üD×??
?
×??
?
` ˉ
^?
V?7y

"￥
^ù?
H
W sé[aV?￥ M t
] ——
ü
^?×?b
? p3±sZ?7
^¨ ±sZ?×??
?
ù?
ü
￥×??b
×??
?
1 }è<￥??
l
2 +!é ×
3 ? ?￥Moé?
4 ? ?￥MoGi
5 ? ?￥ D ? <
J
1 }è<￥??
l

3'÷
?<a
<?D
d
3'÷
?<?
?

3'÷?
a 7? ——??×
á -4/
LCKvá
KDrmb
ù5
#
s
 } ·
×? ￥Hq/ ? e
?} ·
Pá
KvrmKDb
?T
P} ·
??1 ?9é
 ?
?è
| ??M 5} ·
×?b
á

?
x(t) ~ ??è

)1()()(
N
x
rxxftx?==
L
!
í} ·
Hè￥1 ?9é?V Logistic
p
r~%μ9é
q,N~Kvè

?ê
HW} ·
D??è
??1
h(x)=Ex,E~} · <
)()()( xhxfxF?=:
y

Ex
N
x
rxxFtx== )1()()(
} · f ?/
??è

@
?31 p3 x(t),o3?? x(t)×?￥Hq
B¨±sZ?￥
ü?# ×??
)1()(xFx= B¨dL?
1íZ?
F(x)=0￥? x
0
~±sZ?￥
ü?
0
0
0
xxx
xx
≡?=
=

! x(t)
^Z?￥3 ?V x
0

#×￥ ?B′?
?μ,)(lim
0
xtx
t
=
∞→
? x
0
^Z? (1)￥ ×?
ü?
? p x(t),
 x
0
×??￥ZE ——°¤E
)2())((
00
xxxFx?
′
=
(1)￥í
L?Z?
))1(),2((0)(
00
×?xxF?<
′
))1(),2((0)(
00
?×?xxF?>
′
Ex
N
x
rxxFtx== )1()()(
á

?
0),1(
10
=?= x
r
E
Nx
0)( =xF
ü?
ErxFrExF?=
′
=
′
)(,)(
10
×??

0)(,0)(
10
>
′
<
′
< xFxFrE
?×?×?
10
,xx
0)(,0)(
10
<
′
>
′
> xFxFrE ×??×?
10
,xx
r~%μ9é
q
E~} · <
x
0
×?, V¤?×?á
 x
1
×?,??? j
á

?
} ·
×?￥Hq/
e?} · <
Pá
Kv
m3E
)()()( xhxfxF?=
)1()(
N
x
rxxf?=
Exxh =)(
0)( =xF
P￥US x
0
~
ü?
2//
*
0
*
rxhE
m
==
y=rx
h
P
x
0
y
0
y=h(x)=Ex
xN
y=f(x)
P￥:US h~á

)4/,2/(
*
0
*
rNhNxP
m
==
á
Kv
f D h?? P
×?
0
xrE?<
h
m
x
0
*
=N/2
P
*
y=E
*
x
e???è
1Kvè
￥B?
rm
?
} ·
×?￥Hq/ e?} ·
<
PrmKv,
L
!
èg
rNì p
?ê} · <n¨ c
l ? T = ph(x) = pEx | S = cE
cEpExSTR?=?=
?ê
HW ? ?
cE
r
E
pNEESETER=?= )1()()()(
)1(
4
22
2
Np
crN
h
R
=
)/1(
0
rENx?=×?
ü?
p E
P R(E)Kv
)1(
2 pN
cr
E
R
=
p
cN
22
+=
)1(
r
E
Nx
R
R
=
??
è

2
*
r
E =<
)1(
2 pN
cr
E
R
=
} ·
V
>
T} ·  p ? ? R(E)Kv
7b
T} · o p ? ? R(E) > 0
)1(
pN
c
rE
s
=
7
=0
cE
r
E
pNEESETER=?= )1()()()(
R(E)=0
H￥} · <(
"? <) E
s
=2E
R
"? </￥??è

E
s
S(E)
T(E)
0 r
E
)1(
r
E
Nx
s
s
=
p
c
=
E
R
E
*
↓↑
ss
xE,
↓↑ cp,
} ·V
Discussion
2 +!é ×
í
Z (SESE"v ) +!é ×V?
"￥
3
d (?? )
Z +!é ×￥2 
1??Mo?? ?BZ +!v
6B
Z +!9F y
L
!
2??ü6
L ?K?BZ +!v
1- +!9é￥??v
3??Mo±
j
5r?

BZ?i
9F +!￥ . ?b
éB?
L
!
12￥T¨1L? 3￥T¨1è
y

x(t)~JZ +!
 y(t)~YZ +!

gkyxtx ++?= α)(

hylxty +?= β)(

α,β ~ 'Zü6
L ?￥??
k,l ~ Z +!
￥O
g,h ~ 'Z +!é ×￥ . ?b
+!é ×￥2 
t →∞
H￥ x(t) y(t)
±sZ?￥
ü?# ×??
L?è"
±sZ?F
dycxty
byaxtx
+=
+=
)(
)(


￥
ü?# ×??
ü? P
0
(x
0
,y
0
)=(0,0) ~}
Z?
0
0
=+
=+
dycx
byax
￥?
?V P
0

#×￥ ?B′??μ
,)(lim
0
xtx
t
=
∞→
? P
0
^±sZ?￥ ×?
ü?
,)(lim
0
yty
t
=
∞→
:"
?
=
dc
ba
A
+?Z?
0)det( =? IA λ
=
+?=
=++
Aq
dap
qp
det
)(
0
2
λλ
+??
2/)4(
2
2,1
qpp?±?=λ
L?è"
±sZ?F
dycxty
byaxtx
+=
+=
)(
)(


￥
ü?# ×??
+??
2/)4(
2
2,1
qpp?±?=λ
ü? P
0
(0,0)
±sZ?B?3?
T
tt
ecec
21
21
λλ
+
λ
1,2
1μ
μμ
L?
p > 0 O q > 0
ü? P
0
(0,0)×?
p < 0 q < 0
ü? P
0
(0,0)?×?
+?=
++?=
hylxty
gkyxtx
β
α
)(
)(


?
+!é ×
kl
hgl
y
kl
gkh
x
+
=
+
=
αβ
α
αβ
β
00
,
ü?
×??

klAq
p
==
>+==
αβ
βαβα
det
0)(
=
β
α
l
k
A
"
?
ü? (x
0
,y
0
)×?￥Hq 0,0 >> qp
kl>αβ
?￥??3
d
kl>αβ
Z +!×? (
HW s
éa t_μK′ )￥Hq
ü?
kl
hgl
y
kl
gkh
x
+
=
+
=
αβ
α
αβ
β
00
,
+?=
++?=
hylxty
gkyxtx
β
α
)(
)(


?
α,β ~ 'Zü6
L ?￥??
k,l ~ Z +!
￥O
g,h ~ 'Z +!é ×￥ . ?b
1)
Zü6??v?
Z +!O
H +!é ×
?×??5 +!|íK ?fb
2) ? g=h=0,5 x
0
=y
0
=0, αβ > kl/ x(t),y(t)→0,
'?z
#SYV? + Vr?￥ò
üb
3 ? g,h ?1
,'L
ZB
H3
P

H x(t),y(t)
l?y 9×? +!b0,0 >> yx 
4'
P

HBZ (??^?xp ) +!vh, ? x(t)=0
9y
P?Z×? +!
gkyx +=
'io?? ? ( ) %μ?
( ) ￥?Z
? +??òb
0≠k
0≠g
?￥??3
d
α,β ~ 'Zü6
L ?￥??
k,l ~ Z +!
￥O
g,h ~ 'Z +!é ×￥ . ?b
+?=
++?=
hylxty
gkyxtx
β
α
)(
)(


?
Discussion
3 ? ?￥Moé?
B?1 ?ìa?μ
?? ?
3i
ì-W￥
1"Moé? MoGi D ? <
Jb
?
?? ?1? ]B
Jt ?÷
3i bWM
oé?
Hèn￥2 
^é? ? D￥
'
é? ? <￥r?ìa ??￥Kv ?
b
y ?
D
?
í
?? ?Moé?￥V?
sá
3??2 ￥Hqb
=
2
2
1
1
2222
1)(
N
x
N
x
xrtx σ
)1()(
1
1
111
N
x
xrtx?=
=
1
1
111
1)(
N
x
xrtx

?L
!
μJY
?? ?
ì?1
3i
H
M (?V Logistic
p ;
)1()(
2
2
222
N
x
xrtx?=
? ?B 
3i
HYJ9é￥EìT
¨DY￥
??1 ; JYμ]"￥T¨b
?h{J￥'÷7
yY (M? N
2
)
^J
(M? N
1
) ￥ σ
1
b
1
1
>σ
J9é￥Eì
T¨Yv?J
Y￥é? ? <
?
2
2
1
N
x
σ?
?
s
=
2
2
1
1
1
111
1)(
N
x
N
x
xrtx σ
=
2
2
1
1
2222
1)(
N
x
N
x
xrtx σ
￥ t_
H )(),(
21
txtxt ∞→ (
ü?# ×?? )
(=¨ )dL?
(1í )Z?
),()(
),()(
212
211
xxgtx
xxftx
=
=


￥
ü?# ×??
?
ü? P
0
(x
1
0
,x
2
0
) ~ }
Z?
0),(
0),(
21
21
=
=
xxg
xxf
￥?
?V P
0

#×￥ ?B′??μ
,)(lim
0
11
xtx
t
=
∞→
? P
0
^±sZ?￥ ×?
ü?
,)(lim
0
22
xtx
t
=
∞→
 P
0
(x
1
0
,x
2
0
) ×?
?￥ZE ——°¤E
)1(),()(
),()(
212
211
xxgtx
xxftx
=
=


(1)￥í
L?Z?
)2())(,())(,()(
))(,())(,()(
0
22
0
2
0
1
0
11
0
2
0
12
0
22
0
2
0
1
0
11
0
2
0
11
21
21
xxxxgxxxxgtx
xxxxfxxxxftx
xx
xx
+?=
+?=


=
+?=
=++
Aq
gfp
qp
Pxx
det
)(
0
021
2
λλ
ü? P
0
?×? ( 2,1)
p < 0 q < 0
0
21
21
P
xx
xx
gg
ff
A
=
ü? P
0
×? ( 2,1)
p > 0 O q > 0
?
),,0(),0,(
2211
NPNP
ü?
=
≡
=
≡
01),(
01),(
2
2
1
1
22221
2
2
1
1
1
1121
N
x
N
x
xrxxg
N
x
N
x
xrxxf
σ
σ
=
2
2
1
1
1
111
1)(
N
x
N
x
xrtx σ
=
2
2
1
1
2222
1)(
N
x
N
x
xrtx σ
)0,0(,
1
)1(
,
1
)1(
4
21
22
21
11
3
P
NN
P
σσ
σ
σσ
σ
?? σ
1
,σ
2
< 1σ
1
,σ
2
> 1
H P
3
?μil


=
=
2
2
1
12
2
1
222
2
111
2
21
1
1
1
21
21
2
1
2
1
N
x
N
x
r
N
xr
N
xr
N
x
N
x
r
gg
ff
A
xx
xx
σσ
σσ
ü?×
??s
4,3,2,1,det,)(
21
==+?= iAqgfp
i
pi
pxx
≡
≡
2
2
1
1
22221
2
2
1
1
1
1121
1),(
1),(
N
x
N
x
xrxxg
N
x
N
x
xrxxf
σ
σ
ü? P
i
×?Hq p > 0 O q > 0
? ?é?
?￥
ü?#×??
?×?
ü ?
)0,(
11
Np
)1(
221
σrr
p
q
)1(
221
σ rr
),0(
22
Np
211
)1( rr + σ
)1(
121
σ rr
21
22
21
11
3
1
)1(
,
1
)1(
σσ
σ
σσ
σ NN
p
21
2121
1
)1)(1(
σσ
σσ
rr
)0,0(
4
p )(
21
rr +?
21
rr
21
2211
1
)1()1(
σσ
σσ
+? rr
σ
2
>1,
σ
1
>1,
σ
1
<1
σ
2
<1
×?Hq
σ
1
<1,σ
2
<1
P
1
,P
2
^B?? ?i7
6B
'￥
ü?
P
3
^
? ?
i￥
ü? P
1
×?￥Hq σ
1
<1?
2
2
1
1
221
2
2
1
1
1
21
1),(
1),(
N
x
N
x
xx
N
x
N
x
xx
=
=
σψ
σ?
=
2
2
1
1
1
111
1)(
N
x
N
x
xrtx σ
=
2
2
1
1
2222
1)(
N
x
N
x
xrtx σ
ü?×
??￥M
ELs
(1) σ
2
>1,σ
1
<1
12
/σN
21
/σN
1
N
2
N
1
P
1
x
2
x
0
0=?
0=ψ
S
1
S
2
S
3
0,0:
1
>> ψ?S
V ?i?? (t=0)￥ME
L? t_ P
1
(N
1
,0) (t→∞)
P
1
(N
1
,0)
^×?
ü?
0,0:
212
<> xxS

0,0:
211
>> xxS

t ↑→x
1
,x
2
↑
0,0:
213
<< xxS

t ↑→x
1
↑,x
2
↓
t ↑→x
1
,x
2
↓
12
/σN
21
/σN
1
N
2
N?
2
P
1
x
2
x
0
0=?
0=ψ
(2) σ
1
>1,σ
2
<1
1
x
2
x
12
/σN
21
/σN
1
N
2
N
0
3
P
0=?
0=ψ
(3) σ
1
<1,σ
2
<1
P
2
×?
P
3
×?
P
1
P
2
μMEL t_ P
1
μMEL t_ P
2
P
1
×?￥Hq°¤E σ
2
>1
P
1
,P
2
??
( ? )×?
1
x
2
x
12
/σN
21
/σN
1
N
2
N
0
3
P
0=?
0=ψ
(4) σ
1
>1,σ
2
>1
F
D (4)M uY￥ σ
1
<1
P
1
? ×?
2T3
d
P
1
×?￥Hq σ
1
<1,σ
2
>1
1
1
<σ
J9é￥Eì
T¨Yl?J
Y￥é? ? D
?h{J￥'÷7
yY (M? N
2
)
^J
(M? N
1
)￥ σ
1
b
Jr?Kv ?
Y
'
σ
2
>1?J￥é? ? <
P
2
×?￥Hq σ
1
>1,σ
2
<1
P
3
×?￥Hq σ
1
<1,σ
2
<1
Yè σ
1
≈ 1/σ
2
 P
3
×?Hq?
@
Discussion
4 ? ?￥MoGi
JY
? ?￥MoGiμ ???
T
1) J V[?1
3iY?
?1
3i J
YB 
3i
HMo4
Jta\é9éb
2) JY ( V[?1
3i JYB 
3i
HMo4
Jta\é9éb
3) JY (?
?1
3i JYB 
3i
HMo4
Jta\é9éb
?
L
!
J V[?1
3i
M?V Logistic
p ;
JYB 
3i
HY1J4
Jta\é9éb
Y?
?1
3i JYB 
3i
HJ1Y
4
Jta\é9é Y￥9é?
s?'
&
￥EìT¨ (?V Logistic
p )b
=
1
1
1111
1)(
N
x
xrtx

2
2
1
N
x
σ+
Y1J4
Jt
^Jh{￥ σ
1

?
( )1)(
222
= xrtx

+?
1
1
222
1
N
x
xr σ
+?=
2
2
1
1
2222
1)(
N
x
N
x
xrtx σ
J1Y4
Jt
^Yh{￥ σ
2

? ?Gi
?￥
ü?#×??
×?Hq
?×?
1,1
212
<< σσσ
1
,1,1
21
21
<
><
σσ
σσ
ü?
p
q
)0,(
11
NP
)1(
221
σrr
)1(
221
σrr
)0,0(
3
P
21
rr +?
21
rr?
21
22
21
11
2
1
)1(
,
1
)1(
σσ
σ
σσ
σ NN
P
21
2121
1
)1)(1(
σσ
σσ
rr
21
2211
1
)1()1(
σσ
σσ
+? rr
P
2
^JYMoGi7
3￥
ü?
ü? P
2
×?
?￥MEL
21
22
21
11
2
1
)1(
,
1
)1(
σσ
σ
σσ
σ NN
P
),(1)(
2111
2
2
1
1
1
1111
xxxr
N
x
N
x
xrtx?σ =
+?=
),(1)(
2122
2
2
1
1
2222
xxxr
N
x
N
x
xrtx ψσ =
+?=
1
x
2
x
0
21
/σN
1
N
1
S
2
S
3
S
2
P
0=?
0=ψ
4
S
σ
1
<1,σ
2
>1,σ
1
σ
2
<1
.0,0:;0,0:;0,0:;0,0:
214
213
212
211
<<
><
>>
<>
xxS
xxS
xxS
xxS




P
2
×?
+?=
2
2
1
1
1
1111
1)(
N
x
N
x
xrtx σ

J V[?1
3i
+?=
2
2
1
1
2222
1)(
N
x
N
x
xrtx σ
Y?
? ?
3i
2T
3
d
21
22
21
11
2
1
)1(
,
1
)1(
σσ
σ
σσ
σ NN
P
P
2
×?Hq
σ
1
<1,σ
2
>1,σ
1
σ
2
<1
σ
2
>1 ~ JA?1Y4@￥
Jt ——
J1Y4￥
Jt
^Yh{￥ σ
2

σ
1
σ
2
<1 ~ σ
2
>1 -4/ P
2
i￥A1Hq
σ
1
<1 ~ σ
2
>1,σ
1
σ
2
<1￥31 O σ
1
A?@
l?
 σ
2
>1Hq/
P σ
1
σ
2
<1? ?
Discussion
5 ? ?￥ D ? <
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t x(t) y(t)
0 20.0000 4.0000
0.1000 21.2406 3.9651
0.2000 22.5649 3.9405
0.3000 23.9763 3.9269
… … …
5.1000 9.6162 16.7235
5.2000 9.0173 16.2064
… … …
9.5000 18.4750 4.0447
9.6000 19.6136 3.9968
9.7000 20.8311 3.9587
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=


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)(
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=
h ? dt
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y
ayr
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x
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ceyex
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y
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y
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Q
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20
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MEL t_Kì
0 5 10 15 20
0
10
20
30
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2222
1)(
N
x
N
x
xrtx σ
Discussion