The paper deals with the vectorial reaction-diffusion equation \[ (1)\quad u_ t=du_{xx}+f(u);\quad -\infty <x<+\infty, \] where D is a diagonal diffusivity matrix. When D and f satisfy appropriate conditions, the equation (1) has a one-parameter family of periodic travelling wave solutions of the form \[ (2)\quad u=\psi_ 0[x+c(\ell)t;\ell], \] where \(\ell\) is the spatial period and c(\(\ell)\) is the propagation speed of the wave. By means of a sequence of characteristics of the \(\ell\)- periodic wave given by (2), an interesting analysis of the stability and bifurcations of travelling waves is established.