Let　Z　be　a　supercritical　branching　process　with　immigration　at　zero　in　a　random　environment,　where　the　immigration　is　allowed　entering　a　generation　iff　the　previous　generation　is　empty.　The　limit　theorems　for　the　naturally　normalized　population　size　Wn　shall　be　investigated,　under　both　the　annealed　law　and　quenched　law.　In　addition,　in　the　linear　fractional　case,　the　harmonic　moments　of　W　and　the　convergence　rate　of　the　transition　probabilities　p(n)　l,j　of　Z　are　studied.