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Copy this sheet. The copy will be named Reindeer (2). Solve the
equation Nt
$ear N observed Calculate !ambda usi
** ** ** **
*+ +, +-.*- ./2*
*- ,+ ,2./*22+ 2 .2+2*,
*, , 2.,-2 / .222*
* 2- .* * .222,* -* +2.+ +
+ 2* +.+++ -
+ 2, 2+.,*- ,
+2 /-2 /2.+2+
+/ *, *2.2-
+* -/ +2.-2-
++ ,, -,,.2,
+- /, -./,, 2
+, /+ /+ *.*** /
+ ,+, *2.- *+ 22- /2./* +
- 2,+ 2/*.*+- -
- /, /2.+- ,
-2 +/, /-2./
-/ - -+++ *+/.
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= N!ambda"t #or !ambda
n% those three data points. &ow many estimates o# !ambda
will they yield'
*+ + ++ -
+
+
2
2+
/
/+
+,2*
,,
2,+
+-.*-*+.+++,-/--
-,,.2,/*-2
2/*.*+++2
N observed !ambda is ./2* !ambda is .222*
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Copy this sheet. The copy will be named Reindeer (2). Solve the
equation Nt
$ear N observed Calculate !ambda usi
** **
*+
*-
*,
**
+
+
+2
+/
+*
++
+-
+, /+
++
-
-
-2
-/ -
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= N!ambda"t #or !ambda
n% those three data points. &ow many estimates o# !ambda
will they yield'
* *+ + ++ - -+
2
/
*
+
-
,
N observed
N ob
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erved
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0lot N observed vs $ear. Copy wor1sheet and ma1e sure the copy
is named 0eople(2
$ear N observed N hat !ambda N hat rma N hat rNow
*
2, 2 2 2 2
- / / /.*+// /.++,2,
,* * * +.2+/, /.-+,,2+*
, + + -.--2, *./22**2-2* - - .,222 +.*2,-2
2 , , .2/+/* +.//--
2/
+-
,
/
+
2,
3*
3*
(
,+
2
*
-
2
2 /
.
.2
./
.*
.+
.-
.,
.
.
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dN4dt di5erence in years !ambda
./ 2/ .+-+2+/ .+-*
.//// // .2/-2-*+ .22
.,*2+, * .2,-2 .2++
.,-2/,, / .,//+ .,-./////// 2 .+/*,+ .+
./////// 2 .22,+2 .2+
Nt =
!ambda"t =t lo% la =
lo% lam =
lambda =
+ + 2 2+
* + - , 2 2 22
dN4dt
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N6!ambda"t
Nt4Nlo%(nt4n)
4t 6 lo% (nt4n)
"74t6 lo% (Nt4N)
8press this #ormula
with 8cel #unctions
-
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0lot N observed vs $ear. Copy wor1sheet and ma1e sure the copy
is named 0eople(2
$ear N observed N hat !ambda N hat rma N hat rNow
*
2, 2 2 2 2
- / / /.*+// /.++,2,
,* * * +.2+/, /.-+,,2+*
, + + -.--2, *./22**2-2* - - .,222 +.*2,-2
2 , , .2/+/* +.//--
2/
+-
,
/
+
2,
3*
3*
(
,+
2
*
-
2
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dN4dt di5erence in years !ambda
2/ .+-+2+/ .+-*
// .2/-2-*+ .22
* .2,-2 .2++
/ .,//+ .,-2 .+/*,+ .+
2 .22,+2 .2+
Nt =
!ambda"t =t lo% la =
lo% lam =
lambda =
+ + 2 2+
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N6!ambda"t
Nt4Nlo%(nt4n)
4t 6 lo% (nt4n)
"74t6 lo% (Nt4N)
8press this #ormula
with 8cel #unctions
-
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t N N b
slope b .* slope d
.*
2 $3int b 2 .* $3int d
/ .* / .* ./
* * .*
+ + .*- - .*
, , .*
.*
.*
.*
2
/
*
+
-,
2
2 * - 2
2
*
-
2
dN/dt
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N d r = b3d dN4dt rN(3N49)
./ . :;
./ . :;
2 ./ . 2 :;
/ ./ . / :;
* ./ . * :;
+ ./ . + :;- ./ . - :;
, ./ . , :;
./ . :;
./ . :;
./ . :;
N
r .
K
t N
.+,
2 2.2*/
/ /.*+
* *.2+
+ -.*,2
- .22
, 2./,+/
22.2++*
2*.+-/
2,.22
/.*--
2 //.2,
/ /-.-2,
* *.++2
+ **.-
- *.+//2
, +*.,/*,
-.*-*,
--.+*
2 ,/.+-
2 .--,22 .2+/
2/ .,*2
2* .2/
2+ 2.2*
2- /*.-/,*
2, *.,,/
2 -*.**-+
2 .,*+
2 * -
.
.2
./
.*
.+
dN4dt
+ + 2
2
*
-
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/ 2.++*
/ 22.,+
/2 2*+./2+/
// 2,.2-*
/* 2.-*
/+ //.+*+
/- /-+.2//, **.*,/
/ **,.
/ **.2*+
* +*+.+
* -/.*2
*2 ---.-//
*/ ,/-.,
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2
d
b
2+ / /+ * *+
N
N
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t N N b
slope b .2 slope d
3. . .
2 $3int b 2 . $3int d
/ .2 / ., .2
* * .-
+ + .+- - .*
, , ./
.2
.
.
2
/
*
+-
,
2
2 * - 2
3/
32
3
2
/
*
+
dN/dt
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N d r = b3d dN4dt
.2 .
./ .- .-
2 .* .* 2.
/ .+ .2 /.-
* .- . *
+ ., . *- . .- /.-
, . .* 2.
. .2 .-
.
.2 3.2 32
This is not a Time Series
@ynamics not constrained by previous states.
2 *
.+
.
.+
.2
.2+
dN4dt
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- 2
d
b
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t N N b
slope b .* slope d
.* ./
2 $3int b 2 .* $3int d
/ .* / .* .2
* * .*
+ + .*- - .*
, , .*
.*
.*
.*
2
/
*
+
-,
2
2 * - 2
2
*
-
2
*
dN/dt
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N d r = b3d dN4dt rN(3N49)
.2 ./
.+ ./+ /.+ /.+
2 . ./2 -.* -.*
/ . .2 ., .,
* .* .2- .* .*
+ ., .2/ .+ .+- .2 .2 2 2
, .2/ ., . .
.2- .* .2 .2
.2 . . .
./2 .
N
r ./
K 126.667
t N
*.,-
2 .---
/ 2-.,,2+
* /+.--*2
+ *-.*+-
- +,.,+22
, -.,*,-
.2-,
.-,+
.*+
,.*2
2 2.,
/ -./
* .2-,
+ 2.,
- 2/./,/2
, 2*./+,
2+.*
2+.+**
2 2+./*
2 2-.-222 2-./2,
2/ 2-.*/-
2* 2-.++
2+ 2-.++-
2- 2-.+
2, 2-.-+
2 2-.-//
2 2-.-*2+
2 *
.
.2
./
.*
.+
dN4dt
+ + 2
+
+
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/ 2-.-+
/ 2-.-++*
/2 2-.-+
// 2-.--*
/* 2-.--/
/+ 2-.--*2
/- 2-.--+/, 2-.--++
/ 2-.--+
/ 2-.---
* 2-.---/
* 2-.---*
*2 2-.---+
*/ 2-.---+
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- 2
d
b
2+ / /+ * *+
N
N
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t N N b
slope b . slope d
. . ./
2 $3int b 2 .2 $3int d
/ . / ./ .2
* * .*
+ + .+- - .-
, , .,
.
.
.2
2
/
*
+
-,
2
2 * - 2
3*
32
3
3
3-
3*
32
2
dN/dt
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N d r = b3d dN4dt rN(3N49)
.2 .
.+ .- .- .-
2 . .* . .
/ . .2 .- .-
* .*
+ ., 3.2 3 3- .2 3.* 32.* 32.*
, .2/ 3.- 3*.2 3*.2
.2- 3. 3-.* 3-.*
.2 3. 3 3
./2 3.2 32 32
N
r .
K 40
t N
.-
2 .2*,2/
/ .+,
* 2.+*,-
+ /.2*
- *./
, *.,**2
+.*2-
-.2+*,
,./+,
,.22+*
2 .--
/ .***
* 2.2/,
+ 2./
- 2.2
, 22.+
2/./
2*.+2-
2 2*.*2
2 2+.-++*,22 2-./2
2/ 2,.,
2* 2,.,2/
2+ 2.**/
2- 2.+/*
2, 2.,2/
2 /./,/,
2 /.2/
2 *
.+
.
.+
.2
.2+
./
./+
dN4dt
+ + 2
2
/
*
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/ /.**2,*
/ /.-++
/2 /2.*-,
// /2.*--+
/* //./*-
/+ //.2-*
/- /*.2/*,//, /*.-/*
/ /*.
/ /+./2
* /+.-*+2
* /+.*2-/
*2 /-.22+/
*/ /-.*
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- 2
d
b
2+ / /+ * *+
N
N
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t N N b
slope b .* slope d
3. ./ ./
2 $3int b 2 ./ $3int d
/ .* / ./, .2
* * ./-
+ + ./+- - ./*
, , .//
./2
./
./
2
/
*
+
-,
2 2 * - 2
3+
3
3+
+
+
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N d r = b3d dN4dt yield di5erence
.2 ./ 3
.+ ./* /.* 3+.-
2 . ./ - 3/
/ . .2- ,. 3.2
* .* .22 . 3.2
+ ., . - .2 .* .* 3.-
, .2/ . , 32
.2- .- *. 3*.2
.2 .2 . 3,.2
./2 3.2 32 3
N
r ./
K 95
t N
/.**/
2 .*+
/ 2+.+*2
* //.22,
+ *.2+-+
- +.2
, +.+,,+,
-,.+/
,*.//--
,.2,,
*.-2
2 ,.2*
/ .++/-
* .2-+
+ 2./,2+
- /.,++
, /.,+/
*.*/-
*.*2,2
2 *.+,-
2 *.,2***22 *./
2/ *.,/
2* *.,
2+ *./-
2- *.+-
2, *.,,*
2 *.-,
2 *.-,
.
.2
./
.*
.+
dN4dt
yield
di5erence
+ + 2
2
*
-
-
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/ *.-
/ *./2
/2 *.+,,
// *.,
/* *.2
/+ *.-+
/- *./, *./,
/ *.+,
/ *.,
* *.
* *.-
*2 *.
*/ *.*
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2 * - 2
d
b
2+ / /+ * *+
N
N
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s l m lm
.+ .+ 2
2 .+ .2+ 2 .+
/ .+ .2+ 2 .2+
* .+ .-2+ 2 .2+
+ .,+
