Commit c040009e32383226d5ac34f4e03df40f31fee96a

narration added, notation made similar to exercise 2
  • 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/
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*
45Tehtävä 3
46\end_layout
47
48\begin_layout Standard
49Tarkastellaan 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
83Tehdään muuttujanvaihto
84\end_layout
85
86\begin_layout Standard
87\begin_inset Formula \[
88t=\frac{1}{d}\tau\,,\]
89
90\end_inset
91
92ja 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
101Ketjusäännöllä 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
115Jättämällä
116\begin_inset ERT
117status 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
130LatexCommand eqref
131reference "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*}
140d\frac{\partial u}{\partial\tau} & =au-eu^{2}-buv\\
141d\frac{\partial v}{\partial\tau} & =v(cu-d)\,,\end{align*}
142
143\end_inset
144
145ja 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
156Tehdään muunnokset
157\begin_inset Formula \[
158\hat{u}=\frac{c}{d}u\Leftrightarrow u=\frac{d}{c}\hat{u}\]
159
160\end_inset
161
162ja
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
171Malli 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
181eli
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
188Merkitään vielä
189\begin_inset Formula \[
190\hat{a}=\frac{a}{d}\,,\;\hat{e}=\frac{de}{ac}\]
191
192\end_inset
193
194jolloin alkuperäinen malli
195\begin_inset CommandInset ref
196LatexCommand eqref
197reference "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
212Tehdään mallille
213\begin_inset CommandInset ref
214LatexCommand eqref
215reference "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
224Merkitään
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 \[
242A=\left(\begin{array}{cc}
243F_{u} & F_{v}\\
244G_{u} & G_{v}\end{array}\right)=\left(\begin{array}{cc}
245a(1-v-2eu) & -au\\
246v & u-1\end{array}\right)\:.\]
247
248\end_inset
249
250Mallin
251\begin_inset CommandInset ref
252LatexCommand eqref
253reference "eq:malli3final"
254
255\end_inset
256
257 tasapainotilat löydetään ratkaisemalla yhtälöpari
258\begin_inset Formula \begin{align}
259au(1-eu-v) & =0\,,\nonumber \\
260v(u-1) & =0\,.\label{eq:tasap}\end{align}
261
262\end_inset
263
264Yhtälöparin
265\begin_inset CommandInset ref
266LatexCommand eqref
267reference "eq:tasap"
268
269\end_inset
270
271 ratkaisut ovat
272\begin_inset Formula \begin{alignat*}{3}
273u= & 0\,,\qquad\text{ja}\qquad & u= & 1\,,\qquad\text{ja}\qquad & u= & \frac{1}{e}\,,\\
274v= & 0\,, & v= & 1-e\,, & v= & 0\,.\end{alignat*}
275
276\end_inset
277
278Tarkastellaan 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 \[
292A=\left(\begin{array}{cc}
293a & 0\\
2940 & -1\end{array}\right)\:.\]
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
319Pisteessä
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 \[
332A=\left(\begin{array}{cc}
333-ae & -a\\
3341-e & 0\end{array}\right)\;.\]
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 \[
360a^{2}e^{2}>a^{2}e^{2}-4a(1-e)\]
361
362\end_inset
363
364eli
365\begin_inset Formula \[
366e<1.\]
367
368\end_inset
369
370Siis 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
396Tasapainotilassa
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 \[
409A=\left(\begin{array}{cc}
410-a & -a/e\\
4110 & 1/e-1\end{array}\right)\:.\]
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