13. Chaos The Fish in the Pond Problem


The greatest success a mathematician can have is if he/she can match a real physical situation to a mathematical formula.


Einstein achieved this with E = mc2 and the speed of light.


connecting energy with mass


Biologists, trying to work out how real fish in a real pond will occupy the pond, had to find a function which matched the reality of life, hunger, competition, overpopulation etc. in the pond.


Consider a pond. The number of pairs of fish in the pond can vary between 0 (extinction) and 1 (pond is full of fish). A population of 0·1 means the pond is 10% full. A population of 0·5 means the pond is ½ full. The number of fish in the pond can be modelled by the formula:


xnext year = rx(1–x) where x = number of fish in the pond this year


xnext year = number of fish in the pond next year r


= rate of increase (reproductive rate)


Edward Lorenz: American mathematician and meteorologist, (1917–2008)


Lorenz was a meteorologist. His work on weather forecasting led him to accidentally discover the theory of chaos. In 1972 he published a paper entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas? The Chaos Theory was one of the major scientific discoveries of the twentieth century.


Start with the pond 1% full, therefore x = 0·01 Pick a rate of increase, r = 1·2 and investigate what happens.


Today Year 1 Year 2


x = 0·01 . . . . . pond is 1% full x next year = 1·2 (0·01)(1 – 0·01) = 0·01188 . . . pond is 1·18% full


x next year = 1·2 (0·01188) (1 – 0·01188) = 0·01409 . . . pond is 1·4% full 0·01667 0·01967 0·02313 0·02712 0·03166 0·03679 0·04252 0·04886 0·05577 0·06319 0·07103


settles at 0·1666 . . . 16·666% full


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