Composite membranes that exhibit long-term resistance to biofouling comprise a porous support and a crosslinked polyamide discriminating layer having an external surface, the discriminating layer comprising a branched poly(alkylene oxide) (PAO) polymer attached to its external surface. The branched PAO polymer typically has the structure of a molecular comb or brush, and is made by polymerization of a PAO macromonomer of the following formula: RO—[(CHR′)n—O]m—V in which R is hydrogen or a C1-20 aliphatic or aromatic group, V is any group containing a polymerizable site, each R′ is independently hydrogen or a short chain alkyl group, n is an integer of 1-6, and m is an integer of 1 to about 200. The α end group can be either polymerized or copolymerized.