Systems Analysis Modelling SimulationVol. 43, No. 6, June 2003, pp. 793–814

SCALE TRANSFER MODELING: USING

EMERGENT COMPUTATION FOR COUPLING AN

ORDINARY DIFFERENTIAL EQUATION SYSTEM

WITH A REACTIVE AGENT MODEL

RAPHAEL DUBOZ*, ERIC RAMAT and PHILIPPE PREUX

ULCO, Department of Informatics Littoral (LIL) UPRES-JE 2335 BP 719,62228 Calais Cedex, France

(Received 25 July 2002)

This article deals with the coupling of analytical models with individual based models design with the reactiveagents paradigm. Such a coupling of models of different natures is motivated by the need to find a way tomodel scale transfer in large complex systems, i.e. to model how low level of organization can be madeto influence upper level and vice versa. This is a fundamental issue, and more particularly in ecologicalmodeling where models are a real scientific tool of investigation. Individuals and populations are notdescribed at the same scale of time and space but it is known that they act on each others. Based on thisexample, we model individuals in their environment and the population dynamics. While behavior is bestmodeled using an algorithmic framework (the reactive agent paradigm), population dynamics (because ofthe number of interacting entities) is best modeled using numerical models. We propose the use of theconcept of emergent computation as a framework for coupling heterogeneous formalisms. In the sametime, it is crucial to be aware of the consequences of the simplifications and of the choices that are madein the reactive agent model, such as the topology of space and various parameters. In this article, we discussthese issues and our approach on a case study drawn from marine ecology and we show that it is possibleto find classical mathematical functional responses with a reactive agent system. Then, we propose amethodology to deal with the coupling of heterogeneous formalism useful in any kind of system modeling.

Keywords: Coupling; Individual-based model; Zooplankton behavior; Reactive agent model; Ecologicalmodeling; Formalism heterogeneity

1. INTRODUCTION

Modeling natural systems is a very hard task. From molecular interactions tocompetition between populations, numerous entities (i.e. objects of discrete nature)are interacting over time in ecosystems. Futhermore, they operate at different scalesof time and space. In modeling, space scale is related to the size of the environmenton which entities are interacting and time scale is related to the duration of processinvolved in the interactions of entities. At any scale, it is simply impossible to represent

*Corresponding author. E-mail: duboz@lil.univ-littoral.fr

ISSN 0232-9298 print: ISSN 1029-4902 online � 2003 Taylor & Francis Ltd

DOI: 10.1080/0232929031000150355

all entities which are interacting in real system. For instance, if one wants to modelthe ecosystem of an oceanic pool, one must adopt a limited and averaged vision ofthe system. At small scale (the individual scale in ecological modeling), we are interestedin the behavior of entities, how they eat, how they reproduce etc. Furthermore, notconsidering the issue of the choice of scales, a modeler implicitly or explicitly choosesto represent reality at particular level(s) of organization (i.e. the level of detail of entitiesdescription). In natural systems, these different levels are indivisible. Ecosystem modelstend to be a complete formal description of entities and their interactions at fundamen-tal levels of organization (individual, population, ecosystem, biosphere). Furthermore,it is recognized that choices of scale and organization level are fundamental issuesin ecosystem modeling [13]. Indeed, model design depends on the question we wantto answer with the model and the choices of scales and organization level constraint,the qualitative and quantitative choice of the state variables, a particular descriptionof processes acting on state variables and the units of the model. Another fundamentalissue remaining unsolved in ecological modeling is the way and the extent to whichthe individual behavior affect population dynamics [9]. Both issues are stronglylinked to each other and in order to try to address them, we propose the representationof different scales in the same model. This requires mechanisms of action and reactionof one level of organization on the other one. From a systemic point of view [1], we cansay that new properties emerge from the lower level and that the upper level charac-terizes the environment of the lower one. Continuous models are well suited to largescales of time and space while discrete Individual Based Models (IBM) [19,31] arewell suited to individual behavior modeling [15]. Both approaches are formalized bydiscrete or differential equations in most of the cases. In this work, we presentan algorithmic conception of IBM named reactive agent model [10], a part of Multi-Agent Systems (MAS). Agent models are fundamentally discrete. They have norequirements regarding any continuity. It is yet very difficult to prove anythinganalytically about an agent model with regard to its dynamics, its convergence, . . .This paradigm permits the expression of animal behavior with great details and withgreater simplicity than when using analytical modeling. Many works have shownthat MAS can actually be used to model complex systems, either human organizations,or ecosystems, or physical phenomena.We aim at showing that the coupling of MAS with Ordinary Differential Equations

(ODEs) is possible through the conceptual approach we have used in this work (seeFig. 1). It shows that we have to clarify what is emergence, how it is possible to havea positive definition for it (i.e. an operational one).The term ‘‘emergent computation’’ has been introduced in [12] to refer to the global

behavior that emerges from the activity of a set of interacting computational entities(or ‘‘agents’’). In an emergent computation, the global behavior is itself a computation.According to the ‘‘art of programming’’, the tradition in computer science is to designprograms so that the computations they perform are exactly what they are intendedto and thus, flee emergent computations. In sharp contrast, programs that stimulatethe concurrent activity of a certain number of interacting agents have shown moreor less expected global behaviors. Today, these emergent behaviors are actuallysought in such fields as neural networks (see e.g. [23]), insect colony simulation [5],cellular automata [34], genetic algorithms [24], autonomous robot design [2], artificiallife [6] to name a few of them. These emergent properties arise out of nonlinearinteractions between the entities, whereas the traditional programming style completely

794 R. DUBOZ et al.

relies on linear interactions exhibiting well predictable effects. The emerging entityprovides a time series which can be represented in different manners according to itsnature: a constant value, a scalar value function of time, a vector function, . . . a visuallyevolving pattern in a cellular automata (such as Conway’s game of life), or in a MAS(such as Reynolds’ bird flock flight [28]). We want to stress the fact that this emergingentity is a ‘‘mere’’ consequence of the underlying interaction of the entities constitutingthe system. Thus, the simulation of the underlying activities yields these emergententities rather straightforwardly.Classical population models are based on a set of differential equations which

describes the evolution along time, and possibly in space, of a certain set of character-istic variables of the system being modeled. These models can be studied analyticallyup to a certain point, even though we have much greater abilities to study theirasymptotic behavior rather than their transient behavior which is often the part weare really interested in. Many factors are generally parameterizing these equations.These factors are generally constant values which have a certain meaning (the ingestionrate of an organism, its reproduction rate, its carrying capacity, . . . ). It is clear thatthese factors reflect an underlying dynamics which is not described by the set ofequations and that these factors average in some way the effects of this activity inthe equations. For the sake of simplicity, these factors are very generally consideredas constant values, though, at least for some of them, they should be functionalterms of which we do not know enough to consider them explicitly as analyticalfunctions.However, each type of models has its favorite application: numerical models are

perfect to describe the population level or the physical dynamics of an environment,whereas agent models are perfect to describe the individual level. When consideringthe modeling of a system embedding both aspects, one can try to obtain a modeleither purely numerical, or purely agent. However, we feel that a more pragmaticapproach would be suitable, based on the combination of models of each type fordifferent parts of the system to model. The behavior of each living organism is modeledwith a discrete model, while the behavior of its physical environment of the populationdynamics is handled by a numerical model. Addressing the problem in this wayimmediately prompts the issue of coupling models of different natures. We proposeto handle this coupling by way of emergent computation. From the simulation of thebehavior of an organism emerges global characteristics of this behavior. These globalcharacteristics can be instantaneous instead of averaged over their whole range ofviability, over their lifetime. This instantaneous measure can feed a numerical model

FIGURE 1 Conceptual approach for the modeling of scale transfer between two levels of organization innatural systems.

SCALE TRANSFER MODELING 795

that can be thus solved with this ‘‘local’’ value. One step, or a few steps further, a

new instantaneous value can be used to solve the numerical model with this updated

value. This way of coupling provides a connection from the individual level to the

population level. We can go one step further by closing the loop: the solution given

by the resolution of the numerical model can feed back the discrete model: we name

this ‘‘strong’’ coupling.

The case study which is used in this work is the modeling of the behavior of a

small marine crustacean, part of zooplankton named ‘‘copepod’’ (see Fig. 2), a very

important link in marine ecosystem, notably it is the trade off between primary

production (phytoplankton) and commercial fish production.

In the sequel of this article, we first present the model, the dynamics of the copepod

and its environment. In agreement with Grimm [14], we think that a model must be

as simple as possible and it should be fully investigated. For this purpose, issues regarding

the boundary conditions, the representation of the environment, and space dimensions

are investigated. We discuss the effects of these choices on a parameter emerging

from our agent model. The coupling is discussed below and exemplified using a

well-known numerical model [17,32] coupled with a reactive agent model describing

the behavior of a copepod (see Section 2). The reactive agent paradigm applied in a

continuous space imposes us a Lagrangian modeling of individual movements. It is

generally prohibitive with regard to computational load. Our work is also a first step

in the direction of finding a trade off between computer performance and an accurate

representation of scale transfer in biological systems by coupling discrete and continuous

approaches. To conclude, we provide a discussion and describe perspectives opened by

this work.

2. THE MODEL

In this section, we present the reactive agent model of the copepod in its environment.The system to be modeled is composed of a mass of water in which ‘‘patches’’ of cells ofphytoplankton (vegetal plankton) and copepods are immersed. This can be assimilatedto a virtual aquarium. Water is not explicitly represented in the model, we assumethere is no turbulence in the virtual aquarium. For the modeling of copepods andphytoplankton, we use the reactive agent conceptual model, a subclass of the MASapproach. A reactive agent is an entity which actions are performed in responseto external or internal stimuli [10]. An agent is characterized by a set of properties(its size, weight, . . . ) and by a set of methods that describe its actions. An agent isalso characterized by the set of stimuli it is subject to, as well as the set of actions itcan perform. In object oriented programming, agents are instances of classes. Eachclass describes the various characteristics of its instances as well as its behavior(its methods) [18]. Using the classical modeling using differential equations, it is very

FIGURE 2 Copepod Centropages hamatus.

796 R. DUBOZ et al.

difficult to take behavioral rules into account, as well as the discrete nature ofindividuals, and remote actions are quite impossible to stimulate. MAS is a discretemodeling and appears as the ideal framework for the modeling of animal behavior.In the same time, differential equations are perfect to model the continuous threedimension of physical space. Owing to these two points, in this work, living organismsare discrete entities behaving in a continuous three dimension space modeled withdifferential equations. Let us describe the reactive agent model.

2.1. The Copepod

We are interested in the behavior of the copepod. In particular, we consider theresearch and the capture of preys, the movements and the internal processes governingthe ingestion of preys. There is no birth and no mortality in our model regarding theshort time being simulated (a few minutes) to compare with the life cycle of a copepod(months). Moreover, there is no adaptation or learning in the modeled copepod thoughthese features have been described. Again, as argued before, these processes occur atlonger time scale.

Internal Processes

In a majority of articles (see [7,8,33] for instance), the copepod is represented by ananalytical model. These models try to describe each ‘‘process’’ of the organism interms of input flow, output flow and transfer function. Let us describe the process ofingestion and digestion in this manner (see Fig. 3).First, the copepod captures a prey (a particle of phytoplankton). After handling

it for some time, the prey is stored in the gut and the process of digestion begins.The gut transforms the prey into energy, feacal pellets and urine. This transformationis continuous: within each dt, a quantity dq of caught preys is processed (this quantity

is proportional to the quantity already present in the gut). Then, this energy is eitherconsumed (metabolism, digestion, or swimming), or stored (egg production for females,for example).Caparroy [7] proposes a model synthesizing the various models developed before.

This model takes into account the activities of capture and ingestion using fivecoupled differential equations. However, from our point of view, the contributionof the behavior is partially neglected. Indeed, one can put into equations the factthat the activity of nutrition of a copepod is a function of the density of preys, and

the level of turbulence of the environment, and the mode of foraging, and the quantityof food in the gut; however, it is much more difficult to take into account various

FIGURE 3 Processes involved in ingestion and digestion in the copepod (see text for details).

SCALE TRANSFER MODELING 797

factors in the behavior such as the way the copepod perceives its environment, thedifferent sizes of preys, the speed of swimming of the copepod relatively to that ofits prey, and so forth. One usually models these processes by introducing differentcoefficients. For instance, the effect of turbulence is modeled by a more or lessimportant encounter rate coefficient [7,8]. In this work, we use equations developedby Caparroy [7] to model internal processes that govern satiety.The feeding behavior of the copepod is modeled as follows: it captures the particles

of phytoplankton if it has not yet eaten too much. Indeed, the copepod decreases thequantity of food which it absorbs according to its level of satiety, itself directly boundedwith the number of particles of phytoplankton present in the gut. The ‘‘decision’’ ofeating is computed by a probability proportional to the quantity of food in the gut.

Perception

Evidences that copepods can detect the presence and position of remotely locatedpreys have been accumulated since 1980 [4]. Modalities of perception are very complex.We can summarize them in two ways: the detection of the streamline deformations dueto the presence of a prey (mechanoreception), and the detection of chemical traces inthe fluid (chemoreception).In previous works, an agent model has been designed using our framework ‘‘Virtual

Laboratory Environment’’ (VLE) [27]. Two strategies of movement (random andoriented toward food) were experimented. These first simulations with a discretemodel of space in two dimensions confirmed the influence of the phytoplankton distri-bution on the trajectory of the copepod. But with this traditional representation ofspace in the multiagent models, i.e. a discretized space according to a grid wherecells can be square or hexagonal, the modeling of perception geometry of the copepodis quite limited by the geometry of space discretization. Furthermore, the movementsof copepods are fundamentally structured in the three dimensions of space and theinteractions between predators and preys are complex and simplifications related tothe discretization lead to a truncated vision of these interactions.The perception is characterized by a cone (according to the orientation of the

copepod). Perception is defined with an angle (which origin is the center of the copepodand median is its direction vector, corresponding to the perception angle of Table I)and a distance (corresponding to the perception distance of Table I). This defines aperception volume.

TABLE I Value of parameters used in the model

Parameter Value

Perception angle (radian) �Perception distance (mm) 2Catch distance (mm) 0.5Swimming speed (mms�1) 1Initial position (x, y, z) Center of spaceInitial direction angles (radian) xOy¼ 1.8; xOz¼ 1; yOz¼ 1Time step(duration of one iteration in second)

0.05

Average phytoplankton particleconcentration (cellsmm�3)

0.5

798 R. DUBOZ et al.

We have experimented two algorithms for perception. In the first algorithm, the

agent computes whether other agents are located in its perception cone and updates

its trajectory determining a target positioned in space by selecting the nearest particle.

The second algorithm uses the barycenter of perceived particles weighted by the square

of the inverse of the distance between the particle and the copepod. However, the results

obtained with either algorithms are not different. Consequently, we use the first one

regarding to its simplicity of implementation and computing performance.

Movements

At the individual level, movements have to be expressed in the Lagrangian formalism.Furthermore, videos are currently used to observe and to record copepod movements[4]. In a perspective of model validation by confronting real trajectories to simulatedones, it appears interesting to model copepod movements in a continuous space.According to in vivo observations, the copepod basically exhibits three distinctbehaviors: swimming toward food, jumping apparently at random or during attackon preys, and take a sedimentary behavior.

The space in which the copepods are simulated is finite. Thus, we have to deal with

this finiteness and its boundaries. We consider three possibilities to deal with space

boundaries:

1. reflection: at the limits of space, the copepod adopts a reflection trajectory, like aray of light.

2. random bounces: at the limits of space, the copepod updates its trajectorydetermining a target randomly.

3. toroidal space: at the limits of space, the copepod returns exactly at the other sideof space and keeps the same direction.

In Cases 1 and 2, the space is limited. In Case 3, the space is infinite. We could have

chosen another strategy as random generation of direction and position coordinates.

However the choice of random bounces limits random number generation to the

three coordinates of direction. Furthermore as we have said before, we would like in

a close future to make comparisons between real and simulated trajectories. Another

strategy would be the generation of the phytoplankton field around the copepod

while it is moving. This creates a space which is initially infinte. However the computa-

tional cost is very high and it remains difficult to simulate several copepods with this

strategy.

2.2. The Phytoplankton

In our model, the important point regarding phytoplankton is its distribution inthe water. Indeed, the distribution of phytoplankton has been shown to be stronglyheterogeneous at the scale of the copepod [30]. Current results show that this hetero-geneity influences the energy budget of the copepod. By measuring the quantity ofnitrogen ingestion, observations show that outputs relating to the behavior of thecopepod (the excretion rate for example) vary according to the type of distributionof food. For example, a turbulent environment, favorable to an overall mixing,can increase the encounter rate of the copepod with the cells of phytoplankton

SCALE TRANSFER MODELING 799

and increases the ingestion rate [7]. So, the only point that is modeled with regard tophytoplankton is its spatial distribution.We use three different algorithms to generate the spatial distribution of phytoplank-

ton cells:

– a pseudo-random uniform distribution is calculated by the pseudo-random functionof the Linux Cþþ library initialized with the computer clock. This algorithm providesa uniform repartition of particles in space showing no regularity.

– a pseudo-random patches algorithm is computed likewise the pseudo-random uniform:it distributes particles randomly around virtual centers randomly distributed witha standard deviation equal to 1. This provides patches which simulate a heterogeneousdistribution. The random aspect of these algorithms imply multiple runs. The pseudo-random uniform distribution algorithm provides a heterogeneity less importantthan pseudo-random patches algorithm.

– a regular algorithm distributes particles on a regular grid. The size of one cell ofthe grid is proportional to the concentration of phytoplankton and the volume ofspace.

Due to simulation durations which are very short at the level of the copepod and thephytoplankton life cycle, we do not model phytoplankton production: once a particleof phytoplankton has been eaten, it is not replaced by a new one. Furthermore, weconsider only one sort of phytoplankton, its nutritive characteristics being constant.

2.3. Numerical Aspects of the Model

The model is discrete in time. At each time step of simulation, we compute the newvalue of the position and the direction of the copepod. Constant parameters aregiven in Table I. The unit of time of simulation is fixed to the duration correspondingto the time necessary to perform the shortest action, i.e. the handling time of a particleof small phytoplankton by the copepod, 1/20 s [7].

Optimization

The large number of particles to be processed imposes the use of an adapted data struc-ture. In his PhD thesis where he deals with the same kind of problems with computationof water balls trajectory, Servat [29] proposed to partition space to sort particlesaccording to their position. We do the same here. Space is partitioned into a grid.The size of cells of the grid is constant in time and equals the perception distance.Every particle is affected 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 computationcost of the selection of the neighboring cells decreases greatly.

Pseudo-random Number Generator

The use of the pseudo-random number generator is a very important issue in simula-tion. In the sequel, we want to explore the behavior of the model regarding theuse of particular algorithms. Then, it is important to evaluate the quantity ofpseudo-random number generated during a simulation regarding the pseudo-randomnumber generator period. If both numbers are close to each other, the pseudo-random number generator produces artifacts in simulation results. In this work, we

800 R. DUBOZ et al.

use the standard pseudo-random number generator of the Cþþ library. Its periodis 16� 231�1 which is much larger than the estimated number of pseudo-randomnumbers generated during simulations. Indeed, in the sequel, we randomly generatethe position of particles in space (3 coordinates� 32 000 particles in the larger space)and the new direction of the copepod in case of random bounces (3 coordinates� 60006000 bounces for a mean simulation).

2.4. Influence of the Choice of Algorithms on Results

The model having been described, it is legitimate to raise the issue regarding the impactsof the various hypotheses that are used (the algorithms used for the distribution ofthe phytoplankton, the behavior at the boundaries of space, space dimension) onthe dynamics of the simulated system. These issues are thus addressed in the nextparagraphs before going any further in the utilization of the model in the next section.

In classical models of copepod using differential equations, the individual bioener-

getic budget strongly depends on the ingestion rate [8]. This is the reason why we

have decided to consider the number of particles eaten during the simulation duration

by the copepod as a relevant indicator of the effect of boundary conditions, particle

distribution and space size on the simulation. Furthermore, we do not take metabolic

processes into account, neither random bounces or sedimentary behavior in order to

avoid the stochastic aspect of simulation. In absence of perceived particles, the copepod

goes straight on and the simulation stops whenever there are no more phytoplankton

particles in the mass of water or when the copepod eats no longer and can no longer

catch a phytoplankton particle due to the way the simulation is performed.

We perform experiments with different types of distributions. The value of param-

eters used in the model is give in Table I. Three sizes of the volume of the

mass of water (the ‘‘space’’) are taken for the different simulations: 103, 8�103, and

64�103mm3. The concentration of particles is the same for all space size: 0.5

particlesmm�3 which corresponds to a strong concentration [7]. Simulation plan is

described in Table II. The results of simulation are presented with plots of mean

ingestion for each case of Table II and for each space size. Calculating the slope of

this curve, we obtain the instantaneous ingestion rate. We present some results to

discuss the impact of the choice of algorithms.

TABLE II Simulation plan for each space volume (103, 8�103 and 64�103mm3)

Phytoplanktondistribution

Deterministic boundary conditions Random boundaryconditions

(Random bounces)Reflection Toroidal space

Pseudo-randomuniform

30 simulationswith different

initial distributions

30 simulationswith differentinitial distributions

30 simulationswith onedistribution

Pseudo-randompatches

30 simulationswith different

initial distributions

30 simulationswith differentinitial distributions

30 simulationswith onedistribution

Regular One simulation 30 simulations 30 simulations

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In case of random boundary conditions, we perform 30 simulations with different

initial distributions on one hand and 30 simulations with one initial distribution on

the other hand. This permits to separate variability due to distribution from these

due to random bounces. In case of regular distribution and deterministic boundary

condition, there is one simulation because the distribution and the behavior (in this

study) are deterministic.

Impact of Boundary Conditions

Figure 4 presents a comparison between the number of bounces and the cumulativenumber of ingested particles during a particular simulation. We want to emphasizethat ingestion stops whenever the bounce curve becomes linear. This indicates thatthe agent follows the same trajectory over and over. This point has been confirmedby visual observation of these trajectories. The same result is observed for a toroidalspace.

Deterministic boundary conditions seem to induce an artifact in the simulation since

the agent is not able to explore the whole space. In the prespective of finding an optimal

representation of space (one which does not induce any artifact on the ingestion rate),

we have introduced random boundaries.

Figure 5 presents mean ingestion curves in a space of 64�103mm3 with random

bounces as boundary condition, averaged over 30 simulations. For each simulation,

there is a particular particle distribution.

In this case of random bounces, we can note here that nearly all particles are ‘‘eaten’’

by the copepod. As far as there are particles in the whole space, the fact that all particles

are eaten means that the copepod is able to explore the whole space. Figure 7 shows

that the standard deviation, the proportion of remaining particles, decreases dramati-

cally in the case of random bounces.

FIGURE 4 Comparison between the number of bounces and ingestion curves for a particular simulation.

802 R. DUBOZ et al.

Impact of the Algorithm of Distribution

Figure 5 shows that the slope at the origin of the different curves (the ingestion rate)increases with the heterogeneity of particle distribution (the same result is observedfor all space sizes). Is was known that heterogeneity of space distribution affectsbiological systems [13,20], but as far as we are aware of, it has never been shownwith MAS in a continuous space. Furthermore, these results are in agreement with[30]: heterogeneity of the phytoplankton distribution increases the ingestion rate ofthe copepod. Figure 6 plots the ingestion rate for different phytoplankton distributions.By far, the ingestion rate is the largest with a patchy distribution of phytoplankton.These results agree with [7].

FIGURE 6 Ingestion rate at the beginning of simulations showing the crucial role played by the distri-bution of particles. The same results are obtained regardless of the space dimension.

FIGURE 5 Mean cumulative ingestion curves for a 64�103mm3 space dimension: the same results areobserved for other space dimensions.

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All along the simulation, the ingestion 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 of cumulative number of ingested

particles for simulation with random bounces (Fig. 7), we can note that initially,

it increases with time (i.e. with the decrease of prey concentration) and, afterwards,

falls close to zero. The fact that the maximum is encountered more or less early in

the simulation is due to the size of the space.

Here, it is noteworthy 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), the variability principally comes from boundary conditions for

small concentrations.

Impact of Space Size

Figure 7 shows that the choice of the space size has an influence on the efficiency of the(simulated) copepod ingestion: the standard deviation decreases dramatically withthe increase of space size. This is an important result since this means that using arelevant distribution of phytoplankton cells and the optimal space size, we can simulateingestion with a limited impact of space topology on results.

Owing to these results, we choose a space of 64�103mm3, and random bounces as

the optimal choice for the computation of the ingestion rate by the reactive agent

model. In the following, we show how the ingestion rate can be a bridge between

two models of different natures.

FIGURE 7 Standard deviation (in percent of remaining particles) against time of the cumulative number ofingested particles for simulation with random bounces.

804 R. DUBOZ et al.

3. COUPLING

The purpose of this part is the coupling of our reactive agent model with a classicalordinary ODE. Such a coupling permits the coexistence of two organization levelsin the same simulation model and then can lead to study scale transfer, i.e. study theconsequences of individual behavior on population dynamics. This is a fundamentalissue in ecological modeling and we do not pretend to answer it totally here. Wewant to propose a new approach in modeling based on the coupling of an agentmodel with an ODE-based model. In the sequel, we present the model to couple withthe reactive agent, the method and then, we show the possible impact of taking intoaccount the individual scale (heterogeneity of the distribution of preys) on the dynamicsof an ODE system through simulation results.

3.1. The Model

In order to illustrate this approach, we have chosen an extension of the Lotka–Volerraprey–predators model [21,35] namely, the Holling–Tanner model [17,32] (Eq. (1)):

dX=dt ¼ rXð1� X=kÞ � gðXÞY

dY=dt ¼ egðXÞY �mY

(ð1Þ

with

gðXÞ ¼ aX=ðbþ XÞ ð2Þ

where, X is the concentration of preys; Y is the concentration of predators; r is thereproduction rate of preys; k is the carrying capacity; a is the maximum ingestionrate of predators; b is the half saturation constant; e is the transformation coefficientof prey into predator; m is the mortality of predators.

The choice of this model is motivated by pragmatic reasons: before going any further,

we wish to use a model which dynamics is well-known and which is as simple as

possible. This choice is governed by the fact that in this article, we want to propose

and address methodological issues.

This system of equations has some particularities with regard to classical

Lotka–Volerra systems:

– the increase of the amount of preys is a logistic function, which means that the growthof the population of preys is limited by environmental resources,

– the catch efficiency of preys by predators is density dependant (i.e. g(X): thefunctional response of predator to prey density is a Holling’s disc equation [8,3]).

The dynamics of such a system is well known and generate a range of behavior

including stable equilibrium, stable limit cycles and divergent cycles eventually leading

to extinction for particular values of parameters [3]. The Holling–Tanner model

is deterministic. If we introduce stochasticity, it can affect the dynamics in two rather

different ways. First, for populations which exhibit large oscillations, there may be

periods when population sizes are relatively small so that the system may be vulnerable

to extinction due to the effect of demographic stochasticity. Second, because of

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environmental stochasticity, the parameters of the process may vary with time so that

conditions for a stable equilibrium, for instance, may be satisfied in some periods and

not in other ones. As we have shown in the description and analysis of the reactive

agent model, stochasticity remains present in our simulations (random distribution of

particles, random decision of eating particles). Then appears the first issue about our

coupling. What are the consequences of coupling, on one hand a stochastic model

(agent), with on the other hand a deterministic one? Moreover, as the two formalisms

are heterogeneous, a second issue then appears: it is impossible to calculate stable

equilibrium, limit cycles of the system or the stability of the integration schema used

for ODE resolution. Here, the only way to deal with the validity of coupling seems

to be through simulations. We now present the method used for coupling.

3.2. The Method: A Pure Behavioral Model and the Use of Emergent Computation

We have to find common points between the reactive agent model and the ODE,keeping in mind that both models deal with the same system (prey–predator), althoughwith a different viewpoint. The main assumption made here is that the reactiveagent model is able to reproduce the behavior of a process described by a function inthe ODE system. As we described it in Section 2, it is possible to compute the ingestionrate of a copepod with the reactive agent model. In the Holling–Tanner model, theingestion rate is modeled by the Holling’s disc equation. In the following, we showthat the reactive agent model reproduces the density dependence property of theHolling’s disc equation.

In the community studying MAS, computation traces are considered as an epipheno-

menon of the system simulation [25]. When it is possible to infer such a trace with a

mathematical function, we deal with an emergent computation [12]. In other words,

if a mathematical function emerges from simulation traces of a MAS, we face an emer-

gent computation and we assume that it can be used to deal with the transfer across

scales of particular properties. First, we have to identify and discuss mathematical

functions emerging from our reactive agent model.

First Emerging Function

Considering the reactive agent dynamics, we can describe the copepod as an entitywhich moves and gets in contact with particles along time. Measuring the amount ofingested particles, we also trace the encounter dynamics between the agent andparticles. We can do an analogy between our model and a chemical reaction wheremolecules are reacting contacting each other and then producing a new molecule.The kinetic of such a reaction is modeled by monomolecular or Mitscherlisch curves[26] following:

dX=dt ¼ að1� X=kÞ ð3Þ

where X is the reactant concentration, k is the maximum of reactant concentration,and a is the slope at the origin.

Looking at Fig. 5, we can note that the simulation trace of our model can be consid-

ered as a monomolecular curve. Henceforth, we can consider X as the number of

806 R. DUBOZ et al.

ingested particles, a as the ingestion rate of the copepod, and k as the maximum number

of particles. a is usually used to model assimilation in classical approach [8]. a can easily

be simulated by the agent model. At the beginning of a simulation, the assimilation of

particles by the agent is linear. By dividing the number of ingested particles by the time

step, we have the slope at the origin of the monomolecular curve (Eq. (3)). The compu-

tation of a gives a perspective of a coupling between our agent model and an analytical

formalism. The very short time simulated to obtain the value of a (1000 time steps) is

important regarding the issue of obtaining an efficient coupling.

We have shown that the ingestion rate is heterogeneity dependant (Section 2) and

that it is possible to compute it with the reactive agent model. As we said before, we

have to deal with another important property: does the reactive model respect the den-

sity dependence property of ingestion rate?

Second Emergent Function

The Fig. 7 shows the ingestion rate at the beginning of the simulation. From anoptimization perspective, it is noteworthy that the nature of the distribution influencethe ingestion rate at the very beginning of simulations. By averaging the ingestion ratemeasured over 1000 simulation steps for each particular distribution of phytoplanktonand for a wide range of particle concentrations (from 0.2 to 13 preysmm�3), we plotthe evolution of the ingestion rate as a function of the particle density (Fig. 8).For each concentration, 30 simulations are performed in order to take the variabilityof the particle distribution into account.

The general form of the agent simulation results (cross on Fig. 8) shows an increase

of the ingestion rate up to a maximum value as the concentration of preys increases.

This is the classical functional response to the density of preys. Using the classical

least squared method, it is possible to fit g(X) (Eq. (2)) to simulation results.

Parameter values are given on Fig. 8. We can note that the standard deviation of the

ingestion rate for each concentration is smaller in the case of uniform distributions

and for low particle concentrations (see Fig. 8). This is due to the fact that the ingestion

rate is computed considering a short simulation time. At the beginning of a simulation,

FIGURE 8 Cross represent the ingestion rate computed by the reactive agent model. 30 simulations areperformed for each prey density which explains this vertical set of dots for each prey density. The line is thefitted Holling’s disc function. Holling’s disc equation and the value of its parameters are written on the plot.

SCALE TRANSFER MODELING 807

the agent can be situated inside or outside a patch. Therefore, it can ‘‘eat’’ more or lessrapidly.Before going any further, it is important to know if the variation of the ingestion rate

as a function of prey density is significant regarding to inner ingestion rate variability.We adopt a classical variance analysis test (VAT). A sample is composed withingestion rates obtained from 30 simulations. The set of samples is made with simula-tions for each concentration. Results of the analysis of variance are summarizedin Table III.The VAT is highly significant so we can say that our agent model and Eq. (2) are

behaviorally equivalent. Equation (2) can be considered as an epi-phenomenon butits parameters emerge from simulation. In this sense, it is an emergent computation.We have shown that it is possible to find classical functional response with the agent

model. To achieve the coupling, we use the basic idea initiated by Fash et al. [9], calledthe separation of timescales method. The process of ingestion and the populationdynamics do not work at the same timescale. Fash et al. propose a protocol with theassumption that behavioral and population processes can be considered separately.Partially following this protocol, we consider in our modeling:

– processes of the IBM that affect population dynamics (the ingestion rate),– a pure behavioral model (neither birth nor mortality of copepods),– a mean individual (activating satiety, starting with a mean quantity of food in gut),– several simulations of the same model in order to deal with individual responsevariability to the heterogeneity of food distribution.

We can imagine two ways of coupling the analytical model with the agent model(Fig. 9):

– a weak coupling which consists in first obtaining the parameters of g(X) by anemergent computation. Then, we perform the numerical resolution of the systemwith those parameters.

– a strong coupling in which, at each time step of the resolution of Eq. 1, we firstcompute a particular ingestion rate (a in Fig. 9) with the agent model which is theninjected into the ODE system for its resolution.

We assume that a is density and heterogeneity dependent as we have shown before.Figure 9 illustrates the solution that we propose to deal with scale transfer

and must be viewed as the operational answer to conceptual approaches presented inFig. 1. The next paragraph presents the simulation results of this coupling.

TABLE III Results of the analysis of variance of the ingestion rate

Distribution Uniform Patch

Number of measures 600 540Number of samples 20 18Total average 0.25 0.23Variance due toprey conentration

3.6e-3 3.5e-3

Residual variance 5.8e-5 3.9e-4F value(99% of confidence¼ 1.36)

F(19;580) 1890.4 F(107;3132) 277.8

808 R. DUBOZ et al.

3.3. Results of Simulations

We simulate the interaction between phytoplankton and copepod over a large periodof time. Values of parameters of Holling–Tanner equations (Eq. (1)) are:

r ¼ 0:04 h�1, k ¼ 20 indmm�3, e ¼ 0:02 ðdimensionlessÞ,

m ¼ 0:0014 h�1

These value are characteristics of phytoplankton whose cell division can occur

daily (parameter r) and the copepod life cycle lasts one month (parameter m) [8].

We use a weak coupling as described above. Parameters of g(X) are those of Fig. 9,

and results of simulation are given on Fig. 10 and discussed below.

The objective here is to describe the behavior of the Holling–Tanner system taking

into account the heterogeneity of the phytoplankton distribution. The left plot

of Fig. 10 shows us the evolution of prey and predator densities with a uniform distri-

bution of preys. The right plot shows us the evolution of prey and predator densities

with a patchy distribution of preys. We can note two major differences between them:

FIGURE 10 Evolution of prey and predator density using Holling–Tanner equation system for uniformdistribution of preys (left plot) and a patchy distribution of preys (right plot). The continuous line is the preydensity and the dotted line is the predator density. The only parameters are a and b from the Holling’s discequation in the case of a weak coupling (the left plot uses a¼ 0.33 and b¼ 1, and the right plot uses a¼ 0.31and b¼ 0.55) and a in case of a strong coupling. See text for details.

FIGURE 9 Schematic representation of the two ways of coupling of a reactive agent model with a prey–predator model. On the left, weak coupling in which g(X) parameters are computed by the agent modeland subsequently used to solve the ODE system. On the right, strong coupling in which the ingestionrate a is obtained by the agent model at every time step and is used in the ODE system for its resolution.In this latter case, the ingestion rate is instantaneous instead of averaged, and thus changes over time.

SCALE TRANSFER MODELING 809

– the cycle frequency of preys and predators is larger in the case of a uniformdistribution of preys,

– the persistence of a maximum of prey density is longer in the case of a patchydistribution.

We have here the illustration that a property existing at individual scale (the nature of

the distribution of preys) can lead to large changes in upper scale (population

dynamics). Same results are found with a strong coupling with the difference that the

nonlinearity of a (particularly for patchy distribution) introduces stochasticity in the

Holling–Tanner model with problems described at the beginning of this part (see the

model section).

Here, the type of the distribution is constant over time so we can use a weak coupling

which is much faster to compute and which is more stable. If the population model

integrates physics, we would have information on turbulence and consequently on

distribution [30]. In this case, only a strong coupling would be possible because the

heterogeneity of distribution is not stationary.

4. DISCUSSION

In this work, we place our study at the individual level with a pragmatic motivation [16]in the sense that we want to model the interaction between copepod and phytoplanktonin a ‘‘realistic way’’. We have a paradigmatic motivation too since we would like toshed light on the potential contribution of the reactive agent paradigm in theoreticalecology. We can say that this paradigm is another point of view to deal with individualbehavior. We have shown that it can be used in ecological modeling as a powerfultool for the description of interactions between individuals using characteristicssuch as perception geometry, the discrete nature of individuals in a three dimensioncontinuous space, and the heterogeneity of food distribution. Furthermore, it seemsimportant to represent individuals in a continuous space to be more precise withregard to behavior modeling if we want to simulate the perception of its environmentby an organism more precisely. For example, giving various characteristics tophytoplankton cells corresponding to copepod preference, the agent will be ableto ‘‘choose’’, and even to learn during its life span its favorite phytoplankton cells,considering its external–perceivable characteristics. If purely reactive agents are notable to learn anything, closely related agents are able to do it, namely adaptiveagents which, for some of them, mimics animal adaptive behavior. Moreover, thebehavior of our reactive agent model can be explored in two different ways such as:

– storing outputs of model simulations and represent them in some graphical way,– video representation of the agent evolving in space and interacting with particles.

The first method is classical and very powerful because it permits the representation

of the system dynamics for each variable of the model and then analyzing them in

statistical fashion, as an experimenter would do. The second method provides a

visual representation of the agent behavior. It is very useful for validation by specialists

of copepod behavior. It is then possible to experiment behavioral rules of agents in

simulations and compare simulated behaviors to observations (for example by analyz-

ing trajectories). It seems that the agent paradigm is closer to reality and therefore

810 R. DUBOZ et al.

closer to the experimental discourse of ecologists. Furthermore, reactive agent modelsare easy to explore and can improve the cycle of model-creating in theoretical ecology[14].Another aspect of our modeling is that we do not take into account the individual

variability stricto sensus in the sense that we simulate the behavior of an ‘‘average’’copepod. However, stochasticity is present in the digestion function (with an impacton satiety and so on ingestion) and in phytoplankton cells distribution algorithms.Simulating several times the same copepod in different phytoplankton fields is equiva-lent to introduce individual variability in the ingestion rate. Furthermore we can easilyintroduce variability in all the characteristics of the agent (perception distance,speed . . . ) to deal with individual variability, but it was not the goal here.Emergence and self-organization are key issues in life sciences. In most cases, one

cannot infer properties of a large scale system only considering the properties of itselements taken separately. With a nearly infinite computation power, it would bepossible to simulate a system as a whole. Observing the behavior of such a systemat different levels of organization would ideally lead us to the observation of scaletransfer. But it is not realistic nowadays. As we know that in nature, different levelsof organization interact with each others, we have to find operational method to simu-late scale transfers. Earlier researches have shown that algorithmic formalisms such ascellular automata or MAS can generate large scale patterns or scaling laws observed inlarge complex systems [22,36]. Using emergent computations, we have obtainedMitscherlisch and Holling’s disc equations. From a systemic point of view [1], we cansay that the ingestion rate emerges from the model. This leads us to think that it ispossible to find mathematical properties based on an agent formalism in agreementwith [29] and that MAS can provide a way of, or at least, help in modeling multiscalesystems. We have shown that the emergence of mathematical functions can be a way ofcoupling population analytical models (well adapted to large scale modeling) with agentmodeling (well adapted to individual behavior modeling) and then simulate scale-transfer. Furthermore, Holling–Tanner’s model assumptions are that the distributionof prey is homogeneous and the nature of the distribution is stationary. With ourmodel, we have shown that the parameter b (the half saturation constant of Holling’sdisc equation) depends not only on species characteristics, but also on the heterogeneityof prey distribution.Based on these fundamental results, we propose a method by considering the

separation of time scales in the studied system [9], the average behavior of ODE, andthe use of the reactive agent paradigm. We can summarize the method used in thiswork by the following points:

– modeling individuals as reactive agents in a pure behavioral model,– identify upper scale parameters that can be computed by the reactive agent model,– inject those parameters in a classical model of population (population scale),– parameterize the environment of the reactive agent model with the simulation ofthe classical population model.

Are we really dealing with scale transfer? In a sense, the answer is positive becausethe properties like the heterogeneity of discrete particle distribution only exist at theindividual scale and can only be caught by the reactive agent model. With a bottom-up approach and an algorithmic formalism, we have shown that it is possibleto refine classical functional response like prey density dependence. So the emergent

SCALE TRANSFER MODELING 811

computation of ODE parameters in strong coupling can be considered as transfer of

properties from individual to population level.

Another point regards whether it is interesting to integrate heterogeneous formalisms

that are expected to express the same reality. We think that this can improve system

description and understanding. Indeed, in complex system modeling, we often need

several viewpoints to describe the complexity of reality and several viewpoints often

lead to several formalisms. Furthermore, we have shown that different time scales

can be taken into account in this way in a unique modeling. We can easily imagine

that it is possible to do the same in every system where it is interesting to model

processes at different scales and/or with different aspects of continuity. Furthermore,

it can be very useful in the perspective of optimization. Indeed, a major drawback of

IBM is the computational load. By representing the global behavior of such a

system with a set of ODEs which are parameterized by the agent model, we decrease

dramatically the simulation duration.

In this work we emphasize that it is possible to deal with several aspects of a system

which cannot be modeled with only one formalism focusing on the integration of

reactive agent model with ODEs. To deal with the integration, we propose the

following methodology:

– identify parameters or functions of the ODE system that can be obtained bysimulation by way of an agent model,

– build the corresponding agent model as simple as possible, and study its behavior,– use emergent computation to infer functions of the ODE system,– choose between a weak or a strong coupling,– simulate to study the impact of such a coupling on the dynamics of the ODE system.

How to choose between a weak and a strong coupling? In the case of a weak

coupling, we need to know an analytical function and the variation domain of all the

variables implied in the parameterization of the reactive agent model. Indeed, using

emergent computation, we compute the parameters function considering this variation

domain. If we do not have such a function, only a strong coupling, is possible.

Furthermore, as we said before, in the case of strong coupling, stochasticity can

affect the dynamic of the ODEs system. This problem does not appear with a weak

coupling. However, the choice between the two kinds of coupling remains attached

to the particular problem to be solved and stays an open issue.

However, such a coupling uses the concept of multimodeling considering the

principle of hierarchical decomposition defined by Fishwick [11] which consists in

the refinement in the modeling of a subpart of a model. We assume such concepts

coming from modeling in computer science (object modeling, discrete event simulation,

refinement, hierarchical imbrications or decomposition . . . ) can be useful for ecological

modeling. In most of works, heterogeneous hierarchical decomposition of models

implies the dynamics control of a state model (petri net, cellular automaton . . . ) by

the resolution of equations or by other state models [11]. This hierarchical decomposi-

tion does not induce scaling into space or time. In this work, we consider the parame-

terization of the environment of an agent model by the resolution of equations (which is

the reverse of what it is done usually). We focus on two levels of organization but the

method we propose can easily be extended to several levels considering them two by

two.

812 R. DUBOZ et al.

5. CONCLUSION

The main objective of this work is to find ways of coupling continuous and discreteformalisms in the field of modeling, in particular ecological modeling. Before tacklingthe problem of coupling models, we have discussed the impact of some algorithmicchoices on simulation based on an agent model. This experimentation and thediscussion have been performed on a case study dealing with the simulation ofthe behavior of the copepod in a phytoplankton field. From this work, we come tothe conclusion that the choice of the distribution algorithm and the space size beingsimulated are very significant in order to simulate copepod ingestion in a 3D continu-ous space. This work assesses the impact of the assumptions in the model and figuresout what the simplest and still relevant model is in order to simulate the copepodingestion rate in an optimal fashion (accuracy versus computational requirements).Then, we have developed a new approach for the modeling of scale transfer by couplingan agent model with a numerical model. Basically, it consists in the use of emergentcomputations to build a multimodel. In the near future, we will have to assess ourmethod quantitatively. This means that we have to couple a reactive agent modelwith a well studied and validated classical population model and then compare resultswith statistical methods. For instance, taking a population model with explicitspace and physics giving information on turbulence level (i.e. heterogeneity level ofphytoplankton distribution), we can compute an ingestion rate associated to thislevel of turbulence and then feed it back to the population model.

To conclude, we emphasize that, even if MAS are not much used today in the IBM

field, learning agents can be used in the future to model the adaptation of the behavior

of living organisms to their environment during their lifetime to take into account

the learnings a living organism is performing during its lifetime. Evidences that even

very ‘‘simple’’ organisms such as copepods adapt and learn during their lifetime have

been reported, making this kind of modeling important to perform relevant simulation

of living organism behavior. Furthermore, the behavior of organisms can change

qualitatively during their lifetime (maturation). These qualitative changes can be

modeled with agent models. Both issues, learning and maturation, can be handled

naturally with an agent model whereas numerical models may not be used so easily,

in particular to deal with learning during lifetime.

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