Population Biology

Population Biology

Stepping up in the level of mathematical complexity, we can look at a model taken from the study of biological populations. One formula for predicting the changes in the population of a species is:

p_n=r_pn-1(1-p_n-1).

This states that the population in any given cycle is dependent on the population in the previous cycle multiplied by a certain factor. The factor in this case is simply 1-p. The population itself is scaled to a value between 0 and 1 so 1-p always returns a suitable value. The starting value of p is not important - any value between 0 and 1 will do, for example 0.2. The value of r can be any value from 0 to infinity, though you should restrict yourself to values between 0 and 4. The first part of the spreadsheet shows how to set up the model for r=2.5.

The formula in B6 uses absolute references from the starting values in A4 and A6. The formula in B7 uses an absolute reference to the constant r in A6 and relative references to the population cycles in B6, B7, etc. The formula in B6 is copied into B7:B18.

The formulae in column B are repeated in columns D, F and H for different values of r - 2.6, 3.0, 3.5, and so on. D6 will contain a value for r such as 2.6, D7 will contain =$C$6*$A$4*(1-$A$4), D8 will contain =$C$6*D6*(1-D6), and so on.

_{The output values in columns B, D, F and H would
be better represented by a chart. The populations modeled with values of r set
to 2.5, 2.6 and 2.7 quickly stabilise at a constant value but the population
with r=3.6 fluctuates wildly. The interesting thing here is to find the point at
which the wild or chaotic fluctuations begin. You can use the spreadsheet model
to enter new values in cell G6 and observe the behaviour of Series 4 on the
chart. At different values above r=2 .8 the population either stabilises on 2
alternate values or fluctuates wildly, never settling on any value.
This chaotic pattern was first observed by
the biologist Robert May.

The line chart was produced by holding down the
Control key and selecting the data in columns B, D, F and H; holding the Control
key allows a multiple selection of cells.
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