Ws17 - Math 464 Worksheet 17 Consider the Population model from Worksheet 16 x = 0.02x 0.01x2 0.01xy y = 0.03y 0.01y 2 0.02xy Part I Here are some

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Math 464, Worksheet 17 Consider the Population model from Worksheet 16: Δ x = 0 . 02 x - 0 . 01 x 2 - 0 . 01 xy Δ y = 0 . 03 y - 0 . 01 y 2 - 0 . 02 xy Part I. Here are some basic facts we already know: If the population starts out at x (0) = y (0) = 1 then it never changes. In other words, x ( t ) = 1 and y ( t ) = 1 is an equilibrium solution. If the population starts out elsewhere, it seems that one or the other species goes extinct. The diFerence equations are non-linear. Therefore, we cannot write this equation as a matrix equation, and we can’t use eigenvectors or eigenvalues to predict where solutions go. Except that we can. Here’s how: 1. Con±rm that x = y = 1 is an equilibrium by plugging into the right hand side of the diFerence equations. 2. Compute the Jacobian of the system. This is a two-by-two matrix J = b Δ x ∂x Δ x ∂y Δ y ∂x Δ y ∂y B 3. Evaluate J at the point x = 1, y = 1. 4. Compute eigenvalues and eigenvectors for
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This note was uploaded on 10/12/2011 for the course MATH 464 taught by Professor Dougbullock during the Fall '08 term at Boise State.

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Ws17 - Math 464 Worksheet 17 Consider the Population model from Worksheet 16 x = 0.02x 0.01x2 0.01xy y = 0.03y 0.01y 2 0.02xy Part I Here are some

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