Chiwata, Yuhei

Nakagawa, Hiroyuki

Nishimori, Hiraku

Tao, Tomomi

Yamasaki, Masato

Kyoto University Research Information Repository

Title化学走性によるトレイルの形成とその機能 (力学系理論と複雑系の数理)Author(s)Tao, Tomomi; Nakagawa, Hiroyuki; Yamasaki, Masato;Chiwata, Yuhei; Nishimori, HirakuCitation数理解析研究所講究録 (2002), 1244: 137-142Issue Date2002-01URL http://hdl.handle.net/2433/41683RightType Departmental Bulletin PaperTextversionpublisherKyoto University( ) ( )( ) ( ) ( ) *T.Tao(Shizuoka Univ.),H.Nakagawa(Osaka Pref.Univ.),M.Yamasaki(Ibaraki Univ.),Y.Chiwata(Kyoto Univ.),H.Nishimori(Osaka Pref.Univ.)AbstractInspired by the foraging behavior of ants, asimple model for the adaptivetrail formation is proposed[l]. In the simulation, several types of trails are seento be made up depending on the spatial and temporal schedule of feeding torealize the the efficient foraging according to each regime.1IntroductionSo far, various studies have been made on the collective behavior of social insects[2, 3, 4, 5, 6, 7, 8, 9, 10]. Specially, the trail formation of ants [2, 3, 4, 5, 6, 7, 8, 9, 10]have called awide interest as one of the remarkably synergetic behavior fulfilled bypresumably innocent individuals $[11, 12]$ . Still, many points are left unconfirmedon the trail formation process, like the detailed ingredient of almost all types ofchemicals(pheromone) $[2, 3]$ secreted by individual ants and their roles in each $\mathrm{s}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{t}\succ$biological regime [2, 3, 4, 5, 6, 7].In spite of such inadequate understanding of the real counterpart, various mod-els on ants [7, 8, 9] have been proposed to imitate their behaviors using heuristicapproaches. Thence, possible scenarios for the formation process of trails and thebranching process of them have been introduced [7, 8, 9]. However, such approacheshave serious disadvantage that each of them is not exclusively reliable as reflectingthe reality. Yet, if we focus our attention on investigating the general feature of thetrail formation beyond that of individual species, such approaches partially based onunconfirmed assumptions have acertain meaning. Prom such apoint of view, settingasimple set of rules several parts of which are following previous studies [7, 8, 9],we investigate the flexible foraging by ants. Particularly, we focus on the relationbetween the geometry of trails and foraging efficiency under temporally and spatiallyunsteady feeding.2ModelThe Model consists of i)a triangular $2\mathrm{D}$ lattice with periodic boundary conditionon which all the dynamics proceeds, $\mathrm{i}\mathrm{i}$)movable agents (we call them ’ants’)each of1 $\mathrm{e}$-mail:nishimor@ms osakafu-u.ac.jp1244 2002 137-142137Fig. 1Figure 1: Each ant is, at each time step, facingone of the six nearest sites. This facing direction$\bullet \mathrm{p}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{i}\mathrm{n}\mathrm{I}\mathrm{k}\mathrm{o}\mathrm{e}\mathrm{x}\mathrm{t}\Re\eta$vious time step. The possible moving directionscorresponds to the moving direction in the pre-$\ovalbox{\tt\small REJECT}^{\wedge}.\mu \mathrm{v}\mathrm{i}\alpha \mathrm{l}\mathrm{s}\mathrm{s}\varphi \mathrm{s}\mathrm{i}|\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{h}\mathrm{e}$in each time step are: the forward $(=\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{g})$ di-rection and its neighboring (right and left) di-rections. To choose one of them, astochas-tic rule with the weight, $P_{\alpha}=exp(-\Delta^{\alpha}/T)/Z$is adopted where $Z$ is the normalization fac-tor. Here $\alpha$ indicates forward, right, or leftdirection relative to the facing direction, and$\Delta^{\alpha}\equiv\rho_{\beta}^{\alpha}-\rho\rho(\mathrm{x}, t)$ is the gradients of pheromonedensity to the candidate directions where $\beta$ isthe index to indicate the recruit phermone(formode-II ants) or the foot pheromone(for modeIII ants). Note the above weight is not applica-ble for ants in mode I which is making randomwalk independnt of pheromone field.which is, at each time step, located at one site and facing in the direction of one of sixnearest sites(Figl). $\mathrm{i}\mathrm{i}\mathrm{i}$)the nest situated at one site in the lattice(Fig2), and $\mathrm{i}\mathrm{v}$)foodslocated at two corners of an equilateral triangular the 3rd corner of which is the nestsite(Fig2). The amount of foods varies with time, i.e., decreases as taken away by antsor increases by the additional supply at every constant time. On each site $\mathrm{x}$ at timetime step $n$ , two kinds of field variables, $\rho_{\mathrm{r}ec}(\mathrm{x}, n):\mathrm{t}\mathrm{h}\mathrm{e}$ amount of recruit pheromoneand pfoot(x, $n$) $:\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$ of foot pheromone $[2, 3]$ , are allocated. Both of these pheromonesare chemicals emitted by individual ants according to their temporal modes whichconsist of: i)the inactivated mode(mode $\mathrm{I}$), $\mathrm{i}\mathrm{i})\mathrm{t}\mathrm{h}\mathrm{e}$ activated mode without food (mode-$\mathrm{I}\mathrm{I})$ and $\mathrm{i}\mathrm{i}\mathrm{i}$) the activated mode with food (mode-Ill).More concrete descriptions are the foUowings:[mode-I]After leaving the nest until an ant reaches on asite with $\rho_{tec}(\mathrm{x}, n)>\rho_{rec-c}$,he keep within the mode-I and walk randomly while emitting acertain amount offoot pheromone at corresponding sites.[mode-II]An ant is excited into the mode-II on reaching the site with $\rho_{tec}(\mathrm{x}, n)>$$prec-c$ , thereafter he walks perceiving the local gradient of recruit pheromone den-sity(in fig.1) while keeping to emit the foot pheromone at the corresponding site. Inthis mode, if he fails to keep himself along the $\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s}\rho_{rec}(\mathrm{x}, n)>\rho_{rec-c}$ his mode turnsback to the previous mode-I.[mode-Ill]On arriving at the food, regardless of his previous mode, he changes intothe mode III in which mode, perceiving the local gradient of foot pheromone he walksback to the nest. In this mode the recruit pheromone are kept to be emitted at eachsite of the walk.In addition, pheromone evaporates in the air with time, also diffuses to the nearestsites. $\mathrm{N}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{l}\mathrm{y},\mathrm{i}\mathrm{f}$without additional emission by ants, the pheromone density at site$\mathrm{x}$ decays as$\rho_{\beta}(\mathrm{x}, n+1)-\rho_{\beta}(\mathrm{x}, n)=-A\rho_{\beta}(\mathrm{x}, n)+D(<<\rho_{\beta}(\mathrm{x}, n)>>-\rho_{\beta}(\mathrm{x}, n))$ (1)where $<<>>\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}$ the average over the nearest sites of x. Aand D are, respectively138Fig.2Figure 2: The method of feeding.Feeding sites are located at two cor-ners of an equilateral triangular the3rd corner of which is the nest site,The amount $M$ of foods are sup-plied, by turn, from one of thesesites to the other at every feedinginterval $T$ . After each feeding event,they decreases as taken away byants.the evaporation rate and diffusion constant, and $\beta$ is the index to indicate the recruitpheromone(for mode-II ants) or the foot pheromone(for mode-Ill ants). Furtherdetails of the rules are explained in fig. 1.3SimulationsUnder these rules, we investigate the, seemingly, synergetic foraging behavior of antsunder temporally and spatially varying feeding. All the simulations are performed ona150 $\cross 150$ triangular lattice with periodic boundary condition. At the initial timestep 500 ants are located at the nest, and no pheromone are distributed in the field,thence, all ants are simultaneously released from the nest. Note the unit time in oursimulation corresponds to one Monte Cairo step within which totally 500 randomlychosen ants will sequentially take one step. Acertain amount $M$ of foods are supplied,by turn, from one site to the other site of food at regular intervals T. (hereafter wecall $T$ ’the interval of food supply’). The amount of supplied foods at these sites offoods decrease as ants bring them away before the next supply.4ResultsUnder aproper combination of fixed variables on ants (i.e.,total number of ants,evaporation rates and diffusion constants of pheromones, the location of the nest,and, energy supply at the sites of foods and nest, the critical energy to continue theforaging behavior), with afinite variation range of control parameters $\{M, T\}$ onfeeding, formation processes of trails are seen along which ants make the much moreefficient foraging than the case without them. In such regime we $\mathrm{s}\mathrm{e}\mathrm{e}:\mathrm{i}$)the geometriesof trails and $\mathrm{i}\mathrm{i}$ )the accompanying foraging efficiency.Now, before showing the several characteristic geometries of trails and the condi-tions for the appearance of each of them, we introduce (or reconfirm) three time scaleswhich are relevant for the trail formation: i) $T:\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{l}$ of food suPPly, $\mathrm{i}\mathrm{i}$ ) $T’:\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}$time of food at each site of food after each event of supply. This is, in term of acontrolparameter, roughly proportional to $M$ , that is, $T’\simeq aM$ where $a$ is constant depend-ing on ants. Note $T’$ itself is not acontrollable variable in the simulation becaus139Figure 3: The relation betweenemergent trail patterns and thecombinations of feeding schedule pa-rameters, $\{M,T\}$ . Here darknessin each figure means ants densityaveraged over sufficient longer timethan $T$ , the interval between succes-sive feedings. The trail geometriesare, while fluctuating (or switching)with time, roughly categorized intothree types after averaged over suf-ficiently longer period than $T:\mathrm{i}$)V-shaped trail, $\mathrm{i}\mathrm{i}$ ) $\mathrm{Y}$-shaped trail, and$\mathrm{i}\mathrm{i}\mathrm{i})\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}$ trail.Fig.3of the $\mathrm{Y}$-shaped trail gradually approaches to the nest and finaly $\mathrm{V}$-shaped(in timeaverage) trail is realized.To investigate the relation between the efficiency of the foraging and the trailgeometry, the total amount $F_{eff}$ of foods carried by ants into the nest per unit timeis measured under various $M$ with $T$ fixed. Fig.4 is the case for $T=1\mathrm{O}\mathrm{O}$ . Here,monotonically increasing relation between $M$ and $F_{eff}$ is seen until $F_{eff}$ reaches thesaturation value at which most ants is involving in the foraging along the straighttrail to one site of food. In the $M-F_{eff}$ relation there seems four characteristiczones indicated by symbols $A$ , $B$ , $C$ and $D$ in fig.4. Zone $D$ is the saturation phasementioned above. In zones $A$ and $C$ are seen the almost linear $M-F_{eff}$ relationswhereas the inclination within $A$ is approximately twice as that within $C$ . Theseinclinations correspond, respectively, to the foraging tactics with the $\mathrm{Y}$ shape trailand that with the straight trail, while acrossover tactics appears at the zone $\mathrm{B}$ wherethe trail shape is varying from the $\mathrm{Y}$-shaped trail tactics to the straight trail tacticsdepending on time. In such away the change of trail geometry is accompanied withthat of foraging efficiency. Especially, across the zone $\mathrm{B}$ ants are forced to switchtheir tactics from one to the other. At the zone $\mathrm{A}$ , the ’total transport capacity’ ofants exceeds the total supply of foods, thus trails will extend to both sites of foodsto take all of them. On the contrary, at the zone $\mathrm{D}$ the ’total transport capacity’ ofants can not afford to cover both sites of foods, in other words, foods at each site aremore than enough for all ants. Also in the zone $\mathrm{C}$ , ants are insisting to only one siteof food as long as the food does not run out from one site, and, in the zone $\mathrm{B}$ wherethe food of each site becomes insufficient to be maintained steadily, ants come toexplore the other sites and $\mathrm{Y}$-shaped tactics appears though irregularly varying intoand back from the straight tactics. In the present stage it is not simply concludedthat the crossover zone $\mathrm{C}$ in our simulation falls on the most efficiently range for theswitching ffom $\mathrm{Y}$-shaped trail to the straight trail. However it is remarkable thatthrough the above switching of trail geometry ants will flexibly change tactics fromeffective one for inadequate environment(with $\mathrm{Y}$-shaped trails) to that for abundantenvironment(with straight trail140Figure 4: The simbols $\cross \mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}$ thethe relation between the amount offood supply $M$ at each feeding andFeff total amount of foods carriedby ants into the nest per unit time.The figure at the bottom is the insetof the above. There are four charc-tersitic zones $\mathrm{A},\mathrm{B},\mathrm{C}$ and $\mathrm{D}$ along theFeff $-M$ relation. Each zone isaccompanied by each foraging tac-tics: at the zone Ais realized the$\mathrm{Y}$-shaped trail, zones $\mathrm{C}$ and $\mathrm{D}$ cor-responding to the straight trail, andwithin the zone $\mathrm{B}$ the $\mathrm{Y}$-shaped trailand straight trail to each character-stic geometry of trail $\mathrm{Y}$-shaped trail,shown in fig.3. It means ants flexi-bly make trail patterns according tothe regime of feeding to realize effi-cient foraging.The similar is the transition from the $\mathrm{V}$-shaped trail to the $\mathrm{Y}$-shaped trail underthe variation of $T$ with fixed value of $M/T$. This transition also is accompanied bythe switching of the foraging efficiency, and connects two extreme cases, i)the casefor $T>>\tau$ where $\mathrm{V}$-shaped trails is formed as the superposition of the temporallyoptimized two straight trails. $\mathrm{i}\mathrm{i}$)the case for $T<\tau$ where $\mathrm{Y}$-shaped trail is recognized.The details of these cases are left to be reported.5DiscussionStill many problems remain to be solved; Firstly, the more precise relation betweentrail geometries and their foraging efficiency should be investigated which includes thedetailed discussion on the location of crossover zone B. Also awider class of studiesvarying other parameters than the present case. Secondly the comparison of ’our’ antsto real ants is required though several tough problems should be overcome, like thespecification of pheromones relevant to the trail formation. However, in this stage, itwould have acertain meaning to discuss possible mechanism how the adaptive tacticsemerges among the innocent agents according to dynamically changing environment.The authors thank the participants of aresearch project “Creation and Sustenanceof Diversity” organized by International Institute of Advanced Studies for fruitfuldiscussions.References[1] T.Tao, H.Nakagawa, M.Yamasaki, Y.Chiwata and H.Nishimori, ’Adaptive trailformation under dynamic feeding’, preprint(2001141[2] B.H\"olldobler’’, E.O.Wilson, The Ants (Belknap, Cambrige,1990)[3] W.H.Gotwald jr. Army Ants (Cornell University press,[4] H\"olldobler B. M\"oglich M. Insectes SOciava27237(1980)[5] J.W.Haefner and T.O.Crist, J. Theor. Biol 166299 (1994).[6] J.L.Deneubourg, S.Goss, N.Pranks, and J.M.Pasteels, J. Insect Behavior, 2, 719(1989).[7] J.L.Deneubourg, S.Goss, Collective patterns and decision-making Ethology Ecol-ogy&Evolution, 1, 295 (1989).[8] D.Helbing, F.Schweitzer, P.Moln\"ar, Active walker model for the formation ofhuman and animal trail systems Phys. Rev. E, 56, 2527 (1997).[9] F.Schweitzer,K.Lao, and F.Family, BioSystem 41, 153 (1997).[10] E.Bonabeau, M.Dorigo, G.Theraulaz, Swarm Intelligence (Oxford Universitypress, Oxford, 1999)[11] H.Haken, Advanced Synergetics, 2nd ed. (Springer, Berlin,1987)[12] G.Nicolis, and I.Prigogine. Self-Organization in Nonequilibrium Systems. FromDissipative Structures to Order Through Fluctuations (Wiley,New York,1977)142

化学走性によるトレイルの形成とその機能 (力学系理論と複雑系の数理)

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化学走性によるトレイルの形成とその機能 (力学系理論と複雑系の数理)

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