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B compete for resources a n 1 a n k 1 a n k 3 a n b n

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b compete for resources a n + 1 = a n + k 1 a n - k 3 a n b n b n + 1 = b n + k 2 b n - k 4 a n b n Predator-prey – species b eats species a a n + 1 = a n + k 1 a n - k 3 a n b n b n + 1 = b n - k 2 b n + k 4 a n b n War! a n + 1 = a n - k 3 b n b n + 1 = b n - k 4 a n Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Population analysis There are lots of questions we might want to ask about how these models behave, e.g.: What is the long-time behaviour? How sensitive are the solutions to the initial conditions? Can we have sustainable hunting/farming? Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Long-time behaviour Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Malthus Recall the simplest model we looked at, the Malthus model: a n + 1 = a n + ka n = ( 1 + k ) a n = ra n How does its behaviour depend on r ? r = 0 a n + 1 = 0 r = 1 a n + 1 = a n r < 0 oscillatory | r | < 1 decay | r | > 1 growth Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Malthus What is the equilibrium value? At equilibrium: a n + 1 = a n r = 1 or a n = 0 Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Savings account Back to our savings account. Same as Malthus! Include regular withdrawls: a n + 1 = ra n + b Equilibrium: a n + 1 = a n a n = b 1 - r Phil Hasnip Mathematical Modelling
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Introduction Discrete systems Population analysis Savings account Equilibrium: a n = b 1 - r r = 1 no equilibrium Otherwise an equilibrium a exists Are the equilibria all the same?
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