Homework08 - Math 404 - Homework 7 Example Student...

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Unformatted text preview: Math 404 - Homework 7 Example Student Solutions March 11, 2011 S: 6.5.2 Consider the system ¨ x = x- x 2 . a) Find and classify the equilibrium points. b) Sketch the phase portrait. c) Find an equation for the homoclinic orbit that separates closed and non- closed trajectories. Answer a) ˙ x = y ˙ y = x- x 2 Equilibrium points: (0,0), (1,0) F = 1 1- 2 x For (0,0): F = 0 1 1 0 τ = 0, Δ =- 1 Then its a saddle point with vectors 1 1 and- 1 1 with eigenvalues 1 and -1 respectively. For (1,0): F = 1- 1 0 τ = 0, Δ = 1 Then its a nonlinear center b) 1 Figure 1: Drawing of the phase portrait. c) dy dx = x- x 2 y ydy = ( x- x 2 ) dx y 2 2 = x 2 2- x 3 3 + c where c is a constant y 2 = x 2- 2 3 x 3 + c , take initial condition (0,0), so c = 0 Therefore equation is: y 2 = x 2- 2 3 x 3 2 S: 6.5.6 (Epidemic model revisited) In Exercise 3.7.6, you analyzed the Kermack- McKendrick model of an epidemic by reducing it to a certain first-order system. In this problem you’ll see how much easier the analysis becomes in the phase plane. As before, let x ( t ) ≥ 0 denote the size of the healthy population and y ( t ) ≥ 0 denote the size of the sick population. The the model is ˙ x =- kxy, ˙ y = kxy- ly where k,l > 0. The equation for z ( t ), the number of deaths plays no role in the x , y dynamics so we omit it. a) Find and classify the fixed points. b) Sketch the nullclines and the vector field. c) Find a conserved quantity for the system. d) Plot the phase portrait. What happens as t → ∞ ? e) Let ( x ,y ) be the initial condition. An epidemic is said to occur if y ( t ) increases initially. Under what conditions does an epidemic occur? Answer a) ˙ x =- kxy ˙ y = kxy- ly ˙ x = 0 when x = 0 or y = 0 ˙ y = 0 when x = l k or y = 0 Fixed points: On observation of these differential equations, we can see that when y = 0, ˙ x = 0 and ˙ y = 0. Therefore, the entire y-axis consists of fixed points. These are then classified as non-isolated fixed points. b) x nullcline: 0 =- kxy ⇒ x = 0 and y = 0. y nullcline: 0 = kxy- ly ⇒ x = l k . c) dy dx = dy dt dx dt = kxy- ly- kxy = l kx- 1 3 Figure 2: Plot of the phase portrait and the nullclines. dy dx = l kx- 1 Z dy = Z l kx- 1 dx y = l * ln ( x ) k- x C = y + x- l * ln ( x ) k d) As t → ∞ , it is clear to see from the above phase portrait that the number of sick people eventually goes to zero and a certain healthy amount of the population is left. e) In order to determine the condition when an epidemic occurs, we must find where ˙ y > 0. We set ˙ y = 0 to find the point where the epidemic occurs. kxy- ly = 0 x = l k Therefore, x must be greater than l k for an epidemic to occur. 4 Figure 3: Plot of the phase portrait....
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This note was uploaded on 03/13/2011 for the course MATH 404 taught by Professor Staff during the Spring '08 term at University of Michigan.

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Homework08 - Math 404 - Homework 7 Example Student...

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