Before revision:  Inspired by May's analysis of the stability of random ecological communities, we analyze the threshold for reaction-diffusion systems whose linearized dynamics near a homogeneous fixed point are given by a random matrix .
After revision:  Inspired by May's analysis of the stability of random ecological communities, we analyze the probability distribution of the diffusive threshold in reaction-diffusion systems whose linearized dynamics near a homogeneous fixed point are given by a random matrix .