Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15777/doc.pdf Дата изменения: Mon Oct 29 13:07:08 2007 Дата индексирования: Mon Oct 1 20:38:04 2012 Кодировка:

­ .. . (. ) . _. , . .

GROUPS AND ALGEBRAS LIE - TOOLS FOR MODELLING ECOSYSTEMS Yakovenko G.N. (Dolgoprudny) The modification of system Lotka ­ Volterra considered. The modification enables to take into account natural unpredictable influences on ecosystem. The algebraic models, allowing to describe are investigated also dynamics of system to within the description of a variable state. 1. . [1]. . -, , : , , . . uk (t ) t.
165


-, , , ­ . , , .. , , . . 2. 2.1 [2].

& xk =


i =1

r

kl ( x)ul (t ) ,

k = 1, n ,

u U R

r

,



kl

(2:1) ( x) (2.2)

rank kl ( x) = min {n, r

}




i =1

r

ci kl ( x) = 0, cl = const cl = 0, l = 1, n

{

}

(2.3) (2.4)

Xi, X j =
Xl =


k =1

r

k Cik X k , Cij = const , i, j , k = 1, r j




k =1

n

kl ( x)

, l = 1, r , xk

(2.5)

X i , X j ­ (2.5): n X i , X j = { X i kj ( x) - X j kj ( x)} xk k =1

(2.6)

166


.. -- -10, 2002, .165-175

(2.4) ,


k =1

r

uk X k , uk = const Xk k

Cij [3]. , (2.1) r ­ xk = g k ( x10 , ..., xn 0 , v1 , ..., vr ) , (2,7) (2.1) ul(t) = const [3]. (2.7) (2.5) k Cij , (2.4). (2.1) (2.7). 2.1 [4]. {u (t ), t} R ­ x(t) (2.1), u(t): x(0) = x0 x(t). x0 x (t ) (2.7), . . {u (t ), t} v1 , ..., vr , (2.7) x0 x (t ) . x L ­ . 2.2. L ­
n

& xk =


i =1

r

kl ( x)ul (t ) , k = 1, n , u U R

r

(2,8)

r = n . (2.1) r n L ­ (2.8). r > n . 2.2 [4]. (2.1) r > n . (2.1)

167


& xn +i =


i =1

r



n + il

( x)ul (t ) , i = 1, r - n

(2.9)

, (2.1), (2.9) ­ k L - (2.8) Cij , (2.4). (2.9) [4]. 3. , L ­ % x = h( x) -

% . x (2.8)

& % xk =


i =1

r

%% kl ( x)ul (t ) , k = 1, n , u U R
k

r

(3,1)

%% kl ( x) (2.2) ­ (2.4), Cij (2.4), x . (2.8), (3.1), % x x , , n k x , Cij U ul(t):
r & xk = kl ( x)ul (t ) i =1 k % b x x {n, Cij , U r & xk = kl ( x)ul (t ) % %% i =1

-

}

(3.2)

­ : k Cij o

( ( ( X 1 , ..., X n [3] , X l
168


.. -- -10, 2002, .165-175

(3.2) .

{

k n, Cij , U }

. , x: , k .. , Cij , (3.2) : , , , .. 4. , (ki ­ ):
& x1 x1 0 & x2 0 x u (t ) + 2 = M M M 1 & xn 0 0 0 u (t ) + x2 2 M 0
k c1 x1k1 +1 x2 k1 k2 x2 u (t ) + c2 x1 M n M k c x k1 x2 n1
2

+1

2

k K xn n k K xn n un +1 (t ) MM k K xn n +1

(4.1)

xi > 0 ­ , ci , ki ­ , ui(t) ­ . un+1(t) 0 (4.1) ­ . n = 1 (4.1) ­ . n = 2 (4.1) ­ ­ [1]. (4.1) (2.5):

X 1 = x1
X
n +1

, X 2 = x2 , ..., X n = xn , x1 x2 xn
k
n n

= c1 x1k1 +1 x 2k2 L x
kn +1
n

+ c2 x1k1 +1 x x1

k2 +1
2

Lx

k
n

n

+ ... + x2

(4.2)

+ cn x1k1 +1 x 2k2 L x

xn
n+1]

(2.6) [Xi;Xj] = 0; i, j = 1, n , [Xi; X = ki X
n+1;

i = 1, n

(4.3)

169


. . (2.4) (4.1)
n Cin+1 = ki , i = 1, n +1

1

(4.4)

(4.3) ,

x1 > 0, ..., xn > 0,


i =1

n

ci 0,


i =1

n

ki 0

(4.5)

(2.2) ­ (2.4) , .. (4.1) . k1 6 = 0

x1 > 0, ..., xn > 0,


i =1

n

ci 0, k1 0

(4.5)

(4.1) n + 1 = r > n 2.2, (2.9) xn+1 ( ) (4.1) L ­ :
& x1 x1 0 & x2 0 x = u1 (t ) + 2 M M M & xn 0 0 0 u (t ) + x2 2 M 0 k +1 k x2 c1 x1 1 k1 k2 x2 u (t ) + c2 x1 n M M k k x2 cn x1 1 k k1 x2 x1
2

+1

2

2

k K xn n k K xn n un +1 (t ) MM kn +1 K xn k K xn n

(4.6)

L­ (4.6) (4.1) (4.2):

X 1 = x1
X
n +1

, X 2 = x2 , ..., X n = xn , x1 x2 xn
k
n n

= c1 x1k1 +1 x 2k2 L x
kn +1
n

+ c2 x1k1 +1 x x1

k2 +1
2

Lx

k
n

n

+ ... + x2

(4.7)

+ cn x1k1 +1 x 2k2 L x

+ x1k1 x 2k2 L x nkn xn xn

+1

(4.7) (4.3), (4.4), (4.2).
1



k Cij ,

i < j .

170


.. -- -10, 2002, .165-175

{

n + 1, C , U } (. . 3), C (4.4), U = R
k ij
k ij

L­

(4.6)






n +1

.

5. L­ k (4.4) Cij [5]. (4.7): (n+1)­ , k Cij . « » (4.3)

( k1 0 , . (4.5))

1 X 1 , X j = 0, k1 k1 X i - ki X 1 , X j

1 X1, X n k1 = 0, [ k1 X

+1

i

= X n +1 , - ki X 1 , X n

+1

]

= 0,

i = 2, n, j = 1, n


Y=

1 X 1 , Yi = k1 X i - ki X 1 , i = 2, n, Yn +1 = X k1

n +1

(5.1)
n-2

( Xi Yj k1 « »

0)

Y1 , Y j = 0,

[Y1

, Yn

+1

]

= Yn +1 , Yi , Y j = 0, i = 2, n, j = 1, n

.. (5.1) (. (4.4)) C1nn++11 = 1 (5.2) Xi Yj . ui(t) % ui (t ) :
% % % % % u1 = k11 u1 - k2u2 - ... - knun , u2 = k2u2 , ..., un = kn un , u
171
n +1

% = kn +1u

n +1

(5.3)


0 ). % ui(t) ui (t ) L­
( k1 (4.6)
& x1 & x2 M & xn & xn
+1

n-2



=

x1 k1 - k2 x1 0 k1 x2 % % M u1 (t ) + M u2 (t ) + ... + 0 0 0 0
2

(5.4)

+

k +1 - kn x1 k x2 c1 x1 1 x2 k1 k2 cx x2 % M un (t ) + 2 1 M M k1 xn k k x2 cn x1 1 0 k k1 x2 x1

+1

2

2

k K xn n k K xn n % un +1 (t ) MM k K xn n +1 kn K xn

L- (5.4) Yj (. (4.7), (5.1)). 6. L- (5.1)

{

k n + 1, Cij , U } = {n + 1, C1nn++11 = 1, R

n +1

}

.

(6.1)

(5.2) , n- [5] Y2 , ..., Yn +1 . , , Y2 , ..., Yn +1 «» ­ Yk = yk [3]. , (. (4.5))

172


.. -- -10, 2002, .165-175
k k % y1 = ln( x1k1 x2 2 ...xn n ) - c1 xn +1 ,

y2 = M yn = y
n +1

1 % ln x2 - c2 xn +1 , k1
(6.2)

1 % ln xn - cn xn +1 , k1 = 1 e k % c x x 2 ...xn n
k1 k 11 2

(

% c1 xn

+1

-1 .

)



% c1 = c1k1 + ... + cn k % c2 =

n

1 1 % c2 ,..., cn = cn k1 k1

(6.3)

(6.2) (5.4) L­
& y1 & y2 M & yn & y
n +1



=

1 0 0 0 0 1 0 0 % % % % M u1 (t ) + M u2 (t ) + ... + M un (t ) + M un +1 (t ) 0 0 1 0 0 0 1 - yn +1

(6.4)

Yj yk

Y1 =

-y y1

n +1

, Y2 = , ..., Yn = , Yn +1 = yn +1 y2 yn yn

+1

(6.5) (2.4) , xk. L­ (4.6) (5.4) (6.5) .
173


(5.3), (6.2), (6.3) (4.1) (4.6). , (6.4) (2.7) : y1 = y10 + v1 , y2 = y20 + v2 ,..., yn = yn + vn , yn +1 = ( yn +10 + vn +1 )e- v1 (6.6) 2.1 (6.4), % {u (t ), t} (6.6). (6.6) (6.2) xi , (4.6). % . ui (t ) -

% Y2 , ...Yn +1 , .. ui (t ) 0 .
(6.4) ­ [6]: y1 = const . , , (5.3), (6.2), (6.3) : (4.6) % u1 (t ) = k1u1 (t ) + ... + knun (t ) 0 , (4.6) ui(t) ­ k k y1 = ln( x1k1 x2 2 ...xn n ) - (c1k1 + ... + c1kn ) xn +1 = const . (4.6) . 7. (4.1) ( xixj) , (4.1) (n+1)­ , {u (t ), t} . (4.3) (4.2). (4.1) (6.4). (6.4) , (4.1). (6.4),
174


.. -- -10, 2002, .165-175

, . ( 02-01-00697) ( 0015-96137). . 1. .. . 1. ­ : " ", 2002, 232 . 2. .. // ­ -. . . . . . 10, . 1/ . .. , .. , .. . ­ : , 2002. ­ . 101 ­107. 3. .. . .: , 1978. 400 . 4. .. : . .: . , 1997. 96 . 5. . . .: , 1964. 356 . 6. .. : , // . . . / , 1984. . 62. .10 ­ 20.

175