Modelling Biological Systems in Their Physical Environment

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 of the copepod (Seuront 1999). Current results show that Multimodelling, coupling, multi-agent systems, ecology, this heterogeneity influences the energy budget of the zooplankton behaviour. copepod. By measuring the quantity of nitrogen ingestion, the observations show that the output of the behaviour of ABSTRACT the copepod (the excretion rate for example) varies according to the type of distribution of food. For example, a Our current work deals with the coupling of analytical turbulent environment, favourable to an overall mixing, models with individual based models. Such a coupling of increases the encounter rate of the copepod with the cells of models of different natures is motivated by the need to phytoplankton and increases the ingestion rate (Caparroy study the impact of the behaviour of biological organism in and Carlotti 1996). As argued elsewhere by different their physical environment. While behaviour is best authors putting forward the concept of individual-based modelled using an algorithmic language, the dynamics of modelling (IBM), the study of this behaviour is physical parameters is best modelled using numerical incompatible with an analytical approach. So, we use a models. In the same time, it is crucial to be aware of the multi-agent system (MAS) approach (Ferber 1995). consequences of the simplifications and of the choices that are made in the model, such as the topology of space and Coupling models is in keeping with multimodelling various parameters. In this paper, we discuss theses issues approach. A multimodel is considered as a composition of and exemplify our approach on a case study drawn from different homogenous or heterogeneous submodels at marine biology. We conclude that the emergence of several abstraction level (Fishwick, 1995). A lot of papers mathematical properties from agent models can be a way of deal with coupling circulation models with IBM based on coupling discrete and analytical formalisms. analytical formalisms (Miller et al. 1998; Carlotti 1998...). Based on differential equations, They estimate the density INTRODUCTION of phytoplankton in terms of nitrogen concentration for example. If we want to couple these models with agent The main objective of our work is modelling a part of models, we have to use a rule of conversion to know how marine ecosystem. This ecosystem is considered as an much particles of phytoplankton are present in the space. example of complex system. In this work, we are modelling To simplify, one can make the assumption that all particles a small crustacean, part of the family of zooplanktons are identical. If the additional assumption is made that the named "copepod" (see Figure 1), a very important link in particles of phytoplankton are distributed uniformly in the marine ecosystem. We aim at showing that the space, then one ends-up with an algorithm that converts a consequences of the active behaviour of food foraging of continuous representation (concentration of nitrogen) into a the copepod are significant on its viability. It is yet discrete one (number of phytoplankton particles per space controversial in the marine biology community whether the cell). The question to be raised relates to the validity of copepod actually seeks its food or if it simply catches the these assumptions and their consequences. The food that falls under its jaws thanks to the turbulence. transformation that we apply to the particles of phytoplankton can be applied to any discrete parameter Figure n° 1 : Copepod Centropages hamatus 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 Researchers have shown that the distribution of (Grimm 1999) and then simulate scale transfer in ecological phytoplankton is strongly heterogeneous in the environment models (it is recognised that scales and choices of representation level are fundamental issues in ecosystems already present in the gut). Then, this energy is either modelling (Frontier and Pinchod-Viale 1995; Coquillard consumed (metabolism, digestion, or swimming), or stored and Hill 1997). (egg production for females, for example).

At the individual level, movements have to be expressed in Analytical model the lagrangian formalism. If we adopt a traditional representation of space in the multi-agent models, i.e. a (Caparroy and Carlotti 1996) propose a model synthesizing discretized space according to a grid which cells can be the various models developed earlier. This model takes into square or hexagonal, then the coupling of a physical model account the activities of capture and ingestion using five of movement (model of micro-turbulence for example) with coupled differential equations. However, from our point of a discrete model based on agents is difficult if not nearly view, the contribution of the behaviour is partially impossible (see section 2). The use of a lagrangian point of neglected. Indeed, one can put into equations the fact that view is generally prohibitive with regards to computing the activity of nutrition of a copepod is a function of the performance. Our work is a first step in the direction of density of preys, the level of turbulence of the environment, finding the optimal choices in order to conciliate computer the mode of foraging and the quantity of food in the gut; performance and a accurate representation of the biological however, it is much more difficult to take into account system for coupling discrete and continuous approaches. various factors within the behaviour such as the way the In the sequel we first present the model, the dynamics of copepod perceives its environment, the different sizes of agents and some results of simulation in 2D discrete space. preys, the speed of swimming of the copepod relatively to The conclusions of section 2 leads us to a new model. This that of its prey, and so forth... model is still based on the concept of multi-agents but the agents are located in a 3 dimensional continuous space. Agent model Problems regarding the boundary conditions and the choice of a particular distribution algorithm and space dimensions The system to be modelled is composed of a mass of water of the model are examined. We discuss the effects of these in which "patches'' of phytoplankton and copepods are choices on a parameter emerging from our MAS model and immersed. Each agent is located on a cell of a grid then draw some conclusions and perspectives open by this representing the space and it is characterised by its work. properties among which its behaviour. The copepod has various characteristics (such as its THE DISCRETE 2D MODEL: PRESENTATION, "weight'' expressed in nitrogen, the volume of its gut, its SUMMARY OF RESULTS AND PROBLEMS speed of swimming...). Its behaviour, the principal object of the study for the biologist, is defined by a Petri network In a majority of papers (see (Caparroy and Carlotti 1996; (Peterson 1981) or an algorithmic language to allow the Carlotti et al. 2000; Tiselius and Jonsson 1990; Bundy et al. description of sophisticated behaviours. In our model, the 1993) for instance), the copepod is represented by an copepod perceives cells of phytoplankton. The perception is analytical model. These models try to describe each characterised by the sector (according to the orientation of "process'' of the organism in terms of input flow, output the copepod) and the distance within which cells of flow and transfer function (Caparroy and Carlotti 1996). phytoplankton are actually perceived. According to in vivo and experimental observations, the copepod basically exhibits two distinct behaviours: either swimming towards Metabolism food, or jumping at random. These behaviours influence Usable energy Swimming directly the capture of the particles of phytoplankton. Thus, Digestion we focus exclusively on this process and we leave aside the Egg Production digestion, though we do not totally disregard it to obtain a copepod complete model. The algorithm that models the dynamics of moves of the copepod is divided into two parts: Faecal Expelled faecal 1- Normal swimming: Preys Preys in the gut pellets pellets -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 Figure n°2: Processes involved in ingestion and digestion in time, it can capture one particle of phytoplankton, the copepod (see text for details) -if no food is found, the copepod continues to Let us describe the process of ingestion and digestion of swim to reach the next cell, phytoplankton (see Fig. 2). First, the copepod captures a -at the end of the t1 units of time, the copepod prey (a particle of phytoplankton). After handling it for chooses a neighbouring cell to be explored and proceeds some times, the prey is stored in the gut and enters the there. process of digestion. The gut transforms its contents in 2 - Random jumps: energy and feacal pellets. This transformation is -as soon as a cycle of t2 units of time is elapsed, the continuous: within each dt, a quantity dq of caught preys is copepod performs a jump without considering what processed (this quantity is proportional to the quantity 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. Figure n°3 Quantity of preys (pg of nitrogen) in the gut of The unit of time of simulation is fixed to the duration the copepod against time corresponding to the time necessary to perform the shortest action, i.e. the handling time of a particle of phytoplankton The simulation accords to the principal results stated in by the copepod, 1/20 s (Caparroy and Carlotti 1996). (Caparroy and Carlotti 1996) for a uniform distribution of phytoplankton and random-walk swimming. It remains to Simulation perform comparisons with results of in vivo experiments. However, these in vivo experiments remain technically We have developed an agent model using our framework difficult to realize for the moment. "Virtual Laboratory Environment" (VLE) (Ramat and Preux 2000). Initially, we consider only one copepod at a Discussion time since the aim of this study is the foraging behaviour of copepods. The size of the copepod (1 mm) is used as the On a purely experimental basis, turbulence was integrated basic length for the discretization of the environment. For into the discrete model with 2 dimensions that made it the moment, the environment is considered as two possible to show that one could not be satisfied with an dimensional and split into cells of 1 mm2. Each cell is dealt expression of space in discrete terms (a grid). The model by a spatial agent. The particles of phytoplankton are very used for the demonstration is an atmospheric model of numerous (from 10 particles per litre to 108 particles per turbulence (the well-known model of Lorenz (Lorenz litter which yields a maximum of 102 particles per cell). The 1963). This model has the advantage of being simple and environment is composed of a 2D grid of 1600 square cells yields intermittencies in the speed field (expressed in two (40mmx40mm). Each cell is connected to its 8 neighbours. dimensions). However, this model of turbulence is not The particles of phytoplankton are distributed either realistic in our case. It would be necessary to use models of heterogeneously by patches following a multifractal turbulence at micro-scale. Nevertheless, all these models distribution according to (Seuront et al. 1999) (see Fig. 3), produce a speed vector field at each point of space. As long or homogeneously, the concentration being the same in as the particles of phytoplankton are subject to this speed each cell. In both cases, the average density is identical in field, it is necessary to express the coordinates of the the whole environment (2 particles / mm2). particles with precision. It is no longer relevant to merely We define two types of copepods according to their strategy give the number of phytoplankton particles in each cell of of swimming: random-walk or oriented towards food. In the the spatial grid. We now need to characterize each particle second case, the probability that a cell is chosen as the next with its coordinates in continuous space. cell to move into is proportional to the quantity of food it To deal with this change, as far as each particle of holds: the more food, the more likely the copepod moves phytoplankton is located with its continuous coordinates, into it. each time the turbulence model has computed a new speed At each step of simulation, we measure the attributes field, the location of each phytoplankton particle is defining the internal state of the copepod: the energy computed with regards to the discrete space so that the contained in the gut expressed in pg of nitrogen, its usable model of the copepod is not changed. However, this trick is energy, and the number of captured particles of not sufficient: the copepod remains in a discrete space, it phytoplankton. 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 Initial position (x,y,z) Centre of space centre of the copepod and median is its direction vector, Initial direction angles (radian) xOy = 1.8 x Oz = 1 yOz=1 corresponding to the perception angle of table I) and a Time step (duration of one 0.05 distance (corresponding to the perception distance of table iteration in second) I). The precision of the angle and the distance are related to Particle concentration (cells.mm-3) 0.5 the step of discretization of space (1mm in our model); we have to define a perception volume. Table II : Simulation plan for each space size (103mm3, 8.103 mm3, 64.103mm3) The transformation of our model leads to the following questions: what are the impacts of the algorithms used for Deterministic boundary the distribution of the phytoplankton (now located in space conditions Random boundary with precision), the behaviour at the boundaries of space, Distribution condition Toroidal and the space size on the output of the system (here we reflection (Random bounces) focus on the number of “eaten” particles)? In the next space section, we describe the 3D model before addressing these Thirty Thirty Thirty Thirty issues. Pseudo- simulations simulations simulations simulations random with with with one with different uniform different different THE NEW 3D MODEL distribution distributions distributions distributions Thirty Thirty With regards to particular issues implied by the Thirty Thirty Pseudo- simulations simulations transformation of continuous variables into discrete ones, simulations simulations random with with with one with different we build a new model. Space is now continuous and 3D. patches different different distribution distributions Necessary transformations are described below. distributions distributions One One Regular Thirty simulations Description simulation simulation

We have made simplifications on the 2D model. In classical models of copepod, individual bioenergetic budget strongly Phytoplankton distribution algorithms depends of the ingestion rate (i.e. the number of eaten particles per unit of time)(Carlotti et al. 2000). This is the We use 3 algorithms to generate the spatial distribution of reason why we have decided to consider the number of phytoplankton cells. particles eaten by the copepod as a sufficient indicator of The pseudo-random uniform distribution is calculated by the effect of boundary conditions, particle distribution and the pseudo-random function of the C++ library initialised space size on the system. Furthermore, we do not take the with the computer clock. This algorithm provides a uniform metabolic processes into account in order to speed up the repartition of particles in space showing no regularity. execution of model and amplify the ingestion rate. The time The pseudo-random patches algorithm is computed step is 1/20s according to the handling time of a likewise the pseudo-random uniform: it distributes particles phytoplankton cell by the copepod. There are no random randomly around virtual centres randomly distributed, with walk and the simulation stops when there are no more a standard deviation equal to 1. This provides patches phytoplankton particles in the water or when the copepod which simulate a heterogeneous distribution. The random do not eat anymore. The copepod is characterised by its aspect of these algorithms imply multiple runs. The position in space and its direction angle. The chosen pseudo-random uniform distribution algorithm provides a geometry for perception is a cone defined by its angle and heterogeneity less important than pseudo-random patches length which is the perception distance. algorithm. The only parameters are the size of space and the boundary The regular algorithm distributes particles on a regular grid. conditions. We perform experiments with different types of distribution. Values of parameters used in the model are Perception algorithm indicated in table I. Three sizes of space are taken for the different simulations: 103 mm3, 8.103 mm3, 64.103 mm3. The The agent computes whether other agents are located in its concentration of particles is the same for all space size: 0.5 perception cone and updates its trajectory determining a particles.mm-3 which corresponds to a mean concentration target positioned in space selecting the nearest particle. (Caparroy and Carlotti 1996). Simulations plans are Another algorithm has been simulated where the target is described in table II. determined by the barycenter of perceived particles and weighted by the square of the inverse of the distance Table I : Values of parameters use in the model between particles and copepod. With this algorithm, results Parameter Value are not different from those obtained with the first one Perception angle (radian) PI which is simpler and faster. So we decided to use the first Perception distance (mm) 2 algorithm. Catch distance (mm) 0.5 Swimming speed (mm.s-1) 1 Boundary conditions Figure n°4: Standard deviation of the percent of remaining particles against time for deterministic bounces and each We consider three possibilities: dimension of space: 1- Reflection: at the limits of space, the copepod the same type of results are observed for a toroidal space. adopts a reflection trajectory, like a ray of light. 2- Toroidal space: at the limits of space, the copepod Figure 5 presents a comparison between the number of goes exactly to the other side of space. bounces and ingestion curves for a particular simulation. 3- Random bounces: at the limits of space, the We can note that ingestion stops when the bounce curve copepod goes to a new cell selected at random. becomes linear. This indicates that the agent follows the same trajectory over and over. This point has been Optimization confirmed by visual observation of these trajectories.

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 Fig n°5: Comparison between the number of bounces and curve, we obtain the ingestion rate (i.e. the number of ingestion curves for a particular simulation particles eaten per unit of time). We present here some results to discuss the impact of algorithmic choices on Deterministic boundary conditions seem to induce an results. artefact in the simulation as far as the agent is not able to explore the whole space. In the perspective of finding the Boundary condition impacts optimal representation of space (the one which do not induce artefact on ingestion rate), we have introduced Figure 4 shows that standard deviation is large for random boundaries. Furthermore, we have to find a simulations with deterministic bounces. We express boundary condition that permits us to do only one standard deviation of the percent of remaining particles of simulation to express ingestion rate in order to decrease the mean curve in order to compare simulations for simulation duration in the perspective of coupling models. Figure 6 presents examples of ingestion curves in a space of different space size. In each case of space or distribution, 3 3 the agent does not seem able to eat all particles (standard 64.10 mm with this boundary condition averaged over 30 deviation remain constant at the end of simulations) and the simulation with its own particles distribution. 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°6 : mean ingestion curve for a 64.103 mm3 space dimension: the same type of results are observed for others dimensions. 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 Figure 8: standard deviation (in percent of remaining rate of the copepod. Figure 7 plots the ingestion rate for particles) against time for simulations with random different phytoplancton distributions. They also agree with bounces. (Caparroy and Carlotti 1996). 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.

Space size impacts

Figure 8 and figure 4 show that the choice of space size has an influence on the efficiency of copepod ingestion: 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 Figure 7: ingestion rate at the beginning of simulation model which continuous variables are discretized to interact showing the crucial role played by particles distribution. with an agent-based model Same results are observed for other space sizes. CONCLUSION AND PERSPECTIVES All along the simulation, the assimilation rate has a high frequency variability due to the heterogeneity of particle The main objective of this work is to find ways of coupling distribution. This high frequency variability was smoothed continuous and discrete formalism in ecological modeling. by the method of moving average. We have discussed the impact of some algorithmic choices If we look at the standard deviation of mean curves for on simulation. From this work, we come to the conclusion simulation with random bounces (figure 8), we can note that the choice of the distribution algorithm and space size that it increases with time (i.e. with the decrease of prey of the model are very significant in order to simulate concentration) and falls close to zero. The fact that the copepod ingestion in 3D space. The aim of the work is to maximum is encountered more or less early is due to the assess the impact of the assumption made in the model and size of space 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 Frontier, S. and Pinchod-Viale, D 1995. “Ecosystèmes. Structure (Carlotti and al. 2000). Finding Mitscherlisch type curves fonctionnement évolution”. Masson (Eds.). leads us to think that it is possible to find mathematical Grimm, V. 1999. “Ten years of individual-based modelling in properties based on an agent formalism in agreement with ecology: what we have learned and what could we learn in the future ?”. In Ecological Modelling, 115, 129-148. (Servat 2000). This is an important issue in the perspective Le Page, C. and Curry, P. 1996. “How spatial heterogeneity of coupling analytical model with agent ones. We think that influences population dynamics: Simulations in SeaLab”. In the emergence of mathematical properties from the agent Adaptative Behavior, vol.4, No 3 / 4, 255-288. model can be a way of coupling population analytical Le Page, C.; Bousquet, F.; Bakam, I.; Bah, A.; and Baron, C. models (well adapted for large scale modeling) with agent 2000. “CORMAS: A multiagent simulation toolkit to model modeling (well adapted for individual behaviour modeling). natural and social dynamics at multiple scales”, Presented at For example, taking a mathematical circulation and the workshop “the ecology of scales“ Wageningen population model giving information on tubulence level (Netherlands). (i.e. heterogeneity level of phytoplancton distribution), we Lorenz, N.E. 1963. “Deterministic Nonperiodic Flow“. In Journal of the Atmospheric Sciences, vol.20, 130-141. can compute an ingestion rate associated to this turbulence Miller, C. B.; Lynch, D. R.; Carlotti, F.; Gentleman, W.; and level and then give it back to the population model. Lewis V. W. 1998. “Coupling of individual-based population Representation of different scales in the same model model of Calanus Finmarchicus to a circulation model for the requires to find mechanisms of action and reaction of one Georges Bank region”. In Fisheries Oceanography; 7: 3 / 4, level of organization on other. From a systemic point of 219-234. view (Bertalanffy 1963), we can say that the ingestion rate Pavé, A. 1994. “Modélisation en biologie et en écologie”. (Aléas emerges from our modelling and, in agreement with (Le eds.). Page and al. 2000), we think that SMA can be a way of James L. Peterson 1981. “Petri Net Theory and the Modelling of Systems”. Prentice-Hall, Englewood Cliffs, New Jersey. modelling multi-scale systems. Ramat E. and Preux P. 2000. “Virtual Laboratory Environment (VLE) : un environnement multi-agents pour la modélisation In the close future, we will integrate micro-scale turbulence et la simulation d'écosystèmes”. In proceedings of Journées to show its effects on individuals by coupling a physical Françaises d'Intelligence Artificielle Distribuée et Systèmes model with our agent and continue the studies of Multi-Agents 2000 (Hermès Eds.). algorithmic choices on simulation results. Servat, D. 2000 “Modélisation de dynamique de flux par agents. Application aux proccessus de ruissellement, infiltration et Furthermore, it seems important to represent individuals in érosion”. PhD these, Paris VI. a continuous space in order to be more precise with regards Seuront, L.; Schmitt, F.; Lagadeuc Y.; Schertzer; and D. Lovejoy, S. 1999. “Universal multifractal analysis as a tool to to behaviour modelling if we want to simulate the characterize multiscale intermittent patterns. Example of perception more precisely. For example, given a weight of phytoplankton distribution in turbulent coastal waters”. In phytoplancton cells (for a barycentric perception, see Journal of Plankton Research. section 3), the copepod agent will be able to “choose” cells Tiselius, P. and Jonsson, P. R. 1990. “Foraging behaviour of six by calculating its weight according to particular attributes calanoid copepods : observations and hydrodynamic analysis”. like species, cell size, etc... In Marine ecology progress series, vol.66, 23-33

REFERENCES AUTHORS BIOGRAPHY

Bundy, M. H.; Gross, T. F.; Coughlin, D. J.; and Strickler J. R. RAPHAËL DUBOZ* was born on March 30th, 1973 in 1993. “Quantifying copepod searching efficiency using Besançon, France. After a master of Marine Environment swimming pattern and perceptive ability”. In Bulletin of Sciences, he decided to study computer science at the Marine Science, vol.53, n°1, 15-28. university of Marseille II in 1998 and received his DEA of Bertalanffy L. 1963. Théorie générale des systèmes, Dunod Ecological Marine Modelling at the university of Paris VI (French eds. 1993). in 1999. He is currently preparing his PhD in computer Caparroy, P.; Carlotti, F. 1996. “A model for Acartia tonsa : effect of turbulence and consequences for the related physiological science at the Laboratoire d'Informatique du Littoral of the processes”. In Journal of Plankton Research, vol.18, n°11, Université du Littoral Côte d'Opale in Calais. He is doing 2139-2177. research in Ecological Modelling and Multi Agents Carlotti, F. 1998. “A lagrangian ensemble model of Calanus Systems. Finmarchicus coupled with a 1-D ecosystem model”. In Fisheries oceanography, 7 : 3 / 4, 191-204. ÉRIC RAMAT was born on April 3rd, 1970 in Angouleme, Carlotti, F.; J. Giske; and F. Werner 2000. “Modelling France. He received his PhD in computer science from Uni- zooplankton dynamics”. In ICES Zooplancton Methodology versity of Tours in 1997. He is currently assistant professor manual, Academic press , 571-644. of computer science at the Laboratoire d'Informatique du Coquillard P. and Hill D.R.C 1997. “Modélisation et simulation d'écosystèmes. Des modèles déterministes aux simulation à Littoral of the Université du Littoral Côte d'Opale in Calais. évènements discrets”. Masson (Eds.). He is doing research in object and agent Ferber, J. 1995. “Les systèmes multi-agents, vers une intelligence modeling and simulation from ecology. collective”. InterEditions (Eds.). Fishwick, P.A. 1995. “Simulation Model Desing and Execution”. PHILIPPE PREUX was born on July 23rd, 1966 in Bohain Prentice Hall (Eds.). 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'Informa- tique 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 Nord- Pas de Calais, France under contract number: 00 46 0147