There's no stable equilibrium. The ones on the edges are constantly incentivized to move towards the center, but at some point the guy in the middle is better off moving anywhere else. Kind of like the "divide the dollar" game.
I doubt that position is important if the 3 are directly adjacent one another, or at least not as important as say the appearance of your store front or your prices or your quality or your reputation or your display of well known logos ....
If I walk all the way down the beach 3m further to get to a nicer looking icecream shop is nothing. Indeed I think I'd look at the price of the icecreams first, then the ones people are leaving with and make a cost analysis, then probably choose the one selling pistachio flavoured icecream.
Actually, the ones on the edges should spread out a bit, they will reduce the competition for those coming from the ends of the beach who will now have to walk further to get to another shack, and they still split the difference between themselves and the next shack. How MUCH they should spread out and what more competition will do is unstable.
Sure there is equilibrium. Differentiation on price points, quality, service and length of lines will balance things out.
That said, the difference between a third ice cream cart and a 30th ice cream cart in the area is great. But then, the market should correct itself so that the optimal number of ice cream sellers are available.
