while the remaining 99.998 72% of
the rectangle
represents an enormous quantity
of
unoccupied "empty space"
In the ocean, real-world populations
of red-tide dinoflagellates such as
Karenia
brevis
which constitute one of nature's
quintessential examples
of environmental calamities arising
as a result of population
explosions
generate severe red-tides and their associated
fish-kills when their populations reach
concentrations of
100,000 - to- 1,000,000
or more K. brevis cells per liter
Because each cell releases, on an on-going basis, small amounts of poisonous “brevetoxins,” the accumulation of such toxins reaches calamitous levels within the aqueous environment in
which the population resides.
In one liter* water samples
from a red-tide
* 61.024 cubic inches of water-
which are represented by
the
mostly empty rectangle in the illustration above
the entire population of K.
brevis cells
responsible for the deadly calamity
physically occupy
less than
two one-thousandths of one percent
of the total volume that
appears to remain
theoretically-available to them
In the illustration above, one
million such cells all
together could physically-occupy the area denoted by dot
Visitors wishing details of the
mathematics of this critique can find step-by-step
calculations by clicking on "Supporting
Mathematics" in our navigation bar.
This means that the above dinoflagellate
population manages to visit calamity upon
themselves and the aqueous
environment
in which they reside
even though they physically-occupy
less than two one-thousandths of one percent
of the volume that seems to remain theoretically-available to them
In other words, despite an apparently
enormous amount of open space,
and despite the fact that the Karenia
brevis population occupies
a
volumetrically-insignificant
portion of the available “open space,”
they have, by their combined overpopulation
and their
production of unseen, invisible, and calamitous wastes
catastrophically-damaged the watery surroundings
in which they live
Similarly, two classic studies
of reindeer herds (reported by Scheffer, 1951; and Klein, 1968) each showed that a
period of exponential growth in each
population was followed by a collapse in which a catastrophic 99% die-off took place.
More recently, calculations* show
that in each case, these two reindeer herds physically-occupied
less than one-tenth of one percent of the area theoretically-available to them
at the time of the collapse.
