I would like to know what $model = $this->repository->show($id); does and return;
In the meanwhile:
I understand that you building out a Rest API, and it could mean that the show method is bring back a json. The $model->delete() function won't work.
You may need to do this on your destroy method
public function destroy($id)
{
$model = Activity::find($id);
if($model->exists)
{
return $this->respondNotFound();
}
$model->delete();
return $this->respondNoContent();
}