I have always assumed that top predators are controlled by Nature. Too many predators - too few prey items.
I looked for some science but found little
"Researchers have traditionally assumed that the densities of the largest of predators are determined by the availability of their prey supply, but recent studies seem to contradict a bottom-up control." is from a recent study of large carnivores like lions and wolves,
I could find no specific references to (peregrine) Predator/Prey ratios. But there is plenty about their breeding ratios Urban/Rural and there are references like "Prey species diversity was lower in urban environments, supporting findings from other studies that bird diversity tends to decrease with urbanisation (e.g. Blair 1996; Clergeau et al. 1998;..." to be found.
It's all very well not seeing a Peregrine very often and seemingly being plagued by pigeons - but that's not science. Where's the science?