sol=NDSolve[{Derivative[1,0][x][t,z]==((A*x[t,z])/((B+x[t,z])* (1+y[t,z]/E)))+K*Derivative[0,2][x][t,z],   Derivative[1,0][y][t,z]==-((A*x[t,z])/((B+x[t,z])*  (1+y[t,z]/E)))+M*Derivative[0,2][y][t,z], Derivative[0,1][x][t,0]==-(A*(Cs-x[t,0]-y[t,0])*vs)/((15*K*3.142*DR^2)*(B+ (Cs-x[t,0]-y[t,0]))*(1+y[t,0]/E)), Derivative[0,1][x][t,10]==0,  Derivative[0,1][y][t,0]==-(A*(Cs-x[t,0]-y[t,0])*vs)/((15*M*3.142*DR^2)*(B+ (Cs-x[t,0]-y[t,0]))*(1+y[t,0]/E)), Derivative[0,1][y][t,10]==0, x[0,z]==C1, y[0,z]==C2},{x[t,z],y[t,z]},{t,0,900},{z,0,10}]; 

Note: A,B,E,K,M,Cs,vs, DR, C1, and C2 are the constants.

Running NDSolve as above generates this Warning: NDSolve::ibcinc: Warning: boundary and initial conditions are inconsistent. >> Your help and suggestions are greatly appreciated!