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

Explore the possibilities for reaction rates versus concentrations with this non-linear differential equation

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

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

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

Which line graph, equations and physical processes go together?

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

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

Match the descriptions of physical processes to these differential equations.

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

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

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

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

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

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

Build up the concept of the Taylor series

This problem explores the biology behind Rudolph's glowing red nose.

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Make an accurate diagram of the solar system and explore the concept of a grand conjunction.

Get some practice using big and small numbers in chemistry.

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

Work out the numerical values for these physical quantities.

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?

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.

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

Use vectors and matrices to explore the symmetries of crystals.

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

Which units would you choose best to fit these situations?

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

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

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

Can you make matrices which will fix one lucky vector and crush another to zero?

Which of these infinitely deep vessels will eventually full up?

Explore the meaning of the scalar and vector cross products and see how the two are related.

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

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

Starting with two basic vector steps, which destinations can you reach on a vector walk?

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?

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

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

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.

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

In this short problem, try to find the location of the roots of some unusual functions by finding where they change sign.

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

How do you choose your planting levels to minimise the total loss at harvest time?