Tema_6 - Universidad de La Laguna

!"#$!%&/!-0($&'( )*+,&%!*"(-.
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! "#$%&' () *# +,-./&(#(! !" #$%&!'("(
() m )*+)#')* N1, . . . , Nm ,$-).!"(" +$. )/ *'*0
1)%"2
dNi
= Fi(N1, . . . , Nm)
dt
1 ≤ i ≤ m,
3 #$!4,&."#'5! () )6&'/'-.'$ N ∗ = (N1∗, . . . , Nm∗ )2
∗)=0
Fi(N1∗, . . . , Nm
1 ≤ i ≤ m.
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/'-.'$2
Ni = xi + Ni∗
dxi
∗ +x )
= Fi(x1 + N1∗, . . . , Nm
m
dt
m
X
∂Fi
(N ∗)xj + t.o.s.
=
j=1 ∂xj
=
:#&"#'5! /'!)"/';"("2
m
X
dxi
aij xj
=
dt
j=1
m
X
aij xj + t.o.s.
j=1
dx
= Ax.
dt
• A
.
! "!#$%& '(
/!/(
'(
*0
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aij 1 aji2
m
X
dxi
aisxs = · · · + aij xj + · · ·
=
dt
s=1
-
m
X
dxj
=
ajr xr = · · · + ajixi + · · ·
dt
r=1
aij = (3()#*
(0/()%(
j
0*4$(
7
8
7
77
78
8
87
88
−
↑ σ(aji)
−+
−0
• ++
! 5!$%!)%6, (0/()%(
−
+−
0−
−−
← σ(aij )
"+#+! %0"*1 0%"4%*0%01 )**/($!)%6,-
• −−
)*"/(#%)%6,-
• +−
/$(0!9'(/$('!'*$-
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∂F
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i
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
#
"

dN1
N1 + b12N2


=
r
N
1
−

1 1

 dt
K1
#
"


N2 + b21N1
dN2


=
r
N
1
−

2 2

dt
K2
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+# N1, N2 $'& %&"% #%& $% -%(./0'('&, *! %&"% #%
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• b12
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• b21
<$#2%7&#'7!*#=! #97>
Ni
,
ui =
Ki
τ = r1 t
r2
ρ=
r1

du1


= u1(1 − u1 − a12u2)

dτ
du


 2 = ρu2(1 − u2 − a21u1)
dτ
b N
• a12 = 12 2
K1
,
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%./#' *( 01 +.# $! )() )' #',$#%.% ) %+.! /*'%
+(#( *( '%+', ' 02
b N
• a21 = 21 1
K2
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+(#( *( '%+', ' -2
!"! #$%&'&()&*+
• (u∗1, u∗2) = (0, 0) !"#$%#&$ ' ()* '+* *, %# *• (u∗1, u∗2) = (1, 0) !"#$%#&$ ' () *, %# .• (u∗1, u∗2) = (0, 1) !"#$%#&$ ' () *, %# /• 0+ !#*" $%#)
(u∗1, u∗2)
=
Ã
1 − a12
1 − a21
,
1 − a12a21 1 − a12a21
(+ 12 3 12# 3 4
a12 > 1
&
a21 > 1
a12 < 1
&
a21 < 1.
&
!
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• (0, 0) ! "# !$%&' (#)*)+,
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* '% !. -" 2 !)&/ '% 3 !0. /% '% -%.%-"*%*
* !).)/$ * 4!$%,
(1, 0) =
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"# !$%&' !" a
21 < 1
.
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12 < 1
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"
#

N1
N2
dN1


= r 1 N1 1 −
+b12


 dt
K1
K1
"
#


N2
N1
dN2


= r 2 N2 1 −
+b12


dt
K2
K2
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Ni
,
ui =
Ki
τ = r1 t
r2
ρ=
r1

du1



= u1(1 − u1 + a12u2)

dτ

du2



= ρu2(1 − u2 + a21u1)
dτ
,
• 234#*#5(#'& #.%&)!5*%& (0, 0)6 (1, 0)6 (0, 1)+
•
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1 + a12 1 + a21
∗
∗
,
,
(u1, u2) =
δ
δ
('.() δ = 1 − a12a211 2,#.-3-#*/4).-)
,# δ > 01
µ
¶
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5"/.(' δ > 0 .' 6/7 *')+#,-).*#/ 7 $/, (',
),8)*#), )+8$'-/. ). -#)48' 9.#-'1
•
! "#$%#&'($)*+'# #,-).('#&)#/ ,#.0-)#+*,
%*&)12)-(,! !" #$%&'()! *+,()-('"
N ∗(t + T ) = N ∗(t)
-+ &! #(#.+/"
dNi
= fi(N1, . . . , Nm)
dt
t∈R
i = 1, . . . , m
#+ -('+ 0$,1(."%/+!.+ "#(!.).('"/+!.+2 +#."1%+3
#( +# +#."1%+ 4 #( "-+/5# ∃η > 0 ."% 6&+ *","
.$-$ |N0 − N ∗(0)| < η +7(#.+ τ > 0 '$!8
lı́m |N (t) − N ∗(t)| = 0.
t→∞
9:;: <$%&'($!+# *+,()-('"# +! *%"!$:
3*#&*$( 4 =>$(!'",?@A+!-(7$!B! !" N (t) #$%&'
i = f (N , N ).
()*+ ,! %" !(&"()*+ C 1 -%"+" dN
1
2
i
dt
i = 1, 2/ ) N (t) ∈ K -"0" t ≥ 0. ,$+,! K
!# &+ ($1-"(2$ #)+ -&+2$# ,! !3&)%)40)$. !+'
2$+(!# !5)#2! &+" #$%&()*+ -!0)*,)(" N ∗(t) !+
K 2"% 3&!6
|N (t) − N ∗(t)| → 0
t → o∞.
•
!"#$ #%&!'( )*!+','!")&(+'( (*')-#(.
!"! #$%&'(%&)*+$ ,-./&0($ ( 1* +21&'&3-&)!
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,"-&'&" +! !.#".&*$!/ C k 0 k ≥ 20 -1+&-!$/&*$"'!/
dNi
#3*$4"-*/
dt = fi(λ, N1 , . . . , Nm)0 λ ∈ R2
%#!5
67
∂f
µ(λ) ± iω(λ) ∈ σ(A(λ))0 A(λ) = ∂N
(λ, N ∗)2
87
µ(0) = 00 ω(0) > 00 σ(A(0)) ∩ {<z = 0} =
±ω(0)i2
97
dµ
dλ (0)
6= 02
:$;*$.!/ !<&/;!$ ,#$.&*$!/ λ(s)0 T (s)0N (t, s)
3!)&=+&." +! 3!)&*+* T (s) > =)(&;" Γ(s)0 |s| < ε0
> #$ !$;*)$* U ⊂ Rm +! N ∗ ;"'!/ %#!5
&7
N (t, s) /*'#.&=$ 3!)&=+&."
2π 2
λ = λ(s) .*$ T (0) = ω(0)
.*))!/3*$+&!$;! "
&&7 Γ(s) !/ '" ?$&." =)(&;" +! 3!)&*+* T (s) .*$1
;!$&+" !$ U 2
! "#$%&'($ )*&$(+,&-*&.(./* /$0#1(%/*#/$!

µ
¶
N
dN



= rN 1 −
− P N R(N )

dt
K
µ
¶

hP
dP



= kP 1 −
dt
N
R(N ) !"#$!"%& '$()*+(&, -!, -!# !-&-+ & ,&
.& *&)*/( !( !, (01! + -! # !"&"2 3#)*+(!"
#& & N R(N )4 a) = AN 4 b) = AN/(N + B)4
c) = AN 2/(N 2 + B 2)4 d) = a[1 − e−aN ]2
!"# $%&'('%
)*
&#)*(#+
%(!*",%!'-#+.

dN



= rN

N
1−
K
µ
¶
− P N R(N )
dt

dP



= kP (−d + eN R(N ))
dt
/% #01'2, 34

¶
µ
dN
AN
N



−P
= rN 1 −

dt
K
N +B
µ
¶

hP
dP



= kP 1 −
.
dt
N
5)'&*,+'#,%('6%1'2,.
N
u= ,
K
a=
A
,
rh
h
v = P,
K
k
b= ,
r
τ = rt
d=
B
.
K

du
v



= u (1 − u) − a

dt
v+d
µ
¶

u
dv



= bv 1 −
.
dt
v
!"#$#%&#' (u∗, v∗)(
u∗2 + (a + d − 1)u∗ − d = 0.
u∗ = v ∗,
u∗
=
(1 − a − d) +
)*+%#$#,+,(
u
A=

,-*
∗
·
au∗
(u∗ +d)2
q
(1 − a − d)2 + 4d
2

¸
−au∗
u∗ +d 
−1
b
"
.
−b
#
.

ad
∗ > 0.
A=
+
1
bu
(u∗ + d)2
(u∗, v ∗) =
-)*+%$- )# .&+/+A < 0
#0-)*+%$- )# .&+/+A > 0.

