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**Unformatted text preview: **CS1132 Fall 2011 Assignment 1 Adhere to the Code of Academic Integrity. You may discuss background issues and general strategies with others and seek help from course staff, but the implementations that you submit must be your own. In particular, you may discuss general ideas with others but you may not work out the detailed solutions with others. It is never OK for you to see or hear another student’s code and it is never OK to copy code from published/Internet sources. If you feel that you cannot complete the assignment on your own, seek help from the course staff. When submitting your assignment, follow the instructions summarized in Section 4 of this document. Do not use the break or return statement in any homework or test in CS1132. 1 Population Dynamics (The following description is taken from Chapter 8.1 of ”Mathematical Models in Population Biology and Epidemiology” by Brauer and Catillo-Chavez, 2001.) Consider a population that is divided into a finite number of age classes labeled from 0 to m . One method of describing the number of members in each age class as a function of time is by using a linear discrete-time model for population growth. In such a model, we let α j,n denote the number of members in the j ’th class at the n ’th time. We assume that the length of time spent in each age class is the same. Then α j,n +1 , the number of members in the j ’th age class at the ( n + 1)st time, is equal to α j- 1 ,n minus the number of members of this age cohort who die before entering the next age class. We assume that the probability of survival from one age class to the next depends only on age. Let p j be the probability that a member of the j ’th age class survives until the ( j + 1)st age class. All new members recruited into the population are assumed to come from a birth process, with fecundity depending only on age. Assume that there are constants β ,β 1 ,...,β m such that α ,n +1 = β α ,n + β 1 α 1 ,n + ··· + β m α m,n . If we define ~α n = α ,n α 1 ,n . . . α m,n and define the Leslie matrix to be A = β β 1 β 2 ... β m- 1 β m p ... p 1 ... . . . . . . . . . . . . . . . . . . ... p m- 1 then the change in the population through time can be described by the vector difference equation ~α n +1 = A~α n . In the above equation, A~α n is the multiplication of a matrix and a vector, which results in a vector. This operation will be explained below. Let P n be equal to the total population at time n . Then the vector ~α n P n gives the fraction of the population in each age class at time n ....

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