SPECIALIST MATHEMATICS

Modelling with Differential Equations: The Logistic Curve.

Logistic Growth

In a population showing exponential growth the individuals are not limited by food or disease. However, in most real populations both food and disease become important as conditions become crowded. There is an upper limit to the number of individuals the environment can support. Ecologists refer to this as the "carrying capacity" of the environment. Populations in this kind of environment show what is known as logistic growth.

The Logistic Differential Equation.

, where k is the birth rate and F is the carrying capacity.

It is a common model of population growth where the rate of reproduction is proportional to the existing population and the amount of available resources (such as food), all else being equal. The solution curve has a characteristic ‘S’ shape’ with F being a limiting value on the population size.

Example 1:

a) Assuming the initial population of 10, birth rate of and the carrying capacity , solve the DE: and sketch the solution curve.

desolve(p’=1.4*p*(1-p/800) and p(0)=10,t,p)
Hint: To see the solution as a function of ‘e’ set your calculator to Exact Mode.
To sketch the graph, redefine p in function of x. /
Graph the solution equation.
The initial part of the curve is exponential; the rate of growth accelerates as it approaches the midpoint of the curve. At the midpoint, the growth rate begins to decelerate but continues to grow until it reaches an asymptote, the carrying capacity. /

b) Now vary the birth rate with the same carrying capacity of 800. Compare 3 graphs. Initial conditions unchanged. Solve one of the equations algebraically. Comment.

c) Now vary the carrying capacity keeping the birth rate constant at Sketch the three graphs on the same set of axes. Initial conditions unchanged. Comment.

d) Go to the following website and play with the applet by varying the birth rate and the carrying capacity as in Experiments 1-4.

http://www.otherwise.com/population/logistic.html

Example 2: Fitting the logistic regression line using TI Nspire.

This type of curve is frequently used to model biological growth patterns where there is an initial exponential growth period followed by a levelling off as more of the population is infected or as the food supply or some other factor limits further growth.

The following data shows New Cases of AIDS in the United States:

Year / Number of cases
1980 / 50
1981 / 500
1982 / 1000
1983 / 2000
1984 / 4000
1985 / 7000
1986 / 13000
1987 / 21000
1988 / 31000
1989 / 34000
1990 / 41000
1991 / 44000
1992 / 45000
1993 / 46000
1994 / 46500
1995 / 46550

Enter the data on your calculator in List & Spreadsheet, plot the scatter plot and find the logistic regression line:

SOLUTIONS:

Example 1 b.
The bigger the birth rate, the faster the population is approaching the limiting value of the carrying capacity. /
Example 1 c.
The initial population growth is similar, mainly effected by the birth rate, with populations approaching their limiting values of the carrying capacities. /

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Solving DE using Calculus:

Type II DE, flip both sides:

Resolve into partial fractions:

and antidifferentiate:

Substitute initial conditions and express as one log:

Finally,

It is an exponential function, and

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