The population eats a quantity of vegetation less than or equal to the quantity that grew that year.
Balancing feedback mechanisms bring the stock of both deer and and vegetation into an equilibrium. Deer and vegetation can persist on a continuous level.
The deer population have balanced births and deaths. The total number of deer is able to survive on the annual "interest" growth of vegetation without eating the "principal".
That the numbers are kept within a certain limit to ensure the vegetative resources will be able to sustain the deer in the long term
This means there is a balance between the birth and death rates of the deer and the vegetation regeneration to support  the deer population. 
birth rate equals death rate of the herd and the size of the herd is such that only the interest or regeneratable growth is eaten by the herd every year
That the rate of growth of the herd is equal to the long term carrying capacity of the land enclosure.
They can continue to exist without causing gradual degradation of vegetation.
That the consumption of resources matches the replacement of resources.
Deer population is within a limit allowing natural capital to regenerate itself. 
The quantity of deer in the park, their reproduction rate and their death rate balances with the ability of the vegetation to regenerate and keep the deer fed in the future.
That the deer population in the park  (including births and deaths) exactly matches the new (regenerated) grazing capacity
It remains at a level that does not cause a reduction in the level of vegetation, so the vegetation is able to regenerate fully.
The deer population could remain stable indefinitely 
The deer population will be the same over time in the deer park.
Taking into account the birth and death rate, the deer will not eat vegetation at a faster rate than it regrows.
The deer population in the deer park is sustainable when the herd can reproduce and live out a natural life span without overgrazing and thus destroying the vegetation for the future.
The deer population will not collapse suddenly.
There are enough renewable resources to maintain the deer population today and in the future.
The population of the deer remains at a stable number that does not cause overgrazing
birth and death rate are equal
The birth rate and death rate of the deer and the regrowth of the vegetation are in balance.
The population can continue indefinitely at that number.
The deer population is staying about the same number, with deaths and births equal and enough food for all. The vegetation/food supply is not being depleted over time,  but keeps growing enough to keep the deer population healthy.
