[SOLVED] Evolutionary-Dynamics Homework 1

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).

1. (a)  Determine the equilibrium points x∗ of the system.
2. (b)  Arethepointsstableforr=0.5,r=1.5,r=2.5?
3. (c)  Confirm this by numerically iterating the difference equation. Hint: Plot the value x for a series of time steps.
4. (d)  Examine the stability and behaviour for r = 3.5. Hint: Plot the Poincaré section of xt against xt−1.
5. (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