                                                                              .txt
***    M(N,K)  1-4
      : 1(N,K), M2(N,K), M3(N,K), N4(N,K) 
     ALFA,BETA,GAMA -  .

      
*********************************************************************

*** 1(N,K)
 :

0<=t<=tMAX, i=1,2,...,n,

dx(i)/dt = ALFA/[1 + BETA*z(i)] - x(i),



z(1)   = x(N)^GAMA   + x(N-1)^GAMA +...+ x(N-K+3)^GAMA + x(N-K+2)^GAMA
z(2)   = x(1)^GAMA   + x(N)^GAMA   +...+ x(N-K+4)^GAMA + x(N-K+3)^GAMA
......................................................................
z(N-1) = x(N-1)^GAMA + x(N-2)^GAMA +...+ x(N-K+2)^GAMA + x(N-K+1)^GAMA


*** 2(N,K)
 :

0<=t<=tMAX, i=1,2,...,n,

dx(i)/dt = ALFA/Z(i) - x(i),



Z(1) = [1 + BETA*x(N)^GAMA]*[1 + BETA*x(N-1)^GAMA]*...*
                           *[1 + BETA*x(N-K+3)^GAMA]*[1 + BETA*x(N-K+2)^GAMA]
Z(2) = [1 + BETA*x(1)^GAMA]*[1 + BETA*x(N)^GAMA]*...*
                           *[1 + BETA*x(N-K+4)^GAMA]*[1 + BETA*x(N-K+3)^GAMA]
.............................................................................
Z(N) = [1 + BETA*x(N-1)^GAMA]*[1 + BETA*x(N-2)^GAMA]*...*
                           *[1 + BETA*x(N-K+2)^GAMA]*[1 + BETA*x(N-K+1)^GAMA]


*** 3(N,K)
 :

0<=t<=tMAX, i=1,2,...,n,

dx(i)/dt = ALFA/[1 + BETA*z(i)] - x(i),



z(1)   = x(N)^GAMA   * x(N-1)^GAMA *...* x(N-K+3)^GAMA * x(N-K+2)^GAMA
z(2)   = x(1)^GAMA   * x(N)^GAMA   *...* x(N-K+4)^GAMA * x(N-K+3)^GAMA
......................................................................
z(N-1) = x(N-1)^GAMA * x(N-2)^GAMA *...* x(N-K+2)^GAMA * x(N-K+1)^GAMA


*** 4(N,K)
 :

0<=t<=tMAX, i=1,2,...,n,

dx(i)/dt = ALFA*R(i) - x(i),



R(1) = 1/[1 + BETA*x(N)^GAMA] + 1/[1 + BETA*x(N-1)^GAMA] +...+
                    + 1/[1 + BETA*x(N-K+3)^GAMA] + 1/[1 + BETA*x(N-K+2)^GAMA]
R(2) = 1/[1 + BETA*x(1)^GAMA] + 1/[1 + BETA*x(N)^GAMA] +...+
                    + 1/[1 + BETA*x(N-K+4)^GAMA] + 1/[1 + BETA*x(N-K+3)^GAMA]
.............................................................................
R(N) = 1/[1 + BETA*x(N-1)^GAMA] + 1/[1 + BETA*x(N-2)^GAMA] +...+
                    + 1/[1 + BETA*x(N-K+2)^GAMA] + 1/[1 + BETA*x(N-K+1)^GAMA]

*** :  x(i) = x0(i)  t = 0 

         ALFA,
BETA, GAMA  tMAX     ,     -
   ,      .    

******************************************************************************************    



