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INSTRUCTOR:
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MATH 148, SPRING 2016
COMMON EXAM II (PART 2) - VERSION A
FIRST NAME:
DIRECTIONS:
1. The use of a calculator, laptop or computer is prohibited.
2. Present your solutions in the space provided.
Show all your work neatly and concisely and clearly indicate your final answer . You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it.
THE AGGIE CODE OF HONOR
“An Aggie does not lie, cheat or steal, or tolerate those who do.”
Signature:
DO NOT WRITE BELOW!
Question Points Awarded
1
2
3
4
Part 1
Points
12
12
12
12
52
100
1

1. Suppose that a population is divided into three age classes and that 80% of females of age zero and 50% of females of age one survive until the end of the next breeding season. Moreover, assume that females of age one have an average of 4 female offspring and females of age two have an average of 3 female offspring. Currently, the population consists of 100 females of age zero, 500 females of age one, and 50 females of age two.
(a) (8 pts) Find the Leslie matrix, L , and initial population distribution, N (0).
(b) (4 pts) Find the population distribution at the end of the next breeding season, N (1).
2

2. The spread of an infectious disease, such as influenza, is often modeled using the following autonomous differential equation: dI
= βI ( N − I ) − γI, dt where I denotes the number of infected people, N is the total population size, β > 0 is a transmission rate, and
γ > 0 is the rate at which people recover from infection.
(a) (5 pts) Find all equilibria of this differential equation (in terms of N , β , and γ ).
(b) (5 pts) Find the eigenvalues for each of the equilibria found in part (a).
(c) (2 pts) Suppose that β = 0 .
001, N = 5000, and γ = 4. Use your answers from parts (a) and (b) to find the number of infected people in the long-run, lim t →∞
I ( t ).
3

3. Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 2 units of food A and 3 units of food B, each individual of species 2 consumes 3 units of food A and 4 units of food B, and each individual of species 3 consumes 1 unit of food A and 2 units of food B. Each day, 300 units of food A and 500 units of food B are supplied.
(a) (4 pts) Find a system of linear equations describing this situation.
(Let x , y , and z denote the numbers of species 1, 2, and 3, respectively.)
(b) (8 pts) How many individuals of each species can be reared together? Is there more than one solution?
4

4. Consider the matrix
A =
2 2
1 3
.
(a) (8 pts) Show that ~
1
= h 2 , − 1 i and ~
2
= h 1 , 1 i are linearly independent eigenvectors of A .
(b) (2 pts) Represent the vector ~ = h 4 , 1 i as a linear combination of the eigenvectors ~
1 and ~
2
.
(c) (2 pts) Use the eigenvalues and eigenvectors of A to compute A 5 ~ .
5