Chapter 3: Simulating transition: an introduction to spatio-temporel models
Entrées d’index
Keywords : attractor, complexity, diffusion, interaction, migration, modelling, probability, simulation, transition
Texte intégral
Introduction
1Migration – as well as the colonisation of new territories that may have resulted from it – has constituted one of the responses of groups of humans to political, social or environmental changes. We propose in this chapter, as a preamble to the first TransMonDyn transition (‘Leaving Africa’ in 70,000 B.C.E.), to consider the contribution of spatio-temporal modelling to the study of this question. In doing so, we will confine ourselves deliberately to the case of colonisation of new empty (i.e. unpopulated) territories, and we will not take account of possible interactions between allochthon and autochthon populations.
2Moreover, we will purposely limit ourselves, in exploring these colonisation processes through modelling, to the interactions between the populations and their environment, the latter being characterised by a collection of opportunities and constraints.
3The procedure decided on here is typical of analysis of dynamic spatial systems, which consists in formulating hypotheses, then simulating them in order to recover configurations observed elsewhere, instead of studying at the outset colonisation strategies consistent with the configurations observed. This chapter will therefore be the occasion for discussing the fundamentals and the contributions of such a procedure within an interdisciplinary framework.
4In the thematic context engaged here, groups of humans – fairly stable structures consisting of thirty or so hunter-gatherers – move to an unknown territory void of all human population. In this context, the colonisation of a space is defined by the passage from an initial state characterised by the absence of human presence to a state in which by far the greater part of this space is explored by human groups, which live there permanently in different ‘habitable’ areas. The transition may, however, take different forms. In fact, transition, as it is defined in TransMonDyn (cf. chapter 2), implies a change in the structure of the settlement system – for example, a displacement of population (mobility) and/or a demographic evolution. One can imagine, on this basis, different dynamics of colonisation combining these two processes. For example, an initial colonisation by a very rapid movement on the part of the first human groups may be followed, once these are stabilised, by a sparser and more prolonged diffusion, thanks to a phase of demographic growth (figure 1.a.). Inversely, one can also imagine a slower colonisation by relatively static groups or individuals, who are simply ‘poured’ one after another, thanks to very strong demographic growth (figure 1.b). Other forms may, of course, be imagined, given the complexity of interactions between demographic evolution, human mobility and geographic environment.
Figure 1. Two possible forms of colonisation

A at the top and B at the bottom.
The points correspond to previous situations.
1.a : form of colonisation propelled principally by rapid mobility of groups
1.b : form of colonisation propelled principally by slow mobility of groups
5The models presented in this chapter engage in diverse ways the question of the factors favouring one or another form of transition. With regard to the colonisation of an empty space, multiple spatialised variables have been proposed in literature. Thus, one may focus on the speed of the colonisation of the space over time, on the demographic dynamic of the population1, but also on the heterogeneity of languages2 and cultures3. Certain of these studies have taken up a single one of these variables, while others have concentrated on developing a comprehensive approach4. The way in which the environment is described also varies with the perspective adopted. It may be highly stylised (the model of Young5) or, by contrast, extremely detailed6 7.
6The modelling objectives may also differ, according to whether the models aim at reproducing the state of the dynamic system at different dates8 or concentrate on the ‘triggering factors’ of the migration9 10.
7Finally, the methods of resolving equations are likewise variable. Each dimension (variable, space, time) may be quantified in a discrete or a continuous manner. Thus, one passes from partial differential equations, in which everything is continuous, to cellular automata and/or multi-agent systems, in which everything is discrete. These different formalisms may be combined in order to study the same situation.
8Two families of models may serve as a point of departure. The first, derived from pioneering work on the ecology of populations, includes deterministic continuous mathematical models, based on partial differential equations, while the second, more recent, groups together discrete models (space, time, variables) and stochastic ones (that is, the transitions between states are random). In the first case, the model formalises the evolution of local population density without taking account of individual trajectories. This is an approach of the ‘field’ type, with reference to a continuous field (electric field, temperature, etc.). In the second case, one follows the trajectory of each ‘individual’ (particles in physics, individuals in ecology of populations, etc.), an approach which amounts to accounting for the diversity of individual trajectories. Often, models of the first kind can led to an analytic, or numerical, solution requiring a certain technical mastery of the handling of equations, which rapidly become technical, while those of the second kind thrive on calculations and depend on simulations. These represent two rich worlds of modelling. It is notable that a large proportion of the mathematical modelling of spatialised dynamics has been dedicated to understanding the links between the evolution of local density and the history of individual trajectories – two different angles of approach to the same dynamic11 12.
9After presenting each of these families, we compare two of their realisations (with the Fisher-Skellam model for the first family and that of Young for the second). We will them present the ColoDyn model, developed in the framework of the TransMonDyn project, which draws on these two families of models.
10This chapter will emphasise comparisons intervening at different levels: comparisons between different models (mathematical/simulation, discrete/continuous) and confrontation of the results of simulations with different parameter values.
A continuous reference model: The Fisher-Skellam model
11To model a settlement dynamic over a long period implies two fundamental processes: the first describes the demographic growth of the population, while the second focuses on the movements of this population in space. The standard model in population ecology is the Fisher-Skellam model, which belongs to the family of models known as ‘reaction-diffusion’. These models have been much studied, notably in chemistry. Here the ‘reaction’ component corresponds to the local demography, and the ‘diffusion’ component corresponds to the migrations. In its most classic form, the model is written as

12in which one recognises the sum of one reaction term and one diffusion term using the symbol Δ which represents the Laplacian, the diffusion operator.
13In general, if the initial state corresponds to an empty space where one finds only a ‘seed’ formed by a population localised in a single point, evolution takes place by means of local increase accompanied by dispersion, to the point where the size of the population in this locale is equal to the place’s population capacity. When the population exceeds the local carrying capacity, a colonisation front is created whose movement entails an increase in the region occupied, until the population attains its maximum size. One speaks then of the ‘travelling wave solution’.
14The NetLogo model ‘M1_FS-Young’ makes it possible to familiarise oneself with this mathematical model by simulating it and to understand its behaviour intuitively. While the Fisher-Skellam equation is mathematically continuous in space and time, to calculate its numerical solution one can put in place the implementation of a discrete evolution in space and time. Space is therefore divided into square cells of the same size, and a time step corresponds in the model to an elementary unit of time during which the cells modify their population as a function of an ‘iteration’ of the equation (1) (that is, an application at each unit of time of the ‘reaction’ term and the ‘diffusion’ term for each cell).
15The presentations below illustrate the fact that the model produces a ‘field’ of population, starting from the initial site, the form and speed of whose front (but also of the barycentre) depend on the parameters α (rate of growth) and D (speed of ‘diffusion’). The values of the parameters corresponding to the images of figure 2 have been chosen to show the diffusion over time (figure 2.a) and the effect of parameter D (Figure 2.b) – the value of the parameter α being constant – on the speed of diffusion and the spatial structure obtained, starting from a ‘seed’ of local population located at the bottom left of the space.
Figure 2: Examples of ‘patterns’ (configurations) of colonisation obtained with the Fisher-Skellam model for different combinations of parameters

(the brighter the cells, the larger the population they contain; the black cells are unoccupied; ‘ticks’ correspond to the number of time steps elapsed in the model).
16Starting from this simple base, numerous refinements are possible. For example, in his 2009 article, James Steele13 describes the introduction, in the model’s formulation itself, of effects linked to low population densities or to the competition between multiple populations, but also to the existence of spatial and temporal heterogeneities (anisotropic spatial diffusion or temporal effect of delay, for example). These different evolutions of the basic model, which are fully justified, considering the complexity of the phenomenon of colonisation being modelled, nevertheless entail a rapid complexification of the initial model. It is notably in response to this difficulty that other approaches have been proposed, such as the Young model.
A discrete model of reference: the Young model
17In 2002, the physicist David Young proposed, in the journal American Anthropologist14, an ‘alternative’ theoretical model which, although centred on individual behaviour, nevertheless remains founded on the same two processes of demographic growth and mobility.
18Simpler in its formalisation than the Fisher-Skellam model presented above, this model, which describes the settlement dynamic of a space by groups of hunter-gatherers, is likewise based on the two processes of demographic evolution and mobility. The groups are represented in the model by computational agents as understood in multi-agent systems15. For simplicity, we will call them ‘groups’ here, the context making it possible to determine whether we are dealing with computer science agents or with human groups which have actually colonised vacant territories. The processes function at the level of each group.
19The first process is divided into three successive steps, with a parameter attached to each one:
‘reproduction’ (parameter proba-birth), that is, creation of another group in the same place (in a random manner),
‘death’ (parameter proba-death), that is, disappearance of the group (in a random manner),
‘death by overpopulation’ (parameter proba-overcrowding), that is, disappearance of the group as a result of overpopulation
20For each of these three steps, a drawing of lots is effected at the level of the group to determine if the event described takes place. The parameter serves in calculating the threshold of this event’s occurrence. Thus, three numbers are drawn at random using a uniform probability distribution – p1, p2 and p3 – whose value is contained between 0 and 1:
‘reproduction’ occurs if p1 < proba-birth
‘death’ occurs if p2 < proba-death
‘death by overpopulation’ occurs if p3 < proba-overcrowding x (Ni - 1), with Ni designating the number of agents present in cell i.
21The second process, the ‘migration’, is treated in a single step, with which the parameter ‘proba-move’ is associated. This step signifies that the group will move to a neighbouring cell, chosen at random. As for the first process, a number p4 is randomly drawn between 0 and 1, and the step ‘migration’ takes place if p4 < proba-move.
22Figure 3 shows the multiple destinies possible for a group situated in the cell where N groups are found: shown are the possible states at the end of the four steps applied to it, the first three relating to the probability that the agent-group will reproduce itself or disappear (death and death by overpopulation), the fourth to the probability that it will move. At the conclusion of the first three steps, six states are possible: the group born in the first step is the sole surviving (a) and (b); the initial group and the one born in the first step co-exist (c); no group has survived (d) and (e); the initial group remains alone (f). Subsequently, the initial group is subjected to a draw by lots to know if it will move to one of the neighbouring cells. In the case where a supplementary group has been created in the first step (represented by a white circle in figure 3), it cannot move at this stage. The paths followed each possess a probability which can be calculated as a function of the parameters of the model. With regard to implementation of the Fisher-Skellam model, it is important to emphasise that the ‘actions’ are not calculated at the level of the cells on average, but rather group by group.
Figure 3: possible number of groups on one cell, at the end of one iteration of a group in the Young model.

23One may readily imagine that the final result is not wholly independent of the order of execution of these four steps. This point is not explicitly addressed, however, in Young’s article, something which renders its reproducibility difficult. The NetLogo implementation of it that we propose here thus corresponds to one interpretation of Young’s article, the interpretation which seems to us closest to the description provided by the author.
24The precise description of a model, of its ontology, of its hypotheses, but also of its implementation and its behaviour within the space of its parameters, thus comprises essential points, which guarantee inscription of the work undertaken within a logic of reproducibility (cf. chapter 15), coherent with the idea of a cumulative process of knowledge construction in the human sciences16.
25One may question the ‘greater simplicity’ of Young’s model by comparison with that of Fisher-Skellam. Its simplicity is defended by the author in terms both of formalism (formulation on the basis of rules versus the equations of the Fisher-Skellam model), and also of description of the processes involved. Undeniably, divided and formulated in this way, the four steps identified appear simple and make sense from the point of view of the large mechanisms classically brought to bear to analyse the colonisation of new territories. Still, the link between their formulation on the microscopic level (that of the behaviour of each agent) and the statistical signatures produced on the macroscopic level is not immediately accessible. This micro-macro passage is a classic operation, although technically difficult (it is analytically unknown except for a small number of models). It consists in writing the microscopic process stochastically (that is, randomly) and deducing from it the master equation which governs the evolution of the densities.
26If the calculations of the model are performed on the level of the groups, the choice has been made to represent the results of a simulation in the form of population density by cell (figure 4) in a way similar to the representation adopted for figure 2 (the Fisher-Skellam model). The much less ‘smooth’ appearance of the population ‘field’ stems from the discrete nature of the model, here with a very small population of agents (10 units to start with). Taking into account a greater number of agents would tend to bring it closer visually to the Fisher-Skellam model (due to the law of large numbers) but would require much greater calculation times and would not necessarily be adapted to the thematic context that concerns us (small populations on a continent).
Figure 4: Influence of the parameter ‘proba-overcrowding’ in Young’s model on the rhythm of colonisation

(the brighter the cells, the greater the population they contain).
27The parameter ‘proba-overcrowding’ (death by overpopulation) was introduced to attempt to account for the limit on population growth imposed by the environment. The author relies on a standard work in ecology17 to explain that this model is equivalent to a model of logistic growth18 in which the population will cease to grow after having reached a threshold K (maximum capacity of the space).
28Such a link of equivalence between microscopic process and macroscopic law is far from neutral and deserves to be explored further. Indeed, an implementation in Netlogo of Young’s model limited to the processes only of birth and mortality by overpopulation (with values of zero for the parameters ‘proba-death’ and ‘proba-move’) reveals that this mechanism is able to produce, for certain combinations of parameters, an S-shaped curve characteristic of logistic growth. Analytical work on the basic model makes it possible to go farther in exploring this possible equivalence. The mathematical formalisation of the expected19 number of groups obtained starting from an initial population N appears difficult. Nevertheless, according to certain hypotheses, including the fact that the counting of the population is re-evaluated only at the end of the interaction, one can show that the two formulations are equivalent20 with:
α = 1+ proba-birth – proba-death + proba-overcrowding
and
K = 1 + (1 + proba-birth – proba-death) / proba-overcrowding
29Thus, the maximum capacity K, as formalised in the Fisher-Skellam model, possesses an equivalent in the Young model, as the author affirms in his article. In other words, this microscopic model, whose rules are very simple to understand, leads to a result which is not at all intuitive: the ‘reception capacity’ by cell is not independent of the parameters linked to the birth-rate (proba-birth) and death-rate (proba-death). This suggests that the macroscopic effects of rules functioning on the microscopic level should be interpreted with caution, even when it seems possible to interpret them simply. More generally, this kind of systemic effect, resulting from interactions between agents in multi-agent models, constitutes one of the strengths of this modelling paradigm, since highly sophisticated processes can be reproduced with few different rules operating at the elementary level, although it is not simple to explore the possible multiple combinations of rules or to interpret the rules obtained.
Comparison of the two models
30Analysis of the colonisation of a new continent may initially be undertaken through a multi-dimensional prism: the fact that the colonisation in the end occupies the whole territory is merely a single aspect. To give an example, as has been previously seen, one can attempt to understand the factors determining the speed at which the colonisation takes place (i.e. what proportion of the space is colonised after a given period of time).
31In order to compare the two models, those of Fisher-Skellam and Young, we put in place an experimental protocol aimed at determining the influence of the different parameters of the two models on the form of the transition described in figure 1. Four indicators were the object of a systematic calculation for each repetition21 using the two models:
the final population on the continent: in the case of the Fisher-Skellam model, this is the sum of the local populations n and, in that of Young, of the total number of groups;
32the speed of population growth: number of time steps needed to attain 95%22 of the population counted at the end of the simulation;
the spatial extension of the colonisation: in the Fisher-Skellam model, this is the distance between the cells at the end of the simulation and the departure cells, weighted by the local population n and, in that of Young, the average distance of the groups at the departure point;
the speed of the spatial progression of the colonisation front: the number of time steps needed for the colonisation front to attain 95% of the distance measured at the end of the simulation.
33To give an example, figure 5 illustrates the influence of the parameters expressing a limit of capacity of demographic growth (parameter K for the Fisher-Skellam and ‘proba-overcrowding’ for that of Young) for the total population at the end of the simulation. In keeping with what might be expected intuitively, the two parameters exert very important influence on this output indicator.
Figure 5: Influence of the parameter associated with the capacity of the space in the Fisher-Skellam model (left) and that of Young (right) on the final population occupying the continent (output indicator).

34We have explored, in a systematic manner, the influence of the parameters of the two models on the output indicators: the results of this study are presented in tables 1 and 2. It will be noted that repetitions of the simulations in the framework of Young’s model are necessary, since it is stochastic23 (hence random), in contrast with the deterministic Fisher-Skellam model, with which each combination of parameters produces a unique result. For example, the parameters of growth (α) and diffusion (D) of the Fisher-Skellam model, as well as the parameters of birth (proba-birth) and migration (proba-move) of Young’s model, have a positive impact on the rates both of demographic growth and of advance of the colonisation front. On the other hand, the simulations also show that the quantity of the population stock at the outset of the model does not influence the indicators at the conclusion, whatever the configuration of the other parameters.
Table 1: Synthesis of the influence of the parameters of the Fisher-Skellam model on the output indicators concerning colonisation of the empty continent
Effect of the parameter (intensity, direction) | Reference value | Total population after transition | Rate of demographic growth | Rate of advance of colonisation |
K (maximum capacity) | 50 | strong, positive | - | - |
α (rate of growth) | 0,1 | - | moderate, positive | moderate, positive |
D (diffusion parameter) | 0,1 | - | moderate, positive | moderate, positive |
(the cells marked with ‘ – ’ signify that no impact was observed; the impacts in bold type call attention to the strongest effects).
Table 2: Synthesis of the influence of the parameters of Young’s model on the output indicators concerning colonisation of the empty continent
Effect of the parameter (intensity, direction) | Reference value | Total population after transition | Rate of demographic growth | Rate of advance of colonisation |
proba-overcrowding: (probability of death by overpopulation) | 0,15 | strong, negative | strong, negative | strong, negative |
proba-death (probability of dying) | 0,05 | strong, negative | moderate, negative | moderate, negative |
proba-birth (probability of reproducing) | 0,3 | strong, positive | moderate, positive | moderate, positive |
proba-move (probability of undertaking a migration) | 0,3 | moderate, positive | strong, positive | strong, positive |
(the impacts in bold type emphasise the strongest effects; the impacts underlined are those for which the thematic links seem to us counter-intuitive).
35The main lessons which one can draw at this stage are that the Fisher-Skellam model possesses parameters which are linked in more explicit fashion to separate processes (the maximum capacity K affecting the demography, while the two others – α et D – influence the rate of growth of the population and the advance of the colonisation). By contrast, in Young’s model, the different probabilities, a priori easier to interpret at the microscopic level, produce much more tangled effects at the macroscopic level.
36There exists another important qualitative difference between the two models: in the Fisher-Skellam model, the transition is explicitly inscribed in the initial equations and must necessarily take place. The only questions concern the rhythm and the final state at the outcome of this transition. Conversely, in Young’s model, all is not determined at the initialisation: for the same set of parameters, a simulation may issue – or not – in a colonisation of the continent. The first fluctuations at the very start of the colonisation are fundamental, when the population is small and hence susceptible, if several successive negative lots are drawn, of being totally extinguished. The stochastic nature of Young’s model does not therefore merely induce a ‘blur’ at the margin of the results but also possibly a qualitative uncertainty in the interpretative framework of the transition that interests us.
37In order to test the influence on the results of Young’s model of the order of execution of the steps in implementing it, we proceeded to a comparison of two sequences (cf. Table 3) among the 24 orders possible for connecting the four steps. The results presented at the end of the table show the numerical results to be highly sensitive to the order of the sequences, something which once again suggests a need for prudence in manipulating multi-agent models whose implementation has not been specified in sufficient detail.
Table 3: Effect on the results of the order of execution of steps in the implementation of Young’s model
Order | Order A | Order B |
| Reproduction Death Death by overpopulation Migration | Migration Death by overpopulation Death Reproduction |
Number of replications | 100 | 100 |
Number of ‘failures’ (i.e. population zero at the end) | 3 | 1 |
Population at the end: average (standard deviation) | 3870 (686) | 6500 (661) |
Average distance of the groups at starting point: average (standard deviation) | 44,6 (7,9) | 45,6 (4,6) |
(the values of the parameters are the reference values indicated in figure 4).
38The analysis undertaken reveals the great influence of the order of execution of the steps of the Young model on the results (the population after 2,500 time steps varies significantly, practically by a factor of two, depending on the order of the steps) – a fact that raises two questions: First that of the reproducibility of the model proposed in Young’s article, secondly that of the choice of the specified order of the steps in a model of this kind. This challenging issue forms part of the questions which the modellers must address: they must either show that the order has little influence on the results or make a choice based on precise criteria. In the present case, what Young wished to do remains unknown, and the order followed was that in which the rules are presented in the article (sequence ‘A’).
39Beyond their differences, these two models rest on a common base: endogenous demographic growth constrained by a limited resource and a step-by-step dispersion of the population. In this basic form, and applied, as was the case here, to homogenous spaces, these two models are inscribed in the quadrant A of the ‘horseshoe’ (figure 6)24, a grid developed for comparing models. This grid shows an intersection between the level of abstraction (particular-stylised) and the simplification (KISS- KIDS) of models in order better to categorise them. The particular-stylised axis determines the position of the modeller with respect to the observed object in which he/she is interested and which he/she seeks to model: ‘a precise spatial organisation, observed in a given place at a given moment, or a typical organisation, simplified, which one observes repeatedly in time and/or space’.25 The KISS/KIDS axis, for its part, distinguishes approaches privileging the reproduction of observed macroscopic structures (KISS) from approaches focussing more on the underlying mechanisms (KIDS). The Fisher-Skellam and Young models, since their objective is to reproduce observed ‘patterns’ (forms) of colonisation on the basis of very general processes, are identically positioned in this grid.
Figure 6: The models of Young, Fisher-Skellam and ColoDyn according to the so-called ‘horseshoe’ chart.

Proposition of a new model in TransMonDyn: the ColoDyn model
40The ColoDyn model was conceptualised in the framework of the TransMonDyn project26. It takes its inspiration fundamentally from the models of Fisher-Skellam and Young; it is positioned, by contrast, more in the B quadrant of the ‘horseshoe’ (figure 6). In fact, this model introduces a more dynamic vision, one more closely linked with the interactions between populations and environment. Thus, in it the environment is a supplier of a resource which is renewed in dynamic fashion following the removal of elements by the human groups.
41The context in which this new theoretical model is proposed corresponds to an epoch prior to the Neolithic revolution, an epoch during which the groups of humans were not able to establish places for the perennial storage of resources. Fundamentally, then, this is a model of survival in a local environment whose resources may be exhausted.
42As in Young’s model, the agents of Colodyn are groups, defined by their state (living or dead). These agents are situated in cells possessing resources and evolving in a dynamic manner. It follows that, in contrast with the models of Fisher-Skellam and Young, the geographic environment is heterogenous and dynamic.
43The agents have the capacity to move from one cell to another (what we will call a migration). Finally, the agents are characterised by a quantity which we term ‘energy’ and which corresponds to the difference between what is ‘accumulated’ by exploitation of the resources present in the environment and what is ‘spent’ as a result of the migrations effected.
44More precisely, in the model a migration implies an ‘expense’ of energy which depends on the distance to cover and the cost of the migration (parameter M). On the other hand, exploitation of the resource is effected in a nearby area surrounding the group (which corresponds to the idea of territory), as is shown in figure 7. The quantity of resources exploited locally at each time step (parameter H: Human pressure on the environment) conditions the rate at which the energy level of the agents increases and the local resources diminish. It should be noted that a cell belonging to the exploitation zone of a number of groups will be exploited more intensively.
Figure 7: The exploitation zones of human groups and their possible overlap.

The resource Φ(g) is exploited by a group g in the cell. NB: The extent of the compass of exploitation practised by the human groups is fixed arbitrarily here, but it has been tested by simulation.
45From the point of view of the cells, the level of resources evolves as a function of the exploitation by the groups, as we have just seen, and also of the natural growth of the resources. Indeed, the resource of each cell grows at each time step according to a linear threshold function whose slope is defined by the parameter R, which measures the rhythm of ‘regeneration of the resource’. This way of representing space is very different from that of the two earlier models and constitutes a deliberate modelling choice, which anchors the Colodyn model in the register ‘KIDS’.
46As far as migration is concerned, figure 8 presents the decision rule adopted by each group as a function of its environment. The group first considers whether it ‘must’ leave its cell (because of locally insufficient resources), and if this is the case, either it possesses enough energy to move or does not, in which case it disappears. On the other hand, if the local resources in its environment are sufficient, it remains in place and exploits the resource available.
47The driving force of the migration is therefore less deterministic than in the Fisher-Skellam model and less random than in that of Young.
Figure 8: Diagram of the activity of the ‘displacement’ component of the model

48The demography, for its part, is managed by way of the variable ‘energy level’ attached to each group. On the one hand, when the energy level of a group reaches zero, the group disappears. On the other hand, when it reaches twice the initial value assigned to the group (parameter of the model), the group divides into two groups, each of which possesses a level of energy equivalent to half of this value. The ‘former’ group remains in the same place, and the new group created is then induced to move in order to find a zone in which to install itself. Finally, three strategies of group displacement were implemented for purposes of comparison (figure 9).
Figure 9: The three displacement strategies of the human groups.

In each figure, the arrows correspond to the possible destination cells, and the red point corresponds to the cell chosen.
49Table 4 identifies all of the parameters, principal and secondary, and the initial conditions used in the ColoDyn model, as well as the management of time and the sequencing of actions.
Table 4: List of parameters of the ColoDyn model
Principal parameters | Values |
R: Resource regeneration (resources gained in one time stop for each cell) H: Human pressure (human pressure on the environment) M: Migration cost (energy expended for one displacement) | [1; 25] [30; 150] [10; 120] |
Displacement rule | ‘Minimisation of distance’ ‘Minimisation of distance and competition between groups’ ‘Maximisation of the resource’ |
Secondary parameters |
|
Distance-exploitation: radius (in numbers of cells) of the territory of exploitation of each group | 2 |
Initial energy : Initial energy of each group | 1000 |
Resource-min: in the strategy ‘Minimisation of distance’, the nearest cell satisfying the condition ‘resource available’ > ‘Resource-min’ is chosen | 10 |
Beta: parameter used in the strategy ‘Maximisation of the resource’. The group seeks its next localisation in a radius equivalent to ‘beta x Distance-exploitation’ | 2 |
Initial conditions |
|
Distribution-of-Resources | ‘Homogenous’ ‘Gradient’ ‘Random’ ‘U-shaped’ |
P_max-resource: maximum resource of each cell |
|
50The principal parameters appear in the Netlogo interface, available online27, and may therefore be modified by the user. The other parameters of the model have fixed values, also consultable on the internet site. This choice was made on account of their central character in the ontology of the model, as well as their influence on the simulation results. Indeed, the principal parameters Resource regeneration (R), Human pressure (H), Migration cost (M) and ‘Displacement rule’ make it possible, employing their respective values, to obtain clearly differentiated colonisation configurations fairly typical of those described in the literature on the subject.
Figure 10: Colonisation configurations obtained by varying the three parameters {R, H, M}

(Initial centre at lower left; Displacement rules = ‘Minimisation of distance with competition’; Distribution of resources of ‘gradient’ type for figures a to d and of ‘Homogenous’ type for figure 10e)
51A continuous wave (figure 10a) is thus obtained with a rapid rate of resource regeneration (R), weak pressure of the human groups on the environment (H) and a low migration cost (M). Reducing the regeneration rate and increasing the human pressure make it possible to pass to a colonisation front with abandonment (figure 10b), while increasing the migration cost on the latter basis completely transforms the forms obtained in a diffuse colonisation (figure 10c). Finally, the conjunction of moderate pressure, a slow rate of regeneration and a low cost of migration makes it possible to obtain successive waves of colonisation moving outward from an epicentre (figure 10d) which does not necessarily correspond to the initial centre of settlement in the case of a homogenous distribution (isotropic) of the resource (figure 10e). The spatial organisation of the resource also plays a role in the dynamic of the model.
52These different examples effectively illustrate the objective that governed the creation of ColoDyn: to consider the forms of colonisation observed as resulting from dynamic interaction between human groups and their local environment. In this spirit, the model has, above all, an exploratory and heuristic aim. Its graphic interface (cf. note 27) allows for the easy elaboration of scenarios (defined as combinations of principal parameters and initial conditions) and for testing, by simulation, their influence on the forms of colonisation produced.
53A certain number of numerical indicators are also proposed in the Netlogo interface available online, in order to characterise the forms of colonisation obtained. Among these indicators, local density (average of the number of groups within a close radius around each group) and the global density of the groups (number of groups in relation to the area of the colonisation front) present an interesting property: combining them makes it possible to differentiate, in a relatively effective way (figure 11), the three large forms of colonisation obtained (continuous front, front with abandonment, diffuse colonisation), thereby opening the way to the realisation of more systematic experimental plans.
Figure 11: Characterisation of the principal forms of colonisation revealed by the two macroscopic indicators defined

Discussion
54With the rapid development of the capacities of computer treatment, multi-agent models are revealed as tools increasingly adapted to the exploration of social interactions in all their richness. Besides their conceptual simplicity – since the description of the behaviour of agents is ‘natural’, according to Eric Bonabeau28 (i.e. easily understandable, and thus subject to critique) – they present the advantage of applying a formalisation by way of the non-coordinated actions of agents engaged in interactions, a feature permitting exploration of very rich situations, by virtue of both the nature of the objects present in the model (which may be heterogenous in nature) and the diversity of the interactions between agents which may be tested.
55To give an example, if one wishes to modify the formalisation of the dynamic of the settlement system to take account of the heterogeneity of the environment (for instance, limited local resources, but also specific elements such as mountains, water-courses, etc.) the multi-agent procedure is simple: one has only to implement new rules for interaction between the groups and the specific elements of the environment. This could be done, both for Young’s model and the Colodyn model, by the addition of rules operating at the level of each agent and taking into account the local characteristics of the environment. By contrast, for continuous field models such as that of Fisher-Skellam, formalising these modifications is more complex, because it is the effects of this heterogenous environment on the equations governing the dynamic of the fields that must be introduced, and this then necessitates an ability to know the macroscopic effects of these new forms of interactions between population and environment. This is what is proposed, for instance, by Davison, who integrates the influence of the presence of a water-course on the average tendencies of migrations.
56The approach through multi-agent systems also allows for original forms of validation, as in the studies of Tim Kohler (cf. transition 3, chapter 6): a form of plausible behaviour of the population of Pueblo Indians (as it happens, the mutualisation of food production in the villages) is ‘deduced’ from the results of the multi-agent model – that is, that among the different modes of functioning envisaged, only the introduction of such a rule permits explanation of the demographic dynamics actually observed.
57As the discussion of Bonabeau29 illustrates, for the traditional question, ‘can you explain this phenomenon?’ is substituted, ‘can you reproduce this phenomenon?’ – a question at the centre of the generativist manifesto.
58It would be too hasty, however, to deduce from this that multi-agent systems are destined systematically to replace continuous field models, such as that of Fisher-Skellam. Among the factors to consider in choosing between the two approaches, the time needed for calibrating parameters and executing simulations for agent-based models may be found to be impractical, if the aim of the model is limited to the macroscopic signatures. The fine calibration of parameters attached to a number of interconnected processes is an especially delicate point, in view of the large number of combinations to calculate, which may exceed the available capacities of calculation. ‘Intelligent’ techniques for exploring microscopic models have currently been developed to mitigate, in part, these problems of calculation time. Multi-agent simulations seem, therefore, on the way to becoming a supplementary string to the bow of the modeller of population dynamics (and in other scientific domains), which will in the end arbitrate between – or combine– the different possible approaches, in view of the modelling objectives, the data, and the technical means available.
Notes de bas de page
1 Hazelwood Lee, Steele James, ‘Spatial Dynamics of Human Dispersals Constraints on Modelling and Archaeological Validation’, Journal of Archaeological Science, 31 (8), 2004, pp. 669–79.
2 Parisi Domenico, Antinucci Francesco, Natale Francesco, Cecconi Federico . ‘Simulating the Expansion of Farming and the Differentiation of European Languages’, Evolution of Languages, 2008, pp. 1-41.
3 Griffin Arthur F, ‘Emergence of fusion/fission cycling and self-organized criticality from a simulation model of early complex polities’, Journal of Archaeological Science, 38 (4), 2011, pp. 873-883.
4 Griffin A. F, ‘Emergence of…’, op. cit.
5 Young David, ‘A New Space-Time Computer Simulation Method for Human Migration’, American Anthropologist, 104(1), 2002, pp. 138-158.
6 Hazelwood L., Steele J., ‘Spatial Dynamics of…’, op. cit.
7 Davison Kate et al., ‘The role of waterways in the spread of the Neolithic’, Journal of Archaeological Science, 33, 5, 2006, pp. 641-652.
8 Parisi D. et al. ‘Simulating the Expansion …’, op. cit.
9 Young D., ‘A New Space-Time …’, op. cit.
10 Griffin A. F, ‘Emergence of…’, op.cit.
11 Bailey Norman T. J., The elements of stochastic processes with applications to the natural sciences, New York, Wiley, The Wiley Classics Library, 1964.
12 Gardiner Crispin W., Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Berlin, Heidelberg, New York, Tokyo, Springer-Verlag, 1997 (Second edition).
13 Steele James, ‘Human dispersals: mathematical models and the archaeological record’, Human Biology, 81(3), 2009, pp. 121-140.
14 Young D., ‘A New Space-Time …’, op. cit.
15 Joshua M. Epstein, Generative social science: Studies in agent-based computational modeling, Princeton University Press, 2006.
16 Pumain Denise, ‘Cumulativité des connaissances’, Revue européenne des sciences sociales, XLIII-131, 2005, [Online] posted online 04 novembre 2009, consulted 18 février 2016, URL : http://ress.revues.org/357
17 Ricklefs Robert E., Ecology, Newton, Massachusetts, Chiron Press, 1973, p. 504.
18 The logistic function, which corresponds to the first term of the equation 1, describes a growth which is initially slow, then accelerates and finally slows in proportion as the curve approaches a threshold corresponding to the maximum capacity of the environment under consideration.
19 At issue here is the number of groups which one expects to find, on average, starting from a given initial configuration, if one repeats the same simulation a great many times.
20 The demonstration is spelled out on the modelling platform: https://sites.google.com/site/transmondyn/modeles/modeles-pedagogiques
21 The simulation was terminated after 2,500 time steps. At this iteration, practically all the simulations ‘converge’, leading either to complete colonisation of the space or to disappearance of all the groups. All the indicators are therefore calculated after exactly 2,500 time steps.
22 The choice of a threshold of 95% was made after a number of trials to take account of the variability of the indicators around their final value. It is therefore considered that, after having passed this threshold, the system has qualitatively achieved its ‘final regime’ (the population has attained its maximum, for example).
23 The existence of computational constraints, linked particularly with the number of agents to manipulate, limits the number of possible repetitions (only three are effected here). Moreover, in order to shorten the calculation time, we have limited the number of agents in the simulation to 20,000. Certain methods make it possible to free oneself – partially – from these constraints and to explore in far greater detail the behaviour of the model within the space of its parameters (cf. chapter 15). The procedure selected here being essentially heuristic and aimed at giving the reader an intuitive sense of the processes involved and the issues entailed in exploring them, this small number of repetitions is not a disadvantage.
24 Banos Arnaud, Sanders Lena, ‘Modéliser et simuler les systèmes spatiaux en géographie’, in Varenne Franck and Silberstein Marc (eds), Modéliser et simuler. Épistémologies et pratique de la modélisation et de la simulation, Volume 1, Paris, Matériologiques, 2013, pp. 833-863.
25 Banos A. ‘Modéliser et simuler…’, op. cit.
26 This model was created in collaboration with Hélène Mathian and Lena Sanders, and tested in the framework of a summer-school ‘Spatial Structures and Dynamics, Methods and Tools for Exploring Spatial Systems’ organised by the Labex DynamiTe in 2014, URL : http://labex-dynamite.com/fr/ecole-ete-2014-labex-dynamite/
27 https://sites.google.com/site/transmondyn/modeles/modeles-pedagogiques
28 Bonabeau Eric, ‘Agent-based modeling: Methods and techniques for simulating human systems’, PNAS, 99, 2002, pp. 7280–7287.
29 Bonabeau E., ‘Agent-based …’, op.cit.
Auteurs
UMR 6266 CNRS-Université Le Havre Normandie, Le Havre, France.
Université Gustave Eiffel, Laboratoire Ville Mobilité Transport, Marne-la-Vallée, France.
Université Côte d’Azur, CEPAM, France.
Biodiversité Gènes et Communautés, UMR 1202 INRA-Université Bordeaux, France.
Le texte seul est utilisable sous licence Licence OpenEdition Books. Les autres éléments (illustrations, fichiers annexes importés) sont « Tous droits réservés », sauf mention contraire.
Quatre ans de recherche urbaine 2001-2004. Volume 2
Action concertée incitative Ville. Ministère de la Recherche
Émilie Bajolet, Marie-Flore Mattéi et Jean-Marc Rennes (dir.)
2006
Quatre ans de recherche urbaine 2001-2004. Volume I
Action concertée incitative Ville. Ministère de la Recherche
Émilie Bajolet, Marie-Flore Mattéi et Jean-Marc Rennes (dir.)
2006