One organism benefits while the other is killed. Two species need the same resource, which is limited.

Parasitism - two different species where symbiont benefits and the host is harmed or killed.

In this case, you are talking about two different organism relationships. The first one is called parasitism. In the second you are talking about competitive exclusion.

Explanation:

All right, first of all, Parasitism is a form of survival employed by certain organisms in order to survive. It is really common and it can happen in animals, plants, and bacteria.

The second thing you describe is Competitive exclusion which is defined as the competence between two different species that feed on the same source, this is a very nice phenomenon and both species can survive if the source of food is enough. However, if it isn't one f them will perish.