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Matematika v biologii I

Zdenek Pospısil

22. brezna 2011

Uvod

Uvod

Lindenmayerovysystemy

Maticove populacnımodely

Rust homogennıpopulacev diskretnım case

Lekarska diagnostika

Hardyho-Weinberguvzakon

K dalsımu ctenı

Matematika v biologii I – 2 / 20

Lindenmayerovy systemy

Uvod

LindenmayerovysystemySystemy sdiskretnım casem aparalelnımprepisovanım





K dalsımu ctenı


Systemy s diskretnım casem a paralelnım

prepisovanım


Aristid Lindenmayer (1925–1989)


prepisovanım



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)


prepisovanım



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→ )


prepisovanım



6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )

s0 =1


prepisovanım



6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )

s0 =1

s1 =23


prepisovanım



6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )

s0 =1

s1 =23

s2 =224


prepisovanım



6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )

s0 =1

s1 =23

s2 =224

s3 =2254


prepisovanım



6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )

s0 =1

s1 =23

s2 =224

s3 =2254

s4 =22654


prepisovanım



6 7→ 7 7 7→ 8(1) 8 7→ 8 ( 7→ ( ) 7→ )

s0 =1

s1 =23 s5 =227654

s2 =224

s3 =2254

s4 =22654


prepisovanım



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


prepisovanım



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


prepisovanım



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


prepisovanım



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


prepisovanım



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


Abeceda: M , S, +, −, [, ]Axiom: MPravidla: M 7→ S[+M ][−M ]SM S 7→ SS



i = 0



i = 1



i = 2



i = 3



i = 4



i = 5

Maticove populacnı modely

Uvod



Fibonacciovi kralıci

Leslieho populacePopulacestrukturovana podlestadiıObecny maticovymodel




K dalsımu ctenı




Leonardo Pisansky, Fibonacci (1170–1250)



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) = x(t− 1) + x(t− 2)

x(t+ 2) = x(t+ 1) + x(t), x(0) = 1, x(1) = 2






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






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



x(t) – mnozstvı paru juvenilnıch kralıku v t-tem mesıci

y(t) – mnozstvı paru plodnych kralıku v t-tem mesıci



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) – 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)

)



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)

)






(

x

y

)

(t+ 1) =

(

σ1(1− γ) b

σ1γ σ2

)(

x

y

)

(t)






(

x

y

)

(t+ 1) =

(

σ1(1− γ) b

σ1γ σ2

)(

x

y

)

(t)

x(t+ 1) = Ax(t)

Leslieho populace


Patrick Holt Leslie (1900–1972)

Leslieho populace


xi(t) – pocet zen veku i let; presneji veku z intervalu (i− 1, i〉 letpi – podıl zen veku i, ktere prezijı do dalsıho roku

Leslieho populace


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

Leslieho populace


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

Leslieho populace



x1(t+ 1)= f1x1(t) + f2x2(t) + · · ·+ fkxk(t)

xi+1(t+ 1)= pixi(t), i = 1, 2, . . . , k − 1

Leslieho populace



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)

Leslieho populace



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)

Populace strukturovana podle stadiı


Leonard Lefkowitch (1929–2010)



Obojzivelnıci: vajıcko – pulec – dospely jedinec



Obojzivelnıci: vajıcko – pulec – dospely jedinecHmyz: vajıcko – larva – kukla – imago



Obojzivelnıci: vajıcko – pulec – dospely jedinecHmyz: vajıcko – larva – kukla – imago

Rostliny: semeno – mala ruzice – strednı ruzice – velka ruzice – kvetoucı rostlina



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




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




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)

Obecny maticovy model


x(t+ 1) = Ax(t)



x(t+ 1) = Ax(t)

λ1, λ2, . . . , λk – vlastnı hodnoty matice A, λ1 ≥ |λ2| ≥ · · · ≥ |λk|w1,w2, . . . ,wk – prıslusne vlastnı vektory



x(t+ 1) = Ax(t)


Pro jednoduchost predpokladejme, ze λ1 > |λ2| a λi 6= λj pro i 6= j.



x(t+ 1) = Ax(t)



x(0) = c1w1 + c2w2 + · · ·+ ckwk



x(t+ 1) = Ax(t)



x(0) = c1w1 + c2w2 + · · ·+ ckwk

x(1) = c1λ1w1 + c2λ2w2 + · · ·+ ckλkwk



x(t+ 1) = Ax(t)



x(0) = c1w1 + c2w2 + · · ·+ ckwk

x(1) = c1λ1w1 + c2λ2w2 + · · ·+ ckλkwk

x(t) = c1λt1w1 + c2λ

t2w2 + · · ·+ ckλ

tkwk



x(t+ 1) = Ax(t)



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

Rust homogennı populace v diskretnım case

Uvod



Rust homogennıpopulacev diskretnım casePopulace bezomezenıPopulace somezenymi zdroji

Interagujıcı populace



K dalsımu ctenı


Populace bez omezenı


Leonhard Euler (1707–1783), Introductio in analysin infinitorum (1748)



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





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





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





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(t+ 1) = rx(t)

Rekurentnı formule pro geometrickou posloupnost






x(t+ 1) = rx(t)

x(0) = x0 – pocatecnı velikost populace

x(t) = x0rt



Thomas R. Malthus (1766–1834)




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



x(t+ 1) = rx(t), x(0) = x0



x(t+ 1) = rx(t), x(0) = x0

rustovy koeficient

r =x(t+ 1)

x(t)



x(t+ 1) = rx(t), x(0) = x0

rustovy koeficient

r =x(t+ 1)

x(t)

zavisı na velikosti populacer = r

(

x(t))

Populace s omezenymi zdroji


x(t)K

1

r

x(t+ 1)

x(t)



x(t)K

1

r

x(t+ 1)

x(t)Malthus:x(t+ 1) = rx(t)



x(t)K

1

r

x(t+ 1)


Verhulst:

x(t+ 1) =

(

r − (r − 1)x(t)

K

)

x(t)



x(t)K

1

r

x(t+ 1)


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)



x(t)K

1

r

x(t+ 1)


Verhulst:

x(t+ 1) =

(

r − (r − 1)x(t)

K

)

x(t)

Ricker:x(t+ 1) = r1−

x(t)K x(t)


x(t+ 1) =r

1 + (r − 1)x(t)

K

x(t)



x(t)K

1

r

x(t+ 1)


∆x(t) = (r − 1)x(t)

Verhulst:

x(t+ 1) =

(

r − (r − 1)x(t)

K

)

x(t)


x(t)K x(t)


x(t+ 1) =r

1 + (r − 1)x(t)

K

x(t)



x(t)K

1

r

x(t+ 1)


∆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)


x(t)K x(t)


x(t+ 1) =r

1 + (r − 1)x(t)

K

x(t)



x(t)K

1

r

x(t+ 1)


∆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)


x(t)K x(t)


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)



x(t)K

1

r

x(t+ 1)


∆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)


x(t)K x(t)


x(t+ 1) =r

1 + (r − 1)x(t)

K

x(t)

∆x(t) =r − 1

r

(

1−x(t)

K

)

x(t+ 1)



x(t)K

1

r

x(t+ 1)


∆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)


x(t)K x(t)

∆x(t) =(

r1−x(t)K − 1

)

x(t)


x(t+ 1) =r

1 + (r − 1)x(t)

K

x(t)

∆x(t) =r − 1

r

(

1−x(t)

K

)

x(t+ 1)



x(t)K

1

r

x(t+ 1)

x(t)



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)



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)



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)



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)



Rust jedne populace (Rickeruv model)

x(t+ 1) = r1−x(t)K x(t), x(0) = x0




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+ 1) = r1−x(t)K x(t), x(0) = x0


x(t), y(t) – velikosti dvou interagujıcıch populacı v case t




x(t+ 1) = r1−x(t)K x(t), x(0) = x0



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




x(t+ 1) = r1−x(t)K x(t), x(0) = x0



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




x(t+ 1) = r1−x(t)K x(t), x(0) = x0



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




x(t+ 1) = r1−x(t)K x(t), x(0) = x0



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)


Uvod






K dalsımu ctenı



Stav pacienta: ma chorobu A

nema chorobu AC

Test diagnostikuje chorobu +nediagnostikuje chorobu −



nema chorobu AC


P(A) – prevalence nebo incidence chorobyP(+|A) – senzitivita testuP(−|AC) – specificita testu



nema chorobu AC


P(A) – prevalence nebo incidence choroby 0,005P(+|A) – senzitivita testu 0,99P(−|AC) – specificita testu 0,95



nema chorobu AC



P(A|+) =



nema chorobu AC



P(A|+) =

+ − ΣA

AC

Σ



nema chorobu AC



P(A|+) =

+ − ΣA

AC

Σ 1 000 000



nema chorobu AC



P(A|+) =

+ − ΣA 5 000AC

Σ 1 000 000



nema chorobu AC



P(A|+) =

+ − ΣA 5 000AC 995 000Σ 1 000 000



nema chorobu AC



P(A|+) =

+ − ΣA 4 950 5 000AC 995 000Σ 1 000 000



nema chorobu AC



P(A|+) =

+ − ΣA 4 950 50 5 000AC 995 000Σ 1 000 000



nema chorobu AC



P(A|+) =

+ − ΣA 4 950 50 5 000AC 945 250 995 000Σ 1 000 000



nema chorobu AC



P(A|+) =

+ − ΣA 4 950 50 5 000AC 49 750 945 250 995 000Σ 1 000 000



nema chorobu AC



P(A|+) =

+ − ΣA 4 950 50 5 000AC 49 750 945 250 995 000Σ 54 700 945 300 1 000 000



nema chorobu AC



P(A|+) = 0,0905

+ − ΣA 4 950 50 5 000AC 49 750 945 250 995 000Σ 54 700 945 300 1 000 000


Thomas Bayes (1701–1761)


nema chorobu AC



P(A|+) =P(+|A) P(A)

P(+|A) P(A) + P(+|AC) P(AC)




nema chorobu AC



P(A|+) =P(+|A) P(A)

P(+|A) P(A) +(

1− P(−|AC))(

1− P(A))




nema chorobu AC



P(A|+) =P(+|A) P(A)

P(+|A) P(A) +(

1− P(−|AC))(

1− P(A)) = 0,0905




nema chorobu AC



P(A|++) =P(+|A) P(A|+)

P(+|A) P(A|+) +(

1− P(−|AC))(

1− P(A|+)) = 0,663




nema chorobu AC



P(A|+++) =P(+|A) P(A|++)

P(+|A) P(A|++) +(

1− P(−|AC))(

1− P(A|++)) = 0,952

Hardyho-Weinberguv zakon

Uvod






K dalsımu ctenı



Godfrey Harold Hardy (1877–1947) Science (1908)Wilhelm Weinberg (1862–1937) Jahresh. Wuertt. Ver. vaterl. Natkd. (1908)



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gametazygota dospely jedinec-

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dospely jedinec: diploidnı (2 alely v lokusu)gameta: haploidnı (1 alela v lokusu)



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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


Lokus A s alelami a1, a2.


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



p = P(a1) = x+ 12y, q = P(a2) =

12y + z

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Spojenı gamet:

Filialnı generace: P(a1a1) = p2, P(a1a2) = pq + qp = 2pq, P(a2a2) = q2



p = P(a1) = x+ 12y, q = P(a2) =

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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

K dalsımu ctenı

Uvod






K dalsımu ctenı



• 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.

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