You won't find "Fee-Per-Minute" in North America, we have "Ramp Fees" or "Parking Fees" that are daily in nature, if applicable, in addition to Landing Fees. In the U.S. and Canada, you have Airport Contract Signatories, and non-signatories. In addition, you can also have "Primary Signatories", with negotiated lower fees (ie., AA at MIA, or DL in SLC, etc.). For instance, DL and Skywest pay no ramp fees at SLC for parking, even on overnights, and they pay a greatly reduced fee for gates they paid to build. Even reduced landing and passenger pass-through fees can be lower for these.
Non-signatories (charters, low-volume operators) can pay twice as much, or more, than the published contract rate. (Airlines with hubs in a city pretty much hold the city/airport Authorities hostage in contract negotiations in the US)
My current airport has gone to "Shared Use" gates. We have our assigned gates per the contract, that we pay for regardless of use, as well as our floor space. In addition, we can use the SU gates at a cost of $200 per turn, or $400 for an overnight. We generally use them for specials, charters, add-ons, and maintenance waits.
PS-All of these costs are why charters are handled almost exclusively by the FBO's in the US now, no matter how large the airplane. There are exceptions, of course (union contract language)