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Let's think about a hypothetical human population where we're going to track the population
over a series of twenty-year time intervals. So there will be a group of people that are
between zero and 20 years old, a group between 20 and 40, the group from 40 to 60, a group
from 60 to 80, and while there will certain be people older than 80 we're not going to
track people once they are over 80 years old. Since we're tracking for 20 years, it means
that everyone in this group to start if they're still living will be in this group after the
20th time period, and similarly everybody in this group and still living in this group
and so forth. Let's make up some hypothetical data to work with, let's pretend that let's
say there are 100 people in this group to start with, and so let's pretend that when
we tracked them over 20 years, that 98 these 100 people survive. So that means that at
the end of these 20 years there will be 98 of those people who are now in this group,
and then suppose that among those 100 people that we started out with, let's say that there
have been 24 births of new new babies so that would give us 24 new people in this group.
So what I'm showing here with the red numbers is what we have after 20 years based on the
100 that we started out with that were in the 0 to 20 range. Those 100 starters are
responsible for there being 98 survivors in this group after 20 years plus 24 new births
that happen during that time. We now need to translate this discussion into some information
for our population dynamics matrix. We started out with 100 people and we know that after
the first 20 years 98% of them survive and are now in group 2. What to put it another
way, for each one person we start with a group one, we have 98/100 of a person in group 2
after our first 20 year period. But in addition to that we also have 24 new births, so that's
24 new people in group one produced by those 100 that were already there. So for that reason
we need to also enter a .24 right here. As a result of each person that was a group one
to start with, at the end of 20 years we have 24/100 of a person in group 1, that's the
new births, plus the 98/100 which is survivors, but nothing that happens to that group of
people has any effects on groups 3 or 4. Now let's make up some data for the group 2 folks.
Let's say that for each 100 people we start out with in group 2, let's say that 92% of
them survive, so that means after 20 years those 100 people will be 92 people in group
3, and also these folks in group 2 are in heavy child bearing years. So let's suppose
that those 100 people that started out in group 2 during the next 20 year period produce
77 new offspring. So that would be 77 folks, babies that appear in group 1 as a result
of that. To rephrase that, for each 100 people that start inn group 2, at the end of 20 years
92 of them survived and are in group 3, and those 100 people produce 77 new births which
of course occur in group 1. So in our matrix, that says that each person in group 2 produces
92/100 of a person in group 3, starting in group 2, we produce 92/100 of a person in
group 3, but we also produce 77/100 of a person in grip 1. Nobody remains in group 2. And
of course we don't get anybody in group 4. Now let's think about what happens to the
folks in group 3. Let's imagine there being 100 people in group 3 to start with. What
happens to them? Well, let's suppose that 57 of them survive the 20 year period, so
that means that of the 100 that are in the 40-60 age group to start with, the survivors
will be in the 60-80 group after 20 years. We assume there are 57 of them. And since
people in their low 40s are still in child bearing age at the start of that period there
will be a few additional births, so let's say that there were perhaps 4 births that
take place from this group, and that would be 4 folks that pop up back in the first group.
So that gives us additional data for the populations dynamics matrix. Each 1 person that starts
out in group 3 produces 57/100 of a person in group 4, so we put a .57 here starting
in group 3. And also produces 4 new births which take place in group 1, so we have a
.04 back over here. Nobody being produced in groups 2 or 3. And then to finish up. In
the 60-80 age group, anybody that in that to start with is past age 80 by the end of
the 20 year period and doesn't produce anybody new that appears in the picture. So this bottom
row will be all zeros. A person in group 4 doesn't contribute anybody 20 years in the
future that fits in this picture. So we no have our population dynamics matrix. Let's
call it T. And let's suppose we have some initial population distribution that tells
us what the breakdown of the population is to start with. For simplicity we'll suppose
that we start out with a population made up of 1000 people in the 0-20 group, 1000 in
the 20-40 group, same in the 40-60 and 60-80. So if we want to know what the population
distribution looks like after one 20 year period has passed, what do we do? We just
multiply P times T. What would you do if you want to know what the population distribution
would be after 200 years? 200 years would be ten 20-year periods, so you would do P
times T raised to the 10th power.