Simple arbitrage model of a single commodity
As I noted in the comment, buying a major position in a commodity does not qualify as an arbitrage, rather it would be informally referred to as “cornering the market.”

I am unsure whether there is any theoretical work on such strategies. However, from a practical perspective, we can see that it is hard to characterise what would happen.

I will assume that we are talking about a physical commodity; things might be different for a financial asset.

Firstly, going into a market and trying to buy up the supply is difficult. There will be both physical commodities to buy, as well as derivatives (such as futures). How exactly can you get the physical inventories away from other commodities traders and consumers, when they see someone buying up all the other sources? You need to find a way to buy up the commodity ahead of time quietly, and then you need some event (such as a bad harvest) wiping out the holdings of the other commodity traders. Cornering a market has been done, but you need particular circumstances to make it work.

(I would recommend looking up the history of the Hunt Brothers and silver.)

The next problem is your ability to take advantage of your high inventory position. The producers of the commodity will ramp up production (and demand may be destroyed) if you try to only sell at a high price. You could get stuck with a large inventory of a commodity that has fallen in price. If you borrowed to finance your position, you might be forced into distressed selling - and everyone will know about it.

In other words, there are a lot of possibilities, and there is no easy way to characterise what will happen to the price.

Things are different if you control the production of the commodity. That ends up just being te situation of a monopoly (or oligopoly). There is a large literature on monopolies in economics; I am not familiar with it.