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Unformatted text preview: Math 17C
Kouba
Discussion Sheet 7 1.) (Lotka—Voltera PredatorPrey Model) Let N (t) be the prey population at time t and
let P(t) be the predator population at time t. Assume that LN = 6N — 2PN
Elf: ‘—— : PN — 4P
dt a.) Determine all equilibria for this nonlinear system of differential equations. b.) Use the analytical approach (Jacobi Matrix and eigenvalues) to determine (if
possible) the stability of each equilibria. c.) Separate variables and integrate to get an implicit solution (in N and P) for this
system. d.) The solution to part c. is graphed below using MAPLE and assuming that the
integration constant C is —1. Determine the direction of motion of the solution. Assume
that (N, P) = (6, 1) is the initial value, then use this graph to set up a table of values (as
many as you think you need) for pairs (N, P) which satisfy the equation and travel one
complete cycle of this graph. Finally, on the same set of axes, plot N and P as functions of time t. Use the graphs of N and P to descibe the interaction between the predator and
prey species over time. 2.) How many ways can you pick a captain and c0captain from among 12 basketball
players ? 3.) A group consists of 15 people made up of 9 women and 6 men. How many ways can
you pick a committee a.) of 5 people ‘? b.) of 5 people, if exactly 3 must be women ? ) of 5 people, if exactly 3 must be men ? .) of 5 people, if all must be women ? ) of 5 people, if there must be at least one woman on the committee ? 4.) You are to make ordered arrangements of three letters chosen from X, Y, Z, and W.
a.) Determine how many there are by directly listing all possibilities.
b.) Determine how many there are by using a permutation, P(n, k). 5.) There are 6 math, 4 music, 5 biology, and 3 chemistry classes available for you to take.
How many different class schedules are possible if you must pick one of each class ? +++++++++++++++++++++++++++++++++ ”Kids are great. That’s one of the best things about our business, all the kids you get
to meet. It’s a shame they have to grow up to be regular people and come to the games
and call you names.“ former NBA player and current NBA commentator, Charles Barkley ...
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 Summer '09
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