Exercises marked with a ” ” are programming exercises. These can be solved in a programming lan- guage 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)=5×2(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) Arethepointsstableforr=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
(1 point) (1 point) (1 point)
(1 point) (1 point)
(1)
(2 points)
The logistic model for population growth is:
dx(t)
x(t)= K+x0(eλt −1). Hint: Use separation of variables and partial fractions.
x(t) dt =λx(t) 1− K
(a) Show, by direct integration of (1), that the solution is given by: Kx0eλt
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 points)
2
