Write the system of equations and construct the corresponding matrix model for the following exercise. Suppose that one of three different alleles is present in each individual in a population. In each generation the following happens: 9% of individuals carrying allele X mutate to carry allele Y, 7% mutate to allele Z; and the rest remain unchanged; 0.1% of individuals carrying allele Y mutate to carry allele Z and the rest remain unchanged; 80% of individuals carrying allele Z mutate to carry allele X and the rest remain unchanged. Xt + 1 Yt + 1Zt + 1 = Xt YtZt
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Construct the matrix diagram for each of the following models. Mutation Each generation 5$\%$ of individuals carrying allele $X$ mutate to carry allele $Y .$ Using $X_{t}$ and $Y_{t}$ to denote the number of individuals carrying each allele at time $t,$ we have $$\begin{array}{l}{X_{t+1}=0.95 X_{t}} \\ {Y_{t+1}=0.05 X_{t}+Y_{t}}\end{array}$$
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Suppose that as a result of mutation, the number of people carrying allele A and the number of people carrying allele B change in a way described by the equation yt+1 = [[0.95, 0], [0.05, 1]]yt, where yt is the vector whose components are the numbers of individuals carrying each allele in year t. If in the current year the vector is [90.25, 209.75], what was the vector 2 year ago? In the long run (as t -> +inf), what will be the behavior of the vector yt?
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Be sure to show all work and all problem-solving strategies. Give complete explanations for each step. To study how species survive, scientists model their populations by observing the different stages in their life. Scientists consider, for example, the stage at which the animal is fertile, the proportion of the population that reproduces, and the proportion of the young that survive each year. In one study, scientists looked at an animal that is considered immature for the first year of its life, a juvenile for the second year of life, and an adult for the remainder of its life. The following data was collected for this species: Immature Juvenile Adult 0 0 0.4 Immature A = [ 0.1 0 0 ] Juvenile 0 0.3 0.8 Adult [ 600 ] Immature X0 = [ 400 ] Juvenile [ 3500 ] Adult The entries in matrix A represent the percentage of the population that survives to the next year. For example, in the first column, 0% of the immature animals remain immature, 10% become juveniles and, of course, none become adults (that takes two years). Of the juveniles, none go backwards to become immature and 30% become adults. For the adults, 40% reproduce and add to the immature population, none become juveniles, and 80% go on to the next year (still as adults). The matrix X0 represents the number of immature, juvenile, and adult animals in year 0. Let X1 = AX0, X2 = AX1, X3 = AX2, etc. 3. Show that X2 = A^2X0, X3 = A^3X0, etc.
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