- Application: Population dynamics, LOGISTIC MAP x1 -> a*x1*(1.0 - x1)
- LOGISTIC MAP x1 -> a*x1*(1.0 - x1)
Review article from NATURE.
Ricker's original paper
- Bifurcation diagrams.
- Time series (X vs Time) plots; how to recognize stable and oscillating population
- Discrete Predator-prey model.
A cubic map from genetics with multiple attractors:
The cubic map
xn+1 = (1 - a) xn + a xn3
is used in population genetics involving one locus with two alleles.
Set a = 3.2. Trying various initial conditions, find two distinct
period-2 orbits. Provide the Xi-values and the stair-step diagrams of
the two orbits.
Note: Recall that in the logistic map, (almost) all initial
conditions went the same attracting fixed point or periodic orbit.
This is not always the case with all models, as this map demonstrates.
Optional: construct the bifurcation diagram of this map using
several initial conditions.
(Hint: Consider x ranging from -1 to 1, and a ranging from 0 to 4.)
Reference: for more information on this map, see:
T.D. Rogers and D.C. Whitley (1983), "Chaos in the cubic mapping,"
Math. Modelling, 4, 9-25.
- Sensitive dependence on initial conditions:
In the Logistic model, first set a = 2.36. Now take two initial population
sizes and determine the population densities after 70 generations.
Suppose that you make 0.0002 percent error in the determination of
the initial population sizes. Determine the relative percentage error
in the population densities after 70 generations.
Repeat the same experiment with the growth rate a = 3.941.
What are the biological implications of these experiments?
- Verifying calculations in the scientific literature:
Read as much of the Nature article linked above as you can.
This could be difficult reading for you, but do not be discouraged.
In particular, examine Table 3 on page 464 regarding the periodic orbits of the Logistic map.
Try to verify the first three numbers on the 4th row:
period 5(a) orbit appears for a = 3.7382 and disappears at a = 3.7411.
Is the claim in the paper true? Support your answer by drawing stair-step diagrams,
Xi-values, for at least three parameter values: two near the boundaries, and one inside.
Notice that there other parameter windows for which there are period 5 orbits.
If you like, you can explore the rows 5(b) and 5(c).
xn+1 = xn * exp(r*(1- xn/k))
This is a biologically important fisheries model containing
two parameters r (growth rate) and k (carrying capacity).
We assume that both parameters take on non-negative values.
See the link above to the original paper of Ricker.
Construct the bifurcation diagram of the Ricker MAP by fixing k = 1, and varying r.
- By examing your bifurcation diagram,
locate a value of r for which the map has a periodic
orbit of period 3. Draw a stair-step diagram of your
period-3 orbit. What are the values of the population size on this orbit?