We had already been at a safari camp in South Africa, and had seen a lot of cool things, including lions and a leopard. But in South Africa the guides are required to carry guns, and we were usually in a vehicle when we saw large carnivores. Also, the safari camps are part of a large national park, so while the situation is very natural, it's not totally wild.
In Botswana, the situation is different. We were in the Okavango Delta ("swamp", actually, but swamps can be fairly pleasant, it turns out). Our camp (redundantly named "Camp Okavango") was on an island, and we would go out on swamp boats, looking for something interesting. (Zoom out on the Google Map to see the overall shape of the swamp.)
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We had been given a little talk when we first got there, explaining that because the situation was more truly wild, they could promise us lots of cool tropical birds, but we might or might not see bigger things. We also found out that the guides didn't carry guns, because if you know what you're doing, you don't need a gun (in the daytime).
These guides were fantastic. The reeds and papyrus were perhaps 6 feet tall out of the water, and we'd be flying down these channels, when they'd stop because they saw something. At one point, the guide said, "There's dust over that island. Something big is moving there. Let's go see what it is." (I couldn't see anything even after he pointed it out.) He also radioed the camp's other boat to tell them to come (one of their tricks for finding cool things).
When we got out of the boats on the island, the head guide reminded us to do exactly what they said, don't move quickly, speak quietly, don't run, and everything will be fine. If we didn't do what they said, someone could get hurt. Then we headed towards the dust, which even I could see now. After a bit, as we were moving through some trees, we could hear crashing sounds. The guide said, "Ah, it is a herd of buffalo. They are running this way. We are too close."
I thought to myself "I'll do anything you say; please don't let me die." The guides led us up onto this big termite mound, about 6 feet high and maybe 20 feet across. One of them said, "We'll be safe here; the buffalo aren't trying to run us down, and they'll run around this mound, not onto it."
Then a big herd of buffalo came running up; the guides guessed there were over 500 of them. The herd stopped running, and began milling around. After a bit, one of the guides said, "Ah, the reason the buffalo were running is because of the lions." My thought: Oh God. The guide said, "There's one! Ah, and there's another. And another." There were at least six or seven lions circling the outside of the herd, skulking through the underbrush. "You can see that they are hungry, because you can see their ribs showing."
The guides were clearly relaxed and happy; they even high-fived each other, since they had found something cool for us to see. I was utterly terrified. I tried to take a video, but my hands were shaking so badly you can't tell what's going on.
I went over to the head guide and said, "I trust you completely, but I just don't understand. Why is this okay? Why aren't we all about to die?" The guide explained that there were two reasons. One was that people are bony and don't have much meat; as long as there are baby buffalo nearby, the lions would much rather eat them. The other thing is that some people still hunt game in Botswana, and the lions don't know that we don't have guns. We are the other really dangerous animal here besides the lions!
The guide then pointed out that there were pairs of adult buffalo following the lions around, trying to keep track of them. A lion can't beat an adult buffalo; so the buffalo were trying to make sure the lions didn't get any of the baby buffalo. The guide pointed out that some of the other buffalo were watching us, since we were also dangerous predators.
I was so happy that the lions didn't know how pathetically harmless I actually was. I also really appreciated for the first time how good it is to be at the top of the food chain.