Holy cow!  Imagine:

if by 2020 there were 1 billion cows less on this globe! Not only would there be:

far less global warming from superseded methane gaseous outputs, but also from no- longer- necessary transport and oven heating/refrigeration expenses;

 according to a German health report, your own chance of dying:

from cardiovascular disease could shrink from a meat eater's 50% to a vegetarian's 15% , and 

from cancer it would plummet from a meat eater's  24% down to a vegetarian's 8%!

Better still, for vegans the risk of dying from either heart disease or cancer is only 6 %! 

How's that for  a happy & healthy life expectancy! Ain't it worthwhile culling the cows rather than waiting until the cows come home?

Culling is not an issue, because the food chain life cycle is so short.

Animals get tough and inedible very quickly with age. Also, once it reaches full size, feeding an animal to keep it alive is pointless. Poultry is only a few months old. Lamb comes from animals less than 18 months old. Beef, and pork around two years, goat and venison around three years. The animals you see in fields are not destined for old age anyway - the cull has been in progress for centuries.

All food animals are bred specifically for consumption. Nobody is going to breed stock they can't sell. So the population of edible animals will drop as fast as demand - probably much faster than their land can be made productive for direct food crops (via better drainage, extra farm machinery, fertiliser production, seeds, processing and packaging factories).

Dairy cattle are fully productive for up to eight or ten years.

More likely is that we have to cull all the MacDonalds staff.

"...analysis showed that producing a kilogram of beef leads to the emission of greenhouse gases with a warming potential equivalent to 36.4 kilograms of carbon dioxide. It also releases fertilising compounds equivalent to 340 grams of sulphur dioxide and 59 grams of phosphate, and consumes 169 megajoules of energy."

"In other words, a kilogram of beef is responsible for the equivalent of the amount of CO2 emitted by the average European car every 250 kilometres, and burns enough energy to light a 100-watt bulb for nearly 20 days." - New Scientist Magazine

Pets have a carbon footprint the size of a car... so I hear...

It is true that if the birth rate of livestock remained the same, then global vegetarianism - reducing the harvest - would see livestock numbers skyrocket until population numbers were controlled instead by pasture limitations instead of abbatoir activity.

But livestock births are actually carefully controlled by their owners, who allow livestock to breed only if they see a market for them, so the overall population of livestock is regulated by owners so that it is enough to meet the predicted requirements of carnivorous customers.

If the world went vegetarian overnight, there would be a short term increase in livestock numbers, as markets for Beef Wellington collapsed, but the sheep and cows would die out relatively quickly (they are not long lived) and their owners would not breed any more, converting their pastures to tofu vines and lettuce.

Economic feedback is the dominant driver in this case - it controls the growth-harvesting dynamics that the question focuses on.