HW2MMBS12

# HW2MMBS12 - x = a where a is also an arbitrary constant 3(3...

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University of Florida MMB HOMEWORK II Due: Feb 22, 2012 Name: ID #: Instructor: Directions: You have until 5:00 p.m. on the due date to answer the following questions. You must show all your work as neatly and clearly as possible and indicate the Fnal answer clearly. You may use any books and you can work together but each of you must submit a homework. Problem Possible Points 1 5 2 5 All 10 Total 20 1

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(1) Population is modeled by the following equation: N ( t ) = rN ( K - N ) K + aN . The population is subjected to constant-eFort harvesting (1) N ( t ) = rN ( K - N ) K + aN - EN (a) Determine the units of the parameters involved. Derive a non-dimensional form of equation (1). (b) ±ind the equilibria of the model (1). Determine the stability of equilibria. Plot the equilibria as a function of the harvesting parameter E (that is draw a bifur- cation diagram). (c) ±ind the maximum sustainable yield. 2
(2) Find the general form of the solution of the di±erence equation: x n +1 = c - x n where c is an arbitrary constant, and the initial value is given by

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Unformatted text preview: x = a where a is also an arbitrary constant. 3 (3) The following discrete model is given to model population (2) x n +1 = 3 x 2 n x 2 n + 2 (a) Determine the nonnegative equilibria of the model (2). Determine the stabilities of the equilibria. (b) Suppose a fraction a is removed from the population in each generation so that the model becomes x n +1 = 3 x 2 n x 2 n + 2-ax n . For what values of a is there a stable equilibrium only at x ∗ = 0? 4 (4) Given the population model with delay (3) N ′ ( t ) = rN ( t-τ ) N ( t-τ ) + A-dN ( t ) where r, A, d are parameters. (a) Find the equilibria of model (3). (b) Find the characteristic equation of model (3). (c) Find conditions for stability of the equilibria. (d) Use computer algebra system to graph a representative solution of the model which stabilizes to sustained oscillations. Graph N as a function of t . 5...
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## This note was uploaded on 03/20/2012 for the course MAP 4484 taught by Professor Martcheva,m during the Spring '08 term at University of Florida.

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HW2MMBS12 - x = a where a is also an arbitrary constant 3(3...

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