*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.