That is, what do I gain from swimming that I don't from running, and from running that I don't from cycling, etc.
Some of the answer is obvious, with different muscle groups, but I'm curious about the respective cardiovascular effects.
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The differences in cardiovascular improvements between those three activities are going to be minimal in the overall scheme of things. Cardiovascular fitness is different than muscular, in that any stress that raises your heart rate, increases respiratory rate will have improvement effects for the CV system.
What will have much more of an effect is the intensity level and the length of time for your workouts. In that, cycling will have an advantage, because you can go for longer at a higher intensity than you can with running and swimming. Swimming has the advantage of being able to go at a fairly high intensity day after day, as there is no impact, and it is a whole body exercise. Running is pretty high intensity, but because of the impact and the necessity to work up to higher levels, it can't be done as often as the others.
Overall, what will happen is that there will be cellular adaptations for energy conversion and oxygen utilization, your body will develop more capillaries into the working muscle, your circulation will become more efficient, and depending on your body makeup, you may convert some muscle fibers to a different phenotype. This will happen almost regardless of the activity, as long as you are stressing the cardio system.