浙江大学：数学建模课件（刘利刚）：chapter_11

μsZ?
?
μsZ?
?
1
g?ü6?￥ê?
?
2 hg9 ——?
JD?
3 μs?
T￥Eì9é
?
4 ?
M
-sF￥??9é
1
g?ü6?￥ê?
?
v?p Nì/? h
á

C
`
ù
5

DNì￥??
I
1Hq/t_×?
??×?
H??
?|
I
1??
m
P-×?
9Fá
 Nì
h
DNì??
??p
í

DNì￥M
p
ê?
?
x
k
~? k
H


y
k
~? k
H

Nì
)(
kk
xfy =
hf
hn?￥3p1"
3pf
g
x
0
y
0
P
0
f
x
y
0
3á?￥?1" 9f
?f
fD g￥?? P
0
(x
0
,y
0
) ~
ü?
B? x
k
=x
0

5 y
k
=y
0
,
x
k+1
,x
k+2
,…=x
0
,y
k+1
,y
k+2
,…=y
0
)(
1 kk
yhx =
+
)(
1+
=
kk
xgy
x
y
0
f
g
y
0
x
0
P
0
! x
1
ê ? x
0
null→→→→→
32211
xyxyx
0321
PPPP →→→→ null
00
,yyxx
kk
→→
)(
kk
xfy = )(
1 kk
yhx =
+
)(
1+
=
kk
xgy
00
,yyxx
kk
→→
×
×
ê?
?
0321
PPPP →→→→ null
×
P
0
^×?
ü? P
0
^?×?
ü?
P
1
P
2
P
3
P
4
gf
KK >
x
y
0
y
0
x
0
P
0
f
g
x
1
x
2
P
2
y
1
P
1
y
2
P
3
P
4
x
3
y
3
gf
KK <
wL|
q
Z?
?
 P
0
??í¨°Lí
wL
)(
kk
xfy = )0()(
00
>=? αα xxyy
kk
)(
1 kk
yhx =
+
)0()(
001
>?=?
+
ββ yyxx
kk
)(
001
xxxx
kk
=?
+
αβ
)()(
0101
xxxx
k
k
=?
+
αβ
0
xx
k
→
gf
KK <
P
0
×?
)/1( βα<1<αβ
∞→
k
x
gf
KK >
P
0
?×?1>αβ
)/1( βα>
g
K=β/1
f
K=α
Z?
?Dê?
?￥Bá
2T3
d2T3
d
I3 α,β￥cl
x
k
~? k
H


y
k
~? k
H

Nì
)(
00
xxyy
kk
=? α
α ~

h
 1?ê,Nì
h?
)(
001
yyxx
kk
=?
+
β
β ~ Nì
h 1?ê,(/
H
)?￥9

α ~ hn?3p￥
ù??
αl,μ ??ü6×?
β ~
3á?Nì￥
ù??
βl,μ ??ü6×?
1<αβ
ü6×?
2T3
d
ü6?×?
H??￥??÷E
x
y
0
y
0
g
f
1,
P αD
l
? α=0
[??
m
e?Nì?M
3pwLM1
￡
ü
x
y
0
x
0
g
f
2,
P βD
l
? β =0
?wLM1
°
Lü6
L ? e?
?M
?￥w<
3á?5 ?
￡
ü4ú )(
1 kk
yhx =
+
+
=
+
2
1
1
kk
k
yy
hx
3á?? ?-
H
-B
H

￥Nì %?/B
H
￥á
b
]2/)[(
0101
yyyxx
kkk
+=?
+
β
!?f
1
)(
00
xxyy
kk
=? α
3pf
?M
null,2,1,)1(22
012
=+=++
++
kxxxx
kkk
αβαβαβ
=¨L?è"
μsZ?
x
0
1
ü? ù?
ü?×?
' k→∞,x
k
→x
0
￥Hq
?￥w<
012
)1(22 xxxx
kkk
αβαβαβ +=++
++
Z?Y3
kk
k
ccx
2211
λλ +=
(c
1
,c
2
?
SHq?? )
λ
1,2
~+??
'Z? ￥?02
2
=++ αβαβλλ
1
2,1
<λ
ü?×?
' k→∞,x
k
→x
0
￥Hq,
4
8)(
2
2,1
αβαβαβ
λ
±?
=
2
2,1
αβ
λ =
2<αβ
ü?×?Hq
1e ?￥Hq b z

1<αβ
2 hg9 ——?
JD?

?

hg
J
r??hg
"S
?
??
8×·
BMI=w(kg)/l
2
(m
2
),18.5<BMI<25 ~
?è
BMI>25 ~ ?× ; BMI>30 ~ g
,
YV e??
J
a?￥?
?
]
&8
￥-4/
r?h^8×i??/?￥
"S
s

8×M?8
=

o
ê?
?
J


l￡

?8×9F
}??

h{￡

?8×h

?L
!
1
8×9F?1?
l￥￡
 —
—
8000$ 59F8× 1$ X

2
}??￥8×h
?1?8× ——
?
à8×h{ 200$ 5 ~ 320$ 5 (y|7s ),
M?? 70$ X￥|
?h{ 2000$ 5 ~ 3200$ 5

3
??￥8×h
?1?8×
OD?
?
Tμ1

4
1
??Dr B

?8×h
?R?V 1.5
$ X

?
l￡
?1l? 10000$ 5b
hg9
J8× 100$ X

"-
?
l 20000$ 5￡


8×???MbCàhgà 75$ Xb
1
??￥f ?/?
B?
¨
9b
?B¨


?hg 1$ X

?
l￡
?vh

°àr?/K

10000$ 5


?=¨


?
l￡
?/K
hgr?
"S
2
?1F yé?
?=¨
9F?

k?
9b
3
ór?
"Sa??8×￥Z?b
'
?
w(k) ~ ? k? (
 )8×
c(k) ~? k?
l￡

)()1()()1( kwkckwkw βα?++=+
$ 5
$ X /(80001=α
~ }?h{"
(y|7s )
β
1
??f ?￥
¨
hg9
??
J￥}?h{"
?
l 20000$ 5 w=100$ X?M
025.0
1008000
20000
=
×
==
w
cα
β
wcww βα?+=
'
?
$ X8×h{ 20000/100=200$ 5
1
??f ?￥
¨
hg9
?B¨
,w(k)
?h 1$ X,c(k)hà/K 10000$ 5
)()1()()1( kwkckwkw βα?++=+1)1()( =+? kwkw
]1)([
1
)1(?=+ kwkc β
α
kwkw?= )0()(
80001=α
025.0=β
)1(
1
)0()1( kwkc β
αα
β
+?=+
10≤k
k20012000?=
10000=≥
m
C
?B¨
10?,
?h 1$ X
? 10?
8× 90$ X
9,1,0,20012000)1( null=?=+ kkkc

l￡
1
1
??f ?￥
¨
hg9
?=¨


? c(k)? C
m
,w(k)hà 75$ X
)()1()()1( kwkckwkw βα?++=+
'
?
m
Ckwkw αβ +?=+ )()1()1(
])1()1(1[)()1()(
1?
++?++?=+
n
m
n
Ckwnkw ββαβ null
}?¤[ 10000,
8000
1
,025.0 ===
m
Cαβ
50]50)([975.0)( +?=+ kwnkw
n
β
α
β
α
β
mm
n
CC
kw += ])([)1(
?=¨


? c(k)? C
m
,w(k)hà 75$ X
50]50)([975.0)( +?=+ kwnkw
n
nnkwkw p
1pX? 75)(,90)( =+=
50)5090(975.075 +?=
n
19
975.0lg
)40/25lg(
==n
?=¨
19?,
?
l￡
? 10000$ 5,8×?
h
à 75$ Xb)19,,2,1(50975.040)( null=+×= nnw
n
2
?=¨
9F?￥hg9
? '

l
H
$ X8×h{￥￡
 γ ($ 5 ),
? K÷
1?ú (?
) 3￡ (50
ü /s )
7.0 3.0 4.4 2.5 7.9
t~
??
HW (l
H )
)()(
)1()()1(
kwt
kckwkw
αγβ
α
+?
++=+
'
?
)028.0()025.0( =+=
′
→= tαγβββ24,003.0 == tt γαγ '|
β
α
β
α
β
′
+
′
′
=+
mm
n
CC
kwnkw ])([)1()(
6.44)6.4490(972.075 +?=
n
14=n
? γt=24 (
?K÷ 8l
H1?ú 10l
H ),14?' Vb
3
r?
"S8× 75$ Xa???M￥Z?
?
l￡
 c(k)?
è
C

P8× w?M
)()()1()()1( kwtkckwkw αγβα +?++=+
α
αγβ wt
C
)( +
=
wtCww )( αγβα +?+=
)(1500075025.08000 $ 5=××=C???
)(1680075028.08000 $ 5=××=C?? (
=?]- )
3 μs?
T￥Eì9é
?
???
T ￥Eì9é
? (Logistic
? )
)1()(
N
x
rxtx?=null
x(t) ~
?? t
H Y￥
 (| g )
t→∞,x→N,x=N
^×?
ü? (D rvlí1 )
y
k
~
??? k}￥
 (| g )
??
?
T
null,2,1),1(
1
=?=?
+
k
N
y
ryyy
k
kkk
y
*
=N
^
ü?
? y
k
=N,5 y
k+1
,y
k+2
,…=N
)
ü?￥×??
' k→∞,y
k
→N
$
???
TEì9é
?￥
ü?#×??
+
+=
+ kkk
y
Nr
r
yry
)1(
1)1(
1
)1()1(
1
N
y
ryyy
k
kkk
=?
+
kk
y
Nr
r
x
)1( +
=
)2()1(
1 kkk
xbxx?=
+
M

}D
1+=rb:
B¨ (dL? )μsZ?
(2)￥
ü?
br
r
x
1
1
1
*
=
+
=
(1)￥
ü? y
*
=N
)
x
*
￥×??
? ?
M
B¨dL?μsZ?
)1()(
1 kk
xfx =
+ ￥
ü?#×??
(1)￥
ü? x
*
——}
Z? x=f(x)￥?
)2())(()(
***
1
xxxfxfx
kk
′
+=
+(1)￥í
L?Z?
×??
 x*9
^ (2)￥
ü?
1)(
*
<
′
xf
x*
^ (2) (1)￥×?
ü?
1)(
*
>
′
xf
x*
^ (2) (1)￥?×?
ü?
)21()(
**
xbxf?=
′
1)(
*
<
′
xf
0
y
x
xy =
)(xfy =
4/b
*
x 2/1
1
)1()( xbxxfx?==
)1(
1 kkk
xbxx?=
+
￥
ü?#×??
ü?
b
x
1
1
*
=
×??
31 <<b
2/1/11
*
<?= bx
*
xx
k
→

??9

0
x
1
x
1
x
2
x
x
*
×?
21)1( << b
)1)((3
*
>
′
> xfb x
*
?×?
6B
ü
?1 x=0
1+=rb
1)0( >=
′
bf
?×?
b?= 2
)1(
1 kkk
xbxx?=
+
￥
ü?#×??
32)2( << b
3)3( >b
2/1/11
*
>?= bx
01/21
y
4/b
xy =
)(xfy =
0
x
1
x
*
x
2
x
x
y
0
x
xy =
)(xfy =
0
x
1
x
2
x
*
x2/1
1
4/b
*
xx
k
→

?

*
xx
k
→

??1

)1(
1 kkk
xbxx?=
+
′ x
0
=0.2
0.4118100
0.411899
0.411898
0.411897
0.411896
0.411895
0.411894
0.411893
0.411892
0.411891
……
0.37963
0.33662
0.27201
0.20000
b=1.7k
0.6154
0.6154
0.6154
0.6154
0.6154
0.6154
0.6154
0.6154
0.6154
0.6154
…
0.6049
0.6317
0.4160
0.2000
b=2.6
0.8236
0.4794
0.8236
0.4794
0.8236
0.4794
0.8236
0.4794
0.8236
0.4794
…
0.4820
0.8224
0.5280
0.2000
b=3.3
0.8469
0.4327
0.8530
0.4474
0.8469
0.4327
0.8530
0.4474
0.8469
0.4327
…
0.4322
0.8532
0.5520
0.2000
b=3.45
0.8127
0.3548
0.8874
0.5060
0.8278
0.3703
0.8817
0.5405
0.8127
0.3548
…
0.3987
0.8711
0.5680
0.2000
b=3.55
′9
2T
b
x
1
1
*
=b <3,x→
b=3.3,x→
?
K?
b=3.45,x→4?
K?
b=3.55,x→8?
K?
?
ù
l ? ——x
*
?×?f ?￥éB?)
3.3=b
*
212
*
12
,xxxx
kk
→→
+
0?

*
xx
k
→

?

??
ù?
l ?
2?
ù
l ?
(*))())(()(
)2(
12 kkkk
xfxffxfx ===
++
)(
1 kk
xfx =
+
)1()( xbxxf?=
))(( xffx=
)]1(1)[1( xbxxbxb=
b
x
1
1
*
=
b
bbb
x
2
321
2
*
2,1
+
=
(*)￥
ü?
10
*
2
**
1
<<<< xxx)(),(
*
1
*
2
*
2
*
1
xfxxfx ==
x
*
?×?
ù? x
1
*
,x
2
*
￥×??
?
ù
l ?
)()())(())((
*
2
*
1
)2()2(
*
2
*
1
xfxfxfxf
xxxx
′′
=
′
=
′
==
b
bbb
x
2
321
2
*
2,1
+
=
￥×??
2)2(
)]([])([ xfxf
′
=
′
)21)(21())((
*
2
*
1
2
,
)2(
*
2
*
1
xxbxf
xxx
=
′
=
)21()( xbxf?=
′
x
1
*
x
2
*
x
*
b=3.4
y=f
(2)
(x)
y=
x
x
0
1))((
*
2,1
)2(
<
′
xf
449.361 =+<
null
b
*
212
*
12
,xxxx
kk
→→
+
?
ù
l ?￥éB?)
1))'((45.3
*
2,1
)2(
>?> xfb
x
1
*
,x
2
*
(# x
*
)?×?
C 4?
l ?0?
 x
4k
,x
4k+1
,x
4k+2
,x
4k+3
)(
)4(
4 kk
xfx =
+
ü?#×??3ù?
4?
ù
l ?544.3449.3 <<b
Hμ 4?×?
ü?
2
n
?
ù
l ?,n=1,2,… b
n
~ 2
n
?
ù
l ?￥
?
b
0
=3,b
1
=3.449,b
2
=3.544,… n→∞,b
n
→3.57

§C`
b>3.57,?i?
l ?0?

)1(
1 kkk
xbxx?=
+
￥
l ?as′#
§C`
b
4 ?
M
-sF￥??9é
?]
M
-F￥N2
q
μ§
q?]
[I??8
1`
y ?μsZ?
?
)
×? ?/??￥9é
p
L
!Dy

???
M
-vl?s1 n?
M
-F
,i=1,2,…,n
HW ??1
H

éD
M
-FuWM?
,k=1,2,…
? i
M
-F 1I??8 1
H
=￥ N2
q 1 b
i
? i
M
-F 1
H
=￥
μ§
q1 d
i
,i
q 1 s
i
=1-d
i
L
!
D
y

x
i
(k)~
H
k? i
M
-F￥??

)()1(
1
1
kxbkx
i
n
i
i
∑
=
=+
(
!à
 1? b
i
>0)
1,,2,1),()1(
1
==+
+
nikxskx
iii
null
=
00
0
000
1
2
1
121
n
nn
s
s
s
bbbb
L
null
null
null
null
~Leslie ? (L ? )
T
n
kxkxkxkx )](),(),([)(
21
null=
~?
M
-F￥s?_

)()1( kLxkx =+
)0()( xLkx
k
=
???i
H
??
?
M
-F￥s?
×?
s￥
D?
M
nk
k
null,3,2,
1
=≤ λλ? L ?i ??+?? λ
1


T
n
n
ssssss
x
=
1
1
121
2
1
21
1
1
*
,,,,1
λλλ
null
null+?_

? L ?i b
i
,b
i+1
>0,5 nk
k
,,3,2,
1
null=< λλ
)0()( xLkx
k
=
1
1
)],([
= PdiagPL
n
λλ null
P￥? 1

^ x
*
*
1
)(
lim cx
kx
k
k
=
∞→
λ
,c
^? b
i
,s
i
,x(0) %?￥è
O
3
d
1
1
)],([
= PdiagPL
k
n
kk
λλ null
L?
)0()0,0,1(
)(
lim
1
1
xPPdiag
kx
k
k
∞→
= null
λ
*
cx=
×
s ——k sv
???
M
-F￥s?
*
1
)(
lim cx
kx
k
k
=
∞→
λ
~ ???
M
-F￥s?t_×?

x
*
?×?s?,D
Ss?í1b
*
)()1 xckx
k
λ≈
)()1()2 kxkx λ≈+
~ ò
M
-F??
?]B

9h
 λ?%μ9é
q
)()1( kxkx
ii
λ≈+
[]
T
n
ssssssx
121211
*
,,,1
= nullnull
)()1( kLxkx =+
D'
? 1?
3
 λ=1
H
*
)()1( cxkxkx ≈≈+
~ ò
M
-F??
?M
×
s
[ ]
T
n
ssssssx
121211
*
,,,1
= nullnull3
 λ=1
H
**
xLx =
~ 1??8??i
ù
=￥N2
1 1
1
121121
=+++
nn
sssbsbb nullnull
=
00
0
000
1
2
1
121
n
nn
s
s
s
bbbb
L
null
null
null
null
T
n
ssssx ],,,,1[
1211
*
= null,)()4
*
xckx
k
λ≈
1,,2,1),()(
1
=≈
+
nikxskx
iii
null
~i
q s
i
^]B
H
￥ x
i+1
D x
i
-1


D s
i
￥?l 1?

)()1(
1
kxskx
iii
=+
+