Chains of model structures arising from modules of finite Gorenstein dimension

For any integer n≥ 0 and any ring R, \ (PGFₙ, \ Pₙ^⊥ ∩ PGF⊥) proves to be a complete hereditary cotorsion pair in R-Mod, where PGF is the class of PGF modules, introduced by J. {S}aroch and J. {S}\'{t}ov\'{i}{c}ek, and \ PGFₙ is the class of R-modules of PGF dimension ≤ n. For any Artin algebra R, \ (GPₙ, \ Pₙ^⊥ ∩ GP⊥) proves to be a complete and hereditary cotorsion pair in R-Mod, where GPₙ is the class of modules of Gorenstein projective dimension ≤ n. These cotorsion pairs induce two chains of hereditary Hovey triples \ (PGFₙ, \ Pₙ^⊥, \ PGF⊥) and \ (GPₙ, \ Pₙ^⊥, \ GP⊥), and the corresponding homotopy categories in the same chain are the same. It is observed that some complete cotorsion pairs in R-Mod can induce complete cotorsion pairs in some special extension closed subcategories of R-Mod. Then corresponding results in exact categories PGFₙ, \ GPₙ, \ GFₙ, \ PGF<∞, \ GP<∞ and GF<∞, are also obtained. As a byproduct, PGF = GP for a ring R if and only if PGF^⊥ₙ= Pₙ for some n.