I've known for a couple years now that catch rates for Shadow Pokémon in Pokémon Colosseum and XD weren't actually identical to the usual catch rates for Gen III. Now, thanks to Patreon patron Pacguy64 for suggesting it, StarsMmd from Project Pokémon, and a lot of headaches over in-game testing that didn't seem consistent with the ripped data, the Gen III/IV Catch Rate Calculator finally has a Shadow Pokémon checkbox, which you should check when trying to capture Shadow Pokémon in the GameCube games.

The basic rundown is that the actual capture formula is identical to the other Gen III games, except for Colosseum having a provision to make the tutorial Makuhita a guaranteed capture, but the actual catch rate values used for the Shadow Pokémon specifically are overridden to create a more sensible gradient from easier captures early in the game to more difficult captures later - thus, your second capture in Colosseum, Bayleef, Quilava or Croconaw, will be generously bumped up to a catch rate of 180 instead of 45, while lategame Tyranitar gets bumped down from 45 to 10, even lower than the legendary beasts who are at 15. I've implemented these overridden values into the calculator so that it'll use those values if you check the Shadow Pokémon box, and also listed them under the capture rate section on the Gen III/IV Capture Mechanics page.

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