Table 4.1 Symbols and standard values or initial values for the Franks et al. (1986)
model.
(^) VARIABLES AND PARAMETERS VALUES IN STANDARD RUNS
Vm = maximum phytoplankton growth rate 2 d−1
N = nutrient concentration Start at 1.6 μmol N liter−1
Ks = half saturation constant for nutrients 1 μmol N liter−1
P = phytoplankton stock size Start at 0.3 μmol N liter−1
m = phytoplankton mortality rate (apart from grazing)0.1 d−1
Z = zooplankton stock size Start at 0.1 μmol N liter−1
γ = zooplankton growth efficiency 0.3
Rm = maximum zooplankton ration 1.5 d−1
Λ = Ivlev constant 1.0 (μmol N liter−1)−1
d = zooplankton mortality rate 0.2 d−1
(^) It is imagined that the upper water column has just stratified at model initiation and
illumination is great enough to sustain rapid phytoplankton growth, which is limited
only by nutrient (“nitrogen”) availability. Conditions are right for a spring bloom. The
system starts with lots of nutrients and small quantities of phytoplankton and
zooplankton.
(^) Both phytoplankton, P, and zooplankton, Z, are just quantities, and their units
represent their nutrient (as nitrogen) content. Neither has age nor size structure.
Phytoplankton increase as a hyperbolic function of nutrient availability, a relation
represented by the Michaelis–Menten function (Chapter 3), dP/dt = (VmNP/(Ks +
N)).They also have a proportional death rate, −mP, beside that from grazing.
Zooplankton graze as a function of phytoplankton availability, P, with the rate
increasing hyperbolically according to an Ivlev function, dP/dt = −PZRm(1 − e−ΛP), to
an asymptotic value Rm. That is, planktonic grazers will eat more if more food is
offered, but only up to a point. Beyond that amount, their ingestion rate levels off (e.g.
Frost 1972).The model zooplankton die at a rate proportional to their abundance, –kZ
(k for “kill”).
(^) The initial parameter set (modified from Franks et al.1986) produces a strong, brief
bloom that is reduced by grazers (Fig. 4.4a). Nutrients are partially regenerated, and
then, after a few damping oscillations, proportions settle to a steady state. To a limited
extent, the model can be modified to apply more realistic rate parameters and initial
values. Phytoplankton seldom grow at 2.0 d−1; a more realistic rate would be one
doubling per day, that is: Vm = 0.69 d−1. High-temperate North Atlantic waters before