Bystander wrote: N0 = kapp,ave(1-3Ga) - kext,ave(1-3Ga),
and using 10M as a round number for the current species count, we get
kapp,ave - kext,ave = 0.01 - 0.003 species per year. That is, the average NET rate of change in number of species is equal to the difference in the average appearance and extinction rates. Talk about trivial statements of the trivially obvious --- we get no absolute estimates for these rates without appealing to other estimates of the total number of species that have existed, 100-1000 times the currently existing number, or 1-10G appearances in 1-3 Ga, an average appearance rate of 0.3 - 10 species/a. The average extinction rate is then (0.3 - 10) - (0.01 - 0.003).
Approach #2: same game as 1, and subscripting N with number of Ma prior to present to indicate the number of species just prior to the extinction events, using the dates and estimated extinctions from Thomas,
N0 = 0.35N65 + (kapp - kext)65Ma,
... (the eqtns for Triassic, Permian, and Devonian events are left as an exercise for the reader)
N370 = 0.35N440 + (kapp - kext)70Ma,
and, invoking a suspected 6th Cambrian event with a 50% loss as a guess,
N440 = 0.5N600 + (kapp - kext)160M,
and, finally, N600 = 1 + (kapp - kext)(400 to 2400Ma).
This removes extinction losses from the average extinction rate; the suspected Cambrian event's inclusion further reduces the background extinction rate. Working backward through the equations, substituting the last, for N600, into that for N440, and into ta-da, ta-da, we get
N0 = 6 x 10-4 + (130-140 Ma)(kapp - kext). The difference between the appearance and background extinction rates has increased from 0.003 - 0.01 to 0.07 - 0.08/a ; still no real insights regarding magnitudes for the gross appearance and extinction rates.
I've been a little distracted by a discussion on the Social Sciences sub-forum, and am only now getting back to this one; sorry.
Thanks to Evo and Rader for some excellent links and resources; any comments from Bystander? Ivan? Russ?? (seems we've lost Russ).
Bystander, I'm in the slow class today, and I don't really follow what you wrote. So I re-worked it, using just 1Ga, and units that I'm more comfortable with (I can see Andre shaking his head [b(]; he had just begun to form a favourable impression of me

):
Number of species today = Number of species which appeared over the past 1 billion years minus the number which went extinct in this same 1 billion years.
If we assume that there are 10 million species today, then there have been, on average, 0.01 net new species per year, over the past 1 billion years.
Of course, 10 million is almost certainly the difference between two much larger numbers, [/color]{though I can't see how Bystander got one of the numbers to be ~1 billion

}
In approach #2, we plug in estimates of the number of species lost in each of the five mass extinctions:
Number of species today = 35% of those present before the KT event + the number of net new species in the last 65 million years; the latter we assume to be the background new species rate (per year) times the number of years (65 million).
Similarly for the Triassic, Permian, and Devonian mass extinctions, using species survival data from Thomas. [/color]{I'm not sure which 'Thomas' Bystander is referring to - Chris Thomas, of Britain's University of Leeds perhaps? and what are the percentages of species which survived each mass extinction?}
Number of species just before the Devonian mass extinction = 35% of those present before the Ordovician event + the number of net new species in the 70 million years between the Ordovician and Devonian mass extinctions; the latter we assume to be the background new species rate (per year) times the number of years (70 million).
Then, including a postulated early Cambrian event which wiped out 50% of the species at the time:
Number of species just before the Ordovician mass extinction = 50% of those present before the Cambrian event + the number of net new species in the 160 million years between the Cambrian and Ordovician mass extinctions; the latter we assume to be the background new species rate (per year) times the number of years (160 million).
Finally, just before the Cambrian mass extinction:
Number of species just before the Cambrian mass extinction = 1 (the original species!) + the number of net new species in the 400 million years between the origin of species and Cambrian mass extinction; the latter we assume to be the background new species rate (per year) times the number of years (600 million).
The difference between the rate of appearance of species and the rate of extinction of species is then obtained by simple algebra from the above.[/color]
Bystander, please correct anything which is substantially incorrect, in terms of a re-statement of your point.
Nereid
[Edit: formats and added Bystander's conclusion]