M7116 Maticové populační modely
Populace s interní variabilitou
11.5.2015
Populace strukturovaná podle plodnosti
(ni)(t+D = H1-^ v)(ni){t), (ni)(o)
\n2) V    ai7       a2) V12) V722/
A = A((7i, <72, 7, ty?) =
= \ ^ťJi (1 - 7) + cr2 + \j(cti(1 - 7) - cr2)2 + 4ai7(^
A > 1 (7i7^ > (1 - ct2)(l - Oi(l - 7))
Parametry závislé na velikosti složek populace
©c+y-CV :)(:)<«• (:)(oi=G)
ai = ai(nun2) = £ie-siini-si2n2 a2 = ai(nun2) = £2e-S2ini-S22n2 7 = 7(ni,n2) = Te-9ini-92n2 ip = ^(ni,n2) = $e-/ini~/2n2
A — A(cri,cr2,7,^) — A(m,n2) — A(n)
Ao = A(o)
Aoo —   lim A(n)
II TI II —>-00
Parametry závislé na velikosti složek populace
ai = ai(nun2) = £ie-siini-si2n2 a2 = ai(nun2) = £2e-S2ini-S22n2 7 = 7(ni,n2) = Te-9ini-92n2 ip = ^(ni,n2) = $e-/ini~/2n2
A — A(cri,cr2,7,^) — A(m,n2) — A(n)
Ao = A(o)
Aoo —   lim A(n)
II TI I —>-00
Um A(č7l, Č72,7, <p) = ^2
cri —>-0
lim A(c7i, cr2,7, <p) = \ (cn(l -7) + ~ 7)2 + 4ai7^
lim A(c7i, c72,7, <p) = cti
lim A(č7i,č72,7, <p) = cn(l -7)
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adults
0 1000
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o o o o o o o o o o o o o o o o o
3000 _I_
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juveniles 50000
150000 _I
Parametry závislé na velikosti složek populace
©c+y-CV :)(:)<«• (:)(oi=G)
ai = ai(nun2) = £ie-siini-si2n2 a2 = ai(nun2) = Y,2q~ s^ 7 = 7(ni,n2) = Te-9ini-92n2
ip = ^(ni,n2) = $e-/ini~/2n2 I
cd >
A — A(cri,cr2,7,^) — A(m,n2) — A(n) ^
A0 = A(o) = 1.8658 Aoo —   lim   A(n) — 0.1
II TI II —>-00
Si = 0.5, s2 = o.i, r = 0.1, <í> = 50
sn — si2 — 1, sn — si2 = 0, gi = g2 = 0, /i = f2 = 0 a
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Stabilizace populace zvětšením úmrtnosti juvenilních jedinců (infanticidou)
Parametry závislé na velikosti složek populace
n2 J V    ai7       a2) \ri2
7li
n2
(0) =
cti = ai(m,n2) = Sie-Slini-S12n2 a2 = ai(m,n2) = S2e-S2ini-S22n2 7 = 7(ni,n2) = Te'91711-92712
ip = (^(711,712) = $e-/ini-/2n2
A — A(č7i,čJ2,7, <p) — A(m,n2) - A(n)
A0 = A(o) = 1.8658
Aoo      lim   A(n) 1.8221
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10 time
15
20
Ei = 0.5, s2 = o.i, r = 0.1, <í> = 50
sn - si2 - 0, sn — S12 = 1, #i = g2 = 0, /i = /2 = 0
Zpomalení růstu populace zvětšením úmrtnosti plodných jedinců (při velké plodnosti)
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o o
lo
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1 o
15
/
20
time
Parametry závislé na velikosti složek populace
nA {t+i) = H1-
(ty
(0) =
(Ji = ai(m,n2) = Eie-Slini-S12n2 a2 = ai(m,n2) = E2e-S2ini-S22n2 7 = 7(m,n2) = Ve-91"11'9211"2
if = (^(711,712) = $e-/ini-/2n2
A - A(čJi,(72,7,    = A(m,n2) — A(n)
A0 = A(o) = 1.0204
Aoo =   lim  A(n) 0.9837
II TI II —^ OO
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0
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time
15
20
Ei = 0.5, E2 = 0.1, T = 0.1, $ = 10.5
sn = S12 — 0, sn - S12 = 1, gi = gi — 0, /i = /2 = 0
Stabilizace populace zvětšením úmrtnosti plodných jedinců (při malé plodnosti)
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15
20
time
Trajektorie a atraktory
ai = ai(m,n2) = Sie-Slini-Sl2n2 a2 = a1(n1,n2) = ^2e~s^-s^
7 = 7(ni,n2) = re-01711-02712 <p = <p(nun2) = $e-/ini~/2n2
Si = 0.5, s2 = o.i, r = 0.1, $ = 50,
Sil — Si2 — S21 — S22 = 0, gi = #2 = 0, /l = f<2. — 1
Population projection
Attractor
Rovnovážný bod
20
-1-
40
time
"T"
60
20
"T"
40
"T"
60
time
juveniles
Trajektorie a atraktory
n2J V ťJi7
(0) =
ai = cri(m,n2) = Sie-Slini-Sl2n2
7 = 7(711,712) =re-öini_ö2n2
a2 = ai(m,n2) = ^2e~s^-s^ <p = <p(m, n2) = $e-/ini-/2n2
Si = 0.5, £2 = 0.1, r = 0.1, <í> = 500,
Sil — S12 — S2I — S22 = 0, pi = #2 = 0, /l = f2 = 1
Population projection
Attractor
Cyklus délky 4
20
20
—i—
40 time
"T"
60
"T"
40
"T"
60
time
juveniles
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juveniles
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Trajektorie a atraktory
ri2 J V ai7
(t)
(0) =
ai = ai(m,n2) = Sie-Slini-Sl2n2 7 = 7(m,n2) =re-flini-92n2
a2 = cri(m,n2) = S2e-S2ini-S22n2
<P = <p(m,n2) = $e-/ini-/2n2
Ei = 0.5, S2 = 0.1, T = 0.1, $ - 1800,
Sil = S12 — S21 — S22 = 0, gi = 5ř2 = 0, /l = /2 = 1
Podivný atraktor
15.0
15.2
15.4
-1-
15.6
juveniles
15.8
16.0
16.2