A Model for Squirrels
David Witte
Nate Dixon
Duncan Baumgart
M 447 – Mathematical Modeling
December 1st, 2006
Executive Summary
For our independent project, we wanted to model something that was interesting
and available on campus. Our plan was to monitor squirrel movement and population in
the Arboretum during what we researched as the most populated hours of the day for a
period of one week. Our background research on squirrels told us that during the autumn
season at times around dusk is when someone would find squirrels out gathering nuts for
the winter and burying them in the ground. We also discovered from our squirrel
research that they are omnivores, and therefore eat insects, frogs, and small birds. From
this we concluded that the Arboretum was an optimal setting.
To make our model easier to comprehend, we took an aerial view of the
Arboretum from the internet and divided it up into fourteen different regions (states) by
using the multitude of sidewalks as borders for these regions. Our goal was to find the
“best” region to place a squirrel feeder so the squirrels in the Arboretum could survive
during the months they don’t have much food, specifically early springtime. By “best”
we mean the point in a specific region that has the greatest probability of being visited by
a squirrel in the long run. From our initial data and approach of monitoring squirrels, we
ran into difficulties of formulating a Markov Chain, so we had to come up with a more
suitable approach that would lead us to a solution.
Approach and Assumptions
Our goal was to find a good place for a squirrel feeder in the Arboretum. We first
had to do a little research on squirrels to find out their eating habits. The Arboretum has
areas that are full of trees. Also, even though it is populated with students trying to get to
class, the squirrels in Bloomington are not afraid of the Arboretum. This allows us to get
close enough to the squirrels to observe them firsthand.
Our first approach was to monitor squirrel activity over a week period and use
that data to assemble a matrix and figure out the expected number of visits to a particular
place. Their feeding habits were for the fall and they would gather food for winter and
our observations were going to take place during these times of feeding. We also had to
do our observations within a two week period which makes an approximation for the
whole year not a very strong argument.
We had trouble figuring out how to come up with a matrix that would monitor the
movement of the squirrels. When we finally did come up with a matrix, we found it had
no inverse. We tried fooling with the matrix using a formula we constructed:
1
n +1
where “n” equaled the number of un-traveled adjacent areas: given it began in area 'a.' to
get an inverse. We got the inverse, but we realized this was a completely arbitrary
formula and we had to scrap the entire idea of monitoring the squirrel’s activity and just
dealt with the area itself.
Our next approach had to do with taking the adjacent sides to the surrounding
areas in the Arboretum and using those as probabilities to traveled areas. An assumption
we had was in constructing our perimeters and the areas of the individual regions.
The arrows show the line segment we used to calculate linear distances according
to the walk ways throughout the Arboretum. Constructing the areas was not an exact
measurement but more of a very close approximation. This also did not take into account
the number of squirrels in each place and how they traveled, but did deal with the most
accessible places and was an assumption that did not depend on weather and was good all
year round. We came up with a place that would have the easiest accessibility for the
squirrels. For this, we then had to take the fourteen areas and construct a matrix.
The problem here was just a size and time problem. We had a 14 X 14 matrix,
with probabilities we had to calculate, using areas of individual regions and the lengths of
the adjacent sides. The assumptions we had were the probability of staying in a certain
area is given by:
Area of region
Area of region + length of adj acent sides
And the probability of leaving a certain area is given by:
Length of adjacent side
Area of region + length of adj acent sides
After we did this, we had a matrix of probabilities which we could find the inverse by
calling each state an absorbing state, and figuring out what the expected number of visits
to a certain area would be as well as the probability that any state would reach any other
absorbing state. We ran into another problem with this. When we made one of the areas
an absorbing state, the probability matrix “A” had only entries of probability one. This
makes sense because our transition matrix “P” is ergodic, meaning that all states are
accessible to each other, whether it be 1,2,3,4…n steps. Therefore, by making one state
absorbing, we know that all other states will eventually reach that absorbing state, giving
probabilities of 1 for all states in the matrix “A.” We then had to change our approach
yet again and look back at stability vectors.
We are still dealing with our original (14x14) matrix, and to get the stable vectors
we just use the equation:
W*P=W
Where W is the stable vector and P is our transition matrix. We simply raised P to the
100th power to find the stability vector. After doing this, we came up with a fourteen
column stable vector:
W = [.1068209, 0.0896991, 0.029389, 0.055458, .117006, .172481, 0.028156, 0.093085, 0.034956, 0.07764, 0.080485, 0.032695, 0.049877, 0.032252]
The stable vector shows us that the probability of going to region six on the (100th, 101st,
102nd, …) transition exceeds all other regions. From this we concluded that the feeder
goes into region six. To figure out where in region six, we had to call region six an
absorbing state and find our matrix “N” to get the expected number of transitions before
it reaches six. We add up all the rows of “N,” and we assumed that the three rows with
the lowest expected value will determine the place of the feeder in region 6. After we
determined the three regions (10,11,12), we found the midpoints of the adjacent sides to
six, and using the GSP program, we found the a triangle connecting all of the midpoints
of the sides that were adjacent to region 6, and drew median lines through the triangle to
find its centroid. It makes sense to put the feeder at he centroid, because it is the center
of mass. We decided to place the feeder here at the black circle:
Strengths and Weaknesses
The main weakness we had was coming up with a suitable approach. It was
becoming too cold to monitor squirrel activity. During the three to six days we did
monitor, we found a total of one squirrel that stayed in the same area: a hole in the tree.
This was mainly because of the fact it was cold. If we had done these observations in the
beginning of the fall around October, we would have had much more data to work with.
Another weakness was that our assumptions were not based off of our research on
the squirrels. They have eating patterns which we couldn’t monitor and this was also
because of the problem of time. We only had about a week or so to monitor the squirrels.
One can’t make a true data table based off of one week out of a year. This was too
generalized and so we had to go with the area and adjacent sides approach.
A strong part in our approach was using the GSP program to come up with areas
of the regions and the lengths of the sides. Without this, we would have had to find a “to
scale map” and do some tedious calculations. Instead, our model picture is scaled down,
and the areas and lengths of the regions are really close.
Another strength was that the final approach didn’t deal with unpredictable
behavior. Constructing the probabilities using the formulas mentioned above and treating
an adjacent state’s probability as a factor of the size of the accessible region was a major
part of our argument. The construction of the matrix made more sense and also gave us
an inverse, which allowed us to find the expected value.
Conclusion
Our group spent approximately 35 hours on this project due to inputting Maple
code, working with Geometer’s Sketchpad, or finding difficulties in our approach which
led to brainstorming sessions. We tried to find the most suitable approach for this model
that would be applicable to the real world. As we mentioned earlier, our most significant
drawback to our original approach was the squirrel population, or lack thereof. In the
end, an applicable solution was right in front of us, and we were doing a lot more work
than needed. Regardless, modeling is about making assumptions, applying mathematical
data to a real-world situation, and finding shortcomings in approaches, so our group is
pleased of the hard work that was put into this model. We also feel strongly about our
model because most people would have just put the feeder in the center of the Arboretum
without any mathematical data. From this independent project we have gained a better
understanding of how to implement Maple code, the advantages of Geometer’s
Sketchpad, techniques to answer questions using the transition and fundamental matrices,
and that squirrels don’t like us. Given a slightly warmer climate, we believe that there
would be more squirrels to gather data on, therefore enabling us to use our initial
approach.