I will point to the Yellowstone Park and use it as a comparison.
The population of wolves in the area of the US around Yellowstone Park was completely erradicated. Compare this situation to your carnivores. In your case, your race decides to stop hunting. In the case of the wolves, they are no longer able to.
The population of deer and moose around Yellowstone initially boomed, with their primary predator no longer on the scene. The large grazers spread throughout the park, loving the rich growth near the streams and easy access to water and quickly stripped the greenery from this area. Between the eating and the trampling, this resulted in quite a bit of errosion and the destruction of much of the plant growth near streams.
The widespread destruction of the plantlife reduced its viability to support the life of the smaller animals and the amount of soil being washed into the waterways reduced its viability to support a large population of the fish.
In a move towards returning the Yellowstone to a more natural state, a proposal was made to reintroduce to wolf to the area.
As the wolves established themselves, a very interesting change came over the park as a whole. With predators now a threat, the large grazers no longer felt safe in the open area near the water and would only stay near them long enough to get a drink and then retreat back to the forested areas.
With the extensive grazing and trampling reduced near the waterways, saplings, bushes and grasses were able to stablize the soil, which cleared up the water for the fish and gave the small animals a place to florish, which also allowed the smaller carnivores to flourish.
In a similar situation, it has been found that a lack of predators allows herds of animals (wild or domestic) to decimate the environment. It isn't the population of herbivores that is the problem, but the lack of an incentive to keep moving. Herds that are spread out and moving slowly can easily strip an area and begin the desertification of the area. Once the herds are forced to congregate and keep moving (either by carnivores or by human sheperds, herders, or "cowboys"), the land is able to develop a healthy groundcover of plants which is able to hold onto rainwater which greatly reduces the amount of flooding and errosion.
So, in short, if your purely carnivorous species actually loves their prey and wishes to preserve the health of the species and the environment, they will continue to hunt/herd the species.