differential equations, and their solutions when they have simple ones are always
functions, such as the exponential. The absolute amount of change in the light
between two depths depends upon: (i) the amount of light there is to be absorbed, E
(principal); (ii) the fractional rate at which it is absorbed per meter, k (in this case a
rate for negative interest); and (iii) the thickness of the absorbing layer, z (“time” at
interest). We write:
(^) Differential equations of this sort are called “first order” (they involve first
derivatives), “separable” differential equations (using this as an example, they can be
rearranged to have dE and all functions of E on one side and dz and all functions of z
on the other). They are “solved” by rearranging, then integrating:
(^) The integral of dE/E is natural log E, ln E. The integral of dz is z. Thus, the integrals
become:
(^) And, finally, taking differences and antilogarithms, the “solution” is E
z = E 0 e
−kz, in
which Ez is the intensity remaining at depth z relative to the just-below-surface
intensity of E 0.
(^) Populations above a reasonably small size grow exponentially, statistically exactly
so if reproduction isn’t synchronized in some way. When it is, then the exponential
pattern appears for counts at equal intervals measured in reproductive cycles. If we
use N for population numbers, then we get Nt = Noert, and we talk of “r” as the rate of
population increase (or decrease if its sign is negative). Both birth (b) and death (d)
can occur as exponential functions, so we can write: r = (b − d). If N(t) and b are
known, then you can solve for d, or conversely.
(^) Examples of the exponential function will continue to appear in this book. Please
practice with it, using a calculator. Get a very clear understanding of its
characteristics.
(^) Problem 1: It is desired to get an idea of the growth rate of some phytoplankton
cells. A few are inoculated into a jar of sterilized seawater enriched with various
fertilizer compounds (nitrate, phosphate, etc.). At the end of two days there were 200
cells per ml. At the end of four days there were 800. What is the exponential rate of
growth? What is the doubling time? What is the formula relating doubling time to the
exponential rate of growth?
(^) Problem 2: A “simple” extension of the exponential function is the logistic