Geoscience Reference
In-Depth Information
mortality. Consequently, the time between the two sample surveys should be
short and during a season when animals are not migrating.
To understand the mathematical basis of the method, suppose that the two
types of individuals are labeled
X
and
Y
. For two sample survey times,
i
= 1
and 2, let
x
i
= the number of type
X
individuals in the population at time
i
,
y
i
= the number of type
Y
individuals in the population at time
i
,
N
i
=
x
i
+
y
i
, the total population size at time
i
,
p
i
=
x
i
/
N
i
, the fraction of the population that is type
X
at time
i
,
r
x
=
x
1
−
x
2
, the number of type
X
individuals removed between times
1 and 2,
r
y
=
y
1
−
y
2
, the number of type
Y
individuals removed between times
1 and 2, and
r
=
r
x
+
r
y
=
N
1
−
N
2
is the total number of individuals removed (which
follows from the previous definitions).
This means that the proportion of type
X
after the removal is
p
2
= (
x
1
−
r
x
)/
(
N
1
−
r
) = (
p
1
N
1
−
r
x
)/(
N
1
−
r
). Solving for
N
1
gives
N
1
= (
r
x
−
rp
2
)/(
p
1
−
p
2
).
Hence, we can obtain estimates of
p
1
and
p
2
from a sample survey of the
population before and after removals of
r
x
and
r
y
animals. If
ˆ
1
and
ˆ
2
are
estimates of
p
1
and
p
2
, respectively, then
N
1
,
x
1
, and
N
2
can be estimated by
ˆ
ˆ
)/(
ˆ
ˆ
)
Nr
=−
(
rp pp
−
2
,
(5.7)
1
x
2
1
ˆ
ˆ
ˆ
xpN
1
,
(5.8)
1
1
and
ˆ
ˆ
NNr
=−
,
(5.9)
2
1
respectively, assuming that
r
x
and
r
y
are known. From these, other param-
eters can be estimated. For example, the number of type
Y
animals at time 1
can be estimated by subtracting Equation (5.8) from Equation (5.7).
This formulation of the situation allows for one or both of the two types
of individuals to be removed. It also permits negative removals, so that new
individuals of one or both types can be added. Finally, the exploitation rate,
u
=
r
/
N
1
, of harvested animals can be estimated, which is important for man-
agement of a population.
