Model Predator-Prey
September 20, 2022 | Author: Anonymous | Category: N/A
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D STMIK STIE A M : 2@. M . A . P . N , . P , . H , . : : , , , . . P . T . T . B , . T , . T , . : : , , , .
P . M . B 20 . S 20 , , 40 , . J 36 . D 750
19 (C, 2004 :344). P . D . M . P . P . P . (F A, 1992 : 1) D . D
, / . D . B , L, . D . D . B . A 1) , 2) .
1 , 2 , ..... (C, 2004:702). 2. 2 : S ɺ = ( , ) ɺ = ( , ) (2.2) (F L, 1982:287). 3: J ) = 0 (
: ɺ = − ,
= ( )
(C, 2004 : 494)
ɺ =
(2.3)
T (2.3) − = 0, = 0 . J (0,0) (2.3). J (2.2) , : ɺ = ɺ = + + (2.4)
1. 1 : S , , 2 (F L, 1982:132). A . B
,,, . J ≠ 0 (0,0) (2.4) : (2.5) λ 2 − ( − )λ + ( − ) = 0
: 1 = 1 ( , 1 , 2 , ..., ) 2 = 2 ( , 1 , 2 , ..., ) (2.1)
(2.5) . . T (0,0) (2.4) , , (2.5) . . T (0,0) (2.4) , ( ) (2.5) (F L, 1982:293). S , . T . S
⋮
= ( , 1 , 2 , ..., )
D 1 , 2 , .....
,
1 = 1 ( ), 2 = 2 ( ), ... = ( ) ,
,
42
λ 1 λ 2 . S : 1 . T (0,0) (2.4) , ,
2 N. 1 N 2011
, . . S (2.2) : ɺ = + + ( , ) (2.6) ɺ = + + ( , )
L :
, , , , P, Q : . , , ≠0 . P(,) Q(,) (,) : ( , ) ( , ) = ( , )→( 0 ,0 ) 2 + 2 ( , )→(0 ,0) 2 + 2 =0
: ɺ = + ɺ = +
(2.7)
D :
K : 1. I , 2. M , . A : . K P
3. P
. M P , . 3. M S , . A
, . (B, 2002:1). J . I . S : a. R: , , . b. D /:
, 2 1 . P . 4. S , . B , : 1) ? 2)
), ( .
? 3) (
2 . T (0,0) (2.6) (0,0) (2.7) . . T (0,0) (2.6) (0,0) (2.7) . T (2.6) (0,0) (2.7). (F L, 1982:294).
J CACH ISSN: 20860382
43
) (B, 2002:1517). 4. M . M . M : = 1 − (2.8) K , . K . K K . S : 0 ( ) = (2.9) 0 + ( − 0 ) − M
, = 0 = K. T . B G 2.1.
K. H H G, (2.10) K, ( − τ ) . B 1 − (2.10) τ (2.10) . M (2.10) H. P H . S . , . M ( ) ( ) ( ) = ( ) − , . = M
( ) = ( ) +
(2.10) ( ) ( − τ ) + = ( ( ) + ) 1 −
(2.11)
( ) − ( ) ( − τ ) − ( − τ ) . = K () K, ( ) ( − τ ) . S ( ) = − ( − τ ) . (2.12) (2.12), (2.12). P ( ) = λτ (2.12)
λ λτ = − ( 0 , ≠ λ + − λτ = 0 .
G 2.1. G M L P (2.9) K= 100 ,=1 L K A MM (0)= 10, (0)=30, (0)=80, (0) = 120 (0) = 150
5. M ( ) ( − τ ) (2.10) = ( ) 1 − , τ . S
44
λτ
λ −τ )
(2.13) 1 1 M > 0 τ > 0 τ ≤ (2.13) − λτ M ( λ ) = λ + . D
λ
.
A
( λ ) λτ . K ( λ ) = λ + −
2 N. 1 N 2011
− λτ '( λ ) = 1 − τ λ∗ =
1
τ
( τ )
( λ ) O
,
"( λ ) = τ 2 − λτ . I ( λ ). S (λ∗ ) =
1
( ( τ ) + 1 )
τ
1 τ = τ = , 1
τ <
1
(2.13) 1
1
( τ ) τ (2.13) . 1 J τ > , ( λ • ) > 0 , (2.13). K . J λ = ρ + ω ,
, τ =
, λ =
1
< τ . S
( λ )
. M (2.13) λ = ρ + ω = 0 ρ ≥ ≥ 0 . K λ =
(2.13) > 0 ( ω > (2.14)) π 0 < ωτ = τ − ρτ ω τ 0 . J J 1 π < τ < 2 (2.13) . M
( λ ) = λ + − λτ .
D
(2.13) λ . 1 − λτ M '( λ ) = 1 − τ λ• = ( τ ) τ ( λ ) . S
"( λ ) = τ 2 − λτ . I ( λ ) . F ( λ ) 1 • ( λ ) = τ ( ( τ ) + 1 ) > 0
J CACH ISSN: 20860382
G 2.2. G M L P (2.10) K= 100, =1 T N P τ = 1.5,τ = 1.935 τ = 2.5 6. D , , . M L (L, 1932 ; , 1926) C (2004 :760) , , : K . , . T
45
. S . A , .
) . D , ( β ). D ( β )
M L . : 1. P 2. P 3. P 4. T ( , ) S
( β ). P = − + β P β
. D ( ) . M () () : = 1 − − α (2.15) = − + β P : K = = α β
= = =
M : D , , , : = − P (). () (
46
. B ,
. D β . B , , . P , . = D , . K . J : = (1 − ) D , ).D ( α ( α ) ) . B , ( α ). P : = (1 − ) − α A : 1. T (2.15) E 0 = (0,0), E1 = (K,0) E* =(*,*) =
, ( β − ) . A αβ β
2 N. 1 N 2011
β − − > 0. M J
α β
(2.15) 2 − −α = β
τ
− + β −α
2. M J E * − β = β − α
−α
β
0
3. P M J
(λ ) = λ 2 + 4. M =
β
β
λ+
( β − ) β
=
( β − ) β
J
=
− ±
2 − 4
2 λ 1,2 K P Q , P Q . H E * . K β − − > 0 E* . 7. () . M ( − τ ), τ . B M 1974: ( ) ( − τ ) = ( ) 1 − − α ( ) ( ) (2.16) ( ) = −( ) + β ( ) ( ) , ,τ ,α , β . P (2.16) : K = = =
J CACH ISSN: 20860382
= =
= , , . T . L : 3 M β − > 0 τ ± (2.16) : 3π /2 / 2 2 π π / 2 2 π τ + = . τ − = + + , ω+ ω + ω− ω − . D ,
τ ∈ 0,τ + ) ∪ τ − ,τ + ∪ ... ∪ τ − ,τ + ,
(
0
0
1
)
(
−1
)
∗ (3.2) , , τ ∈ τ 0+ ,τ 0− ) ∪ (τ 1+ ,τ 1− ) ∪ ... ∪ (τ + −1 ,τ − − 1 ) ,
∗ . O τ = τ ± , =0,1,2, D ∗ τ = 0 . M 3 . (R λ ) (R λ ) 0 τ τ =τ τ τ =τ
+
−
− λτ P λ 2 + λ + = 0 λ − λτ λ 2λ + τ τ λ − λτ + λ −τ τ − λ = 0,
λ
(2λ + (1 − λτ ) λτ ) τ = λ 2 λτ . −
−
λ τ
A , −1
λ . M : τ λ τ
−1
=
2λ λτ + (1 − λτ )
λ 2
λτ
= 2λ
+ − τ
λ 2
λ
47
− λτ λ 2 + λ + = 0 −λ λτ λτ = 2 = λ + M
λ τ
−1
2
=
−λ +
τ (λ + ) λ λ 2
2
−
O
λ −1 (R λ ) = R τ λ =ω τ λ =ω −1 2 λ + λ =ω + R 4 2 λ + λ λ = ω −1 = + 4 2 2 −ω + ω − ω
= R
ω 4 − 2 2 2 2 ω (ω − )
= =
(ω 4 − 2 )
K ω 4 − 2 = 2ω 4 − ( 2 + 2 )ω 2 M (R λ ) 4 2 2 = (2ω − ( + 2 )ω ) τ λ =ω 2
2
= (2ω − ( + 2 ))
J . (S, 2006:4.10 2006:4.10)) 3.1 B M 1974: ( ) ( − τ ) = ( ) 1 − − α ( ) ( ) (3.1) ( ) = −( ) + β ( ) ( ) , ,τ ,α , β . P (3.1) : K = = = α = β =
τ
= D
. P,
48
, , , : = − P () (). T . D ( ) , ( β ). D ( β ) . B , ( β ). ). P
= − + β P β
. D β . S . D , . P , . = D , . K . J : = (1 − ) D , , ( α ). ). D (
α ) . B ,
2 N. 1 N 2011
( α ). P : = (1 − ) − α K ( τ ) : ( − τ ) (1 = − ) − α J : ( ) ( − τ ) = ( ) 1 − − α ( ) ( ) ( ) = −( ) + β ( ) ( ) 3.2 B , M (1974) : ( ) ( − τ ) = ( ) 1 − − α ( ) ( ) (3.2) ( ) = −( ) + β ( ) ( ) D , ,τ ,α , β . M (3.2) . J τ , τ . D . T (3.2) 2 : 1. T (0,0)
− 2 − α = 0 =
= 0 :
=0 − + β = 0
=
2 −
α − = α β − = α ( β − ) = αβ J ( β − ) , . β αβ D E*, . E* (3.2), . M ( ) = ( ) − * ( ) = ( ) − ∗ . M (3.2) :
ɺ( ) =
( − τ ) + ∗ ( ( ) + ) 1 − ∗
( ) ( ( ) + ) ɺ( ) = − ( ( ) + ) + β ( ( ) + ) (( ) + ) − α ( ) +
∗
∗
∗
∗
β
β
∗
ɺ( ) = ( ) + ∗ − −
∗( − τ ) −
( )( − τ )
∗( ) −
( ∗ )2 ∗
∗
∗
− α ( )( ) − α ( ) − α ( ) − α = 0
∗ ∗ ɺ( ) = −( ) − + β ( )( ) + β ( ) ∗
=
=0
∗
2. T
∗
∗
+ β ( ) + β
S , : ɺ( ) = − ∗( − τ ) − α ∗( ) ɺ( ) = β ∗( ) . M ∗ (3.2) . D J :
J CACH ISSN: 20860382
49
∂ 1 ∂ = ∂ 2 ∂ M
1
ω 4 − ( 2 + 2 )ω 2 + 2 = 0,
∂ ∂ 2 ∂
∂ 1
= ɺ( )
P : λ 0 − ∗ − λτ (λ − ) = − 0 λ β ∗
2
(3.6) τ , : ωτ = = 0
−α 0 ∗
λ + ∗ − λτ α ∗ = ∗ λ − β
ωτ = −1 0
∗ − λτ ∗ ∗ = λ + λ + αβ J : − λτ (3.3) λ 2 + λ + = 0 =
λ 1,2 =
D π / 2 2 π +
τ =
τ − =
+
ω+ 3π /2 /2
ω−
ω + +
,
2 π
ω −
, =0,1,2,
(3.9)
=0.1. M :
τ = 0 : λ 2 + λ + = 0 − ±
ωτ − = 270 + 2 π
S (3.2) =1, K = 200, α = 0.15 , =1 β
∗
= αβ ∗ ∗ .
ωτ + = 90 + 2 π
2
}
ω ± . S τ ± ω ±2
0
β
}
{
J : ∗ −λτ ∗ − −α =
A : 1 ω ±2 = ( 2 + 2 ) ± ( 2 + 2 )2 − 42 , 2 1 (3.8) ω ±2 = ( 2 + 2 ) ± 4 + 4 2 . 2 D (3.7)
{
1 = ɺ( )
∗
(3.7)
(3.3) (3.4)
2
− 4
. (3.5) 2 K P Q , (3.4) . S τ ≠ 0, λ = ω ,ω > 0, (3.4) : 2 − ωτ −ω + ω + = 0, 2
−ω + ω (ωτ ) + ω (ωτ ) + = 0.
D , : 2 (3.6) −ω + ω (ωτ ) + = 0,
G 3.1 G P D D P (()) (0)=9 τ = 0.5
) = 0. ω (ωτ K : 2 2ω 2 2 (ωτ ) = ω 4 − 2ω 2 + 2 2 2 ω ( ωτ ) = 0.
K , :
2ω 2 2 (ωτ ) + 2ω 2 2 (ωτ ) = ω 4 − 2ω 2 + 2 2ω 2 = ω4 − 2ω 2 + 2
K :
50
2 N. 1 N 2011
D (3.9) :
τ 0+ = 1.57080
τ 0− = 5.23599
τ 1+ = 7.85398
τ 1− = 12.21730
τ 2+ = 14.13717
τ 2− = 19.19862
τ 3+ = 20.42035 +
τ 3− = 26.17994
−
τ = 26.70354 τ 5+ = 32.98672 4
τ = 33.16126 τ 5− = 40.14257 4
B B 3 (0,1.57 .57080 080)) ∪ (5.2359 5.23599, 9, .85398) ∪ τ ∈ ∈ (0,1
(12.2173 (12. 21730,14137 0,1413717) 17) ∪( 9.19862,20.42035) ∪ G 3 3..2 G G P P D P (()) (0)=9 τ = 0.5 D D τ = 0.5 : 1 2
(()) 9,00000 11,16000
(()) 5,00000 5,58000
3 . . . 400
12,91882 . . . 9,99372
7,20870 . . . 6,65449
D (3.1) (3.2) τ = 0.5 (10;6.3 (10;6.33) 3) .
G G 3. 3.3 3 G G J J P P P P τ = 2.7 D (3.3) τ = 2. 2.7 7 (10; (10;6. 6.33 33)) .
J CACH ISSN: 2086038
(26.17 (26 .17994,2 994,26.7 6.703 0354) 54) ∪ ( 3.16126,32.98672) (10; (10;6 6.33) .. S S (1.57080,5.23 080,5.23599) 599) ∪ (7.8 398,12.21730) ∪ τ ∈ ∈ (1.57 .42035 035 ∪ 26.179 26.17994) 94) ∪ (14.13717,19.1 (14.1 3717,19.19882) 9882)∪ (2 .42 (26.703 (26 .70354,331 54,331612 6126) 6) ∪ (3 .98672,40.14257)
(10; (10;6 6.33) . , .
G G 3.4 3.4 G G P P D P P ((( () ))) T T P
51
2. A A P P M P . T T . . O O . . M M . . T
G 3 3..5 G G P P D P P ( (( ()) )) P P τ = 0.5 A , , : 1
9,00000
9,00
2 3 . . . 280
11,16000 12,91882 . . . 10,04294
10,85 12,28 . . . 10,00
D D (10;6.33) , 235. 235. S S (10; (10;6. 6.33 33)) , 251, 251, . B , : 1. A P M P . T T . O . . T T .
52
. S ( β − ) , . K αβ β . D τ ∈ ∈ (0,1 (0,1.57 .57080 080)) ∪ (5.2 5.235 359 99,7.85398 ) ∪ (12.21730,1413717) (19.19862,20.42035)∪ (26.17994,26.70354)
(33.16126,32.98672)
(1 (10;6.33) .. S S (1.57080,5. 7080,5.2359 23599) 9) ∪ (7..85398,12.21730) ∪ τ ∈ ∈ (1.5 (14.13717,19.19882)∪ (20. (20.42035 42035∪ 26.17994) 26.17994)∪ (26.70354,331 (26.7 0354,3316126) 6126) ∪(32.98672,40.14257)
(1 (10;6.33) .
1 A, M . 2008. T A AA, AA, ://... ://... D 7 J 2009 . 2 A K K , , A A F F . . 1 93. . B : D F. . 3 A M. J. 1990. . T P : S. 4 A M, M . 1971. . B : D F P P 5 A. 2009. M,:// M, :// .../ /2.. ./44// /2.. D 19 M 2008. 6 A, F. 1992. . J : E 7 B. 2002. . M . M : MM P.
2 N. 1 N 2011
8 B. 1999. P P D: L , ://../. ://../. D 7 J 2009
13 N, C. 2004. . . N J: P E
9 B. 2002. . M: . M: MM P
14 R, C. 2004. .. J: E.
10 F L. 1998. , E K. T S.
15 S, M. Q. 2002. . . J: L H.
J : E. 11 K, S. 2007. () . . S . M: IN M. 12 M, JD. 2002. . . N : S
J CACH ISSN: 20860382
16 S, K. 2008. M F B, ://.../. =&=&=38& =34. D 14 J 2009. 17 T, S. 2006. . . M. D M H 18 , H. 2009. M R A S, ://.. ///010.. /// 010.. D 7 J 2009.
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