The paper deals with the blow-up rate of positive solutions to the system u1=uxx+ul11 Vl12, Vl=Vxx+ul21Vl22 with boundary conditions Ux(1, t)=(Up11Up12) (1, t) and Ux(1, t)=(Up21Up22)(1,t). Under some assumptions on the matrices L=(Lij) and P=(pij) and on the initial data u0, v0, the solution (u,v) blows up at finite time T, and we prove that max X∈[0,1] V(x,t) goes to infinity as (T-t)a1/2 (resp. (T-t)a2/2 where ai＜0 are the solutions of (P-Id)(a1,a2)t=(-1,-1)t.