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

Previewing pages 1, 2 of 7 page document View the full content.
View Full Document

Module IV - Population dynamics



Previewing pages 1, 2 of actual document.

View the full content.
View Full Document
View Full Document

Module IV - Population dynamics

93 views


Pages:
7
School:
Arizona State University
Course:
Mat 294 - Problem-Solving Seminar

Unformatted text preview:

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



View Full Document

Access the best Study Guides, Lecture Notes and Practice Exams

Loading Unlocking...
Login

Join to view Module IV - Population dynamics and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Module IV - Population dynamics and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?