Homework D4: Multi-Variable Dynamical Systems

#1: Consider a population of squirrels, broken into age groups of 0-year-olds
(i.e. between birth and .9999 years old), 1-year-olds, 2-year-olds, and
3-year-olds.  0-year-olds don't have babies, 1-year-olds have an average
of 1 baby each, 2-year-olds have an average of 0.8 babies each, and 3-year-olds
don't have babies.  The chance of surviving from 0 to 1 is 0.7, from 1 to 2
is 0.9, from 2 to 3 is 0.6, and 3-year-olds never survive to 4 years old.
The initial population is 1000, spread equally over each age group.

Using your favorite software,
track the population trajectory over an appropriate number of years.
  (choosing a good # of years is part of the exercise!)
Graph the total population each year.
Also, graph the proportion of total population for each age group as
time goes on (e.g.  we start at 25% of the population being 0-year-olds,
25% 1-year-olds, etc.)--does that reach a steady state?

Compute the % growth rate for the whole population.

Now, suppose a minor famine hits that lowers the birth rates to .5 for
1-year-olds and .4 for 2-year-olds, but it doesn't change the survival
probabilities.  Again, show the population trajectories and the
percent for each age category.

Compute the % growth rate for the whole population in the famine case
as well.

Optional: if you know what eigenvalues are, compute the largest eigenvalue
for the matrix in both the growth and famine cases.  Compare to the % growth rates.

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#2: Program the wolf-vs-moose model :
        delta m = +0.07 * m + -0.0028 * m * w
        delta w = -0.10 * w + +0.0001 * m * w
a) Starting with 1001 moose and 25 wolves, generate about 300 years
of data and produce a phase-plane plot
(best to use dots with connecting line segments).

b) Produce a trajectory plot (time on the x-axis, # of each type of animal
on the y-axis).  Try to get both animals nicely visible on the same plot.
One way to do this is to make another column called "wolves*40" that multiplies
the actual # of wolves by 40, to make it comparable to the # of moose.
Another way is to right-click on the Wolves data series on the plot, do
Format Data Series->Series Options->Secondary Axis.


c) Reproduce parts (a) and (b) but starting with 563 moose and 20 wolves.
These were the populations in 1959, according to
http://emunix.emich.edu/~aross15/coursepack3419/wolf-moose/Wolf_Moose_numbers.xls
(originally from  http://www.isleroyalewolf.org/data/data/womoabund.html but that seems broken now)

d) Produce a phase-plane plot of the actual populations using that
data file from the web.  Comment on how well it matches the phase-plane
portraits that our model produced.

Phase-plane: x=#moose, y=#wolves
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