In this paper, we study the Cauchy problem of a cubic autocatalytic chemical reaction system u1,t = u1,xx − uα1 uβ2, u2,t = du2,xx+ uα1 uβ2 with non-negative initial data, where the exponents α,β satisfy 1<α,β<2, α+β = 3 and 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|→∞. We show the exact large time behaviour of solutions which is universal.