The authors deal with the travelling waves for \[ \begin{cases} \partial_tu=\partial^2_xu+f(u,v)\\ \partial_tv=g(u,v)\end{cases} \] and analyse how the missing diffusivity changes the qualitative picture. The authors are interested in front solutions that connect the states \(S_-=(0,1)\) with \(S_+=(1,0)\), that is \[ \begin{aligned} & (u,v)(x)\to S_-\text{ for }x\to-\infty\\ & (u,v)(x)\to S_+\text{ for }x\to +\infty.\end{aligned} \] The authors study the stability of the blocked waves. It turns out that the stationary fronts are unstable and that a sublinear penetration may occur. Their global analysis shows that, with an arbitrary small perturbation of the front, an invasion of the \(V\)-component at a logarithmic rate is possible.