Homework08

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

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 18

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online