ECOLOGICAL AND MATHEMATICAL ASPECTS OF A PLANKTON COMMUNITIES MODEL
Jean-Marc Ginoux *, Bruno Rossetto, Jean-Louis Jamet
Université de Toulon et du Var, Equipes ISO-EBMA, Laboratoire PROTEE, LA GARDE cedex, France
* ginoux@univ-tln.fr , rossetto@univ-tln.fr, jamet@univ-tln.fr
Abstract
In this article we studied an ecological model transcribing the behavior of a trophic chain with three levels of interaction. The values of
the parameters of the model lead us to regard it as a slow fast autonomous dynamic system. We could establish the equation of the slow
surface on which the attractor is leaned, thus connecting the three variables in the slow field in the form of a state equation.
Key words : Plankton modeling, Slow-Fast Dynamics
Rapp. Comm. int. Mer Médit., 37,2004
364
The ecological point of view
Since 1995, seasonal variations of the plankton community
(phytoplankton and zooplankton) in Toulon Bay (Var, France) have
been studied [1, 2]. Plankton communities differently affected by
anthropogenic inputs of Toulon area have been studied regularly in
two sites (Little Bay, polluted, and Large Bay, non polluted).
The average of nitrate and orthophosphate concentrations and the
abundance and diversity of plankton communities during annual
cycles in these bays exhibit inherent features as the pattern shape of
limit cycles as well as the apparent period of intrinsic evolution of
phytoplankton and zooplankton separately considered.
So, we have searched a model able to transcribe the observed
behaviour of plankton communities and to fit with our data.
Among the different autonomous predator-prey models taken under
consideration in the literature [3-10], we searched the most simple and
consistentmodel with functional responses for growing, predation
and mortality susceptible to lead to a limit cycle and to fit with our
data. To this aim, we were interested by the Rosenzweig-Mac Arthur
model [11].
Modelling
The Rosenzweig-Mac Arthur model [11] for a three trophic level
interaction is involving nutrients (nitrates and orthophosphates, N),
planktonic algae (phytoplankton, P) and herbivorous zooplankton (Z).
It’s the so-called NPZ model. It is composed of limited functional
responses: a logistic prey (N), a Holling type II predator (P), and a
Holling type II top-predator (Z). Predator’s per-capita predation rate
has the Holling type II form [10]. All parameters used are chosen in a
biological range.
Note that the Rosenzweig-Mac Arthur model was developed from
the previous works of Volterra [4] and Lotka [5]. The addition of
limited functional responses made it possible to lead to the model of
Rosenzweig-Mac Arthur. Nevertheless, if only one limitation term is
removed, there is no more limit cycle.
Moreover, as the turn-over of each variable is in a ratio of ten, we
have been lead to focus our interest on slow-fast autonomous dynamic
systems [12, 13, 14].
Mathematical aspects
Under certain conditions, the dimensionless system is singularly
perturbated with three times scales. The rates of change for the prey,
the predator and top-predator range from fast to intermediate to slow,
respectively [12, 13]. We give the equation of a slow manifold on
which the attractor lies. Since, a state equation relying the three
variables is established.
Some simplications could be brought to the model of Rosenzweig-
Mac Arthur in order to make a study in two dimensions of it.
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