Your IP address is sent to the other clients to which you connect. I believe it also gets recorded by the connection servers as an active client. Your IP is then passed around to other clients looking for network connections. When you first connect to the network your IP address is "fresh" and is handed out more often because it's more likely you're still connected. After a while your address drops further and further down the list and is eventually no longer handed out. That's why the hits gradually drop off over the course of the next few days.

This has nothing specifically to do with LimeWire. It's the way gnutella is designed to work.