Homework 1
Math 441, Fall 2021
1. (2.1.1-2.1.3) Interpret ẋ = sinx as a flow on the line.
(a) Find all the fixed points of the flow.
(b) At which points x does the flow have greatest velocity to the right?
(c) Find the flow’s acceleration ẍ as a function of x, and find the points where the flow has maxi-
mum positive acceleration.
2. (2.2.3) Analyze the following equations graphically; sketch the vector field on the real line, find all
the fixed points, classify their stability, and sketch the graph of x(t) for different initial conditions.
ẋ = x− x3.
3. (2.2.8) (Working backwards, from flows to equations) Given an equation ẋ = f(x), we know how
to sketch the corresponding flow on the real line. Here you are asked to solve the opposite problem:
For the phase portrait shown in Figure 1, find an equation that is consistent with it. (There are an
infinite number of correct answers—and wrong ones too.)
4. (2.2.9) (Backwards again, now from solutions to equations) Find an equation ẋ = f(x) whose
solutions x(t) are consistent with those shown in Figure 2.
5. (2.3.4) (The Allee effect) For certain species of organisms, the effective growth rate Ṅ/N is highest
at intermediate N . This is called the Allee effect (Edelstein–Keshet 1988). For example, imagine
that it is too hard to find mates when N is very small, and there is too much competition for food
and other resources when N is large.
(a) Show that Ṅ/N = r− a(N − b)2 provides an example of the Allee effect, if r, a, and b satisfy
certain constraints, to be determined.
(b) Find all the fixed points of the system and classify their stability.
(c) Sketch the solutions N(t) for different initial conditions with r = b = 1 and a = 2.
(d) Compare the solutions N(t) to those found for the logistic equation. What are the qualitative
differences, if any?
6. (2.4.4) Use linear stability analysis to classify the fixed points of the following systems. If linear
stability analysis fails because f ′(x∗) = 0, use a graphical argument to decide the stability.
ẋ = x2(6− x).