Problem 1: Discrete time
Suppose you have a difference equation xt+1 = f(xt) of a discrete time model with
f(x) = 5x2(1−x).
(a) Determine the equilibrium points x∗ of the system.
(b) Which of the equilibrium points x∗ are stable?
Problem 2: Logistic difference equation
In a discrete time model for population growth, the value x (number of cells divided by the maximum
number supported by the habitat) at time t + 1 is calculated from the value at time t according to the
difference equation
xt+1 = r xt(1−xt).
(a) Determine the equilibrium points x∗ of the system.
(b) Are the points stable for r = 0.5, r = 1.5, r = 2.5?
(c) Confirm this by numerically iterating the difference equation. §
(1 point)
Hint: Plot the Poincaré section of xt against xt−1.
(e) What happens for r = 3.9? §
)
Problem 3: Logistic growth in continuous time
The logistic model for population growth is:
dx(t)
dt
= λx(t) 1− x
(t)
K
(1)
(a) Show, by direct integration of (1), that the solution is given by:
x(t) =
Kx0eλt
K +x0(eλt −1)
.
Hint: Use separation of variables and partial fractions.
1(b) Find the equilibrium points of the system and discuss their stability.
(1 point)
Hint: Consider the cases λ > 0 and λ < 0.
(c) Numerically integrate to demonstrate the results above for K = 1. §
2
