5.2 Developing models 211
The equation below is a very simplistic
mathematical model for a fish farm. It is called
the Beverton–Holt model and considers the
effect of various reproduction rates and the
maximum capacity of the farm.n Rn
t nR K
t
t+ 1 =(( 11 +−)/ )
For the non-mathematical, this equation may
look rather frightening, but don’t worry, we
are not going to do any hard algebra and the
examples you will encounter in examinations
will use much simpler mathematics.
In the equation, R is the reproduction rate
(R = 1.5 means the population, if unlimited,
would rise by 50% every year – this allows for
both births and deaths). K is the maximum
capacity of the fish farm. nt is the population in
the current year and nt + 1 the population in the
next year. We can try putting some numbers
into this equation and looking at what
happens. We assume that the initial population
(n 0 ) is 1000 fish and the maximum capacity is
10,000 (beyond this the fish would die of
starvation or overcrowding), then look at how
the population increases year by year for three
different values of R. The results are shown in
the graph below:5.2 Developing models
In Chapter 3.11 we looked at how models can
be used by governments, industry and so on to
carry out ‘what if?’ analyses and look at how
changes to an environment can affect various
factors. In this chapter, modelling is taken
further. Questions may involve the application
of more complex models or the development of
a model for a given situation. In the longer,
multiple question items which may be
encountered in A Level examinations or some
admissions tests, the individual questions
usually increase in complexity, either by asking
for a deeper analysis or by introducing new
situations or conditions.
The activities in this section are progressive,
starting with the application of a model which
is provided and progressing through simple
linear models to the development of a non-
linear model. The final activity is harder than
those that would be encountered in an A Level
examination and will be useful for those
intending to take university admissions tests or
those wishing to prepare themselves better by
tackling harder questions. Candidates will
find a range of questions of appropriate
difficulty in past papers.
The first example shows how models can be
useful in real situations.
R = 1.5
R = 2.0
R = 2.51 23456789101112131415161718192021222324250200040006000800010,00012,000YearPopulation