  New Horizons observations suggest that CH4 on Pluto has a complex history,
involving reservoirs of different composition, thickness and stability
controlled by volatile processes occurring on different timescales. In order to
interpret these observations, we use a Pluto volatile transport model able to
simulate the cycles of N2 and CH4 ices over millions of years. By assuming
fixed solid mixing ratios, we explore how changes in surface albedos,
emissivities and thermal inertias impact volatile transport. This work is
therefore a direct and natural continuation of the work by Bertrand et al.
(2018), which only explored the N2 cycles. Results show that bright CH4
deposits can create cold traps for N2 ice outside Sputnik Planitia, leading to
a strong coupling between the N2 and CH4 cycles. Depending on the assumed
albedo for CH4 ice, the model predicts CH4 ice accumulation (1) at the same
equatorial latitudes where the Bladed Terrain Deposits are observed, supporting
the idea that these CH4-rich deposits are massive and perennial, or (2) at
mid-latitudes (25{\deg}N-70{\deg}N), forming a thick mantle which is consistent
with New Horizons observations. In our simulations, both CH4 ice reservoirs are
not in an equilibrium state and either one can dominate the other over long
timescales, depending on the assumptions made for the CH4 albedo. This suggests
that long-term volatile transport exists between the observed reservoirs. The
model also reproduces the formation of N2 deposits at mid-latitudes and in the
equatorial depressions surrounding the Bladed Terrain, as observed by New
Horizons. At the poles, only seasonal CH4 and N2 deposits are obtained in
Pluto's current orbital configuration. Finally, we show that Pluto's atmosphere
always contained, over the last astronomical cycles, enough gaseous CH4 to
absorb most of the incoming Lyman-flux.
