TOWARDS A COUPLING OF CONTINUOUS AND DISCRETE FORMALISMS
IN ECOLOGICAL MODELLING INFLUENCES OF THE CHOICE OF ALGORITHMS ON RESULTS
Raphaël Duboz
Eric Ramat
Philippe Preux
Laboratoire d'Informatique du Littoral (LIL)
UPRES-JE 2335
BP 719,
62228 Calais, France
E-mail : {duboz, ramat, preux}@lil.univ-littoral.fr
web : http://www-lil.univ-littoral.fr
KEYWORDS
Multimodelling, coupling, multi-agent systems, ecology,
zooplankton behaviour.
ABSTRACT
Our current work deals with the coupling of analytical
models with individual based models. Such a coupling of
models of different natures is motivated by the need to
study the impact of the behaviour of biological organism in
their physical environment. While behaviour is best
modelled using an algorithmic language, the dynamics of
physical parameters is best modelled using numerical
models. In the same time, it is crucial to be aware of the
consequences of the simplifications and of the choices that
are made in the model, such as the topology of space and
various parameters. In this paper, we discuss theses issues
and exemplify our approach on a case study drawn from
marine biology. We conclude that the emergence of
mathematical properties from agent models can be a way of
coupling discrete and analytical formalisms.
INTRODUCTION
The main objective of our work is modelling a part of
marine ecosystem. This ecosystem is considered as an
example of complex system. In this work, we are modelling
a small crustacean, part of the family of zooplanktons
named "copepod" (see Figure 1), a very important link in
marine ecosystem. We aim at showing that the
consequences of the active behaviour of food foraging of
the copepod are significant on its viability. It is yet
controversial in the marine biology community whether the
copepod actually seeks its food or if it simply catches the
food that falls under its jaws thanks to the turbulence.
Figure n° 1 : Copepod Centropages hamatus
Researchers have shown that the distribution of
phytoplankton is strongly heterogeneous in the environment
of the copepod (Seuront 1999). Current results show that
this heterogeneity influences the energy budget of the
copepod. By measuring the quantity of nitrogen ingestion,
the observations show that the output of the behaviour of
the copepod (the excretion rate for example) varies
according to the type of distribution of food. For example, a
turbulent environment, favourable to an overall mixing,
increases the encounter rate of the copepod with the cells of
phytoplankton and increases the ingestion rate (Caparroy
and Carlotti 1996). As argued elsewhere by different
authors putting forward the concept of individual-based
modelling (IBM), the study of this behaviour is
incompatible with an analytical approach. So, we use a
multi-agent system (MAS) approach (Ferber 1995).
Coupling models is in keeping with multimodelling
approach. A multimodel is considered as a composition of
different homogenous or heterogeneous submodels at
several abstraction level (Fishwick, 1995). A lot of papers
deal with coupling circulation models with IBM based on
analytical formalisms (Miller et al. 1998; Carlotti 1998...).
Based on differential equations, They estimate the density
of phytoplankton in terms of nitrogen concentration for
example. If we want to couple these models with agent
models, we have to use a rule of conversion to know how
much particles of phytoplankton are present in the space.
To simplify, one can make the assumption that all particles
are identical. If the additional assumption is made that the
particles of phytoplankton are distributed uniformly in the
space, then one ends-up with an algorithm that converts a
continuous representation (concentration of nitrogen) into a
discrete one (number of phytoplankton particles per space
cell). The question to be raised relates to the validity of
these assumptions and their consequences. The
transformation that we apply to the particles of
phytoplankton can be applied to any discrete parameter
which is defined by a continuous value in the model
(principally individuals). These last remarks are very
important in the perspective of coupling continuous models,
well suited to large scale of time and space, with individual
based models well suited to individual behaviour modelling
(Grimm 1999) and then simulate scale transfer in ecological
models (it is recognised that scales and choices of
representation level are fundamental issues in ecosystems
modelling (Frontier and Pinchod-Viale 1995; Coquillard
and Hill 1997).
already present in the gut). Then, this energy is either
consumed (metabolism, digestion, or swimming), or stored
(egg production for females, for example).
At the individual level, movements have to be expressed in
the lagrangian formalism. If we adopt a traditional
representation of space in the multi-agent models, i.e. a
discretized space according to a grid which cells can be
square or hexagonal, then the coupling of a physical model
of movement (model of micro-turbulence for example) with
a discrete model based on agents is difficult if not nearly
impossible (see section 2). The use of a lagrangian point of
view is generally prohibitive with regards to computing
performance. Our work is a first step in the direction of
finding the optimal choices in order to conciliate computer
performance and a accurate representation of the biological
system for coupling discrete and continuous approaches.
In the sequel we first present the model, the dynamics of
agents and some results of simulation in 2D discrete space.
The conclusions of section 2 leads us to a new model. This
model is still based on the concept of multi-agents but the
agents are located in a 3 dimensional continuous space.
Problems regarding the boundary conditions and the choice
of a particular distribution algorithm and space dimensions
of the model are examined. We discuss the effects of these
choices on a parameter emerging from our MAS model and
then draw some conclusions and perspectives open by this
work.
(Caparroy and Carlotti 1996) propose a model synthesizing
the various models developed earlier. This model takes into
account the activities of capture and ingestion using five
coupled differential equations. However, from our point of
view, the contribution of the behaviour is partially
neglected. Indeed, one can put into equations the fact that
the activity of nutrition of a copepod is a function of the
density of preys, the level of turbulence of the environment,
the mode of foraging and the quantity of food in the gut;
however, it is much more difficult to take into account
various factors within the behaviour such as the way the
copepod perceives its environment, the different sizes of
preys, the speed of swimming of the copepod relatively to
that of its prey, and so forth...
THE DISCRETE 2D MODEL: PRESENTATION,
SUMMARY OF RESULTS AND PROBLEMS
In a majority of papers (see (Caparroy and Carlotti 1996;
Carlotti et al. 2000; Tiselius and Jonsson 1990; Bundy et al.
1993) for instance), the copepod is represented by an
analytical model. These models try to describe each
"process'' of the organism in terms of input flow, output
flow and transfer function (Caparroy and Carlotti 1996).
Metabolism
Swimming
Digestion
Egg Production
Usable energy
copepod
Preys
Preys in the gut
Faecal
pellets
Expelled faecal
pellets
Figure n°2: Processes involved in ingestion and digestion in
the copepod (see text for details)
Let us describe the process of ingestion and digestion of
phytoplankton (see Fig. 2). First, the copepod captures a
prey (a particle of phytoplankton). After handling it for
some times, the prey is stored in the gut and enters the
process of digestion. The gut transforms its contents in
energy and feacal pellets. This transformation is
continuous: within each dt, a quantity dq of caught preys is
processed (this quantity is proportional to the quantity
Analytical model
Agent model
The system to be modelled is composed of a mass of water
in which "patches'' of phytoplankton and copepods are
immersed. Each agent is located on a cell of a grid
representing the space and it is characterised by its
properties among which its behaviour.
The copepod has various characteristics (such as its
"weight'' expressed in nitrogen, the volume of its gut, its
speed of swimming...). Its behaviour, the principal object of
the study for the biologist, is defined by a Petri network
(Peterson 1981) or an algorithmic language to allow the
description of sophisticated behaviours. In our model, the
copepod perceives cells of phytoplankton. The perception is
characterised by the sector (according to the orientation of
the copepod) and the distance within which cells of
phytoplankton are actually perceived. According to in vivo
and experimental observations, the copepod basically
exhibits two distinct behaviours: either swimming towards
food, or jumping at random. These behaviours influence
directly the capture of the particles of phytoplankton. Thus,
we focus exclusively on this process and we leave aside the
digestion, though we do not totally disregard it to obtain a
complete model. The algorithm that models the dynamics
of moves of the copepod is divided into two parts:
1- Normal swimming:
-during t1 units of time (the time for the copepod to
cross a cell of the grid), the copepod explores the cell where
it is and whether food is available there; within each unit of
time, it can capture one particle of phytoplankton,
-if no food is found, the copepod continues to
swim to reach the next cell,
-at the end of the t1 units of time, the copepod
chooses a neighbouring cell to be explored and proceeds
there.
2 - Random jumps:
-as soon as a cycle of t2 units of time is elapsed,
the copepod performs a jump without considering what
surrounds it.
The way the copepod moves is a function of the copepod
behavioural strategy. In a cell, the copepod captures the
particles of phytoplankton if it has not yet eaten too much.
Indeed, the copepod decreases the quantity of food which it
absorbs according to its level of satiety, itself directly
bounded with the number of particles of phytoplankton
present in the gut.
The mass of water constitutes the environment in which the
other entities evolve and move. For the moment, it is not
conceivable to model each particle with one agent due to
the computational cost. The solution which is adopted
consists in defining an attribute "Number of particles'' in
each spatial agents (i.e. a cell of the grid). The management
of food is delegated to the spatial agents, that is to the
environment.
The unit of time of simulation is fixed to the duration
corresponding to the time necessary to perform the shortest
action, i.e. the handling time of a particle of phytoplankton
by the copepod, 1/20 s (Caparroy and Carlotti 1996).
Simulation
We have developed an agent model using our framework
"Virtual Laboratory Environment" (VLE) (Ramat and
Preux 2000). Initially, we consider only one copepod at a
time since the aim of this study is the foraging behaviour of
copepods. The size of the copepod (1 mm) is used as the
basic length for the discretization of the environment. For
the moment, the environment is considered as two
dimensional and split into cells of 1 mm2. Each cell is dealt
by a spatial agent. The particles of phytoplankton are very
numerous (from 10 particles per litre to 10 8 particles per
litter which yields a maximum of 102 particles per cell).
The environment is composed of a 2D grid of 1600 square
cells (40mmx40mm). Each cell is connected to its 8
neighbours. The particles of phytoplankton are distributed
either heterogeneously by patches following a multifractal
distribution according to (Seuront et al. 1999) (see Fig. 3),
or homogeneously, the concentration being the same in
each cell. In both cases, the average density is identical in
the whole environment (2 particles / mm2).
We define two types of copepods according to their strategy
of swimming: random-walk or oriented towards food. In the
second case, the probability that a cell is chosen as the next
cell to move into is proportional to the quantity of food it
holds: the more food, the more likely the copepod moves
into it.
At each step of simulation, we measure the attributes
defining the internal state of the copepod: the energy
contained in the gut expressed in pg of nitrogen, its usable
energy, and the number of captured particles of
phytoplankton.
Figure n°3 Quantity of preys (pg of nitrogen) in the gut of
the copepod against time
The simulation accords to the principal results stated in
(Caparroy and Carlotti 1996) for a uniform distribution of
phytoplankton and random-walk swimming. It remains to
perform comparisons with results of in vivo experiments.
However, these in vivo experiments remain technically
difficult to realize for the moment.
Discussion
On a purely experimental basis, turbulence was integrated
into the discrete model with 2 dimensions that made it
possible to show that one could not be satisfied with an
expression of space in discrete terms (a grid). The model
used for the demonstration is an atmospheric model of
turbulence (the well-known model of Lorenz (Lorenz
1963). This model has the advantage of being simple and
yields intermittencies in the speed field (expressed in two
dimensions). However, this model of turbulence is not
realistic in our case. It would be necessary to use models of
turbulence at micro-scale. Nevertheless, all these models
produce a speed vector field at each point of space. As long
as the particles of phytoplankton are subject to this speed
field, it is necessary to express the coordinates of the
particles with precision. It is no longer relevant to merely
give the number of phytoplankton particles in each cell of
the spatial grid. We now need to characterize each particle
with its coordinates in continuous space.
To deal with this change, as far as each particle of
phytoplankton is located with its continuous coordinates,
each time the turbulence model has computed a new speed
field, the location of each phytoplankton particle is
computed with regards to the discrete space so that the
model of the copepod is not changed. However, this trick is
not sufficient: the copepod remains in a discrete space, it
cannot be subject to the turbulence in the same way as the
particles of phytoplankton.
We have argued for the need to use a continuous space but
this has consequences. The mechanisms of perception of
the copepod are not explicitly expressed in a discrete 2D
space (determination of a set of perceived cells and
consequently the perception of what they contain).
Perception is redefined with an angle (which origin is the
centre of the copepod and median is its direction vector,
corresponding to the perception angle of table I) and a
distance (corresponding to the perception distance of table
I). The precision of the angle and the distance are related to
the step of discretization of space (1mm in our model); we
have to define a perception volume.
The transformation of our model leads to the following
questions: what are the impacts of the algorithms used for
the distribution of the phytoplankton (now located in space
with precision), the behaviour at the boundaries of space,
and the space size on the output of the system (here we
focus on the number of “eaten” particles)? In the next
section, we describe the 3D model before addressing these
issues.
THE NEW 3D MODEL
With regards to particular issues implied by the
transformation of continuous variables into discrete ones,
we build a new model. Space is now continuous and 3D.
Necessary transformations are described below.
Table II : Simulation plan for each space size
(103mm3, 8.103 mm3, 64.103mm3)
Deterministic boundary
conditions
Distribution
reflection
Pseudorandom
uniform
Pseudorandom
patches
Regular
Toroidal
space
Thirty
Thirty
simulations simulations
with
with
different
different
distributions distributions
Thirty
Thirty
simulations simulations
with
with
different
different
distributions distributions
One
One
simulation simulation
Random boundary
condition
(Random bounces)
Thirty
Thirty
simulations simulations
with one with different
distribution distributions
Thirty
Thirty
simulations simulations
with one with different
distribution distributions
Thirty simulations
Description
We have made simplifications on the 2D model. In classical
models of copepod, individual bioenergetic budget strongly
depends of the ingestion rate (i.e. the number of eaten
particles per unit of time)(Carlotti et al. 2000). This is the
reason why we have decided to consider the number of
particles eaten by the copepod as a sufficient indicator of
the effect of boundary conditions, particle distribution and
space size on the system. Furthermore, we do not take the
metabolic processes into account in order to speed up the
execution of model and amplify the ingestion rate. The time
step is 1/20s according to the handling time of a
phytoplankton cell by the copepod. There are no random
walk and the simulation stops when there are no more
phytoplankton particles in the water or when the copepod
do not eat anymore. The copepod is characterised by its
position in space and its direction angle. The chosen
geometry for perception is a cone defined by its angle and
length which is the perception distance.
The only parameters are the size of space and the boundary
conditions. We perform experiments with different types of
distribution. Values of parameters used in the model are
indicated in table I. Three sizes of space are taken for the
different simulations: 103 mm3, 8.103 mm3, 64.103 mm3.
The concentration of particles is the same for all space size:
0.5 particles.mm-3 which corresponds to a mean
concentration (Caparroy and Carlotti 1996). Simulations
plans are described in table II.
Table I : Values of parameters use in the model
Parameter
Value
Perception angle (radian)
PI
Perception distance (mm)
2
Catch distance (mm)
0.5
Swimming speed (mm.s-1)
1
Initial position (x,y,z)
Centre of space
Initial direction angles (radian) xOy = 1.8 x Oz = 1 yOz=1
Time step (duration of one
0.05
iteration in second)
-3
Particle concentration (cells.mm )
0.5
Phytoplankton distribution algorithms
We use 3 algorithms to generate the spatial distribution of
phytoplankton cells.
The pseudo-random uniform distribution is calculated by
the pseudo-random function of the C++ library initialised
with the computer clock. This algorithm provides a uniform
repartition of particles in space showing no regularity.
The pseudo-random patches algorithm is computed
likewise the pseudo-random uniform: it distributes particles
randomly around virtual centres randomly distributed, with
a standard deviation equal to 1. This provides patches
which simulate a heterogeneous distribution. The random
aspect of these algorithms imply multiple runs. The
pseudo-random uniform distribution algorithm provides a
heterogeneity less important than pseudo-random patches
algorithm.
The regular algorithm distributes particles on a regular grid.
Perception algorithm
The agent computes whether other agents are located in its
perception cone and updates its trajectory determining a
target positioned in space selecting the nearest particle.
Another algorithm has been simulated where the target is
determined by the barycenter of perceived particles and
weighted by the square of the inverse of the distance
between particles and copepod. With this algorithm, results
are not different from those obtained with the first one
which is simpler and faster. So we decided to use the first
algorithm.
Boundary conditions
We consider three possibilities:
1- Reflection: at the limits of space, the copepod
adopts a reflection trajectory, like a ray of light.
2- Toroidal space: at the limits of space, the copepod
goes exactly to the other side of space.
3- Random bounces: at the limits of space, the
copepod goes to a new cell selected at random.
Optimization
The high number of particles to be processed imposes the
use of an adapted data structure. In his Ph.D. thesis, D.
Servat (Servat 2000), proposed to partition space to sort
particles according to their position. We do the same here.
Space is partitioned into a grid. The size of cells of the grid
is greater or equal than the perception distance. Every
particle is assigned to a grid cell according to its position.
At each time step, an attribute of the copepod refers to the
cell which contains it. Hence, the computation cost of the
selection of the neighbouring cells decreases dramatically.
RESULTS AND DISCUSSION
We plot the curve of mean ingestion for each case of table
II and for each space size. Calculating the derivative of this
curve, we obtain the ingestion rate (i.e. the number of
particles eaten per unit of time). We present here some
results to discuss the impact of algorithmic choices on
results.
Boundary condition impacts
Figure 4 shows that standard deviation is large for
simulations with deterministic bounces. We express
standard deviation of the percent of remaining particles of
the mean curve in order to compare simulations for
different space size. In each case of space or distribution,
the agent does not seem able to eat all particles (standard
deviation remain constant at the end of simulations) and the
importance of standard deviation indicate simulation is very
sensitive to particles distribution when use deterministic
bounces. The same is observed for a toroidal space.
Fig n°5: Comparison between the number of bounces and
ingestion curves for a particular simulation
Deterministic boundary conditions seem to induce an
artefact in the simulation as far as the agent is not able to
explore the whole space. In the perspective of finding the
optimal representation of space (the one which do not
induce artefact on ingestion rate), we have introduced
random boundaries. Furthermore, we have to find a
boundary condition that permits us to do only one
simulation to express ingestion rate in order to decrease
simulation duration in the perspective of coupling models.
Figure 6 presents examples of ingestion curves in a space of
64.103mm3 with this boundary condition averaged over 30
simulation with its own particles distribution.
Fig n°6 : mean ingestion curve for a 64.10 3 mm3 space
dimension:
the same type of results are observed for others dimensions.
Figure n°4: Standard deviation of the percent of remaining
particles against time for deterministic bounces and each
dimension of space:
the same type of results are observed for a toroidal space.
Figure 5 presents a comparison between the number of
bounces and ingestion curves for a particular simulation.
We can note that ingestion stops when the bounce curve
becomes linear. This indicates that the agent follows the
same trajectory over and over. This point has been
confirmed by visual observation of these trajectories.
We can note here that nearly all particles are “eaten” by the
copepod. This means that the copepod is able to explore
the whole space. Figure 8 shows that standard deviation
decreases dramatically in case of random bounces.
Distribution algorithm impacts
Figure 6 shows that the assimilation speed increases with
the heterogeneity of particle distribution (the same result is
observed for other space sizes). It was known that
heterogeneity of space distribution affects biological
systems (Frontier and Pinchod-Viale 1995; Le Page 1996),
but as far as we are aware of, it has never been shown with
SMA in continuous space. Furthermore, these results are in
agreement with (Seuront et al. 1999) results: heterogeneity
of the phytoplankton distribution increases the ingestion
rate of the copepod. Figure 7 plots the ingestion rate for
different phytoplancton distributions. They also agree with
(Caparroy and Carlotti 1996).
standard deviation decreases dramatically with the increase
of space size. This is an important result since this means
that using a relevant distribution of phytoplankton cells and
the optimal space size, we can simulate ingestion rate with
only one run. This opens up the way to couple an analytical
model which continuous variables are discretized to interact
with an agent-based model
CONCLUSION AND PERSPECTIVES
Figure 7: ingestion rate at the beginning of simulation
showing the crucial role played by particles distribution.
Same results are observed for other space sizes.
All along the simulation, the assimilation rate has a high
frequency variability due to the heterogeneity of particle
distribution. This high frequency variability was smoothed
by the method of moving average.
If we look at the standard deviation of mean curves for
simulation with random bounces (figure 8), we can note
that it increases with time (i.e. with the decrease of prey
concentration) and falls close to zero. The fact that the
maximum is encountered more or less early is due to the
size of space
Figure 8: standard deviation (in percent of remaining
particles) against time for simulations with random
bounces.
It is interesting to note here that for each space size, curves
are rather similar. This means that, for a particular space
size and a particular distribution algorithm (here patch
distribution), variability comes principally from boundary
conditions for small concentrations.
The main objective of this work is to find ways of coupling
continuous and discrete formalism in ecological modeling.
We have discussed the impact of some algorithmic choices
on simulation. From this work, we come to the conclusion
that the choice of the distribution algorithm and space size
of the model are very significant in order to simulate
copepod ingestion in 3D space. The aim of the work is to
assess the impact of the assumption made in the model and
figure out what the simplest and still relevant model would
be.
The plots of figure 6 are monomolecular or Mitscherlisch
curves type (Pavé 1994) with the following mathematical
form:
dx / dt = a (1- x / k )
(where x is the particles number, K the maximum number
of particles and a the origin slope). It is interesting to note
that a mathematical property emerges from the agent
formalism (here perception, move and eat). It can be a way
to study emergence. The factor a can be considered as the
assimilation rate of the copepod (Figure 7 shows the
ingestion rate at the beginning of the simulation); this term
is usually used to compute assimilation in classical models
(Carlotti and al. 2000). Finding Mitscherlisch type curves
leads us to think that it is possible to find mathematical
properties based on an agent formalism in agreement with
(Servat 2000). This is an important issue in the perspective
of coupling analytical model with agent ones. We think that
the emergence of mathematical properties from the agent
model can be a way of coupling population analytical
models (well adapted for large scale modeling) with agent
modeling (well adapted for individual behaviour modeling).
For example, taking a mathematical circulation and
population model giving information on tubulence level
(i.e. heterogeneity level of phytoplancton distribution), we
can compute an ingestion rate associated to this turbulence
level and then give it back to the population model.
Representation of different scales in the same model
requires to find mechanisms of action and reaction of one
level of organization on other. From a systemic point of
view (Bertalanffy 1963), we can say that the ingestion rate
emerges from our modelling and, in agreement with (Le
Page and al. 2000), we think that SMA can be a way of
modelling multi-scale systems.
Space size impacts
In the close future, we will integrate micro-scale turbulence
to show its effects on individuals by coupling a physical
model with our agent and continue the studies of
algorithmic choices on simulation results.
Figure 8 and figure 4 show that the choice of space size has
an influence on the efficiency of copepod ingestion:
Furthermore, it seems important to represent individuals in
a continuous space in order to be more precise with regards
to behaviour modelling if we want to simulate the
perception more precisely. For example, given a weight of
phytoplancton cells (for a barycentric perception, see
section 3), the copepod agent will be able to “choose” cells
by calculating its weight according to particular attributes
like species, cell size, etc...
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Tiselius, P. and Jonsson, P. R. 1990. “Foraging behaviour of six
calanoid copepods : observations and hydrodynamic analysis”.
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RAPHAËL DUBOZ* was born on March 30th, 1973 in
Besançon, France. After a master of Marine Environment
Sciences, he decided to study computer science at the
university of Marseille II in 1998 and received his DEA of
Ecological Marine Modelling at the university of Paris VI
in 1999. He is currently preparing his PhD in computer
science at the Laboratoire d'Informatique du Littoral of the
Université du Littoral Côte d'Opale in Calais. He is doing
research in Ecological Modelling and Multi Agents
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AUTHORS BIOGRAPHY
ÉRIC RAMAT was born on April 3rd, 1970 in Angouleme,
France. He received his PhD in computer science from
University of Tours in 1997. He is currently assistant
professor of computer science at the Laboratoire
d'Informatique du Littoral of the Université du Littoral Côte
d'Opale in Calais. He is doing research in object and agent
modeling and simulation from ecology.
PHILIPPE PREUX was born on July 23rd, 1966 in Bohain
en Vermandois, France. He received his PhD in computer
science from University of Lille 1 in 1991. He is currently
professor of computer science at the Laboratoire
d'Informatique du Littoral of the Université du Littoral Côte
d'Opale in Calais. He is doing research in artificial
intelligence.
*This work is supported by the Conseil Régional du NordPas de Calais, France under contract number: 00 46 0147