Dynamic System In Biology

Transcription

1 Compuaional Siene and Engineering Dnami Ssem In Biolog Yang Cao Deparmen of Compuer Siene hp://ourses.s.v.edu/~s644

2 Ouline Compuaional Siene and Engineering Single Speies opulaion Model Malhus Model Logisi Model Two speies Model Compeiion Model redaor and re Model hase lo and Dnami Ssem Sohasi Model and Simulaion Loka Model Brusselaor Model

3 Malhus Model Compuaional Siene and Engineering I SAID ha populaion, when unheked, inreased in a geomerial raio, and subsisene for man in an arihmeial raio Thomas Malhus

4 Malhus Model Compuaional Siene and Engineering The reproduion rae is proporional o he populaion k Solve i we have e k The populaion in he Unied Saes in ear 6 79 is.9. The orresponding populaion in ear 8 is Wih a daa fiing, we obain:.9 6 e.7 79

5 Compuaional Siene and Engineering Logisi opulaion Model Developed b Belgian mahemaiian ierre Verhuls 88 in 88 The rae of populaion inrease ma be limied, i.e., i ma depend on populaion densi k where k k m k m m m k m k e e e The soluion is

6 Compuaional Siene and Engineering Logisi opulaion Model k m m m k m k e e e The soluion of he Logisi model Wih a daa fiing.4, 97 6 k m

7 Model of wo speies Compeiion Compuaional Siene and Engineering Le he populaion of wo speies be and, and he ompee in he same environmen. If here is no ompeiion, he populaion of X will saisf & r N Wih he ompeiion, & For anoher speies, here is a similar equaion r α N The phsial meaning of and an be undersood as: α he he & r α resoure resoure β eah eah N β X speies Y speies onsume onsume. Thus we have αβ

8 Sae Dnamis lo vs hase lo Compuaional Siene and Engineering Sae Dnamis lo: sae vs ime, hase lo: he sae spae, use arrow o represen he angen veor The phase plo reveals he geomeri proper of a dnami ssem represened b a pair of ODEs. &. &.

9 Sae Dnamis lo vs hase lo Compuaional Siene and Engineering Eample: from differen iniial value, he rajeor follow he direion of he arrows and reahes o is equilibrium sae

10 Sae Dnamis lo vs hase lo Compuaional Siene and Engineering However, a sligh hange of parameers make a big differene in phase plo and lead o a differen onlusion &. &. 9

11 Compuaional Siene and Engineering Sae Dnamis lo vs hase lo 9.. & &

12 Sae Dnamis lo vs hase lo Compuaional Siene and Engineering & & r α N β N The sign of he derivaives are deided b wo values N A dire analsis hrough he phase plo α and α N α r If N > α, X speies will win. N If N < αn, Y speies will win.

13 Model of wo speies redaor and re Compuaional Siene and Engineering Loka-Volerra Model The simples model of predaor-pre ineraions developed independenl b Loka 95 and Volerra 96 Anona s observaion on Shark s populaion during world war I.

14 Model of wo speies redaor and re Compuaional Siene and Engineering Assumpion: The predaor speies is oall dependen on a single pre speies as is onl food suppl, The pre speies has an unlimied food suppl, and here is no hrea o he pre oher han he speifi predaor. Le X represen he pre and Y represen he predaor, wihou he predaor, he Malhus model an be applied However, beause of he predaor, r has o be modified For he predaor, he siuaion is jus he opposie. Thus we ge he ODEs for his model & a & a b & d & a b & d

15 hase lo Analsis & & a b d Compuaional Siene and Engineering There are wo orresponding equilibrium poins:,, or d a b,, a b,,, d

16 Malab Simulaion Resul Compuaional Siene and Engineering Based on eample: &. &.4

17 Effe of arameers Compuaional Siene and Engineering The soluion of he LV predaor-pre model is, d a b where a : b : : d : he naural reproduion rae for he pre he killing rae beause of he predaor he naural deah rae for he predaor he reproduion rae beause of he pre Quesion: Wh he shark raio inreases during world war I?

18 arameer Analsis * *, d a b Compuaional Siene and Engineering When fishing is inrodued in he model, heir effe will be inrease he deah rae of he predaor and derease he reproduion rae for he pre. Thus e, a a e,

19 Compuaional Siene and Engineering Sohasi Modeling Loka reaions: Z Y Y Y X X X A Lead o ODEs A & & The sohasi simulaion generaes ineresing rajeories..,, A

20 Differen Dnami Behavior Compuaional Siene and Engineering

21 Compuaional Siene and Engineering Brusselaor D Y Y Y X C Y X B X A 4 B B A 4 & & 5..5, 5, 5, 4 B A Lead o ODEs 5.., 5, 5, 4 B A Bifuraion happens around he ondiion: 4 4 A B J. Tson s 97, 974 paper

22 Compuaional Siene and Engineering Oregonaor X Z E D Y Z Y Y C B Y X Y X A 5 4 Ez C z C A Ez A & & & 6.6, 4,.,, 5 4 E C A

23 Oregonaor Compuaional Siene and Engineering A X Y X Y B C Y Y Y 4 D E Z 5 X Z A,., C 4, 4.6, 5E 6

24 Compuaional Siene and Engineering Thanks! Quesions? lao is m friend, Arisole is m friend, bu m bes friend is ruh --- Newon