- 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.
- Period-doubling everywhere:
In the Logistic MAP, set the parameter a = 3.83. Fix Start time= 1500, Stop Time=2100
and initial condition 0.208. Draw the Stair-Step diagram to see the period-3 solution.
Check the numbers in Xi-Values view to make sure that the numbers repeat every fourth iteration.
Now, change the value of the parameter a carefully to make this solution to bifurcate to a
period-6 solution. Change a a bit more to obtain a period-12 solution.
Submit your parameter values, Xi values, and the Stair step diagrams.
- Sensitive dependence on initial conditions:
In the Logistic model, first set a = 2.65. Now take two initial population
sizes and determine the population densities after 80 generations.
Suppose that you make 0.0003 percent error in the determination of
the initial population sizes. Determine the relative percentage error
in the population densities after 80 generations.
Repeat the same experiment with the growth rate a = 3.91.
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 following link for further info on this model.
and also the link above to the original paper of Ricker.
Construct the bifurcation diagram of the Ricker MAP by fixing k = 1, and varying r.
(see Figure 5 on the link above.)
- 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?