HW3MMBS12 - MMB University of Florida HOMEWORK III Due Name...

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University of Florida MMB HOMEWORK III Due: March 16, 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 final 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 integro-differential equation with distributed delay: (1) N prime ( t ) = rN ( t ) parenleftbigg 1 - 1 K integraldisplay 0 N ( t - u ) p ( u ) du parenrightbigg . where p ( u ) is given by the following function p ( u ) = 1 τ e - u τ which has an average delay integraldisplay 0 up ( u ) du = τ. This function p ( u ) is called weak generic delay kernel . (a) Show that all equilibria of model (1) are N * 1 = 0 and N * 2 = K . (b) Show that the equation linearized about N * 2 = K ( N ( t ) = N * 2 + v ( t )) is given by v prime ( t ) = - r integraldisplay 0 v ( t - u ) p ( u ) du.
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