matematika v biologii i - masaryk universitypospisil/files/matematika_v_biologii_i.pdf ·...
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![Page 1: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/1.jpg)
Matematika v biologii I
Zdenek Pospısil
22. brezna 2011
![Page 2: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/2.jpg)
Uvod
Uvod
Lindenmayerovysystemy
Maticove populacnımodely
Rust homogennıpopulacev diskretnım case
Lekarska diagnostika
Hardyho-Weinberguvzakon
K dalsımu ctenı
Matematika v biologii I – 2 / 20
![Page 3: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/3.jpg)
Lindenmayerovy systemy
Uvod
LindenmayerovysystemySystemy sdiskretnım casem aparalelnımprepisovanım
Maticove populacnımodely
Rust homogennıpopulacev diskretnım case
Lekarska diagnostika
Hardyho-Weinberguvzakon
K dalsımu ctenı
Matematika v biologii I – 3 / 20
![Page 4: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/4.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Aristid Lindenmayer (1925–1989)
![Page 5: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/5.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Aristid Lindenmayer (1925–1989)
Abeceda: mnozina A
Stav: konecna posloupnost prvku z A
Axiom: inicialnı stav s0Prepisovacı pravidla: zobrazenı f : A → A ∪A2 ∪ A3 ∪ . . .Stav si+1 vznikne ze stavu si tak, ze kazdy clen x v si se nahradı vyrazem f(x)
![Page 6: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/6.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Aristid Lindenmayer (1925–1989)
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
![Page 7: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/7.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
![Page 8: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/8.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23
![Page 9: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/9.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23
s2 =224
![Page 10: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/10.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23
s2 =224
s3 =2254
![Page 11: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/11.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23
s2 =224
s3 =2254
s4 =22654
![Page 12: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/12.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224
s3 =2254
s4 =22654
![Page 13: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/13.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254
s4 =22654
![Page 14: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/14.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654
![Page 15: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/15.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
![Page 16: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/16.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
![Page 17: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/17.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 18: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/18.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 19: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/19.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 20: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/20.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 21: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/21.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 22: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/22.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 23: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/23.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 24: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/24.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 25: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/25.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 26: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/26.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 27: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/27.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 28: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/28.jpg)
Systemy s diskretnım casem a paralelnım
prepisovanım
Matematika v biologii I – 4 / 20
Abeceda: 1, 2, 3, 4, 5, 6, 7, 8, (, )Axiom: 1Prepisovacı pravidla: 1 7→ 23 2 7→ 2 3 7→ 24 4 7→ 54 5 7→ 6
6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )
s0 =1
s1 =23 s5 =227654
s2 =224 s6 =228(1)7654
s3 =2254 s7 =228(23)8(1)7654
s4 =22654 s8 =228(224)8(23)8(1)7654
s9 =228(2254)8(224)8(23)8(1)7654
s10 =228(22654)8(2254)8(224)8(23)8(1)7654
![Page 29: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/29.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
![Page 30: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/30.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
![Page 31: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/31.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
i = 0
![Page 32: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/32.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
i = 1
![Page 33: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/33.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
i = 2
![Page 34: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/34.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
i = 3
![Page 35: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/35.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
i = 4
![Page 36: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/36.jpg)
Matematika v biologii I – 5 / 20
Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS
i = 5
![Page 37: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/37.jpg)
Maticove populacnı modely
Uvod
Lindenmayerovysystemy
Maticove populacnımodely
Fibonacciovi kralıci
Leslieho populacePopulacestrukturovana podlestadiıObecny maticovymodel
Rust homogennıpopulacev diskretnım case
Lekarska diagnostika
Hardyho-Weinberguvzakon
K dalsımu ctenı
Matematika v biologii I – 6 / 20
![Page 38: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/38.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
Leonardo Pisansky, Fibonacci (1170–1250)
![Page 39: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/39.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
Leonardo Pisansky, Fibonacci (1170–1250):
Kdosi umıstil par kralıku na urcitem mıste, se vsech stran ohrazenemzdı, aby poznal, kolik paru kralıku se pri tom zrodı prubehem roku,jestlize u kralıku je tomu tak, ze par kralıku privede na svet mesıcnejeden par a ze kralıci pocınajı rodit ve dvou mesıcıch sveho veku.
![Page 40: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/40.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
Leonardo Pisansky, Fibonacci (1170–1250):
Kdosi umıstil par kralıku na urcitem mıste, se vsech stran ohrazenemzdı, aby poznal, kolik paru kralıku se pri tom zrodı prubehem roku,jestlize u kralıku je tomu tak, ze par kralıku privede na svet mesıcnejeden par a ze kralıci pocınajı rodit ve dvou mesıcıch sveho veku.
x(t) – pocet paru kralıku v t-tem mesıci
![Page 41: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/41.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
Leonardo Pisansky, Fibonacci (1170–1250):
Kdosi umıstil par kralıku na urcitem mıste, se vsech stran ohrazenemzdı, aby poznal, kolik paru kralıku se pri tom zrodı prubehem roku,jestlize u kralıku je tomu tak, ze par kralıku privede na svet mesıcnejeden par a ze kralıci pocınajı rodit ve dvou mesıcıch sveho veku.
x(t) – pocet paru kralıku v t-tem mesıci
x(t) = x(t− 1) + x(t− 2)
![Page 42: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/42.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
Leonardo Pisansky, Fibonacci (1170–1250):
Kdosi umıstil par kralıku na urcitem mıste, se vsech stran ohrazenemzdı, aby poznal, kolik paru kralıku se pri tom zrodı prubehem roku,jestlize u kralıku je tomu tak, ze par kralıku privede na svet mesıcnejeden par a ze kralıci pocınajı rodit ve dvou mesıcıch sveho veku.
x(t) – pocet paru kralıku v t-tem mesıci
x(t) = x(t− 1) + x(t− 2)
x(t+ 2) = x(t+ 1) + x(t), x(0) = 1, x(1) = 2
![Page 43: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/43.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
Leonardo Pisansky, Fibonacci (1170–1250):
Kdosi umıstil par kralıku na urcitem mıste, se vsech stran ohrazenemzdı, aby poznal, kolik paru kralıku se pri tom zrodı prubehem roku,jestlize u kralıku je tomu tak, ze par kralıku privede na svet mesıcnejeden par a ze kralıci pocınajı rodit ve dvou mesıcıch sveho veku.
x(t) – pocet paru kralıku v t-tem mesıci
x(t) = x(t− 1) + x(t− 2)
x(t+ 2) = x(t+ 1) + x(t), x(0) = 1, x(1) = 2
t 0 1 2 3 4 5 6 7 8 9 10 11 12x(t) 1 2 3 5 8 13 21 34 55 89 144 233 377
![Page 44: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/44.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
Leonardo Pisansky, Fibonacci (1170–1250):
Kdosi umıstil par kralıku na urcitem mıste, se vsech stran ohrazenemzdı, aby poznal, kolik paru kralıku se pri tom zrodı prubehem roku,jestlize u kralıku je tomu tak, ze par kralıku privede na svet mesıcnejeden par a ze kralıci pocınajı rodit ve dvou mesıcıch sveho veku.
x(t) – pocet paru kralıku v t-tem mesıci
x(t) = x(t− 1) + x(t− 2)
x(t+ 2) = x(t+ 1) + x(t), x(0) = 1, x(1) = 2
x(t) =1√5
(
1 +√5
2
)t+1
− 1√5
(
1−√5
2
)t+1
![Page 45: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/45.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem mesıci
y(t) – mnozstvı paru plodnych kralıku v t-tem mesıci
![Page 46: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/46.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem mesıciy(t) – mnozstvı paru plodnych kralıku v t-tem mesıci
x(t+ 1)= y(t)y(t+ 1)=x(t)+ y(t)
![Page 47: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/47.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem mesıciy(t) – mnozstvı paru plodnych kralıku v t-tem mesıci
x(t+ 1)= y(t)y(t+ 1)=x(t)+ y(t)
(
x(t+ 1)y(t+ 1)
)
=
(
0 11 1
)(
x(t)y(t)
)
![Page 48: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/48.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem obdobıy(t) – mnozstvı paru plodnych kralıku v t-tem obdobı
σ1 – podıl juvenilnıch paru, ktere prezijı jedno obdobıσ2 – podıl plodnych paru, ktere prezijı jedno obdobıγ – podıl juvenilnıch paru, ktere behem jednoho obdobı dospejıb – plodnost: prumerny pocet paru ktere ,,vyprodukuje” plodny par za jedno obdobı
![Page 49: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/49.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem obdobıy(t) – mnozstvı paru plodnych kralıku v t-tem obdobı
σ1 – podıl juvenilnıch paru, ktere prezijı jedno obdobıσ2 – podıl plodnych paru, ktere prezijı jedno obdobıγ – podıl juvenilnıch paru, ktere behem jednoho obdobı dospejıb – plodnost: prumerny pocet paru ktere ,,vyprodukuje” plodny par za jedno obdobı
x(t+ 1)= (1− γ)σ1 x(t)+ b y(t)y(t+ 1)= γσ1 x(t)+σ2 y(t)
![Page 50: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/50.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem obdobıy(t) – mnozstvı paru plodnych kralıku v t-tem obdobı
σ1 – podıl juvenilnıch paru, ktere prezijı jedno obdobıσ2 – podıl plodnych paru, ktere prezijı jedno obdobıγ – podıl juvenilnıch paru, ktere behem jednoho obdobı dospejıb – plodnost: prumerny pocet paru ktere ,,vyprodukuje” plodny par za jedno obdobı
x(t+ 1)= (1− γ)σ1 x(t)+ b y(t)y(t+ 1)= γσ1 x(t)+σ2 y(t)
(
x(t+ 1)y(t+ 1)
)
=
(
σ1(1− γ) b
σ1γ σ2
)(
x(t)y(t)
)
![Page 51: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/51.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem obdobıy(t) – mnozstvı paru plodnych kralıku v t-tem obdobı
σ1 – podıl juvenilnıch paru, ktere prezijı jedno obdobıσ2 – podıl plodnych paru, ktere prezijı jedno obdobıγ – podıl juvenilnıch paru, ktere behem jednoho obdobı dospejıb – plodnost: prumerny pocet paru ktere ,,vyprodukuje” plodny par za jedno obdobı
x(t+ 1)= (1− γ)σ1 x(t)+ b y(t)y(t+ 1)= γσ1 x(t)+σ2 y(t)
(
x
y
)
(t+ 1) =
(
σ1(1− γ) b
σ1γ σ2
)(
x
y
)
(t)
![Page 52: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/52.jpg)
Fibonacciovi kralıci
Matematika v biologii I – 7 / 20
x(t) – mnozstvı paru juvenilnıch kralıku v t-tem obdobıy(t) – mnozstvı paru plodnych kralıku v t-tem obdobı
σ1 – podıl juvenilnıch paru, ktere prezijı jedno obdobıσ2 – podıl plodnych paru, ktere prezijı jedno obdobıγ – podıl juvenilnıch paru, ktere behem jednoho obdobı dospejıb – plodnost: prumerny pocet paru ktere ,,vyprodukuje” plodny par za jedno obdobı
x(t+ 1)= (1− γ)σ1 x(t)+ b y(t)y(t+ 1)= γσ1 x(t)+σ2 y(t)
(
x
y
)
(t+ 1) =
(
σ1(1− γ) b
σ1γ σ2
)(
x
y
)
(t)
x(t+ 1) = Ax(t)
![Page 53: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/53.jpg)
Leslieho populace
Matematika v biologii I – 8 / 20
Patrick Holt Leslie (1900–1972)
![Page 54: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/54.jpg)
Leslieho populace
Matematika v biologii I – 8 / 20
xi(t) – pocet zen veku i let; presneji veku z intervalu (i− 1, i〉 letpi – podıl zen veku i, ktere prezijı do dalsıho roku
![Page 55: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/55.jpg)
Leslieho populace
Matematika v biologii I – 8 / 20
xi(t) – pocet zen veku i let; presneji veku z intervalu (i− 1, i〉 letpi – podıl zen veku i, ktere prezijı do dalsıho roku
xi+1(t+ 1)= pixi(t), i = 1, 2, . . . , k − 1
![Page 56: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/56.jpg)
Leslieho populace
Matematika v biologii I – 8 / 20
xi(t) – pocet zen veku i let; presneji veku z intervalu (i− 1, i〉 letpi – podıl zen veku i, ktere prezijı do dalsıho rokufi – ocekavany pocet dcer, ktere behem roku porodı zena veku i let
xi+1(t+ 1)= pixi(t), i = 1, 2, . . . , k − 1
![Page 57: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/57.jpg)
Leslieho populace
Matematika v biologii I – 8 / 20
xi(t) – pocet zen veku i let; presneji veku z intervalu (i− 1, i〉 letpi – podıl zen veku i, ktere prezijı do dalsıho rokufi – ocekavany pocet dcer, ktere behem roku porodı zena veku i let
x1(t+ 1)= f1x1(t) + f2x2(t) + · · ·+ fkxk(t)
xi+1(t+ 1)= pixi(t), i = 1, 2, . . . , k − 1
![Page 58: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/58.jpg)
Leslieho populace
Matematika v biologii I – 8 / 20
xi(t) – pocet zen veku i let; presneji veku z intervalu (i− 1, i〉 letpi – podıl zen veku i, ktere prezijı do dalsıho rokufi – ocekavany pocet dcer, ktere behem roku porodı zena veku i let
x1(t+ 1)= f1x1(t) + f2x2(t) + · · ·+ fkxk(t)
xi+1(t+ 1)= pixi(t), i = 1, 2, . . . , k − 1
x1(t+ 1)x2(t+ 1)x3(t+ 1)
...xk−1(t+ 1)xk(t+ 1)
=
f1 f2 f3 . . . fk−1 fkp1 0 0 . . . 0 00 p2 0 . . . 0 0...
......
. . ....
...0 0 0 . . . 0 00 0 0 . . . pk−1 0
x1(t)x2(t)x3(t)...
xk−1(t)xk(t)
![Page 59: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/59.jpg)
Leslieho populace
Matematika v biologii I – 8 / 20
xi(t) – pocet zen veku i let; presneji veku z intervalu (i− 1, i〉 letpi – podıl zen veku i, ktere prezijı do dalsıho rokufi – ocekavany pocet dcer, ktere behem roku porodı zena veku i let
x1(t+ 1)= f1x1(t) + f2x2(t) + · · ·+ fkxk(t)
xi+1(t+ 1)= pixi(t), i = 1, 2, . . . , k − 1
x1(t+ 1)x2(t+ 1)x3(t+ 1)
...xk−1(t+ 1)xk(t+ 1)
=
f1 f2 f3 . . . fk−1 fkp1 0 0 . . . 0 00 p2 0 . . . 0 0...
......
. . ....
...0 0 0 . . . 0 00 0 0 . . . pk−1 0
x1(t)x2(t)x3(t)...
xk−1(t)xk(t)
x(t+ 1) = Ax(t)
![Page 60: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/60.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
Leonard Lefkowitch (1929–2010)
![Page 61: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/61.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
Obojzivelnıci: vajıcko – pulec – dospely jedinec
![Page 62: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/62.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
Obojzivelnıci: vajıcko – pulec – dospely jedinecHmyz: vajıcko – larva – kukla – imago
![Page 63: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/63.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
Obojzivelnıci: vajıcko – pulec – dospely jedinecHmyz: vajıcko – larva – kukla – imago
Rostliny: semeno – mala ruzice – strednı ruzice – velka ruzice – kvetoucı rostlina
![Page 64: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/64.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
xi(t) – pocet jedincu stadia i v case t (i = 1 – ,,novorozenci”)pii – podıl jedincu stadia i, kterı prezijı obdobı a nepromenı se; 0 ≤ pii ≤ 1pij – podıl jedincu stadia j, kterı se behem obdobı premenı na stadium i; 0 ≤ pij ≤ 1fi – ocekavany pocet ,,novorozencu”, ktere vyprodukuje jedinec stadia i; fi ≥ 0
![Page 65: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/65.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
xi(t) – pocet jedincu stadia i v case t (i = 1 – ,,novorozenci”)pii – podıl jedincu stadia i, kterı prezijı obdobı a nepromenı se; 0 ≤ pii ≤ 1pij – podıl jedincu stadia j, kterı se behem obdobı premenı na stadium i; 0 ≤ pij ≤ 1fi – ocekavany pocet ,,novorozencu”, ktere vyprodukuje jedinec stadia i; fi ≥ 0
k∑
j=1
pij ≤ 1, i = 1, 2, . . . , k
![Page 66: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/66.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
xi(t) – pocet jedincu stadia i v case t (i = 1 – ,,novorozenci”)pii – podıl jedincu stadia i, kterı prezijı obdobı a nepromenı se; 0 ≤ pii ≤ 1pij – podıl jedincu stadia j, kterı se behem obdobı premenı na stadium i; 0 ≤ pij ≤ 1fi – ocekavany pocet ,,novorozencu”, ktere vyprodukuje jedinec stadia i; fi ≥ 0
k∑
j=1
pij ≤ 1, i = 1, 2, . . . , k
x1(t+ 1)=k∑
j=1
fjxj(t),
xi(t+ 1)=k∑
j=1
pijxj(t), i = 2, 3, . . . , k
![Page 67: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/67.jpg)
Populace strukturovana podle stadiı
Matematika v biologii I – 9 / 20
xi(t) – pocet jedincu stadia i v case t (i = 1 – ,,novorozenci”)pii – podıl jedincu stadia i, kterı prezijı obdobı a nepromenı se; 0 ≤ pii ≤ 1pij – podıl jedincu stadia j, kterı se behem obdobı premenı na stadium i; 0 ≤ pij ≤ 1fi – ocekavany pocet ,,novorozencu”, ktere vyprodukuje jedinec stadia i; fi ≥ 0
k∑
j=1
pij ≤ 1, i = 1, 2, . . . , k
x1(t+ 1)=k∑
j=1
fjxj(t),
xi(t+ 1)=k∑
j=1
pijxj(t), i = 2, 3, . . . , k
x(t+ 1) = Ax(t)
![Page 68: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/68.jpg)
Obecny maticovy model
Matematika v biologii I – 10 / 20
x(t+ 1) = Ax(t)
![Page 69: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/69.jpg)
Obecny maticovy model
Matematika v biologii I – 10 / 20
x(t+ 1) = Ax(t)
λ1, λ2, . . . , λk – vlastnı hodnoty matice A, λ1 ≥ |λ2| ≥ · · · ≥ |λk|w1,w2, . . . ,wk – prıslusne vlastnı vektory
![Page 70: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/70.jpg)
Obecny maticovy model
Matematika v biologii I – 10 / 20
x(t+ 1) = Ax(t)
λ1, λ2, . . . , λk – vlastnı hodnoty matice A, λ1 ≥ |λ2| ≥ · · · ≥ |λk|w1,w2, . . . ,wk – prıslusne vlastnı vektory
Pro jednoduchost predpokladejme, ze λ1 > |λ2| a λi 6= λj pro i 6= j.
![Page 71: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/71.jpg)
Obecny maticovy model
Matematika v biologii I – 10 / 20
x(t+ 1) = Ax(t)
λ1, λ2, . . . , λk – vlastnı hodnoty matice A, λ1 ≥ |λ2| ≥ · · · ≥ |λk|w1,w2, . . . ,wk – prıslusne vlastnı vektory
Pro jednoduchost predpokladejme, ze λ1 > |λ2| a λi 6= λj pro i 6= j.
x(0) = c1w1 + c2w2 + · · ·+ ckwk
![Page 72: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/72.jpg)
Obecny maticovy model
Matematika v biologii I – 10 / 20
x(t+ 1) = Ax(t)
λ1, λ2, . . . , λk – vlastnı hodnoty matice A, λ1 ≥ |λ2| ≥ · · · ≥ |λk|w1,w2, . . . ,wk – prıslusne vlastnı vektory
Pro jednoduchost predpokladejme, ze λ1 > |λ2| a λi 6= λj pro i 6= j.
x(0) = c1w1 + c2w2 + · · ·+ ckwk
x(1) = c1λ1w1 + c2λ2w2 + · · ·+ ckλkwk
![Page 73: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/73.jpg)
Obecny maticovy model
Matematika v biologii I – 10 / 20
x(t+ 1) = Ax(t)
λ1, λ2, . . . , λk – vlastnı hodnoty matice A, λ1 ≥ |λ2| ≥ · · · ≥ |λk|w1,w2, . . . ,wk – prıslusne vlastnı vektory
Pro jednoduchost predpokladejme, ze λ1 > |λ2| a λi 6= λj pro i 6= j.
x(0) = c1w1 + c2w2 + · · ·+ ckwk
x(1) = c1λ1w1 + c2λ2w2 + · · ·+ ckλkwk
x(t) = c1λt1w1 + c2λ
t2w2 + · · ·+ ckλ
tkwk
![Page 74: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/74.jpg)
Obecny maticovy model
Matematika v biologii I – 10 / 20
x(t+ 1) = Ax(t)
λ1, λ2, . . . , λk – vlastnı hodnoty matice A, λ1 ≥ |λ2| ≥ · · · ≥ |λk|w1,w2, . . . ,wk – prıslusne vlastnı vektory
Pro jednoduchost predpokladejme, ze λ1 > |λ2| a λi 6= λj pro i 6= j.
x(0) = c1w1 + c2w2 + · · ·+ ckwk
x(1) = c1λ1w1 + c2λ2w2 + · · ·+ ckλkwk
x(t) = c1λt1w1 + c2λ
t2w2 + · · ·+ ckλ
tkwk
1
λt1
x(t) = c1w1 + c2
(
λ2
λ1
)t
w2 + · · ·+ ck
(
λk
λ1
)t
wk
![Page 75: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/75.jpg)
Rust homogennı populace v diskretnım case
Uvod
Lindenmayerovysystemy
Maticove populacnımodely
Rust homogennıpopulacev diskretnım casePopulace bezomezenıPopulace somezenymi zdroji
Interagujıcı populace
Lekarska diagnostika
Hardyho-Weinberguvzakon
K dalsımu ctenı
Matematika v biologii I – 11 / 20
![Page 76: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/76.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
Leonhard Euler (1707–1783), Introductio in analysin infinitorum (1748)
![Page 77: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/77.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
![Page 78: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/78.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
![Page 79: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/79.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0
![Page 80: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/80.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t) = (1− d+ b)x(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0
![Page 81: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/81.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t) = (1− d+ b)x(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0r = 1− d+ b – rustovy koeficient, r ≥ 0
![Page 82: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/82.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t) = (1− d+ b)x(t) = rx(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0r = 1− d+ b – rustovy koeficient, r ≥ 0
![Page 83: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/83.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t) = (1− d+ b)x(t) = rx(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0r = 1− d+ b – rustovy koeficient, r ≥ 0
x(t+ 1) = rx(t)
![Page 84: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/84.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t) = (1− d+ b)x(t) = rx(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0r = 1− d+ b – rustovy koeficient, r ≥ 0
x(t+ 1) = rx(t)
Rekurentnı formule pro geometrickou posloupnost
![Page 85: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/85.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t) = (1− d+ b)x(t) = rx(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0r = 1− d+ b – rustovy koeficient, r ≥ 0
x(t+ 1) = rx(t)
x(0) = x0 – pocatecnı velikost populace
x(t) = x0rt
![Page 86: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/86.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
Thomas R. Malthus (1766–1834)
x(t) – velikost populace v case t, ktery plyne v ,,prirozenych” jednotkach
x(t+ 1) = x(t)− uhynulı+ narozenı
x(t+ 1) = x(t)− dx(t) + bx(t) = (1− d+ b)x(t) = rx(t)d – umrtnost (pravdepodobnost umrtı behem casove jednotky), d ∈ 〈0, 1〉b – porodnost (prumerny pocet potomku jedince), b ≥ 0r = 1− d+ b – rustovy koeficient, r ≥ 0
x(t+ 1) = rx(t)
x(0) = x0 – pocatecnı velikost populace
x(t) = x0rt
r > 1, tj. b > d, populace roste
r = 1, tj. b = d, populace ma konstatntnı velikost
r < 1, tj. b < d, populace vymıra
![Page 87: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/87.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t+ 1) = rx(t), x(0) = x0
![Page 88: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/88.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t+ 1) = rx(t), x(0) = x0
rustovy koeficient
r =x(t+ 1)
x(t)
![Page 89: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/89.jpg)
Populace bez omezenı
Matematika v biologii I – 12 / 20
x(t+ 1) = rx(t), x(0) = x0
rustovy koeficient
r =x(t+ 1)
x(t)
zavisı na velikosti populacer = r
(
x(t))
![Page 90: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/90.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)
![Page 91: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/91.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
![Page 92: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/92.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
![Page 93: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/93.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
![Page 94: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/94.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
Ricker:x(t+ 1) = r1−
x(t)K x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
![Page 95: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/95.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
∆x(t) = (r − 1)x(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
Ricker:x(t+ 1) = r1−
x(t)K x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
![Page 96: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/96.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
∆x(t) = (r − 1)x(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
∆x(t) = (r − 1)
(
1−x(t)
K
)
x(t)
Ricker:x(t+ 1) = r1−
x(t)K x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
![Page 97: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/97.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
∆x(t) = (r − 1)x(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
∆x(t) = (r − 1)
(
1−x(t)
K
)
x(t)
Ricker:x(t+ 1) = r1−
x(t)K x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
∆x(t) =(r − 1)
1 + (r − 1)x(t)
K
(
1−x(t)
K
)
x(t)
![Page 98: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/98.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
∆x(t) = (r − 1)x(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
∆x(t) = (r − 1)
(
1−x(t)
K
)
x(t)
Ricker:x(t+ 1) = r1−
x(t)K x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
∆x(t) =r − 1
r
(
1−x(t)
K
)
x(t+ 1)
![Page 99: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/99.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
∆x(t) = (r − 1)x(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
∆x(t) = (r − 1)
(
1−x(t)
K
)
x(t)
Ricker:x(t+ 1) = r1−
x(t)K x(t)
∆x(t) =(
r1−x(t)K − 1
)
x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
∆x(t) =r − 1
r
(
1−x(t)
K
)
x(t+ 1)
![Page 100: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/100.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)Malthus:x(t+ 1) = rx(t)
∆x(t) = (r − 1)x(t)
Verhulst:
x(t+ 1) =
(
r − (r − 1)x(t)
K
)
x(t)
∆x(t) = (r − 1)
(
1−x(t)
K
)
x(t)
Ricker:x(t+ 1) = r1−
x(t)K x(t)
∆x(t) =(
r1−x(t)K − 1
)
x(t)
Beverton-Holt, Pielou:
x(t+ 1) =r
1 + (r − 1)x(t)
K
x(t)
∆x(t) =r − 1
r
(
1−x(t)
K
)
x(t+ 1)
![Page 101: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/101.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)
![Page 102: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/102.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)
Zakladnı rovnice:
x(t+ 1) =r
1 + (r − 1)
(
x(t)
K
)βx(t)
∆x(t) =r − 1
1 + (r − 1)
(
x(t)
K
)β
(
1−
(
x(t)
K
)β)
x(t) =r − 1
r
(
1−
(
x(t)
K
)β)
x(t+ 1)
![Page 103: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/103.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
x(t)K
1
r
x(t+ 1)
x(t)
Rovnice se zpozdenım:
x(t+ 1) =r
1 + (r − 1)
(
x(t− 1)
K
)βx(t)
![Page 104: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/104.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
Warder Clyde Allee (1885–1955)
x(t)Kϑ
1
r
x(t+ 1)
x(t)
Allee:
x(t+ 1) = r4K
(K−ϑ)2
(
1−x(t)K
)
(x−ϑ)x(t)
![Page 105: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/105.jpg)
Populace s omezenymi zdroji
Matematika v biologii I – 13 / 20
Benjamin Gompertz (1779–1865)
x(t)Kϑ
1
r
x(t+ 1)
x(t)
Gompertz:
x(t+ 1) =(
rx(t)−ln rln K
)
x(t)
![Page 106: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/106.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
![Page 107: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/107.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
Rust jedne populace (Rickeruv model)
x(t+ 1) = r1−x(t)K x(t), x(0) = x0
![Page 108: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/108.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
Rust jedne populace (Rickeruv model)
x(t+ 1) = r1−x(t)K x(t), x(0) = x0
rustovy koeficient: r1−x(t)K klesa s rostoucı velikostı populace
![Page 109: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/109.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
Rust jedne populace (Rickeruv model)
x(t+ 1) = r1−x(t)K x(t), x(0) = x0
rustovy koeficient: r1−x(t)K klesa s rostoucı velikostı populace
x(t), y(t) – velikosti dvou interagujıcıch populacı v case t
![Page 110: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/110.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
Rust jedne populace (Rickeruv model)
x(t+ 1) = r1−x(t)K x(t), x(0) = x0
rustovy koeficient: r1−x(t)K klesa s rostoucı velikostı populace
x(t), y(t) – velikosti dvou interagujıcıch populacı v case t
Lotka-Volterra:
x(t+ 1) = r1− x(t)
K1−α12y(t)
1 x(t), x(0) = x0
y(t+ 1) = r1− y(t)
K2−α21x(t)
2 y(t), y(0) = y0
![Page 111: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/111.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
Rust jedne populace (Rickeruv model)
x(t+ 1) = r1−x(t)K x(t), x(0) = x0
rustovy koeficient: r1−x(t)K klesa s rostoucı velikostı populace
x(t), y(t) – velikosti dvou interagujıcıch populacı v case t
Lotka-Volterra:
x(t+ 1) = r1− x(t)
K1−α12y(t)
1 x(t), x(0) = x0
y(t+ 1) = r1− y(t)
K2−α21x(t)
2 y(t), y(0) = y0
α12 > 0, α21 > 0 – konkurence, kompetice
![Page 112: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/112.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
Rust jedne populace (Rickeruv model)
x(t+ 1) = r1−x(t)K x(t), x(0) = x0
rustovy koeficient: r1−x(t)K klesa s rostoucı velikostı populace
x(t), y(t) – velikosti dvou interagujıcıch populacı v case t
Lotka-Volterra:
x(t+ 1) = r1− x(t)
K1−α12y(t)
1 x(t), x(0) = x0
y(t+ 1) = r1− y(t)
K2−α21x(t)
2 y(t), y(0) = y0
α12 > 0, α21 > 0 – konkurence, kompeticeα12 < 0, α21 < 0 – mutualismus, symbioza
![Page 113: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/113.jpg)
Interagujıcı populace
Matematika v biologii I – 14 / 20
Rust jedne populace (Rickeruv model)
x(t+ 1) = r1−x(t)K x(t), x(0) = x0
rustovy koeficient: r1−x(t)K klesa s rostoucı velikostı populace
x(t), y(t) – velikosti dvou interagujıcıch populacı v case t
Lotka-Volterra:
x(t+ 1) = r1− x(t)
K1−α12y(t)
1 x(t), x(0) = x0
y(t+ 1) = r1− y(t)
K2−α21x(t)
2 y(t), y(0) = y0
α12 > 0, α21 > 0 – konkurence, kompeticeα12 < 0, α21 < 0 – mutualismus, symbioza
α12 > 0, α21 < 0 – predace (populace s velikostı x je korist, s velikostı y je dravec)
![Page 114: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/114.jpg)
Lekarska diagnostika
Uvod
Lindenmayerovysystemy
Maticove populacnımodely
Rust homogennıpopulacev diskretnım case
Lekarska diagnostika
Hardyho-Weinberguvzakon
K dalsımu ctenı
Matematika v biologii I – 15 / 20
![Page 115: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/115.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
![Page 116: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/116.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence chorobyP(+|A) – senzitivita testuP(−|AC) – specificita testu
![Page 117: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/117.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
![Page 118: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/118.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
![Page 119: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/119.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA
AC
Σ
![Page 120: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/120.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA
AC
Σ 1 000 000
![Page 121: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/121.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA 5 000AC
Σ 1 000 000
![Page 122: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/122.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA 5 000AC 995 000Σ 1 000 000
![Page 123: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/123.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA 4 950 5 000AC 995 000Σ 1 000 000
![Page 124: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/124.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA 4 950 50 5 000AC 995 000Σ 1 000 000
![Page 125: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/125.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA 4 950 50 5 000AC 945 250 995 000Σ 1 000 000
![Page 126: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/126.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA 4 950 50 5 000AC 49 750 945 250 995 000Σ 1 000 000
![Page 127: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/127.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =
+ − ΣA 4 950 50 5 000AC 49 750 945 250 995 000Σ 54 700 945 300 1 000 000
![Page 128: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/128.jpg)
Matematika v biologii I – 16 / 20
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) = 0,0905
+ − ΣA 4 950 50 5 000AC 49 750 945 250 995 000Σ 54 700 945 300 1 000 000
![Page 129: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/129.jpg)
Matematika v biologii I – 16 / 20
Thomas Bayes (1701–1761)
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =P(+|A) P(A)
P(+|A) P(A) + P(+|AC) P(AC)
![Page 130: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/130.jpg)
Matematika v biologii I – 16 / 20
Thomas Bayes (1701–1761)
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =P(+|A) P(A)
P(+|A) P(A) +(
1− P(−|AC))(
1− P(A))
![Page 131: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/131.jpg)
Matematika v biologii I – 16 / 20
Thomas Bayes (1701–1761)
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+) =P(+|A) P(A)
P(+|A) P(A) +(
1− P(−|AC))(
1− P(A)) = 0,0905
![Page 132: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/132.jpg)
Matematika v biologii I – 16 / 20
Thomas Bayes (1701–1761)
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|++) =P(+|A) P(A|+)
P(+|A) P(A|+) +(
1− P(−|AC))(
1− P(A|+)) = 0,663
![Page 133: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/133.jpg)
Matematika v biologii I – 16 / 20
Thomas Bayes (1701–1761)
Stav pacienta: ma chorobu A
nema chorobu AC
Test diagnostikuje chorobu +nediagnostikuje chorobu −
P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95
P(A|+++) =P(+|A) P(A|++)
P(+|A) P(A|++) +(
1− P(−|AC))(
1− P(A|++)) = 0,952
![Page 134: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/134.jpg)
Hardyho-Weinberguv zakon
Uvod
Lindenmayerovysystemy
Maticove populacnımodely
Rust homogennıpopulacev diskretnım case
Lekarska diagnostika
Hardyho-Weinberguvzakon
K dalsımu ctenı
Matematika v biologii I – 17 / 20
![Page 135: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/135.jpg)
Matematika v biologii I – 18 / 20
Godfrey Harold Hardy (1877–1947) Science (1908)Wilhelm Weinberg (1862–1937) Jahresh. Wuertt. Ver. vaterl. Natkd. (1908)
![Page 136: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/136.jpg)
Matematika v biologii I – 18 / 20
Godfrey Harold Hardy (1877–1947) Science (1908)Wilhelm Weinberg (1862–1937) Jahresh. Wuertt. Ver. vaterl. Natkd. (1908)
dospely samec
dospela samice
gameta
gametazygota dospely jedinec-
-
-
���*HHHj
![Page 137: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/137.jpg)
Matematika v biologii I – 18 / 20
Godfrey Harold Hardy (1877–1947) Science (1908)Wilhelm Weinberg (1862–1937) Jahresh. Wuertt. Ver. vaterl. Natkd. (1908)
dospely samec
dospela samice
gameta
gametazygota dospely jedinec-
-
-
���*HHHj
dospely jedinec: diploidnı (2 alely v lokusu)gameta: haploidnı (1 alela v lokusu)
![Page 138: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/138.jpg)
Matematika v biologii I – 18 / 20
Godfrey Harold Hardy (1877–1947) Science (1908)Wilhelm Weinberg (1862–1937) Jahresh. Wuertt. Ver. vaterl. Natkd. (1908)
dospely samec
dospela samice
gameta
gametazygota dospely jedinec-
-
-
���*HHHj
dospely jedinec: diploidnı (2 alely v lokusu)gameta: haploidnı (1 alela v lokusu)
• plodnost nezavisı na genotypu• prezitı nezavisı na genotypu• alela je do gamety vybrana nahodne• nahodne krızenı
![Page 139: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/139.jpg)
Matematika v biologii I – 18 / 20
Lokus A s alelami a1, a2.
![Page 140: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/140.jpg)
Matematika v biologii I – 18 / 20
Lokus A s alelami a1, a2.Rodicovska generace: x = P(a1a1), y = P(a1a2), z = P(a2a2)
![Page 141: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/141.jpg)
Matematika v biologii I – 18 / 20
Lokus A s alelami a1, a2.Rodicovska generace: x = P(a1a1), y = P(a1a2), z = P(a2a2)
p = P(a1) = x+ 12y, q = P(a2) =
12y + z
![Page 142: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/142.jpg)
Matematika v biologii I – 18 / 20
Lokus A s alelami a1, a2.Rodicovska generace: x = P(a1a1), y = P(a1a2), z = P(a2a2)
p = P(a1) = x+ 12y, q = P(a2) =
12y + z
Vytvorenı gamety:◦
a1a1
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![Page 143: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/143.jpg)
Matematika v biologii I – 18 / 20
Lokus A s alelami a1, a2.Rodicovska generace: x = P(a1a1), y = P(a1a2), z = P(a2a2)
p = P(a1) = x+ 12y, q = P(a2) =
12y + z
Vytvorenı gamety:◦
a1a1
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p q p q
p q
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AA
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HHHH
Spojenı gamet:
![Page 144: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/144.jpg)
Matematika v biologii I – 18 / 20
Lokus A s alelami a1, a2.Rodicovska generace: x = P(a1a1), y = P(a1a2), z = P(a2a2)
p = P(a1) = x+ 12y, q = P(a2) =
12y + z
Vytvorenı gamety:◦
a1a1
�
� a1a2
�
� a2a2
�
�
a1m a1m a2m a2m��
��
AA
AA
���
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x y z
1 112
12
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a1ma1m a1ma2m a2ma1m a2ma2m
a1m a2m
p q p q
p q
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AA
AA
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HHHH
Spojenı gamet:
Filialnı generace: P(a1a1) = p2, P(a1a2) = pq + qp = 2pq, P(a2a2) = q2
![Page 145: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/145.jpg)
Matematika v biologii I – 18 / 20
Lokus A s alelami a1, a2.Rodicovska generace: x = P(a1a1), y = P(a1a2), z = P(a2a2)
p = P(a1) = x+ 12y, q = P(a2) =
12y + z
Vytvorenı gamety:◦
a1a1
�
� a1a2
�
� a2a2
�
�
a1m a1m a2m a2m��
��
AA
AA
���
���
@@@
@@@
x y z
1 112
12
◦
a1ma1m a1ma2m a2ma1m a2ma2m
a1m a2m
p q p q
p q
��
��
AA
AA
��
��
AA
AA
����
HHHH
Spojenı gamet:
Filialnı generace: P(a1a1) = p2, P(a1a2) = pq + qp = 2pq, P(a2a2) = q2
P(a1) = p2 + pq = p(p+ q) = p, P(a2) = pq + q2 = (p+ q)q = q
![Page 146: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/146.jpg)
K dalsımu ctenı
Uvod
Lindenmayerovysystemy
Maticove populacnımodely
Rust homogennıpopulacev diskretnım case
Lekarska diagnostika
Hardyho-Weinberguvzakon
K dalsımu ctenı
Matematika v biologii I – 19 / 20
![Page 147: Matematika v biologii I - Masaryk Universitypospisil/FILES/Matematika_v_biologii_I.pdf · Matematika v biologii I – 3 / 20. Syst´emy s diskr´etn´ım ˇcasem a paraleln´ım pˇrepisov´an´ım](https://reader030.vdocuments.pub/reader030/viewer/2022040321/5e5347ca30985962c3168ba2/html5/thumbnails/147.jpg)
Matematika v biologii I – 20 / 20
• John Maynard Smith. Mathematical Ideas in Biology. Cambridge Univ. Press: Cambridge, 1968.
• Tomas Havranek a kol. Matematika pro biologicke a lekarske vedy Academia: Praha, 1981.
• James Dickson Murray. Mathematical Biology. I An Introduction, II Spatial Models and Biomedical Applications.
Springer: New York-Berlin-Heidelberg, 2001, 2003.
• Hal Caswell. Matrix Population Models. Sinauer: Sunderland MA, 2001.
• Mark Kot. Elements of Mathematical Ecology. Cambridge Univ. Press: Cambridge, 2001.
• Kurt Kreith, Don Chakerian. Iterative Algebra and Dynamic Modeling. Springer: New York, 1999.
• Vojtech Jarosık. Rust a regulace populacı. Academia: Praha, 2005.
• Anthony W.F. Edwards. Foundations of Mathematical Genetics. Cambridge Univ. Press: Cambridge, 1977.
• John A. Adam. Mathematics in Nature. Modeling Patterns in the Natural World. Princeton University Press:Princeton&Oxford, 2003.
• 7 ruznych autoru SSSR. Cıslo a myslenı. Mlada fronta: Praha, 1983.
• Ian Stewart. Cısla prırody. Neskutecna skutecnost matematicke predstavivosti. Archa: Bratislava, 1996.