I'm looking for examples of natural populations that have actually
grown exponentially. Are there cases where censuses have been taken of
invasive species to put good numbers on their growing populations?
(Human populations excluded.)

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Clarification of Question by fasteddie44-ga on 11 Sep 2006 15:41 PDT

Specifically, what I will pay for is references to, or excerpts from,
or texts of three scholarly papers that contain population counts of
invasive species.

just take a look at the history of rabbits in Australia
http://library.thinkquest.org/03oct/00128/en/rabbits/history.htm
from 24 released they reached plague proportions in about 30 years. It
still runs amock.

The fox introduced to australia attempting to control the rabbits
didn't achieve the results desired but they too multyplied like
rabbits.

Or the 26 Tanzanian queen bees introduced in Brazil in 1956, they
mixed with the local European bees producing the Africanized (Killer)
bees, by the mid 90's they had reached the southern United States. It
was thought that they would never pass Texas because of the cold
winters but Climate Change is allowing species to migrate further
north every year.
http://www.invasivespeciesinfo.gov/animals/afrhonbee.shtml

For many other such examples follow this link, just pick and choose 
http://search.nal.usda.gov/query.html?charset=iso-8859-1&ht=0&qp=&qs=-url%3Anal.usda.gov+-url%3Afstea.org&qt=invasive+species&qc=-&pw=100%25&ws=0&la=en&qm=0&st=1&nh=10&lk=1&rf=0&oq=&rq=0&si=1&x=0&y=0

You might also find this article interesting

INVASIVE SPECIES:
The Galápagos Islands Kiss Their Goat Problem Goodbye
Jerry Guo

SANTA CRUZ, GALÁPAGOS ISLANDS--The world's largest eradication
campaign has virtually rid an ecological wonderland of feral goats, a
devastating invader. Next in the crosshairs: cats and rats. (Read
more.)

Full story at http://www.sciencemag.org/cgi/content/full/313/5793/1567?etoc