If half the world's people population suddenly died, whatever caused those deaths would most likely kill off most of the remainder as well. If the ecosystem were to undergo some kind of revolutionary process, say the mass extinction of dinosaurs 65 million years ago, the ecosystem itself would be destroyed. Periodically, mass extinctions have wiped out a good portion of life on Earth; as one species meets oblivion, breaking the web of life, other species follow, forever altering the ecosystem. However, life itself is adaptable and sturdy -- eventually, given enough time and stable conditions after a catastrophe, new species arise and create a new ecosystem.
It is likely that a die-off on the scale you're talking about would have huge impacts on the ecosystem, no matter what kind of organism you're talking about. Some major impacts:
- The population of whatever your organism eats would explode. So lets say the wolf population drops (in the old days). The deer population explodes in the short term.
- In this example, the population of whatever the deer eat would, in the short term, be devastated. So the grass would be overgrazed, for example.
- Other things that compete for the grass would likely have their populations fall as all the "extra" deer ate all the grass.
- Populations of things that compete with "your" organism would go up too. So maybe as there is less competition from wolves, the mountain lion population would go up.
Of course, the deer population would drop again once they ate too much grass and there wasn't enough for all of them and they starved. But the process I've mentioned above would happen in the short term.