There might not be as many pigeons as you think.
For example, R A Fuller, J Tratalos and K. Gaston published a study in 2009 that examined the numbers of birds in Sheffield (which has a population of roughly half a million people). While they found that there were slightly more birds than people overall (estimated at around 603,000 birds in total), the breeding population of feral pigeons was only estimated at around 12,300.
Such populations tend to congregate in very small areas (typically city centres) leading to the belief that the overall population is far greater than it actually is. (e.g. if you're in Fargate, in the centre of Sheffield, you'll probably see lots of pigeons but if you go out to the suburbs, which cover a far greater area, you'll be unlikely to see any).