Why are they fighting? Why is your router aggressively reusing the IPs as soon as the lease expires?
In my house, all of my devices end up with pretty stable IPs (even non-Apple ones) and I haven't done anything special to configure this. I assume my router holds onto previously-assigned IPs in case the device comes back as long as there's still unassigned IPs available to give out to new devices, though I haven't investigated it.