Bees collective dynamics.

1. INTRODUCTION

Social insects are very rudimentary creatures whose solitary behaviour can hardly be called complex or intelligent. Nevertheless, insects as a colony show some remarkable complexity in their global behaviour. This is so-called collective intelligence brilliantly manifesting itself in such different tasks as dynamic division of labour, sorting, selecting the most profitable food source, finding the shortest path, keeping comb or hive thermoregulation, nest relocation, etc. Obviously, insects didn't "decide" to interact with each other in order to adapt to the environment but it is the network of interactions that shapes the pattern of insect's social organization. This social organization is, thus, an emergent property of the network's geometry.

In this paper, we are going to analyze insect's foraging dynamics focusing for concreteness on a honeybee colony.

First, we analyze the transition from individual to collective foraging. Collective behaviour is shown to be emerging property of the colony size. The efficiency of the collective mode compared to the individual one is demonstrated.

Second, we analyze the emergence of collective decision-making. Depending on the level of communication among the individuals, trails to discovered profitable nectar sources can either co-exist in proportion to profitability of these sources or compete with each other. The latter results in selection of most profitable source, the phenomenon emerging due to the enhancement of the individual's communication level.

2. MODEL

Let us give definition of main components and briefly describe the foraging mechanism.

Employed foragers are associated with a particular food source which they are currently exploiting or are "employed" at. They carry with them information about this particular source, its distance and direction from the nest, and the profitability of the source. Employed foragers will share this information with a certain probability. The greater the profitability of a food source, the higher the probability the honeybee will do a waggle dance and share her information with her nestmates.

Unemployed foragers are looking for a food source to exploit. There are two types of unemployed foragers, scouts, who search the environment surrounding the nest (up to a 14 km radius) in search of new food sources, and onlookers who wait in the nest and find a food source through the information shared by employed foragers. The mean number of scouts averaged over conditions is about 10% [Seeley (1995)].

The value of a food source to an insect depends on many factors including its proximity to the nest, richness or concentration of energy, and the ease of extracting this energy. For the simplicity of our model, however, we wish to describe the "profitability" of a food source with a single quantity.

Bee's dance duration is proportional to the nectar source profitability she is dancing for. Onlookers, however, rarely watch an entire dance before alighting to the source indicated, therefore they cannot be taking into account the source profitability as this information is coded in the duration of the dance! How are bees preferentially recruited to more profitable sources? This is a number game. There is a greater probability of onlookers choosing more profitable sources because at any given time more information is circulating about the more profitable sources. Employed foragers share their information with a probability which is proportional to the profitability of the food source, and the sharing of this information through waggle dancing is longer in duration. So at any given moment, the amount of information circulating about a food source will be proportional to the profitability of that source.

The minimal model of honeybee foraging that consists of employed foragers and onlookers has been developed [Tereshko (2000 ; Tereshko and Lee (2002]. Here we consider the influence of scouting too.

The foraging mechanism stages can be described in terms of chemical reactions. Employed and unemployed foragers are denoted by X and Y respectively:

Y + X [right arrow] 2X (1)

Reaction (1) illustrates the autocatalytic nature of the recruitment process. If an employed forager recruits a nestmate by means of a waggle dance to a food source at which it is employed, the recruit will in turn reinforce the trail and recruit other nestmates, and so forth.

An employed forager abandons an unrewarding food source at a rate inversely proportional to that source's quality: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where f is the quality (as jugged by a bee) of a nectar source, or source's profitability.

To account for smaller part of unemployed foragers that is not affected by dancers and scout the surrounding environment independently, rep resents scouting process:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Reaction (3) assumes a scout to eventually discover an attractive source and become employed at it. The probability of reaction (3) is much lower compared to reaction (2) meaning scouting rate [epsilon] is set to be small.

The kinetic equations corresponding to reactions (1-3) take the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where x and y are concentrations, and [D.sub.x] and [D.sub.y] are diffusion constants of employed and unemployed foragers respectively, and f (r, t) is the food landscape. Taking Wright's idea of fitness landscape which assigns a fitness to each point in a genetic space, the food landscape gives a food source quality value to each point in physical space. Numerous factors affect the assessment of food source quality as judged by an insect, such as proximity to the nest, richness (concentration of sucrose solution, ease of food extraction, predator risk. For the sake of simplicity, food sources are characterized by a single value proportional to the insects' characterization of the source, i.e. the larger the value, the "better" the source in the eyes of the insect. Experimental tests have confirmed that this single value may be the net energetic efficiency of the food source [Seeley (1995].

For simplicity of our analysis, we "freeze" the food landscape, i.e. we study system (4) where the nectar sources are constantly being replenished. Briefly, we discuss the case of depleting sources too.

3. TRANSITION FROM INDIVIDUAL TO COLLECTIVE FORAGING

In reality, most waggle dances occur on a small area, 4-18 cm from the entrance to the hive, the "dance floor", and onlookers are able to obtain information about different nectar sources. Employed foragers are assumed to diffuse very slowly, which corresponds to localization of information about every point of the explored environment space. Distribution of this information through the nest can be modelled by mixing of the onlookers. The faster the onlooker's diffusion, the more uniform this distribution is, which means equal accessibility to the information by each individual in the nest.

[FIGURE 1 OMITTED]

In the limit of vanishing [D.sub.x] and high [D.sub.y], model

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Let us to consider simplest case of single nectar source and analyze the colony dynamics depending on its size. In this case, the foragers dynamics is described by simple logistic equation with small constant growth term (determined by scouting):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The only physical attractor the system converges to is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

It is easy to see that [x.sup.*] remains low up to the point C = 1/f + [member of]. Above this point, [x.sup.*] sharply increases and eventually tends to C (at higher values of the latter). The Fig. 1 illustrates this.

The above figure elucidates the transition from individual to collective foraging. Indeed, when C < 1/f + [member of], the employed foragers concentration growths entirely due to the free term of equation (6), i.e. due to scouting. Unlike, at C > 1/f + [member of], the recruitment cycle is the mechanism governing the forager dynamics. Note that we still have an increase of employed foragers caused by the scouting. However, the dominant term contributing to the above increase, (C - 1/f[member of])x, is determined completely by the recruitment.

[FIGURE 2 OMITTED]

In general case of n sources, define colony's (total foragers flow as [summation][x.sub.2]. Then, the flow dynamics is governed by the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Taking into account that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where <1/f> is nothing but averaged (over the set of sources profitability, obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Obviously, all previous results obtained for single source case hold, and at C = <1/f> + [member of] the colony undergoes the individual-collective foraging transition.

To look at forager's spatial distribution, we solve system (4) numerically. We use the explicit method of numerical integration of PDEs when space and time are divided into discrete uniform sub-intervals, and derivatives are replaced by their finite-difference approximations. The numerical integrations are performed on a 2D lattice with the space and time steps chosen to guarantee the stability and convergence of the explicit scheme. We consider two-dimensional (40 by 40) lattice with no-flux boundary conditions. Throughout, the initial concentration and the diffusion of employed foragers are taken to be [x.sub.0] = 0.001 (for every cell and [D.sub.x] = 0.001 respectively. For un employed foragers, initial concentration varies and [D.sub.y] = 10.

[FIGURE 3 OMITTED]

Consider the food landscape with three spatially separated niches where one is more profitable compared to two others (Fig. 2). At very low scouting rate ([member of] = 0.01), for small and large colonies, one has coexistence and almost pure selection of the best source respectively (Fig. 3). At higher scouting rate ([member of] = 0.1), the coexistence for large colonies becomes more profound (Fig. 4). In all cases, the foraging efficiency (represented by the number of employed foragers) undergoes drastic transition with increasing the colony size. In other words, the efficiency sharply increases with switching from individual to collective mode.

4. COLLECTIVE DECISION MAKING

Let us analyze purely collective mode, i.e we disregard the scouts by taking [member of] = 0.

The diffusion of onlookers inside the nest deter mines their access to the collected information, thereby defining the pattern of information map ping over the nest. To study how collective intelligent choice emerges through communication between employed and unemployed foragers, we compare two cases: (i) locally informed onlookers, individually possessing information about one lo cal point of environment space without knowing of any other, and (u) globally informed onlookers possessing information on the entire environment space.

[FIGURE 4 OMITTED]

According to our model, the first case involves a situation where the diffusion of onlookers is so slow that information about every point of environment is localized within the nest. In the limit of vanishing diffusions and for n spatial modes which correspond to the local maxima of the food landscape, system (4) reduces to the system of un coupled equations describing the logistic growth of employed foragers at i-th nectar source [Tereshko (2000); Tereshko and Lee (2002)]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

with [[alpha].sub.i] = [C.sub.i] - 1/[f.sub.i] (12)

to be the reproductive rate of i-th mode.

Every mode associated with a food value exceeding the food quality threshold, i.e. for

[f.sub.i] > 1/[C.sub.0], (13)

converges to the attractor

[x.sup.*.sub.i] = [C.sub.i] - 1/[f.sub.i], i = 1, .., n. )14)

When individuals are only informed at a local level, colony's decisions are thus also made locally: trails to discovered rich nectar sources, i.e. sources satisfying condition (13~, coexist in proportion to profitability of these sources. Note that without communication the employed forager distribution would be complete random.

Let us consider now the opposite case when on lookers have access to all information available on the explored environment. Distribution of this information through the nest can be modelled by mixing of onlookers. The faster the diffusion, the more equal the accessibility to information by each individual in the nest. In the limits of onlooker's full mixing (uniform distribution of in put information) and vanishing [D.sub.x], if only the modes corresponding to local maxima of the food landscape are taken into account, the infinite dimensional system (4) reduces to the system of coupled equations for spatial mode amplitudes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

In this case, the reproductive rate of i-th mode

[[alpha].sub.i] = C - 1/[f.sub.i] (16)

where C is nothing but the colony size. In this case we have no coexistence but selection of the system's fittest mode [Tereshko (2000; Tereshko and Lee (2002)]. All other modes decay in time. If the profitability value of at least one mode exceeds the replication threshold, then the trivial equilibrium

[x.sup.*.sub.i] = 0, i = 1, .., n (17)

loses stability and system (15) converges to the non-trivial attractor

[x.sup.*.sub.m] = C - 1/[f.sub.m], [x.sup.*.sub.i] = 0, i = 1, .., n; i [not equal to] m (18)

where [f.sub.m] > [f.sub.i], which corresponds to selection of the most profitable food source.

A colony of globally informed individuals thus exhibits collective intelligence in a sense that the colony is able to select the most profitable food source in an explored environment.

For depleting food sources, when the profitability of the i-th food source decreases proportionally to a number of foragers exploiting it, the model was able to qualitatively exhibit the switching behaviour. Consider example of environment with two nectar sources of different profitability. The insects first collectively choose the most profitable source, the largest numbers exploiting this source. Being exploited, this source becomes less profitable than other. At this time insects stop congregating at the first source and begin to move en masse to the second source. This cycle continues with the magnitude of the oscillations decaying as the sources become less profitable, and not worth the energy expenditure to exploit. Though individually informed about the profitability of one source, collectively the insects were able to choose the most profitable source, and were able to respond in switching to other sources when they became more profitable.

5. CONCLUSION

Foraging is just a particular mode of a honeybee colony activities. To generalize, we studied the transition from individual learning, exploration and exploitation to collective learning, exploration and exploitation.

The natural selection which created this system of communication can also be seen within the system. Information about different parts of the environment are like species in competition. The fitness of the species is given by the profitability of the food source it describes. Information survives by continuing to circulate within the nest, and is able to reproduce itself by recruiting new foragers who become informed of the food source and come back to the nest and share their information.

We can clearly see how a social insect society can effectively select food sources in a changing environment. Foragers are budgeted according to the amount of information about each source in the environment, which in turn is proportional to the profitability of each of these sources. Thus foragers are distributed widely and in numbers to best exploit the value of the food source. Even when highly profitable sources are found not all the foragers are employed. Scouts still search the environment for new sources, and a core of unemployed foragers is still left at the nest which allows for a quick response (increase in amount of information if a new richer source is found.

The model illustrates that the individual units do not have to posses great cognitive powers to make individual decisions about the environment, they simply respond to the information which they see. Taking these principles, a swarm of inexpensive insect-like robots can be designed [Deneubourg et al. (1991; Werger and Matarie (1996)].

REFERENCES

J.-L. Deneubourg, S. Goss, R. Beckers, and G. Sandini. Collectively self-solving problems. pages 267-278. Plenum Press, 1991.

T.D. Seeley. The Wisdom of the Hive. Harvard University Press, Cambridge, MA, 1995.

V. Tereshko. Reaction-diffusion model of a honeybee colony's foraging behaviour. In et al M. Schoenauer, editor, Parallel Problem Solving from Nature VI, Lecture Notes in Computer Science, volume 1917, pages 807-816. Springer Verlag, Berlin, 2000.

V. Tereshko and T. Lee. How information mapping patterns determine foraging behaviour of a honey bee colony. Open Systems and In formation Dynamics, 9:181-193, 2002.

B.B. Werger and M.J. Matarie. Robotic food chains: Externalization of state and program for minimal agent foraging. pages 625-634. MIT Press, 1996.

Valery Tereshko

School of Computing, University of the West of Scotland, Paisley PA1 2BE, UK

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