difference eqns

# Eg savings account 1 interestmonth invest 1000

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E.g. savings account, 1% interest/month Invest £1000 initially What is balance after a year? Phil Hasnip Mathematical Modelling

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Introduction Discrete systems Population analysis Savings account Define initial amount a 0 = 1000. Then the next value in the sequence is a 1 = a 0 + Δ a 0 , where Δ a 0 is the amount due to the monthly interest. i.e. in this case we have Δ a 0 = 0 . 01 a 0 , and: a 1 = a 0 + 0 . 01 a 0 a 2 = a 1 + 0 . 01 a 1 . . . a 12 = a 11 + 0 . 01 a 11 Phil Hasnip Mathematical Modelling
Introduction Discrete systems Population analysis Discrete systems The savings account example led to a simple series We may have other actions – e.g. regular withdrawals In general we don’t have a precise formula -→ have to fit change to data Phil Hasnip Mathematical Modelling

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Introduction Discrete systems Population analysis Approximating change In practice continuous systems are often modelled as discrete processes Experimental data is usually discrete Often need to guess an approximate form for model and fit to data Phil Hasnip Mathematical Modelling
Introduction Discrete systems Population analysis Population analysis Remember this from the first lecture? We had: Data on population every 10 years -→ discrete changes Had to deduce functional form Simplest was Malthus a n + 1 = a n + ka n Verhulst model saturates – finite carrying capacity a n + 1 = a n + k 1 - a n a a n Phil Hasnip Mathematical Modelling

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Introduction Discrete systems Population analysis Extensions to model Competition – species a and
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