Summary: We consider the asymptotic behaviour of the positive solutions of the generalized Beddington-Holt equation \[ x_{n+k} =ax_{n+k-1} +\sum^{k-1}_{i=0} \frac {b_ix_{n+i-1}} {1+c_ix_{n+i-1} +d_ix_{n+i}},\;n=1,2,3, \dots \] near the zero equilibrium. When \(k=1\), we also consider the asymptotic behaviour of the positive solutions of the generalized Beddington-Holt equation near the positive equilibrium.