Chains of model structures arising from modules of finite Gorenstein dimension

For any integer $n\ge 0$ and any ring $R$, \ $({\mathcal{P}\mathcal{G}\mathcal{F}}_n,{\mathcal{P}}_n^{\perp }\cap {\mathcal{P}\mathcal{G}\mathcal{F}}^{\perp })$ proves to be a complete hereditary cotorsion pair in $R$-Mod, where $\mathcal{P}\mathcal{G}\mathcal{F}$ is the class of PGF modules, introduced by J. Šaroch and J. Štovíček, and \ ${\mathcal{P}\mathcal{G}\mathcal{F}}_n$ is the class of $R$-modules of PGF dimension $\le n$. For any Artin algebra $R$, \ $({\mathcal{G}\mathcal{P}}_n,{\mathcal{P}}_n^{\perp }\cap {\mathcal{G}\mathcal{P}}^{\perp })$ proves to be a complete and hereditary cotorsion pair in $R$-Mod, where ${\mathcal{G}\mathcal{P}}_n$ is the class of modules of Gorenstein projective dimension $\le n$. These cotorsion pairs induce two chains of hereditary Hovey triples \ $({\mathcal{P}\mathcal{G}\mathcal{F}}_n,{\mathcal{P}}_n^{\perp },{\mathcal{P}\mathcal{G}\mathcal{F}}^{\perp })$ and \ $({\mathcal{G}\mathcal{P}}_n,{\mathcal{P}}_n^{\perp },{\mathcal{G}\mathcal{P}}^{\perp })$, 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 ${\mathcal{P}\mathcal{G}\mathcal{F}}_n$, \ ${\mathcal{G}\mathcal{P}}_n$, \ ${\mathcal{G}\mathcal{F}}_n$, \ ${\mathcal{P}\mathcal{G}\mathcal{F}}^{<\mathrm{\infty }}$, \ ${\mathcal{G}\mathcal{P}}^{<\mathrm{\infty }}$ and ${\mathcal{G}\mathcal{F}}^{<\mathrm{\infty }}$, are also obtained. As a byproduct, $\mathcal{P}\mathcal{G}\mathcal{F}=\mathcal{G}\mathcal{P}$ for a ring $R$ if and only if ${\mathcal{P}\mathcal{G}\mathcal{F}}^{\perp }\cap {\mathcal{G}\mathcal{P}}_n={\mathcal{P}}_n$ for some $n$.