In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system

$$ u_{1,t}=u_{1,xx}-u_1u^{2}_2,\qquad u_{2,t}=du_{2,xx}+u_1u^{2}_2 $$

with non-negative initial data, where the constant $d>0$ is the Lewis number. Our purpose is to study the global dynamics of solutions under mild decay of initial data as $|x|\rightarrow\infty$. In particular, we show that, for a substantial class of $L^1$ initial data, the exact large-time behaviour of solutions is characterized by a universal, non-Gaussian spatio-temporal profile, subject to the apparent conservation of total mass.