Hmm my instinct is that I think it would be avoidable, by combining an understanding of the natural world and the animals that live in it which technological/practical solutions.
For example, as raised in the other thread, you can't expect to kill off all the wolves/big cats/etc and then not have an ever-increasing deer population. But by sensitively re-introducing predators (for example, by choosing animals that are shy of people and tend to stay away from urban environments) you can maintain a more natural balance of prey animals.
Interestingly, in a vegan society it would be the prey animals that would be the big problem, as there wouldn't be farm animals for predators to attack (or infect). Culling prey animals is always going to be difficult because they tend to breed very quickly, because they are prey animals and so otherwise they wouldn't survive, which is why - despite vigorous attempts at wiping them out - animals like rabbits continue to thrive. I don't know a lot about farming, but in the UK I would think the biggest problem animals for crops are birds and insects, which aren't currently "culled", so perhaps it wouldn't be a big problem.