The ants’ garden: Qualitative models of complex interactions between populations

e c o l o g i c a l m o d e l l i n g 1 9 4 ( 2 0 0 6 ) 90–101

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The ants’ garden: Qualitative models of complexinteractions between populations

Paulo Sallesa,∗, Bert Bredewegb, Nurit Bensusana

a University of Brasılia, Biological Sciences Institute, Campus Darcy Ribeiro, Brasılia DF 70.904-970, Brazilb University of Amsterdam, Faculty of Science, Human Computer Studies Laboratory, Kruislaan 419 (Matrix I), 1098 VA Amsterdam, TheNetherlands

a r t i c l e i n f o

Article history:

a b s t r a c t

Understanding interactions between populations is an important topic for research, man-

Available online 13 December 2005

Keywords:

Qualitative reasoning

Community ecology

Interactions between populations

Causality

Ants’ garden

agement and education in ecology. However, a number of problems hamper the use of

traditional modelling approaches when addressing complex systems involving three or

more populations. In this paper we describe implemented qualitative models for improving

understanding about the ants’ garden, a complex system consisting of ants, their cultivated

fungi, a virulent parasitic fungus that may attack the garden and bacteria that produce

antibiotics against the parasitic fungus. These models are based on a qualitative theory

of population dynamics and use models about symbiosis, commensalisms, amensalism

and parasitism to create the structure of the ants’ garden. Simulations show the effects of

changes in populations affecting the whole garden behaviour. Finally, we discuss the pos-

sibility of using a qualitative approach for building conceptual models of complex systems,

grounding explanations on explicit representations of the causal influences, implementing

easy to change assumptions, testing different hypotheses and complementing numerical

models.

© 2005 Elsevier B.V. All rights reserved.

1. Introduction

Because few organisms cultivate their own food, fungus-gardening by ants is considered to be a major breakthrough inevolution. It is a symbiosis in which organisms of two differ-ent species (ants from the family Formicidae and fungi mostlyfrom the family Lepiotaceae) benefit each other and creates asystem that can successfully survive in a number of differentenvironments, being the dominant herbivores in the Neotrop-ics. Recent studies (Currie et al., 1999a,b) showed that the ants’garden is far more complex than initially understood. A thirdspecies, the specialized garden parasite fungi of the genusEscovopsis is often involved and may destroy the system, byattacking the cultivated fungi. However, it almost never hap-

∗ Corresponding author. Fax: +55 61 3272 1497.E-mail addresses: [email protected] (P. Salles), [email protected] (B. Bredeweg).

pens because ants carry on their body colonies of bacteria(genus Streptomyces) that produce antibiotics effective to con-trol the growth of Escovopsis. Therefore, the system consists offour species and of complex balance of interactions in whicheventually the ants’ garden survives. The basic configurationof the ant’s garden is shown in Fig. 1.

Interactions between populations have been a hot topic inecological theory and practice. Competition, for instance, isstill seen as a driving force for shaping biological communi-ties. However, traditional modelling approaches, mostly basedin differential and difference equations, are limited in manyaspects. In general, they are difficult to build, very hard to cal-ibrate and almost impossible for non-experts to understandtheir result. Such models are ‘black boxes’, that is, they have

0304-3800/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ecolmodel.2005.10.004

e c o l o g i c a l m o d e l l i n g 1 9 4 ( 2 0 0 6 ) 90–101 91

no a clear representation of the system structure and cannotexplain how a system works (Gillman and Hails, 1997).

Applied to more complex systems, the predictive capacityof numerical population models may be jeopardized by otherproblems. It has been shown that even simple mathematicalmodels may produce complex trajectories, with stable points,cycles and chaotic behaviour (cf. May, 1974, 1976). For exam-ple, modelling interactions among populations with ordinarydifferential equations, Gilpin (1979) demonstrated that chaoscan be observed when there are at least three populations.Cyclic behaviour has been observed in many populations, butthe existence of chaotic behaviour in natural populations isstill an open question (e.g., May, 1974; May and Oster, 1976).

Qualitative reasoning models (Weld and de Kleer, 1990) mayplay an important role in this discussion, particularly for rep-resenting the structure of the system and grounding causalexplanations about the system behaviour in this structure.We here describe models about the ants’ garden developed forimproving understanding about this complex system, imple-mented on the top of models about single population dynam-ics (Salles and Bredeweg, 1997) and interactions between twopopulations (Salles et al., 2003). These models are ‘simpler’than the complex mathematics used to represent populationand community dynamics but powerful enough to supportuseful conclusions about the system behaviour.

We start presenting the ants’ garden and some questionswe expect our models to answer (Section 2), and then discussr(mt(atn

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endemic Escovopsis were only overgrown by this fungus. Actu-ally, even in the presence of ants Escovopsis may overgrow gar-dens, both in the field and in laboratory (Currie et al., 1999b).

What factors may control Escovopsis growth and keep theants’ garden working? Further studies discovered a thirdmutualist, bacteria of the genus Streptomyces. These bacte-ria have been largely used in the pharmaceutical industryto produce antibiotics for human use. Currie et al. (1999a)demonstrate that they produce antibiotics that specificallysuppress the growth of Escovopsis. Further studies indicatethat Streptomyces produce metabolites (vitamins, amino acids)that may enhance the growth of the cultivated fungi (Currieet al., 1999a).

A modelling approach may provide a representation of thesystem structure to support explanations about why the sys-tem behaves as it does. Models are intellectual tools thatgive a ‘simplified’ picture of reality. In some way, modelsreflect our knowledge about a system, and provide formalrepresentations of this understanding so that they can beassessed by members of the scientific community (Jørgensenand Bendoricchio, 2001).

Models designed to improve the understanding of the sys-tem are also called conceptual models. According to Grimm(1994), conceptual models should be understandable, manage-able and allow full exploration. These kinds of models can beused for supplying a ‘conceptual framework’ for a researchprogram and for proposing and testing hypotheses, that is, to

elated work on building qualitative models about populationsSection 3). The model building effort (Section 4), the resultant

odels (Section 5) and simulations with two models (Sec-ion 6) are presented next. Finally, we comment related workSection 7) and discuss the potential of qualitative modellingpproaches for representing complex systems with more thanwo populations and how they can complement current tech-iques used in ecological modelling (Section 8).

. How does the ants’ garden system work?

he ants’ gardens have been thought to be ‘monocultures’ freef microbial parasites, but it has been discovered that they areosts of specialized garden parasites belonging to the genusscovopsis. Currie et al. (1999b) noted that these fungi are onlynown from attine gardens and that they are found in manyttine nests. In the absence of the ants, rapidly and almostnvariably Escovopsis overgrow the garden. Instead of multi-le contaminants, after removal of tending ants, gardens with

ig. 1 – The ant’s garden: positive and negative interactions.

demonstrate the ‘consequences of what we believe to be true’.The objective of the work described here is to build mod-

els to improve learners understanding of the ants’ garden.The models described here should be able to answer ques-tions such as What are the mechanisms, at the population level,that keep the ants’ garden functioning, even in the presence of para-sitic fungi? As shown in the following sections, we explore thecausal dependencies in interactions between the four popula-tions to provide answers for this question.

3. Modelling approaches to multi-speciescommunities

Theoretical population ecology became a well-established dis-cipline during the 20’s and 30’s by Lotka, Volterra, Gauseand others. These authors have developed mathematicalapproaches both to field and to experimental population ecol-ogy that still are references for modelling population dynam-ics and interactions between populations (Kingsland, 1991).Although powerful and elegant, differential equations pose anumber of constraints on the way knowledge is represented,and require good quality data. Given that data about ecologicalsystems are often difficult to obtain and available knowledgeis incomplete, qualitatively expressed, building mathematicalecological models is thus not an easy task. Nevertheless, mostof the ecological modelling approaches are based on mathe-matical equations.

In spite of the wide acceptance of mathematical populationecology (cf. May, 1976), during the 1970s these models receivedsome criticisms: they assume a deterministic approach tosystems that are affected by stochastic factors; they are notadequate to describe the behaviour of certain species; they

92 e c o l o g i c a l m o d e l l i n g 1 9 4 ( 2 0 0 6 ) 90–101

oversimplify the problems and so on. Moreover, it has beenshown that the very simplest non-linear difference equationcan possess a rich spectrum of dynamic behaviour, from stablepoints to stable cycles and to a regime in which the behaviour,although fully deterministic, is in many aspects “chaotic”(May, 1976). Further studies on this topic are not conclu-sive and whether or not natural populations present chaoticbehaviour is still an open question (Godfray and Blythe, 1990).

Derived from population dynamics, research on commu-nity dynamics has been assuming that all the species in a com-munity were fluctuating around a stable equilibrium of den-sities of the different populations. It was accepted that morecomplex biological systems (defined in terms of the numberand nature of the individual links in the trophic web) wouldbecome more stable (defined by the tendency of returningto the equilibrium after small perturbations). Investigationsdescribed by May (1973) did not support this idea and fuelleda discussion about whether or not complexity and stability arein fact coupled in biological communities.

It is interesting to build models that predict behaviours thatwere not observed in nature. However, the fact that simple anddeterministic equations can possess dynamical trajectorieswhich look like random noise has disturbing practical impli-cations for the analysis and interpretation of biological data,as discussed by May and Oster (1976). Being black box mod-els (Gillman and Hails, 1997), the use of these approaches isnot very helpful to improve understanding about the structure

pretation, except that the rate is subtracted from the statevariable.

Qualitative proportionalities carry much less informationthan direct influences. According to Forbus (1984), P + (A, Y)represents a situation in which there exists a function thatdetermines A and is increasing monotonic (that is, strictlyincreasing) in its dependence on Y. In algebraic notation, thiscan be represented as A = f(. . ., Y, . . .). In other words, if P + (A, Y)is the only influence on the quantity A, then when Y is chang-ing (increasing or decreasing), A is also changing, in the samedirection. Negative proportionalities are defined in the sameway, except that the function is decreasing monotonic.

Combining more than one direct influences or qualitativeproportionalities is called influence resolution. If their relativestrength is known, there is no ambiguity. If not, either themodeller should include extra knowledge or the reasoningengine will try all the possible combinations, increasing thenumber of states in the simulation.

Causality is represented in both direct influences and pro-portionalities as flowing in one direction: for example, in bothexpressions I + (Y, X) and P + (Z, W), the quantities X and Wcause changes, respectively, in Y and Z, and not the contrary.Combinations of direct influences and proportionalities suf-fice for building causal chains. For example, if I + (Y, X) andP + (A, Y) and the process is active, then Y increases by anamount equals to X and given that Y is increasing, A willincrease as well. A number of examples of this causal rea-

and behaviour of ecological systems.Alternative approaches and conceptual models are

urgently required to enhance understanding about complexecological systems. Qualitative reasoning models (Weld andde Kleer, 1990) are an interesting option in a situation likethis, providing representations of relevant concepts whenmathematical approaches in general are not good options orwhen the lack of good quality numerical data hampers theirapplication. Some basic concepts of qualitative reasoningmodels are presented in the next section.

4. Qualitative reasoning and populationecology models

In this modelling effort, we adopted the ontology providedby the qualitative process theory (Forbus, 1984). This theorydefines a simply notion of physical processes that is a usefullanguage to write domain theories about dynamics. Accord-ingly, processes are assumed to be the only mechanism thatmay cause changes in the system. Two modelling primitivesare used to represent the effects of processes: direct influences(I+; I−) and qualitative proportionalities (P+; P−).

These modelling primitives have both a mathematical anda causal interpretation. The mathematics of direct influencesand proportionalities can be summarized as follows. Directinfluences define the derivative of influenced quantities. Thus,for example, the expression I + (Y, X) reads dY/dt = (. . . + X . . .).If this is the only direct influence on Y, then its derivativewill take the value X and, if X > 0, it will increase by anamount equals X. Compared to differential equation mod-els, Y plays the role of a state variable and X is the rateof the process. Negative direct influences have similar inter-

soning will be given below.Using the ontology provided by the qualitative process

theory, we implemented a qualitative theory of populationdynamics, used as the basis for representing the successionin the communities of the Brazilian Cerrado vegetation (Sallesand Bredeweg, 1997), interactions between populations (Salleset al., 2003) and now to implement models about the ants’ gar-den.

The starting point of the qualitative domain theory of pop-ulation dynamics is the equation:

Number oft = Number oft−1

+ (Born + Immigrated) − (Dead + Emigrated)

where Number of is the number of individuals in the popula-tion at the time points t and t − 1, and Born, Immigrated, Deadand Emigrated are the rates of the basic processes of natality,immigration, mortality and emigration. The qualitative repre-sentation of this equation is:

I + (Number of, Born); I + (Number of, Immigration);

I − (Number of, Dead); I − (Number of, Emigration).

In order to capture the idea that there is a feedback loopinvolving these quantities, so that when the population sizeincreases (or decreases), the rates of the basic processes alsochange (except immigration because it is seldom influencedby the population size), the following three proportionalitieswere implemented:

P + (Born, Number of ); P + (Dead, Number of );

P + (Emigration, Number of ).


For implementing the domain theory we adopt the com-positional modelling approach (Falkenhainer and Forbus, 1991).Accordingly, domain knowledge is encoded as stand alonepartial models called model fragments. A model fragmentoften contains information about objects, relations with otherobjects, quantities and their possible values, inequality state-ments, causal dependencies and descriptions of situationsand processes. Model fragments are automatically combinedby the reasoning engine in order to compose a running model.The model building activity is, in fact, the work of buildinga library of model fragments. These model fragments can bereused, so that the library, in general, supports the construc-tion of more than one model, scaling up the level of complex-ity (considering the number of processes and state variablesincluded in the model). A special type of model fragment is theinitial scenario, used to start a simulation. A scenario contains apartial description of the system, including objects, relations,dependencies and the initial values of relevant quantities.

The models described here were implemented in thedomain independent qualitative simulator GARP (Bredeweg,1992). Associated to GARP we used the graphical tool VisiGarp(Bouwer and Bredeweg, 2001) to inspect the running modelsand the simulations.

Given an initial scenario, the library, and a set of modelfragments specifying domain independent transition rules,GARP produces a simulation. This is done by selecting fromthe library the model fragments that mention the entities andfiiistosodswtsoc(

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{zero, normal, maximum}. The magnitudes of birth rate (Born)and death rate (Dead) are included in the quantity space {zero,plus}. Derivatives of all the quantities have quantity space{minus, zero, plus} meaning that the quantity is decreasing,stable or increasing, respectively.

As typically occurs in Artificial Intelligence, entities arerepresented by using a subtype (isa) hierarchy. In this isa hier-archy entities placed below in the hierarchy inherit attributesfrom the entities above. For example, the models described inthis paper include entities such as ‘population’ and, below it,‘ants’. Assuming that ‘ants’ isa ‘population’, GARP associatesto ‘ants’ the quantity Number of and its quantity space. Thisis a powerful way of encoding knowledge: a domain theoryabout population dynamics defined for the entity ‘population’can be associated to the entities involved in the ants’ garden.

Entities may be physical objects or conceptual objects. Con-cepts are also organized in subtype hierarchies. The modelsdescribed here include conceptual entities such as ‘assump-tions’. Examples are the ‘operating assumptions’ and the ‘sim-plifying assumptions’. Operating assumptions play an impor-tant role in defining a perspective for approaching the sys-tem. For instance, if the operating assumption ‘closed pop-ulation’ is assumed, as done for the models described here,then only natality and mortality processes are considered.Simplifying assumptions do not change the structure of the sys-tem but simplify the simulations, because they reduce thenumber of ambiguities and the number of states in a sim-

t to the conditions stated in the initial scenario. From thenitial scenario, one or more initial states are created. A qual-tative state is a unique description of the structure of theystem combined with a distinctive set of values associatedo the quantities of interest, described by a particular subsetf model fragments. Next, GARP checks the conditions of theystem in that state and proceeds to the influence resolutionf direct influences and proportionalities. As a result, new con-itions are defined and GARP goes through the library again,electing a new set of model fragments that are consistentith the new conditions of the system. In the description of

his new state, some model fragments active in the previoustate may continue to hold, some may be removed and newnes may be included. Details of GARP and the reasoning pro-ess can be found in Bredeweg (1992), Salles and Bredeweg2003, in press) and in Bredeweg et al. (in press).

The backbone of GARP models is the description of the enti-ies involved in the model and definitions of how they areelated. For example, a model may include the entities ‘popu-ation’ and ‘biological entity’ and their relation may be definedy the statement ‘consists of’. In this case, the structure of theodel can be described as ‘population consists of biological

ntity’. Relevant properties of the entities are represented asuantities. For example, the size of the entity ‘population’ cane represented by the quantity Number of.

Numerical values of quantities are abstracted in order toepresent just their most relevant qualitative states. In quali-ative models, these values are included in a set called quantitypace (Forbus, 1984). Any quantity value in a GARP model isharacterized by the tupple 〈magnitude, derivative〉. The formerives an idea of the ‘size’ of the quantity and the latter definests direction of change. We may say that the magnitudes ofumber of assume qualitative values from the quantity space

ulation. For example, models about parasitism may includecorrespondences (Bredeweg, 1992) between the population sizeand the strength of its effect on the other population. This way,Number of = maximum corresponds to an effect with maximumstrength (see below).

The simulation run in GARP can be inspected with the visu-alizing tool VisiGarp (Bouwer and Bredeweg, 2001). The stategraph (or behaviour graph) shows the set of states and statetransitions produced during the simulation. A sequence ofstates is called a behaviour path. It is possible to identify aset of behaviour paths in the state graph. The state where abehaviour path finishes is called an end state. Other aspects ofthe simulation can be explored with VisiGarp. Worth to men-tion here are the dependencies and the so-called value historydiagram. The former presents the causal model, a diagramwith the quantities and the active dependencies (direct influ-ences, proportionalities, correspondences and inequalities)relating them in each state. The value history diagram rep-resents the values of magnitude and derivative each quantityassumes in each state during the simulation. Each sequenceof states and state transitions constitutes a behaviour path; itis possible to follow the changes a quantity goes through dur-ing the simulation. A number of examples of these featuresare presented in the sections below.

In the next section, we present the models about interac-tions between populations used for implementing the ants’garden.

5. The ants’ garden models

Different types of interactions between populations can bedescribed according to the effects each population causes on


Table 1 – Interactions between the four populationsinvolved in the ants’ garden

Ants Cultivatedfungi

Parasiticfungi

Bacteria

Ants ** + ? +Cultivated fungi + ** + ?Parasitic fungi ? − ** ?Bacteria ? + − **

The sign refers to the effects of the population in the row on thepopulation in the column. For example: the fourth row reads asbacteria having an unknown influence (?) on the ants, a positiveeffect (+) on cultivated fungi, and a negative effect (−) on parasiticfungi. The symbol (**) indicates self-interaction.

the growth of the other population (positive, neutral or neg-ative) by combining the signs, respectively {+, 0, −} (Odum,1985). Following this author, we use [(predator, prey) = (+, −)]to represent, for example, predation. The predator increasesdue to the positive influence (+) from the prey, while the latterdecreases due to the negative influence (−) from the predator.

Table 1 presents the relationships between the four speciesinvolved in the ants’ garden system, as inferred from Currie etal. (1999a,b).

The relations between ants and cultivated fungi, andbetween cultivated fungi and parasitic fungi are clearlydefined in the table. However, some interactions are only par-tially defined. They are: [(ants, bacteria) = (?, +)]; [(bacteria, par-asitic fungi) = (?, −)] and [(bacteria, cultivated fungi) = (?, +)]. Forthe sake of simplicity, we assume that these relations are uni-directional, that is, both ants and bacteria are not affected bythe other populations. Therefore, we define the following rela-tions as the minimum set of interactions required to model theants’ garden:

(a) [(ants, cultivated fungi) = (+, +)];(b) [(parasitic fungi, cultivated fungi) = (+, −)];(c) [(ants, bacteria) = (0, +)];(d) [(bacteria, parasitic fungi) = (0, −)];(e) [(bacteria, cultivated fungi) = (0, +)].

Salles et al. (2003) describe six models of interaction types

Fig. 2 – Base model for two interacting populations models(Salles et al., 2003).

‘Non-existing population’. In the former, Number of is greaterthan zero and, in the latter, equal to zero.

The assumption that only natality and mortality pro-cesses may be active is implemented by the model fragment‘Assume open population’. As this assumption applies to allinteractions, only the model fragments ‘Natality’ and ‘Mortal-ity’ are included in the ants’ garden models.

The models of interactions are specifications of the basicarchitecture depicted in Fig. 2.

As shown in Fig. 2, the effect one population can cause onthe other is represented by the quantity Effect. The effects ofpopulation1 on population2 and vice versa are represented,respectively, by Effect1on2 and Effect2on1. These quantities arerenamed to better express each interaction. For example, Bene-fit or Supply in positive interactions and Consumption or Pollutionin negative interactions. The relation between effect and thesize of the influencing population is modelled by the propor-tionality P + (Effect, Number of).

This option for an intermediate quantity (Effect) as an alter-native for creating an influence directly from Number of to theinfluenced population increases the representational capacityof the models, as it allows for modelling a situation in whichEffect either corresponds or not to the size of the populationthat produces the effect. For example, the ants interact bothwith cultivated fungi and with bacteria. Each interaction ismodelled by a specific quantity Effect and their values can bedifferent.

between populations, defined by all the possible combinationsof the signs {+, 0, −}. According to Odum (1985), these com-binations can be used to represent nine types of interactions,given that some pairs apply to more than one type of interac-tion. For example, (+, −) can be used to represent predation,herbivory and parasitism. In order to implement the ants’ gar-den models, we took from Salles et al. (2003) the followingmodels: symbiosis (a), parasitism (b), commensalisms (c, e)and amensalism (d). Recently, these models have been slightlyadapted following discussions with experts. For details seeBredeweg and Salles (in press).

The library for the ants’ garden models consists of 33 modelfragments. Three model fragments are used for a general defi-nition of population. The model fragment ‘Population’ definesthe relation between a set of things (population) and a biologi-cal entity (species): ‘population consists of species’. The quan-tity Number of is introduced and associated to population. Thismodel fragment has two subtypes: ‘Existing population’ and

Given that only processes may change the population size,in the ants’ garden models a population influences the otherpopulation via the basic processes of natality and mortality.A positive interaction, for example, can be implemented asincreasing the birth rate, P + (Born, Effect), decreasing the deathrate, P − (Dead, Effect), or both at the same time. Similarly, neg-ative interactions can decrease the birth rate; increase thedeath rate or both.

Each interaction is typically modelled by using five modelfragments: a model fragment that defines general aspects ofthe interaction, and four subtype model fragments to imple-ment detailed knowledge. Among them, one model frag-ment specifies details of the interaction; two model fragmentsdefine the absence of each population; and one model frag-ment defines some simplifying assumptions for the interac-tion. Extra model fragments may be used to implement extraassumptions for the interactions. Details of each interactionincluded in the ants’ garden models are described below.

To model parasitism, a set of five model fragments are used.The most general is ‘Parasitism’, which introduces the two


populations, being population1 the parasite and population2the host, and the quantities Supply (the effect of the host on theparasite) and Consumption (the effects of parasite on the host).A directed correspondence (Bredeweg, 1992) establishes that themagnitude of Supply takes the same values as the magnitudeof Number of2. The relation between the values of Consumptionand Number of1 is represented in the same way.

The model fragment ‘Parasitism interaction’ implementsthe effects of the interaction: it establishes that Supplyincreases Born1 and decreases Dead1, while Consumptionincreases Dead2. This model fragment also assumes that boththe magnitude and the derivative of Supply have to be greaterthan or equal to, respectively, the magnitude and the deriva-tive of Consumption. This way, consumption is less than or, atmost, equal to supply and there is no food shortage for theparasite.

The conditions for parasite and host populations to goextinct are defined, respectively, by the model fragments ‘Par-asitism without parasite’ and ‘Parasitism without host’. Theformer sets no constraints or consequences for the host pop-ulation due to the disappearance of the parasite population.However, we assume that the parasite cannot exist withoutthe host: if Number of2 goes to zero, then Number of1 will dothe same. Finally, simplifying assumptions for this interactionare encoded in the model fragment ‘Parasitism assumptions’:directed correspondences establish that the derivatives of Bornand Dead take the value of the derivative of Number of. Thisa(i

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standing that bacteria disappear from the ants’ garden ifthey do not receive the benefit from the ants is captured inthe model fragment ‘Commensalisms without producer’. Itestablishes that if Number of1 is zero, then Number of2 alsoshould go to zero. The model fragment ‘Commensalisms with-out affected’ sets no constraints or consequences for popula-tion1 due to the disappearance of population2. Finally, as inparasitism and symbiosis, simplifying assumptions involvingNumber of, Born and Dead are implemented by the model frag-ment ‘Commensalisms assumptions’.

Three model fragments implement constraints on thestrength of the influence of population1 on population2. ‘Com-mensalisms effects magnitude1’ establishes that only whenBenefit has value maximum, Number of2 can go to maximum.‘Commensalisms effects magnitude2’ establishes that if themagnitude of Benefit is normal, then the magnitude of Num-ber of2 has to be below maximum. Finally, ‘Commensalismshigh impact’ implements the ‘commensalisms high impact’assumption, which establishes that the value of Number of2always corresponds to the values of Benefit. This assumption ismore restrictive because the value of the former is completelydetermined by the latter. As a consequence, the number ofstates in a simulation is reduced.

Finally, amensalism is implemented with eight modelfragments in the same way as commensalisms. The modelfragment ‘Amensalism’ introduces the two populations andthe quantity Pollution to represent the effect of popula-

ssumption reduces the number of possible combinationsfor example, Born increasing, Dead decreasing and Number ofncreasing), and therefore reduces the size of the simulations.

Symbiosis is also modelled by five model fragments. ‘Sym-iosis’ is the most general one, and introduces the two popu-

ations (1 and 2) and the quantities Benefit1 and Benefit2. As inarasitism, direct correspondences establish that magnitudesf Benefit take the same magnitudes of Number of in both pop-lations.

The model fragment ‘Symbiosis interaction’ defines howhe effects are implemented: in both populations, Benefitncreases Born and reduces Dead. This model fragment alsomplements correspondences establishing that the maximumalue of Benefit1 corresponds to the maximum value of Bene-t2, and that the derivatives of the two benefits correspond.hese assumptions are useful to reduce ambiguities during

he simulation. The conditions for the absence of each pop-lation are represented in two model fragments, ‘Symbiosisithout symbiont1’ and ‘Symbiosis without symbiont2’. Given

hat we assume the symbiosis as being obligatory for bothnts and cultivated fungi (they cannot survive alone, accordingo Currie et al. (1999a)), when Number of1 is zero, Number of2ill be also zero, and vice versa. Finally, the model fragment

Symbiosis assumptions’ sets for this interaction the sameimplifying assumptions defined for parasitism.

Commensalisms are modelled by a set of eight model frag-ents. ‘Commensalisms’, the most general one, introduces

he two populations and the quantity Benefit, to represent thenfluence of population1 on population2. This quantity corre-ponds to Number of1 and their maximum values are assumedo be equal.

The model fragment ‘Commensalisms interaction’ defineshat Benefit increases Born2 and reduces Dead2. Our under-

tion1 on population2. It is also assumed that the maxi-mum value of Pollution corresponds to the maximum valueof Number of1. The model fragment ‘Amensalism interaction’defines that Pollution decreases Born2 and increases Dead2.The model fragments ‘Amensalism without producer’ and‘Amensalism without affected’ set no constraints or con-sequences for any population due to the disappearance ofthe other population. The same simplifying assumptionsalready described for the other three interactions are imple-mented for amensalism by the model fragment ‘Amensalismassumptions’.

As in commensalisms, three model fragments implementdetails about the strength of the amensalism interaction:‘Amensalism effects magnitude1’ establishes that if Pollutionhas value maximum, then Number of2 has value zero; ‘Amen-salism effects magnitude2’ determines that if Pollution hasvalue normal, then Number of2 has to be smaller than maxi-mum; and ‘Amensalism effects magnitude3’ establishes thatif Pollution has value zero, then Number of2 can assume anyvalue.

It is important to mention that these magnitude assump-tions implemented in commensalisms and amensalism donot produce unknown or unexpected behaviours, but onlycontrol the level of details represented in the simulations. Forexample, when the ‘commensalisms high impact’ assumptionis not introduced in the initial scenario, the simulation hasmore states to express fundamentally the same behaviours,and most of these states show different combinations ofderivative values of the quantities. These results are correct,but superfluous, as they do not give additional insights to theunderstanding of the system.

In the next section we describe three simulations obtainedwith two models of the ants’ garden.


Fig. 3 – Causal dependencies involving four populations (four interactions) in state 3 of simulation 1.

6. Simulations with the ants’ gardenmodels

As mentioned above, building qualitative models is actuallybuilding libraries of model fragments and initial scenarios.From such libraries, often the reasoning engine can automati-cally build a number of simulation models exploring differentaspects of the encoded domain knowledge.

Following the suggestions of Salles and Bredeweg (1997),the ants’ garden library was built incrementally. Starting withsimulation models about the symbiosis between ants and cul-tivated fungi, we included parasitism between cultivated andparasitic fungi and commensalisms to express the relationbetween ants and bacteria. Finally, we introduced commen-salisms between bacteria and cultivated fungi. Adequate ini-tial scenarios can retrieve simulations of each of these steps.

Simulations involving two models of the ants’ garden willbe shown in this section. The first model consists of four pop-ulations linked by four interactions. Only the relation betweenbacteria and cultivated fungi is not considered. Two simu-lations are run with this model, showing the ants’ gardenwithout the parasitic fungi and the four populations withinitial values set to normal, with unknown derivatives. Thesecond model implements the fifth interaction, representingthe interaction between bacteria and cultivated fungi. A sim-ulation with this model will show the outcomes of the four

parasitic fungi is 〈zero, ?〉. Three initial states are derived fromthe initial scenario. While the parasitic fungi are stable inall these initial states, the other three populations may bedecreasing (state 1), stable (state 2) and increasing (state 3).The causal model, as it appears in state 3, is presented in Fig. 3.

The full simulation produces eight states. The state graphof this simulation and the values (magnitude and derivative)of the state variables Number of in these states are shown inFig. 4.

The state graph shows six end states, that is, states inwhich the behaviour path stops: [2, 4, 5, 6, 7, 8]. The behavioursobserved in these end states are, respectively, the following:the three populations are stable in the value normal (2); orin the value maximum (4); cultivated fungi is stable in normalwhile the other two populations are stable in maximum (5); antsand bacteria are stable in normal and cultivated fungi is stablein maximum (6); bacteria is extinct (7); and, finally, the wholesystem is extinct (8). Behaviours expressed by the system andthe six possible behaviour paths are shown in Table 2.

The results of the simulation with this model were as wewould expect for an ants’ garden free of parasitic fungi: differ-ent states of coexistence of the three populations; and, as weassume an obligatory symbiosis between ants and cultivatedfungi, either only the bacteria is extinct or the whole gardenis extinct. Worth to mention is that natality and mortality ofparasitic fungi are not active because this population has mag-nitude zero, a ‘Non-existing population’. In this case, no direct

populations with initial values set to normal, with unknownderivatives. In both models, the ‘commensalisms high impact’assumption is included. Details of each simulation are pre-sented below.

6.1. Simulation 1

The initial scenario describes four populations and four inter-actions between them, with the initial values of Number ofants, cultivated fungi and bacteria set in 〈normal, ?〉, that is,magnitude normal and derivative unknown, while Number of

Table 2 – Behaviours and behaviour paths expressed bythe ants’ garden in simulation 1

Behaviour of the system Behaviour paths

Coexistence of the threepopulations

[2], [3 → 4], [3 → 5] and [3 → 6]

Extinction of bacteria [1 → 8]Extinction of the whole sys-

tem[1 → 7]


Fig. 4 – State graph and value history of the quantities Number of of the four populations (four interactions) in simulation 1.

Fig. 5 – State graph and value history of the quantities Number of of the four populations (four interactions) in simulation 2.

influences are affecting Number of (see Fig. 3). The same con-figuration appears when a population goes extinct during thesimulation (for example, bacteria in state 8). These examplesillustrate one of the most interesting features of qualitativereasoning models: the possibility of including into or remov-ing from the model structural elements during a simulation.

6.2. Simulation 2

The second simulation with this model starts with an ini-tial scenario which assigns to Number of the value 〈normal, ?〉,that is magnitude normal and derivative unknown, for all thepopulations. From this initial scenario, five initial states areproduced. The magnitudes of the state variables are the same(normal), but their derivatives are different: all the populationsare decreasing (state 1); parasitic fungi is decreasing, while theothers are increasing (state 2); all the populations are stable(state 3); parasitic fungi is stable and the others are increasing(state 4); and all the populations are increasing (state 5).

The full simulation consists of 15 states. The state graphand the value history showing magnitudes and derivatives ofthe state variables Number of ants, cultivated fungi, bacteriaand parasitic fungi are depicted in Fig. 5.

The simulation produces nine end states: [3, 6, 7, 8, 9, 10,12, 13, 14]. The behaviours expressed in these end states are,respectively, the following: the four populations are stablewith value normal (3); ants, bacteria and parasitic fungi aresapm

fungi is extinct (8); ants and bacteria are stable in normal, culti-vated fungi is stable in maximum, and parasitic fungi is extinct(9); ants, bacteria and cultivated fungi are stable in normal, andparasitic fungi is extinct (10); ants and cultivated fungi arestable in normal, parasitic fungi and bacteria are extinct (13);bacteria is extinct (14); and, finally, all the garden is extinct(12).

The 17 possible behaviour paths are shown in Table 3.This simulation shows a rich variety of behaviours

expressed by the garden, according to the implementedassumptions: coexistence of the four populations, with dif-ferent population sizes; extinction of parasitic fungi and sta-bilization of the other three populations in different values;extinction of bacteria and of parasitic fungi; extinction of thebacteria; and the extinction of the garden. As in the previ-



Coexistence of the fourpopulations

[3], [4 → 6], [5 → 6] and[2 → 6]

Extinction of parasitic fungi andcoexistence of the otherpopulations

[1 → 10], [2 → 7],[2 → 11 → 7], [2 → 8],[2 → 11 → 8], [2 → 9],[2 → 11 → 9] and [2 → 10]

table in normal, and cultivated fungi is stable in maximum (6);nts, cultivated fungi and bacteria are stable in maximum, andarasitic fungi is extinct (7); ants and bacteria are stable inaximum, cultivated fungi is stable in normal, and parasitic

Extinction of parasitic fungi andbacteria

[1 → 13] and [1 → 15 → 13]

Extinction of bacteria [1 → 14]Extinction of the whole system [1 → 12] and [1 → 15 → 12]


Fig. 6 – Causal dependencies involving four populations (five interactions) in state 5 of simulation 3.

ous simulation, ants and cultivated fungi are always present,except when the whole garden disappears.

6.3. Simulation 3

For this simulation we used a model that includes commen-salisms to represent the fifth interaction, namely betweenbacteria and cultivated fungi. The initial values of all the statevariables Number of are set in 〈normal, ?〉. GARP derives fiveinitial states, with the same values observed in the five ini-tial states of simulation 2. The causal dependencies of thismodel, as they appear in state 5, are depicted in Fig. 6. Notethe effects of bacteria on the other populations, representedby two quantities: Pollution (which affects parasitic fungi) andBenefit (which affects cultivated fungi).

The whole simulation consists of 10 states. The state graphand the values of magnitudes and derivatives of Number of ofthe four populations are depicted in Fig. 7.

The state graph shows that the six end states of the simula-tion are [3, 4, 5, 6, 7, 9]. The behaviours expressed in these endstates are, respectively, the following: the four populations arestable with value normal (3); ants, bacteria and cultivated fungiare in value 〈normal, +〉 and parasitic fungi is stable in normal(4); the four populations are in value 〈normal, +〉 (5); ants, bac-teria and cultivated fungi are stable in maximum, and parasiticfungi is extinct (6); ants, bacteria and cultivated fungi are sta-ble in normal, and parasitic fungi is extinct (7); all the garden

value zero as well. This prevents the bacteria to be extinctalone.

7. Related work

Given the difficulties of using mathematical models whenknowledge is incomplete or numerical data have no good qual-ity or simply do not exist, a number of authors tried alternativeapproaches for modelling multi-species community dynam-ics. Some of these works are commented in this section.

Noble and Slatyer (1980) and Moore and Noble (1990, 1993)proposed an approach for building qualitative models aboutthe dynamics of communities subject to recurrent disturbance(such as fire). This approach is based on a small number ofattributes of the plant’s life history (vital attributes), which canbe used to characterise the potentially dominant species in aparticular community, under different types and frequenciesof disturbance. Simulations typically produce a replacementsequence which depicts the major shifts in composition anddominance of species which occur following a disturbance.These models differ from ours in many aspects. They describethe replacement sequence, but cannot provide causal expla-nations for changes in the system. Also, these models rely on

is extinct (9).The behaviours expressed by the garden and the nine pos-

sible behaviour paths are shown in Table 4.The inclusion of the interaction between bacteria and cul-

tivated fungi increased the possibilities of coexistence, asshown in states 4 and 5. Also, the extinction of the bacte-ria population was not observed, except along with the otherthree populations. This can be explained by the assumptionimplemented by the model fragment ‘Commensalisms with-out producer’ (see Section 5): when bacteria go to value zero,then cultivated fungi also should assume the value zero. How-ever, for the latter to be true, the ants should assume the



Stable coexistence of the fourpopulations

[3]

Three populations increasingand parasitic fungi stable

[4]

The four populations increasing [5]Extinction of parasitic fungi

and coexistence of the otherpopulations

[2 → 6], [2 → 8 → 6], [1 → 7]and [2 → 7]

Extinction of the whole system [1 → 9] and [1 → 10 → 9]


Fig. 7 – State graph and value history of the quantities Number of of the four populations (five interactions) in simulation 3.

mathematical simulations for calculating changes in popula-tion sizes, while our representation is based only in qualitativeknowledge, including magnitudes and derivatives of the quan-tity values.

State-transition modelling is often used to describe com-munity dynamics. For example, McIntosh (2003) presents arule-based modelling language to describe succession in com-munities stressed by fire and grazing using this paradigm.Pivello and Coutinho (1996) also describe a state-transitionmodel about changes in the Brazilian Cerrado vegetationunder influences of fire, wood cutting and grazing. Althoughsuccessful in certain way, these implementations of thestate-transition approach do not explain why things happen,because there is no representation of the underlying mecha-nisms that cause changes in the system. Also, these modelslack flexibility, as they aim at describing specific systems andanswering particular questions and cannot be (easily) changedor combined to others to address more (or less) complex prob-lems.

One of the most important studies on complexity and sta-bility of biological communities was a qualitative analysis ofthe results produced by differential equation models aboutinteractions between populations (May, 1973). May’s intentionwas to investigate what can be said if only the topologicalstructure of the trophic web is known, i.e., knowing only thesigns (−, 0, +) of the interaction between the species and rea-soning with changes over time, that is, with the derivatives oft

csstfbm

teria required for stability analysis and, in these cases, thesigns of the interactions alone are not enough, and the inter-action magnitudes should be taken into account. This workwas inspiring for us, as May reasons with derivatives to drawuseful conclusions. Our representation of derivatives does thesame job, and qualitative values of quantity magnitudes over-come some of the limitations he mentions. Although we arenot able to provide precise answers, qualitative models do notdepend on good quality data and fine tuned parameters andprovide causal explanations for the system behaviour.

Guerrin and Dumas (2001a,b) describe a model represent-ing empirical knowledge of freshwater ecologists on the func-tioning of salmon spawning areas. Their model is based on(single) population dynamics and explore mortality at earlystages, aiming at predicting and explaining the survival rateof fish under various scenarios. Their approach representsprocesses that occur at different time-scales (fast and slow)and introduces a real time dating and duration in a purelyqualitative model. This model was built according to theso-called constraint-based approach to qualitative reasoning(Kuipers, 1994), which provides qualitative representations ofdifferential equations. Accordingly, knowledge is encoded asconstraints and there is no explicit representation of causal-ity. This approach also does not explore the possibility ofcombining partial model fragments, so it is not possible toreuse models and, doing that, to address more complex prob-lems. Finally, our approach allows for easily implementing

he quantities.May showed that the ‘common-sense wisdom’ that more

omplexity means increased stability may not be true. In hisimulations, a less complex community met the conditions fortability, while the more complex was not stable. He points outhat this can be a useful approach for modelling quite complexood webs and to capture the general tendencies of the system,ypassing long and complicated steps required by numericalodels. However, larger populations violate some of the cri-

and changing assumptions that may reduce the complexityof the simulations, a serious problem for the constraint-basedapproach.

8. Discussion and concluding remarks

This paper describes an effort to build qualitative modelsabout the complex interactions between four populations in


the ants’ garden. Current understanding of the ants’ gardenis that ants (Formicidae), cultivated fungi (Lepiotales) and bacte-ria (Streptomyces) established a highly interdependent interac-tion long time ago, in which the bacteria is transmitted fromparent to offspring colonies of ants and the fungus is car-ried by the queens in their mouth when new ant coloniesare founded. Ants provide the fungi with a safe environ-ment and raw material (leaves and organic material). Lepio-tales fungi digest the organic matter and provides (becomes)food for the ants that eat the garden. Another group of fungi(Escovopsis) was found in colonies and may produce high mor-bidity and mortality on cultivated fungus. Normally, they donot overgrow the garden because Streptomyces produce antibi-otics that suppress Escovopsis growth. It is also possible thatthese bacteria produce growth factors used by the cultivatedfungi.

Mathematical models about population dynamics, usuallybuilt as differential and difference equations, may not be use-ful to model this problem. Besides the well known difficultiesto obtain numerical data of good quality for implementing,calibrating and evaluating these models, it has been shownthat such models and their results are difficult to under-stand and can hardly explain why the systems exhibit certainbehaviours.

Qualitative reasoning techniques may be useful to addresscomplex systems like the ants’ garden, as they provide meansto build comprehensive conceptual models using incomplete

Thus, the situation described in the initial scenario in simu-lation 1 is quite common, and the results obtained with thesimulation are plausible.

Coexistence of the four populations was observed in morethan 70% of the bioessays carried out by (Currie et al., 1999a),and complete elimination of the parasitic fungi was observedin 25% of their bioessays. These results were obtained in simu-lations 2 and 3, with stabilization of coexisting populations ofdifferent sizes and extinction of the parasitic fungi population.

However, some aspects of the ants’ garden were notcaptured by our current version of the models. Bacte-ria were present in all the colonies of attine ants stud-ied by Currie et al. (1999a). Therefore, the results obtainedin states 13 and 14 of simulation 2, in which the pop-ulation of bacteria was eliminated, were not reported bythese authors. Given that no specific knowledge about thisrestriction was encoded in the models, our results arecorrect.

Also, according to Currie et al. (1999b), in the absenceof ants and sometimes even in their presence, the parasiticfungi may outgrow the cultivated fungi and destroy the gar-den. This kind of behaviour does not appear in the simula-tions due to the implemented assumption that both mag-nitudes and derivatives of Supply (the effect of the host onthe parasite population) are greater than or equal to, respec-tively, the magnitudes and derivatives of the quantity Con-sumption (the effect of the parasite on the host population).

knowledge. Qualitative models are not sensitive to arbitraryparameters and allow for valid predictions in situations thatmathematical models cannot be used. Some features that maybe interesting include the possibility of developing conceptualmodels with a rich vocabulary to describe objects, quantities,relations, situations, mechanisms of change and conditionsfor processes to start and finish. Moreover, explicitly repre-sented causality is useful to improve understanding of thestructure of ecological systems and to support explanationsabout the systems behaviour.

Our models of the ants’ garden reflect the current under-standing of how populations interact (“the consequences ofwhat we believe is true”, in Grimm’s (1994) words). When infor-mation was not available, we assumed the interactions to beas simple as possible. It happens with the interpretation ofthe relationship between ants and bacteria (commensalisms),bacteria and parasitic fungi (amensalism) and between bac-teria and cultivated fungi (commensalisms). Explicitly repre-sented modelling assumptions may either provide alternativeperspectives for approaching the system or reduce ambigu-ity and the number of states in the behaviour graph. Theseassumptions may easily be changed, and alternative hypothe-ses can be tested. Using an incremental approach to scale upthe complexity of the systems being represented, we devel-oped alternative initial scenarios involving the four popula-tions to fully explore different aspects of the ants’ gardensystem.

The three simulations presented in this paper are rela-tively small and, therefore, easy to be inspected. Most of theresults obtained in these simulations were described by Currieand her co-authors. Contamination of ants’ garden by non-mutualistic fungi was found in 39.7% of the samples studiedby Currie et al. (1999b), being Escovopsis the most abundant.

This assumption, as mentioned in Section 5, keeps the pop-ulation size of parasites at most equal to the size of hosts.Relaxing this assumption will produce states that show theextinction of the tripartite association of ants, cultivated fungiand bacteria. Finally, Currie and collaborators do not mentionany situation in which the whole system disappears. How-ever, it is plausible that this situation happens in the realworld.

Ongoing work includes implementations of alternativehypotheses to answer some open questions. For example, theinteractions between ants and bacteria, and between bac-teria and cultivated fungi may be symbiosis. The idea is totry alternative models in order to make predictions that canbe tested; to provide explanations for behaviours observedin the field; and to include new knowledge about the gar-den. If possible, these qualitative models could be associatedwith mathematical models, providing a conceptual frame-work for building the equations. Numerical data could thenbe used to assess the predictions supported by qualitativemodels. Exercises of these types may help us to better under-stand the functioning of complex systems such as the ants’garden.

Acknowledgements

We would like to thank Waldenor Barbosa da Cruz for draw-ing our attention to this beautiful ecological system and thestudents enrolled in the course ‘Models in Ecology’ (2/2002) atthe University of Brasilia for their enthusiasm and fruitful dis-cussions. Thanks also to the three anonymous reviewers fortheir comments and for pointing out ways for improving thiswork.


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The ants’ garden: Qualitative models of complex interactions between populations