Lamb's Domain (How much of Rapture did Lamb actually control??) :
Map ... http://img62.imageshack.us/img62/4247/lambsterritory.jpg
Sofia Lamb controlled about 1/4 of the City of Rapture at the time of BS2, from near the Atlantic Express Depot to the Greenbelt farming area north of Minerva's Den ( see map ). Other factions had blocked further expansion (destructively with floodings and destroyed connecting structures, and force and threats Lamb couldnt ignore).
You would think that Lamb's Big Sisters could traverse the water gaps at the borders, but there were other Big Sisters who had been conditioned by the more able factions to counter this (and protect THEIR Little Sisters - forcing Lamb's ambitions to seek a source of new Little Sisters elsewhere).
Later threats from Hephaestus (Ryan's old followers) to 'turn off her power' brought a standoff truce and limited Lamb's troublemaking in the rest of the City.
Food to feed Lamb's minions was available from the Greenbelt farming area and large numbers of her 'Family' toiled there. Being cut off physically and ideologically from Neptunes Bounty had lost the staple of fish from that source. But then, low protein diets are perfect for numbing down peoples mental capacity and making them more malleable.
The research center (and the remaining staff) at Fontaine Futuristsics fell into her hands, enabling her warped ADAM experiments (with Gil Alexander and an unknown number of children being the victims of her search for the first 'Utopian').
Technical expertise in Minerva's Den remained out of reach, with simple firepower blocking her attempts to seize it (think bigger Ion Lasers firing at Big Sisters and submarines thru windows). Similarly, expansion into Hephaestus was denied (shouldn't mess with people who can turn your power, heat, water and air off). Lamb tried to infiltrate both, but failed (with some of her 'agents' being pneumo'd back to her from Hephaestus).
When Simon Wales fell fully into his cult organizing roll (and spliced himself into insanity), Lamb lost one source of know-how that could have greatly assisted in improving her section of Rapture (and her subjects living conditions). But then, such things were not that important to her ultimate goal.
Explanation for why the Mk2 AE trackage (the other lines) was not used in Lamb's territory (they do connect more than a few locations in that zone) : The main connecting tracks down below AE Depot were cut/damaged in the faction wars (and the other connection thru the Freightyards had long been flooded). All the AE Mk2 equipment was blocked in at the Depot facilities. The remainning accessible Mk2 train equipment eventually failed from lack of maintenance. Whereas the Mk1 connection to the Depot (The Short Line that you traversed most of in BS2) survived, allowing access to repairs and spare equipment and allowed continuance of the limited train operations.
The Trolley 'street' system in her territory sort-of worked in a limited way - due to a mix of incompetance and Lamb deciding that insane people would be 'safer' walking (and restricting her 'subjects' movements made them easier to control).
Lamb obviously had some submarines (which she used to grab children from the surface) which were largely the smaller privately owned type. For longer distance ones (Neptunes Bounty had the majority of these), a few fell into Lambs control early in the faction wars that began after Ryans death. Wales had access to resources (he had been working on installations for Rapture's defenses for Ryan in the past, before he 'got religion') which we saw some of in BS2 (the torpedo incident).
The existing Rapture infrastructure was still useable by Lamb and the people under her control - the Pneumo still worked where it wasnt cut off by damage, the communications links (propaganda is a first priority for a tyrant) continued to function for Lamb's endless pronouncements, heat/power/air/water still flowed to more than enough habitation space. Garbage disposal wasnt a hight priority, but it was simple enough when it got too thick to push it out of an airlock. It was likely that leaving everything in a state of wreck and ruin was to Lamb's purpose, first to blame Ryan and second to make it easier to get 'The Family' to give themselves up to be killed to achieve her mad delusion.
With Sofia Lamb gone, it would be one more step in the direction of Rapture's road to restoration.