1 1.1 p(x n+1 x n, x n 1, x n 2, ) = p(x n+1 x n ) (x n ) (x n+1 ) * (I Q) 1 ( 1 Q 1 Q n 0(n ) I + Q + Q 2 + = (I Q) ] q q +/. * q

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1 Masato Shimura Example Script Script Example Sheep Example(S heep) Script Reference 17

2 1 1.1 p(x n+1 x n, x n 1, x n 2, ) = p(x n+1 x n ) (x n ) (x n+1 ) * (I Q) 1 ( 1 Q 1 Q n 0(n ) I + Q + Q 2 + = (I Q) ] q q +/. * q q +/. * q +/. * q *1

3 Q q +/. * Example Input type data ALIVE_RATE age_mat_sub0 ALIVE_RATE R Q

4 0 r r 0 r r 0 r 1 r 2 Q = 0 0 r r 2, Q 2 = r 1 r 2, Q 3 =, r 0 + r 0 r 1 + r 0 r 1 r r 1 + r 1 r r Q q=:}."1}.age_mat_sub0 ALIVE_RATE Q I Q i_minus q 1 _ _ _

5 (I Q) 1 %. i_minus q (I Q) 1 I + Q + Q 2 + Q Q n +/"1 %. i_minus }."1 }. age_mat_sub0 ALIVE_RATE alive ALIVE_RATE NB alive 2.2 Script NB Age Average alive=: 3 : (i. # tmp),. tmp=. +/"1 alive0 y alive0=: 3 : %. (=/ i. # tmp) - tmp=. }.}."1 age_mat_sub0 y NB. (I-Q)ˆ-1 age_mat_sub0=: 3 : 0 NB. age_mat_sub ALIVE_RATE Y0=: remove_null y NB. remove no alive(not pass age) RATE0=. 1- }: {:"1 Y0 MAT0=. (SIZE=: 2# (<: # Y0)) $ 0 DIAG=. diag i. SIZE MAT0=. SIZE $ ( RATE0) (DIAG)};MAT0 MAT0=. ({:"1 Y0),. 0,. MAT0,0

6 MAT=. (1,(# MAT0)#0), MAT0 ) remove_null=: 3 : (-. ( +/ "1 y e. 0) e. 2) # y 2.3 Q.Q 2, Q 3 Q.Q 2, Q 3 (I Q) n 2.4 Script markov_loop=: 4 : 0 ANS=. <TMP=. y for_ctr. i. x do. TMP=. TMP +/. * y NB. mp rightside ANS=. ANS,<TMP end. ({@> i. >: x),.,.ans ) i_minus=: 3 : (=/ i.# y)-y

7 3 markov_loop }.}."1 a / > }.("1) 3 markov_loop }.}."1 a markov_loop a

8 2.5 Example Sheep S heep Bradie S heep age_mat_sub {"1 SHEEP (I Q) n %. i_minus }."1 }. age_mat_sub {"1 SHEEP

9 ( 3 0 S heep (i.11),.+/ "1 %. i_minus }."1 } age_mat_sub {"1 SHEEP 3 P.H.Leslie( ) Oxford Oxford Bureau o f Animal Population 1940 Bxernardelli(1941),Lewis(1942),Leslie(1945) 1959 Leslie 1966 J.H. Pollard stocastic version x k+1 x (k+1) = Lx (k) birth a i, i = 1, 2, 3,, n death b i, i = 1, 2, 3,, n Example(S heep) Input DATA:S HEEP (i.12),. SHEEP

10 time t t+1 n1 f1 n1 n2 n3 n-w-1 n-w f2 f3 fw-1 fw pw-2 p1 p2 p3 n2 n3 n-w-1 n-w a 1 a 2 a 3 a n b b L = 0 0 b n mreg age(1)ai bi x 0 = [ ] (1) 1 S HEEP 1 Caughley, data collected by Hicky)

11 3.1.1 leslie_mat0 {1 2 { : SHEEP *2 *3 ". 7j3 ": (10;0) leslie_loop SHEEP *2 2 *3 (10;0), (10;1) 10

12 {char_lf leslie_mat0 {1 2 { : SHEEP j j_ _ j _ j_ j j_ _ _ j _ j_ _ j _ j_ _ e_ % NB. 0-1 age NB. 1-2 age NB. 2-3 age total

13 require plot line,stick plot {:"1 (50;0) leslie_loop SHEEP pd eps /temp/sheep_leslie0.eps S heep {. 10}. DAT age pop(f)pop(m)birth alive(f) alive(m)

14 : :100 a n 0 F 0 0 M 4.1 key T F M plot ;("1) +/("1) (L:0) 200 leslie_human_loop }."1 DAT pd eps /temp/japan_leslie.eps

15 Year Total Female Male e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e7 1.2e8 1e8 8e7 6e7 4e7 2e7 0 T F M (200 )

16 , (Economist July 2008) ( % (NYT imes22/july/2008) PercentageChange(%) World Asia S ubs aharan A f rica Middle East and Northern A f rica Oceania Latin America Northern America Europe Script leslie_loop=:4 : 0 NB. markov chain NB. Usage: e.g. leslie_loop RACCOON/ POP;Fx;Px NB. x is 10;0 //(times to loop); select 0/1 NB. 0 is birth rate of F --> birth // 1 is M+F--> bitrh * 1r2 NB. y is 3 factors NB. Population; f(birth-rate);p(alive-rate) POP F0 P0 =.{ : y if. 2= # x do. TIME SEL =. x else. TIME SEL =. x; 1 end.

17 P0=. }: P0 MAT=. F0, (P0 *=i. # P0),. 0 NB. make Leslie matrix ANS=. < POP for_ctr. i. TIME do. POP=. MAT +/. * POP if. 1= * SEL do. POP=. birth_half POP end. ANS=.ANS,<POP end. TMP=:;("1),. ANS (i.>: TIME),. TMP,. +/"1 TMP ) leslie_mat0=: 3 : 0 F0 P0 =. y NB. Fx;Px F0, (P0 *=i. # P0),. 0 NB. make Leslie matrix ) birth_half=: 3 : 0 (-: {. y), }. y NB. rate of f,m is 0.5:0.5 ) 5 Reference Brain Bradie [numerical Analysys] Pearson

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