This is our collection of tasks on the mathematical theme of 'Population Dynamics' for advanced students and those interested in mathematical modelling.

The probability that a passenger books a flight and does not turn up is 0.05. For an aeroplane with 400 seats how many tickets can be sold so that only 1% of flights are over-booked?

Why MUST these statistical statements probably be at least a little bit wrong?

Invent scenarios which would give rise to these probability density functions.

How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics.

Match the charts of these functions to the charts of their integrals.

Here are several equations from real life. Can you work out which measurements are possible from each equation?

How is the length of time between the birth of an animal and the birth of its great great ... great grandparent distributed?

Simple models which help us to investigate how epidemics grow and die out.

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

Which line graph, equations and physical processes go together?

Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

Get some practice using big and small numbers in chemistry.

Was it possible that this dangerous driving penalty was issued in error?

Use your skill and knowledge to place various scientific lengths in order of size. Can you judge the length of objects with sizes ranging from 1 Angstrom to 1 million km with no wrong attempts?

Get further into power series using the fascinating Bessel's equation.

Formulate and investigate a simple mathematical model for the design of a table mat.

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

Can you sketch these difficult curves, which have uses in mathematical modelling?

Many physical constants are only known to a certain accuracy. Explore the numerical error bounds in the mass of water and its constituents.

Match the descriptions of physical processes to these differential equations.

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

Estimate these curious quantities sufficiently accurately that you can rank them in order of size

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

Build up the concept of the Taylor series

Investigate circuits and record your findings in this simple introduction to truth tables and logic.

In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?

When you change the units, do the numbers get bigger or smaller?

Look at the advanced way of viewing sin and cos through their power series.

Work with numbers big and small to estimate and calculate various quantities in biological contexts.

Work with numbers big and small to estimate and calculate various quantities in physical contexts.

Which units would you choose best to fit these situations?

Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and fill in the blanks in truth tables to record. . . .

Can Jo make a gym bag for her trainers from the piece of fabric she has?

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Explore the properties of matrix transformations with these 10 stimulating questions.

Go on a vector walk and determine which points on the walk are closest to the origin.

Explore the shape of a square after it is transformed by the action of a matrix.