence as the time step is taken smaller than 1 and tends to zero.

Here the number of bacteria at sound barrier time would be:

and the 99% mark is reached after a time of:

Click on the link for Wikipedia's stuff on the Logistic curve

For the theory we have to consider the differential equation: and initial condition c= c₀ at time t= 0. Again we can use the antiderivative trick of thinking of t as a function of c, but this time the gradient function is not a simple reciprocal quadratic, and we have to use a technique like partial fractions to split it into the sum of two reciprocal function 1/c and 1/(m-c) . You can work out the details for yourself! Here is how TI InterActive! can help. First we insert a Math Section Break so that previous definitions of constants such as m no longer apply. Then we can evaluate the definite integral - but this time TII! is rather more fussy than we have been about the choice of variable - we can't use c both for the variable of integration and the upper limit.

which is equivalent to the logistic function l(t) quoted earlier.

If m is much bigger than c₀ then this approximates to the exponential growth:
If m is close to c₀ then this approximates to the constant function .
There is a point of inflection where c'' = 0 but c' &g; 0:
So clearly the point of inflection is where c = m/2, and t =