Ecological
Succession
(Adapted
from Mooney and Swift, A Course In
Mathematical Modeling.)
Background
In
this project we consider modeling the ecological succession of barren dunes to
an oak community using matrices. In
this simple model, it is assumed that there are three states for a specific
ecological region. These states are
bare dunes, which will be labeled BD;
a grass community, which is labeled G;
and an oak community, labeled O.
Data
The
table below gives the transition
probability between each pair of states.
For example, the number 0.2 in the column labeled G and the row labeled BD
means that after each time period, 20% of the grass communities become bare
dunes. Each time period is 50 years,
each ecological region is 500 acres.
|
|
BD |
G |
O |
|
BD |
0.3 |
0.2 |
0.1 |
|
G |
0.6 |
0.5 |
0.1 |
|
O |
0.1 |
0.3 |
0.8 |
Problems
1.
Let
bd(t), g(t), and o(t) be the number of regions which are bare dunes, grass and oak,
respectively. Write down equations for bd(t), g(t), and o(t) in terms of bd(t-1),
g(t-1), and o(t-1).
2.
Convert
the system of linear equations in part 1 into a matrix equation. The matrix is the transition matrix.
3.
There
are 100 regions. Suppose that initially
10% are bare dunes, 10% are grass communities, and 80% are oak
communities. Determine the distribution
of states after 2 time steps, 10 time steps and 50 time steps. Interpret the meaning of these results.
4.
Now
start with some other initial distribution of your choice. Again determine the distributions after 2,
10 and 50 time steps.
5.
Compare
the results in parts 3 and 4. What are
your conclusions?
6.
Change
the table of transition probabilities and repeat parts 1-5. Make sure that (a) the entries in each
column of the table add up to 1 (since all the regions have to become something) and (b) all the entries are
positive (so there is always some chance that a region could move into either
of the other two states). What are your
conclusions? Matrices which satisfy
conditions (a) and (b) are called regular
stochastic matrices, and are extremely important in modeling many kinds of
systems.