Analysis on a generalized Sel’kov–Schnakenberg reaction–diffusion system

Abstract This paper concerns a generalized Sel’kov–Schnakenberg reaction–diffusion system. Criteria for the stability and instability of the unique constant steady state solution are given. Various conditions on the existence and nonexistence of nonconstant steady state solutions are established. In particular, it is proved that the system admits no nonconstant steady state solution provided that d 2 is large enough and 0 p ≤ 1 , while it has nonconstant steady state solution if d 2 is large enough and p > 1 . This implies, when d 2 is large enough, the index p = 1 is the critical value of generating spatial pattern (especially, Turing pattern). Our main results essentially improve those in previous works.