Biology Reference
In-Depth Information
of
n
i
(
k
. To do this, we must consider the ways an individual of
stage
s
j
at time
t
0
can produce an individual in stage
s
i
at time
t
0
+
t
0
+
1
),
i
=
1
,
2
,...,
1. They are:
1.
Movement between stages. An individual moves from stage
s
j
to stage
s
i
.If
the probability for this transition in one unit of time is
p
ji
, then the expected
number of members produced in stage
s
i
by a single member in stage
s
j
in one
unit of time is
p
ji
. Note that
i
and
j
may be the same and
p
jj
is the probability
that an individual in stage
s
j
stays in stage
s
j
after one time interval. In many
models death is not considered to be a stage, and if this is the case, then the
probabilities across all stages
s
i
will not necessarily sum to 1. The probabilities
p
ji
are called
transition probabilities
.
2.
Reproduction. The individual in stage
s
j
produces new offspring, and the aver-
age number of offspring generated in one unit of time by one individual of stage
s
j
that begin life in stage
s
i
is denoted
f
ji
.
Therefore, if we know the following, we can determine the total number of mem-
bers expected in stage
s
i
at time
t
0
+
1:
1.
the number of individuals
s
j
in each stage
j
k
at time
t
0
,
2.
the probability
p
ji
that an individual in each of those stages will become an
individual in stage
s
i
,
=
1
,
2
,...,
k
, after one time interval,
3.
and the average number of offspring in stage
s
i
generated by a single individual
in stage
s
j
, denoted
f
ji
.
Using the terms we defined above, we can express our ideas more precisely. Each
member that is in stage
s
j
at time
t
0
will produce an average of
p
ji
+
i
=
1
,
2
,...
f
ji
members in
stage
s
i
at time
t
0
+
1 . Since there are
n
j
(
t
0
)
members in stage
s
j
at time
t
0
, the stage
s
j
will be expected to result in
n
j
(
t
0
)
[
p
ji
+
f
ji
]
members in stage
s
i
at time
t
0
+
1.
To get
n
i
(
t
0
+
1
)
, that is the total number of members expected in stage
s
i
at time
t
0
+
1, we sum over all stages
s
j
to get
n
i
(
t
0
+
)
=
n
j
(
t
0
)
[
p
ji
+
f
ji
]
.
1
(7.1)
j
This formula will tell us how many individuals we expect in a single life stage
after one time interval has passed. To keep track of the number of individuals in all
life stages simultaneously, we turn to matrix algebra.
7.5
CONSTRUCTING A PROJECTION MATRIX
The framework provided by linear algebra allows us to build a population growth
model that includes all the life stages we have defined and then use that model to
gather information about the possible fate of the population. We can build a matrix that
encompasses transitions among life stages, reproduction, and mortality for a specified
time interval. We can also use the tools of linear algebra to describe properties of the
projectionmatrix that are analogous to the biological characteristics of the population.
Search WWH ::
Custom Search