PSP was beating DS until DS made a comeback.

This keeps getting brought up, but there's not a lot of truth to it.

Japan:

In its first 6 weeks on the market, it amasses a lead of 1.06 million on the PSP.  This includes the first uncontested week of 469K, and a later week where the lead was 289K.

PSP then leads for the next 10 straight weeks, but at a slow rate.  During this time, PSP makes up 197K of the deficit.  At that rate it would've taken a whole year to make up for the difference DS produced in the first month.  During that time it went from having 25.3% of the total DS/PSP market to having 35.6% of the total DS/PSP market.  It leads once again 4 weeks later, but not again for the rest of the year.

In short: Big DS launch boom, followed by small PSP boom, followed by near-permanent DS leads.

US:

Things are much more even here.  In DS's first 4 uncontested months, it gets a userbase of about 1.5 million units.  PSP also sells about 1.5 million in its first 4 months, but since DS has hit a slump while PSP has a strong month 5 and 6, it makes up much of the deficit; it remains behind DS by a couple hundred thousand.  They sell quite evenly from September 2005 through the launch of the DS Lite in June 2006.  Really I think it's easier to look at GBA being the DS's main obstacle in the US rather than PSP.

In short: Big DS launch boom, followed by big PSP launch boom, followed by a year of evenness, followed by DS pulling away.