# ASU MAT 294 - Module IV - Population dynamics (7 pages)

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## Module IV - Population dynamics

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- School:
- Arizona State University
- Course:
- Mat 294 - Problem-Solving Seminar

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Module IV Population dynamics Recurrence relations Example 1 The number of bacteria in a colony triples every hour If a colony begins with 10 bacteria how many will be present after n hours We let a n denote the number of bacteria at the end of n hours Hence a n 3a n 1 where n is a positive integer This equation called the recurrence relation together with the initial condition a 0 10 uniquely determines the sequence a n for all nonnegative integers n The sequence of a recurrence relation is called a solution of the recurrence relation Example 2 Rabbits and the Fibonacci numbers A pair of rabbits does not breed until they are 2 months old After they are 2 months old each pair produces another pair each month Starting with a pair of rabbits find a recurrence relation for the number of pairs of rabbits after n months assuming that no rabbit ever dies Solution At the end of the first month the number of pairs of rabbits is a 1 1 At the end of the second month the number of pairs of rabbits is a 2 1 To find the number of pairs at the end of n months add the number in the previous month a n 1 and the number of newborn pairs a n 2 since each newborn comes from a pair at least two months old Hence a n a n 1 a n 2 n 3 with a 1 1 and a 2 1 is the desired recurrence relation Exercise Assume that the population of the world in 2002 is 6 2 billion and is growing at the rate of 1 3 a year a Set up a recurrence relation for the population of the world n years after 2002 b Find an explicit formula for the population of the world n years after 2002 c What will the population of the world be in 2002 Solving recurrence relations Definition A linear homogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form a n c1 a n 1 c 2 a n 2 c k a n k where c1 c 2 c k are real numbers and c k 0 The recurrence relation in the definition is Linear since the right hand side is a sum of multiples of the previous terms of the sequence Homogeneous since

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