(Eqn. 4.2) (^) is provided in Box 4.2.
(^) The solution settles immediately into boring and unrealistic limit cycles (Fig. 4.2a),
regardless of initial conditions. However, with a density-dependent self-consumption
term added to the grazer equation (2) (variant 2 in Box 4.2), there is a stabilization of
the oscillation. The new equations are:
(Eqn. 4.3)
(^) and
(Eqn. 4.4)
(^) The self-limitation of grazers (by cannibalism, Fig. 4.2b), represented by the
quadratic term in equation (4), is powerful and P and Z quickly become constants. A
surprisingly realistic model can be produced by installing at each time step some
random variation in the phytoplankton growth-rate constant (vary a from 0.52 to 0.86;
variant 3 in Box 4.2). Now the variables sustain oscillations (Fig. 4.2c) something like
those observed in real oligotrophic pelagic ecosystems. The simplicity of the model
has the benefit that the proposed mechanism very likely causes the calculated effect,
not some other aspect of the interaction. In the case of the interaction of real nano-
and picophytoplankton with protozoan grazers, the self-limitation need not be actual
intraspecific cannibalism, only a rising tendency for some microherbivorous species
to feed on others as they become abundant and phytoplankton decrease. Notice,
however, that the basic period of the oscillation without the grazer cannibalism (Fig.
4.2a) has approximately “re-emerged”. Perhaps that, too, is how such interactions
come to oscillate as they do. As in all such modeling, the success of the model only
shows the feasibility of a concept. It does not prove that the same mechanism operates
in the field. That requires tests in the field, or at least contained incubations involving
real organisms from the field.
Box 4.2 Matlab script for Lotka–Volterra type
model of nanophytoplankton–protozoan grazer
interaction
(^)
%PROGRAM STROM.m
%1. Set up vectors of P, Z and T
% filled with NaN (“not a number”) for
% 120 steps/day for 60 days:
ndays = 60; nsteps = 120;
P=ones(nstepsndays,1)NaN; Z=P; T=P;
%2. Set starting values:
% PP=phytoplankton, ZZ=grazers
PP=10.; ZZ=8.;