In the basic predator‐prey model under investigation, growth rates of both species are influenced by both predator and prey population levels. Random dispersal is modelled by diffusion‐like terms in both discrete and continuous situations. Application of a combined successive‐approximations multiple‐scale approach to stability theory shows that under some circumstances uniform conditions will be succeeded by a new steady state wherein predator and prey are more concentrated in certain regions. Under other circumstances (in the discrete case only), although uniform conditions are stable according to linear theory, nevertheless sufficiently large pertubations are destabilizing. The biological implications of the results are discussed.