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- demo2/Harjoitus_2_teht3.lyx 458 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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#### demo2/Harjoitus_2_teht3.lyx

1 | #LyX 1.6.4.2 created this file. For more info see http://www.lyx.org/ | ||
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2 | \lyxformat 345 | ||

3 | \begin_document | ||

4 | \begin_header | ||

5 | \textclass article | ||

6 | \use_default_options true | ||

7 | \language english | ||

8 | \inputencoding auto | ||

9 | \font_roman default | ||

10 | \font_sans default | ||

11 | \font_typewriter default | ||

12 | \font_default_family default | ||

13 | \font_sc false | ||

14 | \font_osf false | ||

15 | \font_sf_scale 100 | ||

16 | \font_tt_scale 100 | ||

17 | |||

18 | \graphics default | ||

19 | \paperfontsize default | ||

20 | \use_hyperref false | ||

21 | \papersize default | ||

22 | \use_geometry false | ||

23 | \use_amsmath 1 | ||

24 | \use_esint 1 | ||

25 | \cite_engine basic | ||

26 | \use_bibtopic false | ||

27 | \paperorientation portrait | ||

28 | \secnumdepth 3 | ||

29 | \tocdepth 3 | ||

30 | \paragraph_separation indent | ||

31 | \defskip medskip | ||

32 | \quotes_language english | ||

33 | \papercolumns 1 | ||

34 | \papersides 1 | ||

35 | \paperpagestyle default | ||

36 | \tracking_changes false | ||

37 | \output_changes false | ||

38 | \author "" | ||

39 | \author "" | ||

40 | \end_header | ||

41 | |||

42 | \begin_body | ||

43 | |||

44 | \begin_layout Section* | ||

45 | Tehtävä 3 | ||

46 | \end_layout | ||

47 | |||

48 | \begin_layout Standard | ||

49 | Tarkastellaan alunperin samanmuotoista peto-saalismallia kuin tehtävässä | ||

50 | 2. | ||

51 | Nyt saaliiden kasvunopeutta rajoitetaan termillä | ||

52 | \begin_inset Formula $-eu^{2}$ | ||

53 | \end_inset | ||

54 | |||

55 | , jolloin saadaan seuraavanlainen malli | ||

56 | \end_layout | ||

57 | |||

58 | \begin_layout Standard | ||

59 | \begin_inset Formula \begin{align} | ||

60 | \frac{\partial u}{\partial t} & =au-eu^{2}-buv\,,\nonumber \\ | ||

61 | \frac{\partial v}{\partial t} & =v(cu-d)\,.\label{eq:malli3}\end{align} | ||

62 | |||

63 | \end_inset | ||

64 | |||

65 | (Huomaa, että parametri | ||

66 | \begin_inset Formula $e$ | ||

67 | \end_inset | ||

68 | |||

69 | ei liity mitenkään tehtävän 2 samannimiseen parametriin, ja että toinen | ||

70 | yhtälö on samaa muotoa kuin tehtävänannossa käytetty | ||

71 | \begin_inset Formula $v_{t}=cv(u-\tilde{d})$ | ||

72 | \end_inset | ||

73 | |||

74 | , missä | ||

75 | \begin_inset Formula $\tilde{d}=d/c$ | ||

76 | \end_inset | ||

77 | |||

78 | . | ||

79 | Tässä käytetään kuitenkin yllä olevaa muotoa.) | ||

80 | \end_layout | ||

81 | |||

82 | \begin_layout Standard | ||

83 | Tehdään muuttujanvaihto | ||

84 | \end_layout | ||

85 | |||

86 | \begin_layout Standard | ||

87 | \begin_inset Formula \[ | ||

88 | t=\frac{1}{d}\tau\,,\] | ||

89 | |||

90 | \end_inset | ||

91 | |||

92 | ja merkitään | ||

93 | \end_layout | ||

94 | |||

95 | \begin_layout Standard | ||

96 | \begin_inset Formula \[ | ||

97 | \tilde{u}(\tau)=u(d\tau)=u(t)\,,\qquad\tilde{v}(\tau)=v(d\tau)=v(t)\,.\] | ||

98 | |||

99 | \end_inset | ||

100 | |||

101 | Ketjusäännöllä saadaan | ||

102 | \end_layout | ||

103 | |||

104 | \begin_layout Standard | ||

105 | \begin_inset Formula \begin{align*} | ||

106 | \frac{\partial u}{\partial t} & =\frac{\partial\tilde{u}}{\partial\tau}\frac{\partial\tau}{\partial t}=d\frac{\partial\tilde{u}}{\partial\tau}\,,\\ | ||

107 | \frac{\partial v}{\partial t} & =\frac{\partial\tilde{v}}{\partial\tau}\frac{\partial\tau}{\partial t}=d\frac{\partial\tilde{v}}{\partial\tau}.\end{align*} | ||

108 | |||

109 | \end_inset | ||

110 | |||

111 | |||

112 | \end_layout | ||

113 | |||

114 | \begin_layout Standard | ||

115 | Jättämällä | ||

116 | \begin_inset ERT | ||

117 | status open | ||

118 | |||

119 | \begin_layout Plain Layout | ||

120 | |||

121 | |||

122 | \backslash | ||

123 | ~{} | ||

124 | \end_layout | ||

125 | |||

126 | \end_inset | ||

127 | |||

128 | -merkit pois malli | ||

129 | \begin_inset CommandInset ref | ||

130 | LatexCommand eqref | ||

131 | reference "eq:malli3" | ||

132 | |||

133 | \end_inset | ||

134 | |||

135 | saadaan muuttujanvaihdon jälkeen muotoon | ||

136 | \end_layout | ||

137 | |||

138 | \begin_layout Standard | ||

139 | \begin_inset Formula \begin{align*} | ||

140 | d\frac{\partial u}{\partial\tau} & =au-eu^{2}-buv\\ | ||

141 | d\frac{\partial v}{\partial\tau} & =v(cu-d)\,,\end{align*} | ||

142 | |||

143 | \end_inset | ||

144 | |||

145 | ja edelleen | ||

146 | \begin_inset Formula \begin{align*} | ||

147 | \frac{\partial u}{\partial\tau} & =\frac{a}{d}u(1-\frac{e}{a}u-\frac{b}{a}v)\\ | ||

148 | \frac{\partial v}{\partial\tau} & =v(\frac{c}{d}u-1)\,.\end{align*} | ||

149 | |||

150 | \end_inset | ||

151 | |||

152 | |||

153 | \end_layout | ||

154 | |||

155 | \begin_layout Standard | ||

156 | Tehdään muunnokset | ||

157 | \begin_inset Formula \[ | ||

158 | \hat{u}=\frac{c}{d}u\Leftrightarrow u=\frac{d}{c}\hat{u}\] | ||

159 | |||

160 | \end_inset | ||

161 | |||

162 | ja | ||

163 | \end_layout | ||

164 | |||

165 | \begin_layout Standard | ||

166 | \begin_inset Formula \[ | ||

167 | \hat{v}=\frac{b}{a}v\Leftrightarrow v=\frac{a}{b}\hat{v}\,.\] | ||

168 | |||

169 | \end_inset | ||

170 | |||

171 | Malli saadaan muunnosten jälkeen muotoon | ||

172 | \end_layout | ||

173 | |||

174 | \begin_layout Standard | ||

175 | \begin_inset Formula \begin{align*} | ||

176 | \frac{d}{c}\frac{\partial\hat{u}}{\partial\tau} & =\frac{a}{d}\frac{d}{c}\hat{u}(1-\frac{e}{a}\frac{d}{c}\hat{u}-\hat{v})\\ | ||

177 | \frac{a}{b}\frac{\partial\hat{v}}{\partial\tau} & =\frac{a}{b}\hat{v}(\hat{u}-1)\,,\end{align*} | ||

178 | |||

179 | \end_inset | ||

180 | |||

181 | eli | ||

182 | \begin_inset Formula \begin{align*} | ||

183 | \frac{\partial\hat{u}}{\partial\tau} & =\frac{a}{d}\hat{u}(1-\frac{de}{ac}\hat{u}-\hat{v})\\ | ||

184 | \frac{\partial\hat{v}}{\partial\tau} & =\hat{v}(\hat{u}-1)\,.\end{align*} | ||

185 | |||

186 | \end_inset | ||

187 | |||

188 | Merkitään vielä | ||

189 | \begin_inset Formula \[ | ||

190 | \hat{a}=\frac{a}{d}\,,\;\hat{e}=\frac{de}{ac}\] | ||

191 | |||

192 | \end_inset | ||

193 | |||

194 | jolloin alkuperäinen malli | ||

195 | \begin_inset CommandInset ref | ||

196 | LatexCommand eqref | ||

197 | reference "eq:malli3" | ||

198 | |||

199 | \end_inset | ||

200 | |||

201 | saa lopulta muodon | ||

202 | \begin_inset Formula \begin{align} | ||

203 | \frac{\partial\hat{u}}{\partial\tau} & =\hat{a}\hat{u}(1-\hat{e}\hat{u}-\hat{v})\nonumber \\ | ||

204 | \frac{\partial\hat{v}}{\partial\tau} & =\hat{v}(\hat{u}-1)\,.\label{eq:malli3final}\end{align} | ||

205 | |||

206 | \end_inset | ||

207 | |||

208 | |||

209 | \end_layout | ||

210 | |||

211 | \begin_layout Standard | ||

212 | Tehdään mallille | ||

213 | \begin_inset CommandInset ref | ||

214 | LatexCommand eqref | ||

215 | reference "eq:malli3final" | ||

216 | |||

217 | \end_inset | ||

218 | |||

219 | stabiilisuusanalyysi laskemalla systeemin tasapainotilat sekä muodostamalla | ||

220 | systeemille stabiilisuusmatriisi. | ||

221 | \end_layout | ||

222 | |||

223 | \begin_layout Standard | ||

224 | Merkitään | ||

225 | \begin_inset Formula $F(u,v)=au(1-eu-v)$ | ||

226 | \end_inset | ||

227 | |||

228 | ja | ||

229 | \begin_inset Formula $G(u,v)=v(u-1)$ | ||

230 | \end_inset | ||

231 | |||

232 | . | ||

233 | Lasketaan ensin stabiilisuusmatriisi | ||

234 | \begin_inset Formula $A$ | ||

235 | \end_inset | ||

236 | |||

237 | : | ||

238 | \end_layout | ||

239 | |||

240 | \begin_layout Standard | ||

241 | \begin_inset Formula \[ | ||

242 | A=\left(\begin{array}{cc} | ||

243 | F_{u} & F_{v}\\ | ||

244 | G_{u} & G_{v}\end{array}\right)=\left(\begin{array}{cc} | ||

245 | a(1-v-2eu) & -au\\ | ||

246 | v & u-1\end{array}\right)\:.\] | ||

247 | |||

248 | \end_inset | ||

249 | |||

250 | Mallin | ||

251 | \begin_inset CommandInset ref | ||

252 | LatexCommand eqref | ||

253 | reference "eq:malli3final" | ||

254 | |||

255 | \end_inset | ||

256 | |||

257 | tasapainotilat löydetään ratkaisemalla yhtälöpari | ||

258 | \begin_inset Formula \begin{align} | ||

259 | au(1-eu-v) & =0\,,\nonumber \\ | ||

260 | v(u-1) & =0\,.\label{eq:tasap}\end{align} | ||

261 | |||

262 | \end_inset | ||

263 | |||

264 | Yhtälöparin | ||

265 | \begin_inset CommandInset ref | ||

266 | LatexCommand eqref | ||

267 | reference "eq:tasap" | ||

268 | |||

269 | \end_inset | ||

270 | |||

271 | ratkaisut ovat | ||

272 | \begin_inset Formula \begin{alignat*}{3} | ||

273 | u= & 0\,,\qquad\text{ja}\qquad & u= & 1\,,\qquad\text{ja}\qquad & u= & \frac{1}{e}\,,\\ | ||

274 | v= & 0\,, & v= & 1-e\,, & v= & 0\,.\end{alignat*} | ||

275 | |||

276 | \end_inset | ||

277 | |||

278 | Tarkastellaan seuraavaksi tasapainotilojen stabiiliutta. | ||

279 | Stabiilisuusmatriisi | ||

280 | \begin_inset Formula $A$ | ||

281 | \end_inset | ||

282 | |||

283 | pisteessä | ||

284 | \begin_inset Formula $(0,0)$ | ||

285 | \end_inset | ||

286 | |||

287 | on | ||

288 | \end_layout | ||

289 | |||

290 | \begin_layout Standard | ||

291 | \begin_inset Formula \[ | ||

292 | A=\left(\begin{array}{cc} | ||

293 | a & 0\\ | ||

294 | 0 & -1\end{array}\right)\:.\] | ||

295 | |||

296 | \end_inset | ||

297 | |||

298 | |||

299 | \begin_inset Formula $A$ | ||

300 | \end_inset | ||

301 | |||

302 | :n ominaisarvot ovat | ||

303 | \begin_inset Formula $a$ | ||

304 | \end_inset | ||

305 | |||

306 | ja -1. | ||

307 | Koska | ||

308 | \begin_inset Formula $a>0$ | ||

309 | \end_inset | ||

310 | |||

311 | , on tila | ||

312 | \begin_inset Formula $(0,0)$ | ||

313 | \end_inset | ||

314 | |||

315 | epästabiili. | ||

316 | \end_layout | ||

317 | |||

318 | \begin_layout Standard | ||

319 | Pisteessä | ||

320 | \begin_inset Formula $(1,1-e)$ | ||

321 | \end_inset | ||

322 | |||

323 | |||

324 | \begin_inset Formula $A$ | ||

325 | \end_inset | ||

326 | |||

327 | on | ||

328 | \end_layout | ||

329 | |||

330 | \begin_layout Standard | ||

331 | \begin_inset Formula \[ | ||

332 | A=\left(\begin{array}{cc} | ||

333 | -ae & -a\\ | ||

334 | 1-e & 0\end{array}\right)\;.\] | ||

335 | |||

336 | \end_inset | ||

337 | |||

338 | |||

339 | \begin_inset Formula $A$ | ||

340 | \end_inset | ||

341 | |||

342 | :n ominaisarvot ovat | ||

343 | \begin_inset Formula \begin{align*} | ||

344 | \lambda_{1}= & -\frac{ae}{2}+\frac{\sqrt{a^{2}e^{2}-4a(1-e)}}{2}\,,\\ | ||

345 | \lambda_{2}= & -\frac{ae}{2}-\frac{\sqrt{a^{2}e^{2}-4a(1-e)}}{2}\,.\end{align*} | ||

346 | |||

347 | \end_inset | ||

348 | |||

349 | |||

350 | \begin_inset Formula $\lambda_{2}$ | ||

351 | \end_inset | ||

352 | |||

353 | :n reaaliosa on selvästi aina negatiivinen. | ||

354 | |||

355 | \begin_inset Formula $\lambda_{1}$ | ||

356 | \end_inset | ||

357 | |||

358 | :n reaaliosa on negatiivinen, jos | ||

359 | \begin_inset Formula \[ | ||

360 | a^{2}e^{2}>a^{2}e^{2}-4a(1-e)\] | ||

361 | |||

362 | \end_inset | ||

363 | |||

364 | eli | ||

365 | \begin_inset Formula \[ | ||

366 | e<1.\] | ||

367 | |||

368 | \end_inset | ||

369 | |||

370 | Siis tasapainotila | ||

371 | \begin_inset Formula $(1,1-e)$ | ||

372 | \end_inset | ||

373 | |||

374 | on stabiili, kun | ||

375 | \begin_inset Formula $0<e<1$ | ||

376 | \end_inset | ||

377 | |||

378 | . | ||

379 | Tasapainotila | ||

380 | \begin_inset Formula $(1,1-e)$ | ||

381 | \end_inset | ||

382 | |||

383 | on epästabiili, kun | ||

384 | \begin_inset Formula $e>1$ | ||

385 | \end_inset | ||

386 | |||

387 | . | ||

388 | Kyseinen tasapainotila ei ole stabiili eikä epästabiili, kun | ||

389 | \begin_inset Formula $e=1$ | ||

390 | \end_inset | ||

391 | |||

392 | . | ||

393 | \end_layout | ||

394 | |||

395 | \begin_layout Standard | ||

396 | Tasapainotilassa | ||

397 | \begin_inset Formula $(1/e,0)$ | ||

398 | \end_inset | ||

399 | |||

400 | stabiilisuusmatrsiisi | ||

401 | \begin_inset Formula $A$ | ||

402 | \end_inset | ||

403 | |||

404 | on | ||

405 | \end_layout | ||

406 | |||

407 | \begin_layout Standard | ||

408 | \begin_inset Formula \[ | ||

409 | A=\left(\begin{array}{cc} | ||

410 | -a & -a/e\\ | ||

411 | 0 & 1/e-1\end{array}\right)\:.\] | ||

412 | |||

413 | \end_inset | ||

414 | |||

415 | |||

416 | \begin_inset Formula $A$ | ||

417 | \end_inset | ||

418 | |||

419 | :n ominaisarvot ovat | ||

420 | \begin_inset Formula $\lambda_{1}=-a$ | ||

421 | \end_inset | ||

422 | |||

423 | ja | ||

424 | \begin_inset Formula $\lambda_{2}=1/e-1$ | ||

425 | \end_inset | ||

426 | |||

427 | . | ||

428 | |||

429 | \begin_inset Formula $\lambda_{1}<0$ | ||

430 | \end_inset | ||

431 | |||

432 | aina, ja | ||

433 | \begin_inset Formula $\lambda_{2}<0$ | ||

434 | \end_inset | ||

435 | |||

436 | , jos | ||

437 | \begin_inset Formula $e>1$ | ||

438 | \end_inset | ||

439 | |||

440 | . | ||

441 | Tasapainotila | ||

442 | \begin_inset Formula $(1/e,0)$ | ||

443 | \end_inset | ||

444 | |||

445 | on epästabiili, kun | ||

446 | \begin_inset Formula $e<1$ | ||

447 | \end_inset | ||

448 | |||

449 | . | ||

450 | Kyseinen tasapainotila ei ole stabiili eikä epästabiili, kun | ||

451 | \begin_inset Formula $e=1$ | ||

452 | \end_inset | ||

453 | |||

454 | . | ||

455 | \end_layout | ||

456 | |||

457 | \end_body | ||

458 | \end_document |