Geoscience Reference
In-Depth Information
the
j
th sample of which
r
j
are recaptured later. Hence, it is expected that
z
j
/(
M
j
-
m
j
) ≈
r
j
/
R
j
, so that
M
j
≈
m
j
+
R
j
z
j
/
r
j
, suggesting the estimator
M
ˆ
j
=
m
j
+
R
j
z
j
/
r
j
(8.1)
for
M
j
, which can be evaluated for
j
= 2, 3, . . . ,
k
- 1 because for these values of
j
all the values on the right-hand side can be obtained from the data.
Once
M
j
is estimated, the population size
N
j
at time
j
can be estimated
because the proportion of marked animals in the
j
th sample should be
approximately equal to the proportion in the population, that is,
m
j
/
n
j
≈
M
j
/
N
j
. Thus, an estimator of
N
j
is
ˆ
ˆ
NnMm
/
j
,
(8.2)
j
j
j
which can be evaluated for
j
= 2, 3, . . . , k - 1.
To estimate ϕ
j
, the survival rate over the period
j
to
j
+ 1, it can be noted that
there are
M
j
+
R
j
-
m
j
marked animals in the population just after the releases
from the
i
ith sample, and that
M
j
+1
of these are still alive at time
j
+ 1. Thus,
ϕ
j
≈
M
j
+1
/(
M
j
+
R
j
-
m
j
), suggesting the survival estimator
·
φ=
ˆ
ˆ
MMRm
/(
+−
)
.
(8.3)
j
j
+
1
j
j
j
M
ˆ
1
because there are
no marked animals in the population just before the time of the first sample.
F i n a l ly,
B
j
, the number of new entries to the population between samples
j
and
j
+ 1 that are still alive at time
j
+ 1, can be estimated by setting it equal
to the estimated difference between the population size at time
j
+ 1 and the
expected number of survivors from time
j
, that is,
This can be evaluated for
j
= 1, 2, . . . , k - 2, taking
=
·
ˆ
ˆ
ˆ
BN NRm
=
φ+−
(
)
,
(8.4)
j
j
+
1
j
j
j
j
which can be evaluated for
j
= 2, 3, . . . , k - 2.
One important feature of Equations (8.1) to (8.4) is that it is not necessary
for
R
j
, the number of animals released from the
j
th sample, to equal the sam-
ple size
n
j
. This means that damaged animals do not need to be released,
and extra animals can be added to the population if desired. Of course, if the
number released does not equal the number captured, then this will change
population sizes from what they would otherwise have been.
The JS estimators are biased for small samples, and infinite estimates are
possible because of divisions by zero. To overcome this problem, approxi-
mately unbiased estimators can be obtained by replacing
M
ˆ
j
and
N
ˆ
j
by
ˆ
*
MmRz
=+ +
( )/( )
r
+
,
(8.5)
j
j
j
j
j
