Family names

You are asked to help with a sociological study on the survival and extinction of surnames.

Consider a closed community with N individuals at time t = 0 with K different surnames. Assume that all the children of any marriage take the father's surname.

Make other assumptions as necessary and set up a model with the aim of answering the following questions.

1.What will be the distribution of surnames after x generations?

2.What is the probability that a particular surname becomes extinct?

3.How long will a particular surname survive on average?

Estimating animal populations

One method of estimating the size of a living population occupying a finite region, e.g. fish in a lake, or squirrels in a wood, is to capture a few individuals and mark or tag them.

Suppose that x animals are caught, marked and then released. Some time later, n animals are caught and y of them are found to be marked. What is the best estimate that we can make of the total number N of animals living in the region? How accurate is this estimate likely to be and on what does the accuracy depend?

Derive 95% confidence limits for N. Develop a simulation model to check your answers. What advice would you give on the choice of values for x and n?

Simulation of population growth

Consider the life of an organism as the sequence birth-childhood-adulthood-old age-death. Suppose that an individual is capable of producing offspring only during adulthood and that the production of offspring can be modelled by a Poisson process. Make suitable assumptions about the duration of the three stages childhood, adulthood and old age.

Construct a simulation model which can be used to study the growth of a population starting from three young males and three young females at time t = 0.

Find the size N ( t ) of the population at a later time t and also find t ( n ), the time that it takes for the population to grow to a size n. You should carry out a number of runs of the model and calculate mean values.

Needle crystals

In many chemical and metallurgical processes a surface film of needle-shaped crystals is formed as a substance cools. The crystals appear to start to grow from random points (‘nuclei’) and continue to grow in directly opposed directions. The position of a nucleus and the direction of growth appear to be totally random. The rate of growth stays reasonably constant. The crystals do not push each other apart when they come into contact; in fact, when a growing end touches another segment, that end stops growing and remains attached to the segment that blocked it (Figure 9.33).

We need to know the mean number of crystal ends per unit area which are growing (unblocked) at time t. No method is known for finding this number exactly. You are asked to help as mathematical modellers.

Figure 9.33

1.Select appropriate variables and parameters for defining the problem.

2.Develop a mathematical model with associated software to find the number n(t) of growing ends at time t.

3.Use your model to find how n(t) depends on the other parameters in the problem.

Car parking

A training manual for car drivers wishes to offer the following advice for parking a car neatly into a space in a line of parked cars standing next to a kerb. For backing in, first overshoot the space by x % of a car length and stand off by y % of a car width (Figure 9.34).

1.You are asked to develop a mathematical model which will enable the author of the manual to substitute appropriate values for x and y.

2.What is the minimum gap length L in order for parking to be possible without driving over the kerb?

3.Find similar answers for the problem when driving into the space front wheels first.

4.After parking neatly and doing your shopping, you return to find that another driver has boxed you in. You may still be able to drive out by suitable manoeuvres. Use your mathematical model to investigate what manoeuvres may be necessary.

Figure 9.34

Overhead projector

Figure 9.35 shows a lecture room with 25 desks and an overhead projector and screen. The students at