  We study the possibility of non-simultaneous blow-up for positive solutions
of a coupled system of two semilinear equations, $u_t = J*u-u+ u^\alpha v^p$,
$v_t =\Delta v^+u^qv^\beta$, $p, q, \alpha, \beta>0$ with homogeneous Dirichlet
boundary conditions and positive initial data. We also give the blow-up rates
for non-simultaneous blow-up.
