Q1 is an IC. I just used that symbol as a placeholder. So all this chip does is gradually brighten and dim each of the LED lines, in overlapping intervals. So red brightens, then as red dims, green brightens, then as green dims, blue brightens, and so on. The IC receives a constant 12 V. There is no part number on the IC and I've replicated the board perfectly in the schematic.
Edit: theres another resistor from pin 1 to pin 4.
Take a look at the PIC 12F675 and similar series, power on pin 1, ground on pin 8, I/O on the rest.
Looks like they're just using something like that and pulse width modulating the outputs for dimming.
No way to know for sure, but I'm going to guess a PIC12F508-E/P -- Pin 1/8 as V+ and V- is a very "microchip" thing to do.
I would guess it’s an ASIC, a very simple one with an oscillator and sone digital logic. There are a lot of such LED driver chips from relatively unknown makers for high volume things like novelty flashing LEDs, Christmas etc lights, flashlights, bicycle lights etc.
