A model for osmotropotactic orientation (I)

A model for osmotropotactic orientation (I)
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  J. theor. Biol. (1992) 158, 359-393 A Model for Osmotropotactie Orientation (I) V. CALENBUHR AND J.-L. DENEUBOURG Service de Chimie-Physique : Unit of Theoretical Behavioural Ecology, C.P. 231, Universit~ Libre de Bruxelles, B-1050 Bruxelles (Received on 9 October 1991, Accepted in revised form on 30 April 1992) A phenomenological model for osmotropotactic behaviour in ants and termites is presented. The model concentrates on the physical chemistry of scent trails and the dynamics between a chemical stimulus and the behavioural response at the level of the individual using a non-linear perception-response function. The model reproduces experimental data and shows that a flexible response, saturation at the level of perception and a certain amount of noise are necessary to obtain trail following. The comparison between theoretical and experimental results reveals what factors are important in general for orientation on scent trails. The same model also describes orientation towards point sources. Various examples for collective patterns, which are based on trail following or aggregation, are known in ants and termites. These patterns emerge from the integration of individual behaviour taking into account interactions between the individuals. The model presented can be used for a group of equally equipped individuals and serves thus as a starting point for the description of complex behaviour. 1. Introduction Numerous invertebrate species use chemical trails to communicate between group or society members. A moving animal lays a 'chemical line' which is followed by conspecifics. This has been shown in molluscs (Chelazzi et al., 1989) and in caterpil- lars (Fitzgerald & Peterson, 1988). Furthermore, quite similar phenomena can be found even at the cellular level (Reichenbach, 1986; Pfistner, 1990; Stevens, 1990). The classical examples of trail laying and trail following--to which this paper is devoted--are given, however, by ants and termites (Leuthold, 1975). They use chemi- cal communication extensively. At the individual level the trail is a means of orienta- tion for the individuals. The integration of individual behaviour in turn leads to patterns of organization on a larger scale. These patterns can occur in the spatial as well as in the temporal domain. Examples are the 'foraging columns' observed in ants and termites (Schneirla, 1940; Jander & Daumer, 1974; Topoff, 1980), collective decisions when faced with the problem of choosing between different food sources or alternative paths (Pasteels et al., 1987a, b; Goss et al., 1989; Beckers et aL, 1990; Beckers et al., in press), interconnection of a set of nests (Aron et al., 1990) and different recruitment strategies (Wilson, 1971; H611dobler & Wilson, 1990). Work on chemical trail communication so far has concentrated mostly on the chemical analysis, synthesis or isolation of the pheromones involved and on the 359 0022-5193/92/190359 + 35 $08.00/0 © 1992 Academic Press Limited  360 V. CALENBUHR AND J.-L. DENEUBOURG biological significance of these substances, see for example, Van Vorhis Key et al. (198 l); Evershed (1982); Morgan (1984) and for a review Eder & Rembold (1987), which contains a lot of references of more recent work. From the theoretical point of view we mention Bossert & Wilson (1963) and Wilson et al. (1969), who discussed possible geometries of active spaces and threshold values for the behavioural response. Bossert (1968) investigated the problem of signal shape and propagation from the point of view of information theory and Mankin & Mayer (1983a, b) discussed receptor activity and possible stimulus-response functions in the spirit of psychophysics. Their model was designed to allow for easier prediction of threshold values of sex pheromones with possible applications in pest control. Not much is known, however, concerning the dynamics of the trail which depends on soil type, temperature, humidity, etc and the behavioural response (Hangartner, 1967; Torgerson & Akre, 1970). This response can be regarded as an output of the chemical stimulus, which is determined by these different elements. Despite all these influences the ants are supposed to function, so to speak, more or less in the same way. A series of behavioural responses is thus needed, that is flexible enough to work under strongly changing external conditions. Another point of view that has been overlooked in the past is that many large scale phenomena can possibly be put down to individual behaviour. In this self- organization approach, large scale phenomena emerge from the interactions between a chemical stimulus and the individuals or between the individuals (communication via chemical trails are accordingly a kind of indirect interaction between individuals). From this point of view the dynamics between a chemical stimulus and the behav- ioural response are the starting point for the comprehension of numerous collective phenomena in connection with chemical trails. The aim of the present model for trail following in ants and termites is two-fold. It should provide a simple but adequate description of individual behaviour which can be used both as a tool for the understanding of more complex collective behaviour and also as a tool for facilitating the interpretation of experimental data on trail following. Hangartner (1967) showed that ants of the species Lasiusfuliginosus use at least two orientational mechanisms for trail following: osmotropotaxis, a bisensor simul- taneous mechanism and klinotaxis, a unisensor successive mechanism. Both mecha- nisms can be active at the same time. He could also show that there are situations, where only one of the mechanisms is turned on (for a review see: Schrne, 1984). Leuthold (1975) carried out the same type of experiment in the termite Schedorhino- termes lamanianus and found the same results. It is believed that these mechanisms are the basis for chemo-orientation in ants and termites in general and for trail following in particular (Leuthold, 1975). In osmotropotaxis it is assumed that the animal perceives the concentration of a pheromone with both antennae and computes the concentration difference. The animal then changes direction towards the higher concentration. Klinotaxis (which is based on a unisensor mechanism) only provides information as to whether a change of direction is necessary, but not as to what direction to choose (Kiihn, 1919; Fraenkel & Gunn, 1940; see Bell & Tobin, 1982, for a more recent discussion on  A MODEL FOR OSMOTROPOTACTIC ORIENTATION (I) 361 chemo-orientation). We are interested in modelling the osmotropotactic component of the animal's orientation, because it is, in contrast to klinotaxis, sufficient for orientation on a chemical trail without requiring supplementary information. There is still a deficit of information concerning the question of how the animals actually react to a chemical stimulus; whether they always change direction by the same angle (all-or-none-behaviour), or else show a more flexible response. It is also not known, how smoothly they can scan the trail and how these questions are related to the physical chemistry of the pheromone (evaporation and diffusion from a trail under different conditions). Further, what are the stochastic properties of the individ- uals and to what degree are these properties due to the chemical stimulus or to a behavioural component? Because of this lack of information a microscopic description, that is a description in neurobiological terms, is almost excluded right from the beginning. On the other hand, in trying to obtain collective patterns emerging from individual behaviour by going through all information processing that is actually performed at the individual level, a phenomenological description seems to be more appropriate. When designing a phenomenological model the question arises, as to "how pheno- menological" the model is supposed to be? How much of the underlying mechanisms are considered and actually built into the model certainly depends on the data avail- able. Further, it is also dependent on the use of the model. Whether it is supposed to be a more or less final description, a kind of "state of the art", or the starting point for a model on a more complex level of organization. It is our aim to devise a model on the level of the individual that can be used (with or without extensions/ modifications) on a higher level of organization in an ensemble of identical individu- als. Therefore, what we want is a simple mapping of an input condition onto an output. As soon as an appropriate mapping has been found, that is one that is compatible with experimental results, it will be taken for granted, no matter what the underlying processes are. Furthermore, the model should be able to allow for modifications to incorporate mechanistic points of view at the individual level if needed and if the necessary data are available. In mathematical models it is possible to test the influence of each parameter separately. It is also possible to test the influence of two different mechanisms (osmotropotaxis and klinotaxis) independently. This has the advantage that hypo- theses based on experimental observations could be tested quantitatively, at least in some cases. In general, however, parameters are coupled in experiment, for example, an animal with a larger span between the antennae might also have longer legs, which corresponds to a larger speed, so that different influences might interfere. This will make quite interesting issues difficult to test. The purpose of the model then is also, besides our aim to develop a simple, realistic basic model for applications in an ensemble of interacting ants, to provide some general insights into what properties are advantageous or disadvantageous for trail following and to provide a setting in which results of (carefully chosen) experiments can be interpreted or which might assist in the design of experiments that are easier to interpret. In section 2 we will present the model and section 3 is devoted to the discussion of some of its general properties in comparison with experimental results. In  362 V. CALENBUHR AND J.-L. DENEUBOURG section 4 we will discuss further the general properties of the model. A comparison of theoretical results and those obtained by experiments designed and carried out to test the models predictions can be found in the second part of this study (Calenbuhr et al., 1992). 2. The Model The model, which can be regarded as a representative of a class of similar models that share common properties, is based on the release of pheromone from an idealized trail and on the behaviour of the individual. A short description of the model and some of its basic properties have already been given elsewhere (Calenbuhr & Deneubourg, 1989, 1990). Since trail following experiments are performed frequently on circular trails, computer simulations were performed for circular and linear trails. 2.1. THE TRAIL EQUATIONS In the experiments, pheromone trails are drawn on filter paper (Van Vorhis Key & Baker, 1982; Pasteels et aL, 1986). The pheromone is adsorbed by the filter paper and released in due course. The conditions met for diffusion are thus continuous release for an infinitely long line source. For an infinitely long line source in a semi-infinite space the diffusion equation has the following solution for instantaneous emission: { r2 } C(r, t) = 2JrDt exp - ~ (1) where r is the radial distance from the trail, Q is the amount of pheromone deposited with units of (g cm-J), D is the diffusion coefficient in (cm 2 sec -~) and t is the diffusion time. The solution of the diffusion equation for a circular line source for instantaneous emission reads as (see Appendix A) C(xp, yp, t) - QRr¢ { 2(irDt)3/2 exp with R2+ 2}/°(z)4Dt (A.1 I) ,2 ., 2.2 ., 2., (,_z2)n I0(z) =1+ ~z +tzz) +[~z)-+...+~ (A.10) (l ) 2 (2 ) ~ (3 ) 2 (n ) 2 where xp and yp denote the cartesian co-ordinates of the antenna and u their radial distance with respect to the centre of the trail with radius R. The argument z of the Bessel function lo(z) is given by .Ru z =--. (A.6) 2Dt  A MODEL FOR OSMOTROPOTACTIC ORIENTATION (I) 363 If we want to take into account continuous emission integration of (1) over the emission time t' yields C(r,t)=I o' M(t')exp{ r2 }dt'. 21rD(t - t') 4D(t - t') (2) After two integrations by substitution and assuming that the pheromone emission rate M(t') is constant (Van Vorhis Key & Baker, 1982), one obtains with C(r, t)= M f~ e-" - --d. (3) 21rDt Jr/ 4,JUfiit u r ~ U = 02 -- (4) 4D(t - t')" The integral is known as an exponential-integral function and has the following solution (Abramowitz & Stegun, 1970) M -:r-~ln--- C(r, t) =~ 4Dt .= i nn J (5) where 7 = Euler's constant. We note that close to the source (a few millimetres) the concentration given by (5) changes by about 20-30% for the duration of the experiment (see Table 1), which is of the order of magnitude of a few minutes (Van Vorhis Key & Baker, 1982). This seems to be a lot at first sight. Considering, however, the fact that ants are able to follow trails that differ by as much as four orders of magnitude in strength (Pasteels et al., 1986) the trail appears to have the same strength for the animal during an experiment. This allows us to approximate the concentration profile near the trail by using the constant-time solution of the diffusion equation for the case of instanta- neous emission in still air for an infinitely long line source in a semi-infinite space, i.e. eqn (1). This approximation has the advantage, that on a short time-scale the dynamics of the evaporation-diffusion process need not be considered, which results in being able to specify the concentration profile by the simple parameter Dt in eqn (1). Although the solutions of the diffusion equation give the pheromone concentration at any point we usually assume that the ants perceive the pheromone on the ground, i.e. at zero-height. 2.2. THE ANTS' BEHAVIOUR.AL EQUATIONS The movement of the animals will be described by a pair of equations of motion embedded in an algorithm. The general notion of osmotropotaxis can be translated
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