MATH 489 Homework 1: Equilibria and Phase Portraits for Differential Equations, Assignments of Differential Equations

Download MATH 489 Homework 1: Equilibria and Phase Portraits for Differential Equations and more Assignments Differential Equations in PDF only on Docsity! MATH 489, Section C13, HW 1. Due date: 02/06/09. Problem 1 The doomsday vs. extinction model x ′ = ax( x T − 1) may need to be modified so that unbounded growth does not occur then x is above the threshold T > 0. The simplest way to do this is to introduce another factor that will have the effect of making dx dt negative when x > K > T . Thus, we consider x ′ = ax( x T − 1)(1 − x K ). 1. Find the equilibria and draw the phase line for this equation. 2. Sketch some representative solutions for initial conditions 0 < x(0) < T , T < x(0) < K, and x(0) > K. (Use the slope field of the equation.) 3. Repeat the previous part for K = T . Problem 2 It is sometimes reasonable to assume that the rate at which fish are caught depends on their population x: the more fish there are, the easier it is to catch them. To include this effect the logistic equation is replaced by x ′ = ax(1 − x K ) − Ex. 1. Find the two equilibria x1 < x2 for this equation if E < a. 2. A sustainable yield xs = Ex2 of the fishery is a rate at which fish can be caught indefinitely. Find xs as a function of E. 3. Determine E so as to maximize xs and thereby find the maximum sus- tainable yield xm. Problem 3 For each of the following differential equations, find all equilibrium solutions and determine whether they are sinks, sources, or shunts. Also, sketch the phase line. (i) x′ = x3 − 4x; (ii) x′ = |1 − x2|. Problem 4 Each of the following families of differential equations depends on a param- eter a. Sketch the corresponding bifurcation diagrams. (i) x′ = x3 − ax; (ii) x′ = x3 − x − a. 1