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Численный алгоритм склеивания решения на поверхности разрыва для систем с клеточной структурой.

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.
.
-
. 2010.
4.
. 156–163.
519.62
. .
, . .
,
. .
.
. .
.
.
-
.
,
MEP2,
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:
,
,
-
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,
,
-
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,
,
,
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-
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.
-
,
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,
,
,
-
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:
,
.
-
,
. .
[1].
,
© . .
, . .
,
. .
, 2010
-
...
,
-
157
X(t)
dX / dt (t )
,
,
.
-
.
.
,
.
'–
F(X), X
'.
Rn
,
'
CF
',
F
-
(1)
',
'.
[2–4].
«
c
-
. .
:
(
);
–
,
»
DF
-
-
,
.
.
-
-
,
-
MEP2 (Modelling Evolution Processes 2)
DF
.
.
.
(1),
-
F ( X ),
1.
(1),
C
C
CF.
–
;
;
;
CF
Ci, i = 1, ..., n.
2.
-
CS,
(1)
.
Ci ,
i = 1, ..., n
–
.
F(t,X).
.
-
,
,
DF
,
F(t,X)
,
n
.
: DF
[5; 6].
-
n
Ci \
i 1
Ci .
i 1
3.
X(t),
-
a<t<b
(1)
,
Y(t):[a,b] Rn,
Y(a) = Y(b) = 0,
b
dX
dt
F( X )
(1)
X(t),
b
( Y ' ( t ), X ( t ))dt
( Y ( t ), F ( X ( t )))dt .
a
a
4.
F.
( , )F.
-
(
) V
X0
V(X0) (
X,
)
F(X)
, ,
. .
158
,
. .
.
F
-
X0
,
X0,
F(X0).
-
F –
D
,
.
Ci, i = 1, ..., n
F
( , ),
,
F
-
,
–
,
0.
V(X0)
. .
-
F
,
[7].
–
F(Xn)
Xn X0.
Sn-1
S(X0) –
5-
-
-
.
F
S(X0).
[8,
S(X0).
F
. 191].
,
F
V’(X0)
S+n-1
. .
W
.
,
-
.
X0 DF,
F.
dir(F,X0)
,
dir(F,X0)
,
DF,
,
F(X0)
X0,
V’(X0)
dir(F,X0)
. .
-
|X – X0| < ,
(F(X), V(X0)) > .
,
( ) 1
,
S(X0).
S+n-1
W
/2,
V
V
S+n-1,
V
V(X0).
5.
-
X(t), a < t < b,
(1)
,
.
X(t)
Ci,
X*(t + h)
,
,
~
X ( t ) V ( X ( t )),
-
X*(t + h)
Cj, j i.
.
X (t ) –
X(t).
.
(
«
)
»,
-
1 (
).
,
-
[1].
,
,
X(t)
F
)
(
.
5
3
X1 = X(t1), t1 = t + h
Ci
,
||X1||,
||X1||,
–
, Xb
DF –
.
X1
.
. . ||X1- Xb||
,
...
2 (
).
n(Xb))
3,
159
(F(X1),
(n(Xb), F(X2)) > 0,
.
: ||F(X)||
a
t
M
b,
X
X2,
.
prDf F(X1)
DF
F
,
-
.
,
. .
. (
n(Xb) –
dF
(X 2)
dX
X2
X2*
Xb.
F(X2)
,
Xb
lim X (t1
r
Ci, Cj,
{ k} –
-
Xb
Xb
–
.
(2)
-
X (t )
Ci
~
X2
X2* = X2 – X2*.
X1
ht F ( X 1 )
(
(2)).
-
~
X2
~
X2
X2:
)ht ( F ( X 1 ) F ( X 2 )) O( ht2 ).
(1
(2)
,
X1
X2* –
X2
X2* –
X2,
,
ht ),
O ( ht2 ).
.
Cj.
,
X ( t1
(4),
(2)
X (t)
X ( t 2 ) C j , ht = t2 – t1, a
Xb
-
(2)
(1).
X2
(4)
X1 ( F ( X 1 )
-
.
,
Xb
.
Xb
(||F(Xb)||> 0).
) ht F ( X 2 ) O( h ).
) F ( X 2 )) ht
(1
-
.
(3)
2
t
X 2 (1
X2
.
(1)
ht F ( X 1 ) O (ht2 ),
X1
Xb,
Xb- = Xb+ = Xb.
(3)
X2:
:
V,
Xb+
X (t )
X(t + r-1h) Cj.
h2 = ( r-1- r)h
Cj
K1 = F(X1),
K2 = F(X1+ h2 K1),
X(t1+ h2) X2 = X1+ (K1+ K2) h2/2.
X1 + h2K1
X2.
Xb
V = (K1 + K2)/2,
ht )
0
X(t)
Xb-
Cj).
lim X ( t1
ht ) .
0
).
3 (
-
.
Xb
,
.
X2
ht + O(ht2),
X2* = R
,
F(X1)
L
R 2M( –1/2).
F(X)
1,
F(X)
,
-
X
X2:
X (t1 ) Ci ,
t1< t2
X2*
b.
X 2*
1/ 2) ht F ( X 1 )
(
) F ( X 2 ) F ( X 2 ) / 2) O ( ht2 ).
ht ((1
F
X2
(0,1).
,
(2).
-
,
.
F(X)
~
F( X 2 ) F( X 2 )
X2
~
L X2
O
~
X2
2
.
. .
160
,
.
O
,
X2
2
,
2
t
O (h ).
X2*:
X 2*
ht (
1) Lht / 2 ( F ( X 1 )
1/ 2) (
F ( X 2 )) O ( ht2 ).
X 2*
,
dx
dt
dy
dt
dx
dt
dy
dt
R = ( -1/2) (F(X1) – F(X2)).
F(X)
||F(X1) – F(X2)|| 2M,
R
–1/2).
.
.
,
ht
–1/2 ht.
,
. .
a11 ( x c1 ) a12 ( y d1 ),
a21( x c1 ) a22 ( y d 1 ), ( x , y ) D1 ,
(5)
a22 ( x c 2 ) a12 ( y d 2 ),
a21( x c2 ) a11 ( y d 2 ), ( x , y ) D2 .
,
a11 = a22 = 0, a12 = a21 = 1
O( ht2 ),
R ht
. .
(
D1
M,
2M ( –
).
c1 < x0
D2.
,
c2 > x0,
d1 = d2,
,
.
(2)
-
X2* = O(ht2).
-
D0,
.
-
.
,
MEP2
(
. 1).
-
MEP2 [9],
.
,
:
,
-
,
.
. 1.
:
(5)
•
;
•
-
. 1,
D1
;
•
•
-
,
D2,
.
-
;
-
–
.
.
,
.
-
,
,
D1: x < x0
,
D2: x
x0
-
.
-
(x,y):
.
,
.
-
...
(
. 1)
(5).
.
-
MEP2
dR
dt
1 u RP
[10; 11].
RCC
R
K
1 hRC
u RP RP P
RPhRP R uCP
RC R
,
CPhCPC
eRC RC R
dC
C
dt
1 hRC RC R
(R),
(C)
R r 1
161
(P).
,
.
1 u RP
:
dP
dt
(6)
uCP CP P
RPhRP R uCP
mC ,
CPhCPC
u RP RPeRP R uCP CPeCPC
mP .
1 u RP RPhRP R uCP CPhCPC
P
.
,
(6)
-1
r
K
0,3
10
C/
0,037 /(
0,025 /(
0,025 /(
1(
1(
3
4
4
0,6 (
0,36 (
0,6 (
-1
0,03
0,0275
RC
RP
CP
uRP
uCP
hRC
hRP
hCP
eRC
eRP
eCP
mC
mP
,
,
,
,
)
)
)
)
)
)
)
)
-1
(6)
. 2
3.
,
-
(R, C, P),
(P, C) (R, C).
uCP = 0
(uRP = 1;
(6))
(uRP = uCP = 1).
-
.
.
.
R* (
-
.
. 4 (A)).
,
,
: 1)
,
,
;
-
2)
,
(7)
C*
.
(uRP = uCP = 1),
eRP
hRP
-
.
eCP
.
hCP
,
(7)
(uRP = 0; uCP = 1).
,
(uRP = 1),
,
R*.
-
«
»
,
(
. 4(B)).
. .
162
(
. 2, 3),
,
. .
,
. .
-
,
,
.
.
.
(6)
–
(
)
,
-
. 5.
,
,
.
-
,
,
-
.
. 2.
(6)
(R, C) (R, P).
: eRP = 0,2, eCP = 0,1, R* = 10,
.
.
. 3.
(6)
: eRP = 0,1, eCP = 0,3, C* = 5,
(P, C)
.
(R, C).
. 4.
. 5.
–
2–
(6)
,
.
: eRP = 0,1, eCP = 0,3, C* = 5,
.
: R(0) = 15, C(0) = 1, P(0) = 7
:1–
...
.
(
).
-
(6).
-
.
.
,
–
–
,
(
),
,
[2] Gear C. W., Osterby O. Solving ordinary differential equations with discontinuities // ACM Trans.
Math. Software. 1984. Vol. 10. P. 23–44.
[3] Piiroinen P. T., Kuznetsov Yu. A. An event-driven
method to simulate Filippov systems with accurate computing of sliding motions // ACM Trans.
Math. Software. 2008. Vol. 34.
13. P. 1–24.
[4] Shampine L. F., Thompson S. Event location for
ordinary differential equations // Computer and
Mathematics with Application. 2000. Vol. 39.
P. 43–54.
[5]
. .,
. .,
. .
//
.
. 2008. . 13.
2. . 70–81.
[6]
. .,
. .
-
[7]
[8]
[9]
(
. 5).
[10]
[1]
.
.
.
1985.
. :
,
163
[11]
//
.
. 2010. . 15.
2. . 56–72.
Dormand J. R., Prince P. J. A family of embedded
Runge-Kutta formulae // J. Comp. Appl. Math.
1980. Vol. 6. P. 19–26.
.,
.,
.
.
. .:
, 1990.
. . MEP2 –
. URL :
http://users.univer.omsk.su/~korobits/mep2 (
: 01.06.2010).
Krivan V. Optimal Intraguild Foraging and Population Stability // Theoretical Population Biology.
2000. Vol. 58. P. 79–94.
Krivan V., Diehl S. Adaptive omnivory and species
coexistance in tri-trophic food webs // Theoretical
Population Biology. 2005. Vol. 67. P. 85–99.
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