{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 20 "Peto- ja saaliskanta" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "Peto- ja saaliskantojen keskin\344ist\344 vuorovaikutusta voidaan mallintaa to isistaan riippuvien differentiaaliyht\344l\366iden avulla. T\344ss\344 tarkastellaan yksinkertaista mallia, joka perustuu ns. " }{TEXT 256 9 "Volterran" }{TEXT -1 1 "\226" }{TEXT 257 6 "Lotkan" }{TEXT -1 1 " \+ " }{TEXT 265 10 "yht\344l\366ihin" }{TEXT -1 84 ". Yht\344l\366t konst ruoidaan seuraavalla logiikalla. Saalisel\344inkantaa kuvaavan funktio n " }{TEXT 266 1 "s" }{TEXT -1 1 "(" }{TEXT 267 1 "t" }{TEXT -1 132 ") muutokset (derivaatan arvot) riippuvat kahdesta tekij\344st\344. Tois aalta kantaan vaikuttaa ymp\344rist\366tekij\366iden muodostama kasvu termi " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT 268 2 " s" }{TEXT -1 1 "(" }{TEXT 269 1 "t" }{TEXT -1 146 "), joka kuvaa populaation (ek sponentiaalista) kasvua olosuhteissa, jossa petokannan vaikutusta ei o le. T\344m\344n lis\344ksi kantaan vaikuttaa petokannan " }{TEXT 270 1 "p" }{TEXT -1 1 "(" }{TEXT 271 1 "t" }{TEXT -1 74 ") aiheuttama kant aa v\344hent\344v\344 tekij\344. Perusyht\344l\366 saaliskannalle on s iis \n" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(s(t),t) = (alpha-beta* p(t))*s(t);" "6#/-%%diffG6$-%\"sG6#%\"tGF**&,&%&alphaG\"\"\"*&%%betaGF .-%\"pG6#F*F.!\"\"F.-F(6#F*F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "jossa " }{XPPEDIT 18 0 "al pha;" "6#%&alphaG" }{TEXT -1 38 " on saalispopulaation kasvukerroin ja " }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 169 " on petopopulaati onsy\366miskerroin. Vastaavasti petokannan yht\344l\366ss\344 tulee ol la saaliskannan puuttumisesta aiheutuva kantaa heikent\344v\344 (ekspo nentiaalinen) v\344henemistekij\344 " }{XPPEDIT 18 0 "gamma;" "6#%&gam maG" }{TEXT -1 1 " " }{TEXT 272 1 "p" }{TEXT -1 1 "(" }{TEXT 273 1 "t " }{TEXT -1 96 ") sek\344 saaliskannan runsaudesta aiheutuva kasvuteki j\344. Niinp\344 perusyht\344l\366 petokannalle on muotoa" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(p(t),t) = (-gamma+delta*beta*s(t))*p(t);" "6#/-%%diffG6$-%\"pG6#%\"tGF**&,&%&ga mmaG!\"\"*(%&deltaG\"\"\"%%betaGF1-%\"sG6#F*F1F1F1-F(6#F*F1" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "miss\344 " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 36 " on saa listajien v\344henemistekij\344 ja " }{XPPEDIT 18 0 "delta;" "6#%&delt aG" }{TEXT -1 65 " on saalistajien tehokerroin saalisel\344inten olles sa saatavilla. \n" }}{PARA 0 "" 0 "" {TEXT -1 177 "Tarkastellaan seura avaksi, millaisen mallin kantojen k\344ytt\344ytymiselle yll\344 esite tyt mallit antavat. Olkoot tarkasteltavina lajeina ilves- ja lumikenk \344j\344niskannat Kanadassa. \n" }}{PARA 256 "" 0 "" {BITMAP 543 197 197 1 "?TMghc=:;JUL:K;yb:fRF:K:;::Z;B:<:KjN>]JNBMSB_lR@mkFahNaE]uokLRaWgNpUqEagP@mrP_ l_pTsViwpLQT]gpwpTsfkiqXUwuor@qTghqIQeyxywwqyXyyyyI:yyy>:;:::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JJ>;;:uRAuBB;CB;kDdJ;Xj=`jAXKD TJ=tJ?dJ?Pj\\Iv[Iv\\Q^]O^ :Yb>_b?ob@eb=SB=Ir:AR:=b:Ar:AB;Ar:Ar:DJ;DJ;@j:DZ^Z>>::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::B:>>ZWn[?F[;Z;V[@FZeB eb@mb;HJ=djAtKJ<\\Z::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::Z:JJ:LK>PJ=@:;B:?r:AJDtJA@kAxJApJ@DkBlJAtj @tJA\\j>lZCvZFvZAv[Iv\\Sn\\JV[AF[EN:?r=Yr=Ob`j>lJDLkCpjDPkD@KBTkFpKELJB TkFdj;XJ@tkCHKI>:::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::>bAYJ:>:=b:Eb:fZIF\\Gf[G>\\C^Z>nZCf[CV Z@^[\\HV[E>\\ON]V>]NV[JF]WN]OV\\OF]W>\\Qn]ZJJ::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::B:; KJZRnZdKGTkDdkHpKJPkElKJ<\\Z::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::Z:>>;;TK[FF[ HF\\FfZBF[BVZA^Z?fZ<^ZC^Z\\SF]Tn]P^]Z>> ;;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::<:;[Xv\\Q^]U>]Z>^Zf]Tf]OF[=FZ;R;EbGrKb;ErYr=Eb;mB?SB=]B?QR>]RaBB;CB;;:::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::B:^Mv\\V>^TV]Q^\\@NZ=VZ^ZB>[D^]ZfZJV\\Jf[GV[Nn]QFZ;F\\Nv[In[C>]Z>>;;::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::Z:JJb\\j@`J=xJ?V[Q^ZKN\\F^Z>v[U>^XNZ=V\\Dv [HvZPn]Z>>;;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::Z:JJlk?TJJ^VV[>tKFdj:@j;Pj;dJ?\\j>xJDDkCxJAtj@lj=LJ>Tj<^\\IN[Dn[GFZ;^] B<=BA;[Z:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::Z:>>kb;;B?wBB;KJZXn\\BVZAvZ=VZ=VZAF[HN\\GV[Ln[B>[Lf[B F[FF[Hn[IN[=nZEFZ;^\\G^[KF\\F^[G^\\INZ=jA^Z>>bAib:OR?eB=GB=SB=ER;OB?YR;ER;GR:ER^ZBB:::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::B:sb=Ib:?B@;CB<^ZBB::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::B:er:AR: ;bA;CB<hJ=Dj?LK:JI\\[?N[GN\\ HVZRn]Z>>;;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::Z:v[:FZ;NZW>^ZbAwZXv[;FZ;vZF^[C^[F^[>N[EV[EN[A^Z=VZ CN:=RIr=sBB;KJ::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::B:MR:?r:AB?wBB\\GF[FN[B^[BNZ>>[B>[?VZDV\\KnZ?fZ>N]Tv\\I>^VnZ=V[ HN\\LF[:v[Z>^Z>>:::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::\\Pn\\MN\\ANZC v[?>[BNZCnZ=NZ=R;QB=ER;IB;sBAiR<;CAYb;EB=_b>Wj? 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NZ=VZ=v:MJ@lJ@`ZCF[CF[GB=SB=R>]jBpj>Dj:FZ=^[KF\\F^[Kj@@;SB=MJZ:R:=j>Xj;FZ>^ZAb=SB;MR_B==B;IB;Dj=HJHJNZSB=Uj@ pJ@B=]jBD[GF\\G^[CNZ=J@R>]J@:]R=XJ@j:Z:>Z:V:IJBTKEDkBlJF[g b?qJJp[OB:>>:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::gRv:`jBn[F^[F^[GF[C:UB=MR`j>`J@l:Ar:?B;SR=Uj=Z;^Z>FZFvZ>vZ;VZ>VZ=f[A^ZAF[C^:=B:;r:WB>Lj:lJEhKJ>^X>^XF[J>^ :>>;;:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZX f\\C^Z>v:Xj;Hj;HJ<`J@`j>lj>F;SRF[>FZ;r:lJ@lJ@@kB@k@vZAvZA F\\Fv:CB;IR>:::PkDPkDj;@j=`ZCF [On]ZbAMr:WRA;cAob?uJBDkH>:::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::JGPkBlj>FZ]B=Ir;Cj=`J@VZ;j>lj@pJ>;;:::dKEPkDTkHtj:Dj@tJB P[Cn[AvZCvZtJBDj>f=;KJ<>;;::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::bAwb@oR?Sr;=B:;R:=r;I R:@J;XZAFZ;j;r;MB=Sr;Ib:;R:?b:CB=Ir;Ir;?b:=R:AJXj:WR:>;;::::V]CNZ=f\\TN]WF[CV]:>>;;::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::wRAqb@or@T;gjDn\\Pf\\ Pb@oR?[R?eJAtJAtjDTkDPkDeZO>\\OV]XN]Tn=>>::::ZPVZHf\\>;CB :::::RAMr:CB=Wb=AB>wB:;CB;;::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::ZW>^ZBB::::::ZWn[C^Z;f<>>:::::ZWF[;R:;R?; C:>>:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::ZWn[;FZO>><<::::::PK:@J:tJI>:::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::JI>:::::::hK;>::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::V ]=NZZ>^Z>>:::::::JBXjD>::::::::wb@uB:>>::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :F[>n]Z>>;;:::::::Z:>>;;:::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::eb?;CB<<::::::::<5:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Laskuje n aluksi on syyt\344 h\344vitt\344\344 mahdollisista aiemmista laskuis ta j\344\344neet muuttujat. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "M\344\344ritell \344\344n ilves- ja j\344niskantoja kuvaava differentiaaliyht\344l\366 ryhm\344 (yksikk\366n\344 tuhat yksil\366\344). (" }{XPPEDIT 18 0 "alp ha;" "6#%&alphaG" }{TEXT -1 8 " = 1.2, " }{XPPEDIT 18 0 "beta;" "6#%%b etaG" }{TEXT -1 9 " = 0.04, " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" } {TEXT -1 8 " = 0.5, " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 7 " = 0.3)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 82 "ryhma:= diff(s(t), t)=s(t)*(1.2-0.04*p(t)), diff(p( t), t)=p(t)*(-0.5 +0.012*s(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Probleeman tuntemattomat funktiot ov at j\344niskantaa kuvaava " }{TEXT 258 1 "s" }{TEXT -1 1 "(" }{TEXT 276 1 "t" }{TEXT -1 27 ") sek\344 ilveskantaa kuvaava " }{TEXT 259 1 " p" }{TEXT -1 1 "(" }{TEXT 275 1 "t" }{TEXT -1 8 "). Aika " }{TEXT 260 1 "t" }{TEXT -1 25 " kuvaa kuluneita vuosia. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "tuntemattom at:= \{s(t), p(t)\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Valitaan kantojen kokoa kuvaavat alkuehdot tar kastelun alkaessa. Ilveksi\344 olkoon 15 000 ja lumikenk\344j\344niksi \344 60 000 yksil\366\344. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "alkuehto:= s(0)=60, p(0)=15; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Ratkaistaan differentiaaliyht\344l\366ryhm\344. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "rtk:= dso lve(\{ryhma,alkuehto\}, tuntemattomat, type=numeric, output=listproced ure);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Poimitaan ratkaisusta kantojen koot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "maarat:= su bs(rtk, [s(t), p(t)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "Piirret\344\344n kuvat saalisel\344in- ja petokantojen vaihtelusta ajan suhteen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot([maarat[1], maara t[2]], 0..80);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 522 "Saamme tulokseksi eritt\344in s\344\344nn\366llises ti vaihtelevat populaatiot, jossa j\344niskanta on merkitty punaisella ja ilveskanta vihre\344ll\344 k\344yr\344ll\344. \n\nTodellisuudessa \+ vain harvoin ymp\344rist\366olot ovat niin tasaiset, ett\344 yll\344 l asketun kaltaista kantojen vaihtelua esiintyisi. Yleens\344 s\344\344 \+ ja muut kannoista riippumattomat ymp\344rist\366tekij\344t vaikuttavat merkitt\344v\344sti populaatioihin. Verrataan kuitenkin saatua tulost a Kanadan lumikenk\344j\344nis- ja ilveskantoihin vuosina 1855 - 1930 \+ (datan alkuper\344n\344 on Hudson's Bay Companyn tilastot). \n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "janis:= [[1855, 76], [1856, 88], [1859, 19], [1861.5, 26], [1862.5, 10], [1863.5, 155], [1865, 15 1], [1867, 20], [1868, 5], [1871.5, 11], [1872.5, 69], [1873, 51], [18 75, 52], [1876, 100], [1877.5, 73], [1881.5, 7], [1882.5, 9], [1884, 4 3], [1885, 130], [1887, 133], [1889.5, 19], [1892, 62], [1893, 65], [1 895, 86], [1897.5, 11], [1899.5, 6], [1901, 12], [1902, 4], [1904, 67] , [1906.5, 24], [1907.5, 23], [1909, 24], [1910, 50], [1912.5, 72], [1 916.5, 3], [1919, 9], [1921, 57], [1923, 77], [1925, 72], [1926, 10], \+ [1928, 4], [1930, 7]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 422 " ilves:= [[1855, 29], [1857, 28], [1861, 4], [1862.5, 4], [1866, 68], [ 1867.5, 72], [1869, 12], [1872, 7], [1873, 33], [1875, 35], [1876, 48] , [1879, 44], [1881, 11], [1882, 11], [1884.5, 50], [1885.5, 78], [188 7, 80], [1889, 39], [1892, 21], [1893, 46], [1895, 52], [1901, 9], [19 03, 20], [1905.5, 62], [1907, 55], [1908, 5], [1911, 14], [1912, 36], \+ [1915, 43], [1917.5, 6], [1920, 6], [1925, 45], [1927, 52], [1930, 19] ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot([janis, ilves], \+ legend=[\"J\344nis\", \"Ilves\"]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "Havaitaan, ett\344 vaikka luon nollisia kannanvaihteluja aiheuttavat my\366s muut tekij\344t, on yhde ks\344n vuoden v\344lein esiintyv\344 maksimi molemmille kannoille var sin s\344\344nn\366llinen. " }{TEXT 261 135 "Volterran\226Lotkan yht \344l\366 n\344ytt\344\344 siis kuvaavan jotain olennaista saalis-peto -suhteessa olevien kantojen keskin\344isest\344 vuorovaikutuksesta" } {TEXT -1 85 ". Toisaalta se ei kerro kovinkaan monesta kannan suuruute en vaikuttavasta tekij\344st\344. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 512 "Tarkastellaan viel\344 peto- ja saaliska ntojen k\344ytt\344ytymist\344 faasitasossa. Tarkoituksena on piirt \344\344 faasitasokuva, jossa x-akseli kuvastaa kannan kasvunopeutta j a y-akseli kannan suuruutta. Periaatteessa kannan kasvunopeus on kanna n kokoa kuvaavan funktion derivaatta. Koska ratkaisu kuitenkin laskett iin numeerisesti, se ei ole funktio jota voitaisiin derivoida. Karkea \+ kuva on sen sijaan mahdollista piirt\344\344 laskemalla erotusosam\344 \344r\344n avulla derivaatta tietyiss\344 pisteiss\344, joiden kautta \+ lopuksi piirret\344n k\344yr\344." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "Lasketaan j\344niskannan koko 100 pistees s\344:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "spisteet:= seq(maarat[1](t/10), t=0..100):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "La sketaan erotusosam\344\344r\344n avulla derivaatalle arvio samoissa pi steiss\344:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "sderivpisteet:= seq((maarat[1](t/10+0.00001)-maara t[1](t/10))/0.00001, t=0..100):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Tehd\344\344n vastaavat operaatiot i lveskannalle:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ppisteet:= seq(maarat[2](t/10), t=0..100):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "pderivpisteet:= seq((maarat[ 2](t/10+0.00001)-maarat[2](t/10))/0.00001, t=0..100):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Piirret \344\344n kantoja kuvaavat kuvaajat samaan kuvaan siten, ett\344 x-aks elilla on kasvunopeus ja y-akselilla on kannan koko:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot([zi p((x, y)->[x, y], [sderivpisteet], [spisteet]), zip((x, y)->[x, y], [p derivpisteet], [ppisteet])]);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 8 "T eht\344vi\344" }}{PARA 0 "" 0 "" {TEXT -1 283 "Mit\344 t\344m\344 faas itason kuvaaja kertoo kantojen k\344ytt\344ytymisest\344? Tarkastele, \+ miten eri parametrien muuttaminen vaikuttaa faasitason kuvaajaan. \n\n Tutki my\366s, onko olemassa alkuehtoja, joiden avulla saavutetaan tas apaino peto- ja saaliskannan v\344lill\344. Vihje: Tarkastele alkuehto a, jossa " }{TEXT 274 1 "x" }{TEXT -1 2 "'(" }{TEXT 262 1 "t" }{TEXT -1 4 ") = " }{TEXT 277 1 "y" }{TEXT -1 2 "'(" }{TEXT 263 1 "t" }{TEXT -1 8 ") = 0. \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 264 24 "JP & SKK & MS 12.07.2001" }{TEXT -1 0 "" }}}}{MARK "0 2 0" 18 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }