Biological Oceanography

(^) where c is a proportionality constant. The maximal rate, V
max, will be attained when
[ES] = [E], that is when Vmax = c[E]. Substituting, we have the Michaelis–Menten
relation:
(^) The graph of V vs. [S] is hyperbolic with asymptote V
max. When V/Vmax = 0.5, then
[S] = ks. Therefore, ks is called the “half-saturation constant”. It can readily be
determined, and it is often used as a simple measure of enzyme substrate affinity. It
can be used to characterize the slope of the initial portion of an ecological relationship
of hyperbolic form. Note that ks is a “backwards” variable: high ks values denote low
affinity, or slow approach to saturation.
(^) The Ivlev (1945) approach is to suppose that with a great plethora of food, animals
will eat some amount Rmax, the maximum ration, and no more. At lesser food
abundance, they will eat fractionally less. The maximum ration is approached
asymptotically. The resulting equation is:
(^) where food density is ρ and Λ is a constant, the Ivlev constant. To derive this,
differentiate with respect to ρ, and then establish an argument for assumptions leading
to the resulting differential equation. It is just as useful to simply examine the
approach of Ration to Rmax as ρ increases (the limit of e−Λρ → 0 as ρ → ∞).
(^) Other functions for hyperbolic relationships are in use, and some will appear later in
the text, for example the hyperbolic tangent function recommended by Jassby and
Platt (1976) to characterize the increase of photosynthetic rate toward an asymptote
with increasing available irradiance. When hyperbolic ecological or physiological
data are strongly variable, it may not matter which deterministic function is used to
represent the central tendency of responses to some forcing variable. The best choice
may depend upon mathematical convenience, say in a numerical model, not on
precision of fit.
(^) Threshold effects are frequent in ecological relationships. It is sometimes found, for
example, that animals won’t feed at all unless there is more than some minimum of
available food. This minimum is a threshold. Threshold effects are readily added to
either the Michaelis–Menten (here restated in terms of food) or Ivlev equations:
(^) In both, ρ
t is a threshold food abundance for feeding. Both of these equations must
be applied only where R ≥ 0; that is, where ρ ≥ ρt use the equation, otherwise R = 0.
Failure to follow this restriction (as in computer code for ecosystem models) will