Any global climate model worth its salt will have a comprehensive carbon cycle submodel, tracking the sources and sinks of carbon in all its forms, between ocean, soil, land, and atmosphere. Some models evolve the amount and type of plants e.g. C1 to project future biome distribution [0]
One such model is aptly named TRIFFID [1]
[0] https://en.m.wikipedia.org/wiki/Carbon_cycle
[1] https://www.nature.com/articles/35041539
Edit: non-paywall link to TRIFFID paper [3]