Bio 481 - Problem Set 5 - Dynamics 2
General Note: a^b stands for a raised to the b power.
- Start with your familiar favorite friend, the exponential map.
Using Vanderplot enter this equation in two separate times (maybe once for
M and once for N). Use any r>1.0. The only thing that should vary is the
initial population size (take N0=0.5 and N1=0.501). Add a value equation
for D and let D=the difference in population size between the populations.
Look at D over time. Look at the natural log of D over time.
- Repeat problem 1, but this time use the logistic map (K=1). Do
this for values of r=2.0, 3.2, 3.6, and 3.8.
- Calculate the inverse of the logistic map. (Take the positive
inverse). You can do this by solving for Nt as a function of Nt+1. Then
take the inverse and enter it in. Using a chaotic logistic equation try
two different starting N0's (in the chaotic regime) and project the population.
What happens? Do you see this senstivie dependence on initial conditions here?
- Take the logistic map and add in a constant predation factor.
Find the equilibrium point for this new equation using Maple (use solve({eq1,
eq2},{var1,var2}) to find this value in terms of r, and P). Prove to yourself
that you have the right value. What is the slope of the curve at this point?
(Hint: the slope is the same as the equation w/o predation, just evaluated
at a different point).
- Under what conditions will the above expression yield an attractor?
When will it yield a repellor?
- Look at the logistic map (Nt+1 vs. Nt) at some different r values with
chaotic dynamics. See if you can find the boundaries of the attractor.
- Do a bifurcation plot of the logistic map. Find and classify the major
bifurcation locations.
- Do a bifurcation plot of the logistic map with constant predation
(bifurcate with P). Find and classify the major bifurcation locations.
- In a region with chaotic dynamics find all the attractors as well as
there respective basins of attraction.
- Did you find a basin boundary collision bifurcation in the eighth
problem? Look at a graph of Nt+1 vs Nt around this point (as you get close
to it and then on the other side). Can you see the basins collide?
