Exercises marked with a "§" are programming exercises. These can be solved in a programming language of your choice. Please make sure to hand in your code along with your answers to these exercises.
Problem 1: Discrete time (tutorial question)
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 = rxt(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. §
Hint: Plot the value x for a series of time steps.
(d) Examine the stability and behaviour for r = 3.5. §
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
λx t 1 (1)
dt K
(a) Show, by direct integration of (1), that the solution is given by:
(2 points)
Kx0eλt x(t)= .
K+x0(eλt −1)
Hint: Use separation of variables and partial fractions.
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(b) Find the equilibrium points of the system and discuss their stability. Hint: Consider the cases λ > 0 and λ < 0.
(c) Numerically integrate to demonstrate the results above for K = 1. §
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