Models of Termite Nest Construction Daniel Ladley Computer Science 2003/2004

Transcription

1 Models of Termite Nest Construction Daniel Ladley Computer Science 2003/2004 The candidate confirms that the work submitted is their own and the appropriate credit has been given where reference has been made to the work of others. I understand that failure to attribute material which is obtained from another source may be considered as plagiarism. (Signature of student)

2 Summary This project aims to investigate the methods used by termites to build nests. The project starts by detailing the field of Swarm Intelligence and its biological background. It then moves on to examine models of collective construction in detail. Two models are developed to investigate collective construction, the first is a reimplementation of a mathematical model by Bonabeau et al. [5], the second a new agent-based model which aims to model elements of the process which were ignored in the mathematical model. The results of these two models are then compared and it is found that there are weaknesses in the mathematical model where processes are not accurately captured. It is found that, despite the introduction of constraints, the agent-based model is able to reproduce the results of the mathematical model whilst not suffering from the same weaknesses. i

3 Acknowledgements Many people have helped make this project what it is. Firstly I d like to thank Seth Bullock both for his invaluable advice and encouragement throughout the project and for all the time he s given over the last two years. I d also like to thank Netta Cohen for her comments on the project as assessor and the Biosystems reading group for fascinating weekly discussions. On a personal note I d like to thank my parents (Kathryn and Ian), my sister (Amie) and my girlfriend (Yiyu) for supporting me throughout my time at the University and for putting up with this strange obsession. ii

4 Contents 1 Aims 1 2 Background Reading Introduction Collective Construction Introduction Ants Wasps Termites Applications Aspects of Swarm Intelligence Foraging behaviour Collective sorting Division of labour Collective transport Summary Models Of Collective Construction Introduction Mathematical model of Construction in Termite Colonies How it worked Evaluation of the Model Extensions Agent-based model of Wasp Building Behaviour How it worked Evaluation of the Model Extensions Summary iii

5 4 Method Introduction Methodology Project Schedule Design of the Agent-based Model World Pheromones Evaporation Diffusion Pheromone Sources Termites Movement Block Placement Wind Paths Implementation of the Agent-based Model Implementation of the Mathematical Model Results Introduction Results from the Mathematical Model Pillars Wind Queen Trails Extensions Summary Results from the Agent-based Model Queen Pillars Wind Single Path Two Paths Summary Discussion Introduction Overview Quantitative differences in the mathematical model iv

6 6.4 Queen Chamber Pillars Wind Paths Crossed Paths Future Work Summary Conclusion 59 Bibliography 61 8 Appendix A What went well What went wrong and what I have learnt from it Appendix B Evaluation Criteria for the models Did the agent-based model capture the behaviour of the real system? Was the physics of the agent-based model plausible? Did the implementation of the mathematical model replicate the original work? Were the extensions of the mathematical model plausible? Criteria for the comparison of the models Was anything learnt from the comparison of the models? Were the methods of comparison scientifically strong? v

7 Chapter 1 Aims Termites create some of the most impressive structures seen in nature, building features such as air conditioning, fungus farms and royal chambers into their huge mounds. Recent research has started to explain how these complicated architectures are constructed, but we have yet to fully understand how communities of simple organisms can collaborate successfully on such grand constructions. What seems clear is that the simple termite cannot be relying on centralised control in the form of either a genetic blueprint or guidance from a single executive such as the termite queen. Rather, we now believe that much of the construction process is co-ordinated through stigmergy, a form of indirect communication via environmental cues that are typically produced as a side-effect of the very activity that requires co-ordination [19]. In the case of termite construction work, as building material is assembled, the local environment of the builders is altered in such a way as to encourage appropriate behaviours. Partially built structures communicate with workers in such a way as to facilitate their own completion. Over time, as building work continues, the messages conveyed by the environment change in subtle ways that allow complicated, heterogeneous architectures to be constructed. In addition to the physical presence or absence of building material, the environment influences termite building behaviour through supporting various diffusive processes. For instance, recently deposited material is a source of a particular kind of cement pheromone. This pheromone attracts termites, and thereby encourages the deposition of more building material at or near its source. The consequent positive feedback concentrates building effort in hot-spots, efficiently amplifying what are initially randomly-placed pieces of building material into well-spaced pillars, and subsequently walled enclosures, which may eventually be covered to form chambers. A second pheromone exuded by the queen termite encourages building activity when encountered at a particular concentration level. The 1

8 pheromone gradient template created by a stationary queen encourages material to be deposited at a characteristic distance from her location, eventually leading to the construction of a royal chamber found at the centre of many termite mounds. Finally, a third trail pheromone, deposited by moving termites, also guides building activity, co-ordinating the formation of galleries and covered walkways. Even wind may affect the structures built by termites, who are thought to be sensitive to air currents and able to make decisions based on direct interaction with the wind. In addition, any wind will disturb pheromone diffusion and influence the structures being built [12, 5]. Whilst developing and exploring models of stigmergic and decentralised construction has direct application in insect biology, there also exists the possibility that improving our understanding of these processes might lead to powerful new engineering and design methodologies. Self-assembling robots, automatic repair of space craft or nuclear reactors, and the construction of nanoscale structures are all examples of engineering challenges that might benefit from or even require a termite-inspired approach. This is due to the extreme scales at which the activity must take place, the inaccessibility of the building site, or simply the savings in time and money that could potentially be achieved. This project aimed to compare the results obtained from a previous model of construction in termite nests by Bonabeau et al. [5] with those obtained from a new agent-based simulation. From this it was possible to gain insight into the construction methods used by termites and to draw conclusions about the limiting factors in this process from the assumptions made by the two models. The objectives of this project were to Create an agent-based simulation of termite constructive behaviour Recreate the model of Bonabeau et al. [5] Perform a detailed comparison of the structures produced by the two models 2

9 Chapter 2 Background Reading 2.1 Introduction In the last ten years the field of swarm intelligence has acquired increasing interest from academics and industry. Many systems and behaviours are of interest to researchers in this area most notably social insects, however, the flocking behaviour of birds [31], the shoaling behaviour of fish [38], patterns in bacteria [1] and synchronisation of fireflies [20] all exhibit similar characteristics. This project focused on the building behaviour of social insects, however, this behaviour has many similarities with other behaviours investigated in this field. These systems can be characterised by many individuals producing an ordered pattern or effect. The way in which this order is maintained is crucial to the field of swarm intelligence. There are five ways in which this control can occur [7]. 1 A leader. Some force may impose order on the system. E.g. a flow of traffic is controlled by a set of lights. In this case the main source of order in the system is the traffic lights, which tell the cars when they may move and when they cannot. Though even in this case there is another source of order in the system, the interaction between cars causing them to slow down and speed up to avoid hitting each other. 2 A blueprint. Every individual in the system may have a blueprint or design of the goal to be achieved. E.g. a construction team building a house [7]. In this case the construction team are given a blueprint for the house which specifies the exact design of the pattern to be formed. Again there is another source of order present, a leader i.e. the construction supervisor who has some measure of control in how the structure is created. 3 A recipe. Every individual in the system may have a recipe, i.e. a number of actions to be formed in a set order to achieve the goal. E.g. a distributed computer program. Every computer in the system 3

10 will be following a specified recipe, i.e. the program. The program stored on every computer gives a list of operations which need to be carried out in order to achieve a task. 4 A template. A template is a guide or mould which the individuals in the system use to construct the final pattern. E.g. a jelly mould. The individual particles of jelly are poured into the mould which constrains their movement and forces them to form into a pattern which is the same shape as the mould. 5 Self-organisation. This is possibly the most interesting of the five methods for imposing order and is the one which swarm intelligence is most concerned with. Self-organisation is different from the other four methods in that the other methods rely on order being imposed on the system by some external force; in self-organisation the order within the system is a direct result of the interaction of the agents within the system. When agents are mentioned they refer to the individuals which make up the system whether biological, such as ants or cells, or non-biological such as grains of sand. Templates, Recipes, Blueprints and Leaders are all ways of imposing order [7], they control what every member of the group can do at any time without necessarily being part of the group. A recipe, for example, tells an agent its next action yet the recipe, without being part of the interacting system of agents even though it controls the action, is not part of the system because it does not contribute directly to the pattern formed. Selforganising system are also characterised by the agents reliance on local information. Agents in these systems lack external sources of order so they only have the information they can gather about their own local environment. In addition, in many cases the agents lack the capacity to use general information as they are too simple to make use of it therefore their decisions are based on the relatively small amount of information available to them at any given time. Self-organisation often leads to the system displaying emergent properties. An emergent property is one that cannot be understood by observing a single agent of the system. Only when many agents are placed together can the emergent property be observed [7]. One example of this is the brain. From the behaviour of a single neuron it is not possible to deduce the emergent effect of consciousness, only when many neurons are connected together into a network can this effect be observed [15]. Often it is difficult to reason where the emergent property comes from resulting in incorrect theories of its origin, for instance for many years it was believed that a single termite controlled the design and construction of a termite mound [9]. The termite mound is so complicated it was believed that some termite must be overseeing the construction. The possibility of it emerging from the interaction of the many individuals in the system was not even considered until relatively recently. Self-organisation is often based on the interaction of positive and negative feedback [7]. In many systems positive feedback is associated with instability, however, in self-organising systems positive feedback is vital for the formation of patterns. Positive feedback is responsible for turning a small break of symmetry into a large pattern through amplification. In most biological systems negative feedback is predominant as it helps to constrain the system within certain parameters, for instance thermoregulation is based on negative feedback. In self-organising systems negative feedback is employed to a lesser extent, instead of maintaining the system in a constant state it prevents the system from getting out of control. An example of the interaction of these two forces is in the construction of pillars in termite 4

11 nests [9]. Initially material will be deposited randomly around the system, at some point the symmetry of the system will be broken by one piece of material being placed on another. Termites are encouraged to place pieces of material on top of other pieces which led to a positive feedback cycle, as more pieces are placed the stimulus to place material increases and more termites start to place material. At this point negative feedback comes into play, as the pillar gets larger it begins to compete with other pillars within its neighbourhood. All pillars will provide a strong stimulus for termites to place material, however, there are only a finite number of termites so at a certain point the number of termites adding to a particular pillar will stabilise as it reaches a stable pattern in relation to its neighbours. The presence of positive and negative feedback was one of the key tests of the model construct in this project. Connected to this use of positive and negative feedback is the concept of stigmergy [19]. The example of termite building provides a good demonstration of stigmergy. The termites are encouraged and directed in their actions by the structure they are building. Information from the environment directs their actions rather than direct communication between the individuals. By altering the structure, therefore, the termite alters the information it can obtain from the environment and so alters the direction of construction. Behaviour is termed stigmergic when the method of communication is through an alteration of the environment which affects agents encountering it later rather than through direct communication between agents. In this case the building behaviour is stigmergic since adding material to the structure affects the building behaviour of termites in the future. There are two particular forms of stigmergy [2], discrete stigmergy such as that present in wasps where a particular configuration of cells may cause the next deposit of material and continuous stigmergy such as that present in ant trail following where the probability of following the trail is dependant on the amount of pheromone present. The behaviours in this chapter are believed to be self-organising. The agents in the two systems seem to meet the aforementioned requirements of self-organisation i.e. use of interactions between agents and use of local information. Later models will be introduced which again only use interaction between agents and local information to recreate the behaviours found in nature giving a good indication that these principles are the ones used in nature. The agent-based model which was designed to demonstrate building in termite colonies also adhered to these principles. This chapter will focus on some of the most interesting classes of behaviour of social insects. The key features of these behaviours were examined along with their implications for the work in this project. The chapter will also detail the engineering applications of these behaviours and the properties that make them desirable. 2.2 Collective Construction Introduction In nature animals build nest structures to give them a selective advantage. The advantages provided by a nest can be of varying sorts. A nest may provide defence against predators, help to maintain temperature or humidity, provide storage for food or provide a place for young to be reared [24]. Social 5

12 insects also build nests for similar reasons; the important difference is the scale. Whereas a bird or a mammal may build a nest which is slightly bigger than itself, social insects build nests which can be many millions of times larger than the individual workers [2]. The members of the colony cooperate to form a structure which is beneficial to the whole colony [33]. The best examples of nests built by social insects are those of ants, wasps and termites. In these cases the building behaviour is mainly self-organised, there is no leader or blueprint (though there are sometimes templates or recipes which work with the self organisation). Stigmergy is very important in these cases where a structure is being formed. The information in the structure is vital for directing the termites future building behaviour. The general characteristics of the structures created by ants, wasps and termites are summarised below. A more in depth consideration of how these groups create structures will be carried out in Chapter Ants Certain species of ants such as Leptothorax build their nests between two very close flat surfaces. The gap is sufficiently small that they can effectively build a two dimensional wall around the nest from grains of sand [16]. From an initially random distribution of grains of sand the ants will build a wall around their central brood. The wall is built at a sufficient radius from the centre of the nest to give every ant in the colony on average 5mm 2 of space[16]. The exact method of construction is unknown, it is believed that the ants are using a template based on the internal volume of the nest, however, there may be self organising and stigmergic effects as well caused by an increased likelihood of placing a grain next to another grain which has already been placed. It is known that the radius is not set as an influx of individuals will cause a spate of rebuilding where the nest walls are moved further from the brood [17]. This shows the adaptability of this construction method, as it is able to respond to change even during its operation Wasps Many species of wasps build nests; of particular interest are the species of social paper wasps. Every species of paper wasp builds a slightly different type of nest [24], however, some commonality may be observed in the structures formed. It is theorised that the major evolutionary pressure on the design of wasp nests is predation by ants and it is by the method used to prevent attack by ants that wasp groups may be split [24, 36]. Firstly those which use a pheromone to prevent ants from attacking the nest. In this case the nests tend to hang from a single thin petiole which is coated in chemical which repels ants [23]. A single petiole is used as multiple petioles would require more chemical to be laid by the wasps and would provide more ways into the nest for ants. From this petiole cells are constructed which are used to rear young wasps. The second defence against ants is to build a defensive sheath around the nest with a single entrance which can easily be defended. In this case multiple petioles are often created to support the extra weight. In both cases building proceeds in a similar manner, first a thin petiole is 6

13 constructed followed by the first two brood cells [13]. After these stages there is no set pattern as wasps deposit material in locations depending on the local configuration of cells and their own internal rules [14] i.e. discrete stigmergy. This shows a combination of an initial recipe i.e. a set order of parts to construct followed by self-organising behaviour Termites Termites produce some of the most impressive structures in nature. A termite mound can be many millions of times the size of an individual termite and take several generations of termites to complete [7]. The colony cooperates to build this structure which consists of many features including a queen chamber, thermoregulatory veins, air conditioning vents, protective structure, fungus gardens and brood chambers. Due to the complexity of the structure formed it was originally believed that one termite, possibly the queen, must be in charge of the building. It quickly became obvious, however, that this could not be the case, a single termite could not possibly communicate with the millions of members of the colony to coordinate the building process [7]. The next method hypothesised was the presence of a genetic blueprint within each of the termites, however, this idea was also rejected, as the construction of the nest is adaptive, i.e. it will change in response to food locations and brood size and the storage requirements of the blueprint would make it unrealistic. In addition for a blueprint to be of any use to the termite it must know exactly where it is in the nest in relation to the blueprint which when comparing the size of the termites to the nest seems unlikely. Currently it is believed that the termite nest is a product of self-organisation. Each worker responds to local stimuli from the structure being created to guide its building behaviour i.e. stigmergy [9, 5]. The individual termite does not need to have any concept of the consequences of an individual action in terms of the structure only that a particular action agrees with its behavioural rules. This model of construction will be elaborated on in the next chapter Applications The study of collective building behaviour has important implications for future uses of robotics. There are many places that require some sort of construction to be carried out that are very difficult for humans to work in, such as on the moon, in a nuclear reactor or inside the human body [34]. In these situations robots may provide a solution. The most obvious way to use robots to achieve these tasks is to design a robot capable of performing the task such as repairing the reactor. The problem with this is that the tasks the robot has to complete are often complicated leading to robots which are also complicated and so have a tendency to break down. Unfortunately, due to the environment that the robot is working in, it is often impossible to fix the robot and let it try again [2, 3]. In some cases the complexity may be reduced by having a human controlling the robot remotely, however, this does not work in cases such as on the moon where the delay in sending and receiving signals makes remote control nearly impossible. 7

14 An alternative approach is to have many cheap and simple robots. These robots may not individually be capable of reasoning a method for completing the task, however, as a group their interactions led to the task being completed. If one, or several of the robots breaks down it does not matter, as the group are still capable of solving the task. Having a group of simple robots leads to the system being error tolerant which is a very desirable feature for these systems. By studying the way in which social insects build structures it is possible to gain an understanding of the types of structures which may be built by groups of simple agents and the types of algorithms which must be used to achieve them. The building algorithms found in nature have many features that would be good to incorporate into robotic algorithms. They are adaptive, for instance if the number of individuals in a Leptothorax nest is changed the ants naturally rebuild the walls [16]. They are robust and fault tolerant, if any of the members of the swarm die the construction process can carry on as before. The algorithms naturally recover from damage to the structure, if a termite mound is damaged the termites naturally fill the hole created [9, 5]. They are also parallel enabling building work to go on in many places at once allowing the work to be completed faster. 2.3 Aspects of Swarm Intelligence Other applications of Swarm Intelligence and their basis in nature will now be examined Foraging behaviour Many ant species forage for food as a group to increase efficiency [10]. They have evolved methods for increasing their efficiency at performing the tasks associated with locating food. Ant colonies have developed a unique way of finding the closest food source to the nest. Initially ants move out from the nest and wander randomly until a food source is found [10]. The ant then returns to the nest by the shortest path it can and as it returns it lays a pheromone trail between the food source and the nest. This pheromone trail encourages other ants to follow the trail and visit the food source whilst laying their own trails in turn. If two food sources are found at approximately the same time one closer than the other then the colony is able to select the closest one. As one food source is closer than the other an ant visiting the closer food source will return to the nest faster than one visiting the more distant food source meaning that the path to the closest food source will be walked by the ant more times in a given amount of time than the path to the far food source. As the path is walked more times the amount of pheromone laid at any point on the path will be higher leading to more ants being recruited on to that path since the probability of an ant following a path increases as the pheromone level increases. As more ants follow the path the pheromone level increases even faster resulting in the ants following the longer path moving to the shorter path when they set out from the nest after returning from the more distant food source. In a short amount of time this leads to the more distant food source being abandoned in favour of the closer one. In addition to calculating the shortest food source this method also allows the shortest path between 8

15 the nest and a food source to be selected [10]. The only weakness with this system is that if a food source is found closer to the nest after a more distant one has been established it will be ignored as the path to the new food source will not be able to compete with the amount of pheromone on the path to the more distant source and so will never become established. A similar process may be observed in the raids of army ants where pheromone is used to reinforce paths to and from the front line [11]. It is believed that this pheromone in connection with the type of prey typically found can explain the varying fractal patterns observed in army ant raids of different species. Ant foraging behaviour is the basis of many algorithms. These algorithms try to gain some of the features of ant foraging behaviour which are deemed beneficial for a particular situation. They come under the title of Ant Colony Optimisation and are considered to be one of the main success stories of swarm intelligence. Ant Colony Optimisation was first tested on the travelling salesman problem [2]. In this problem N cities are connected by routes, each route has a specific time cost; the aim is to visit all cities in the minimum time. In order to solve this problem virtual ants are released on the network of cities. The ants proceed around the network where they generate paths between the nodes. Initially these paths are random, however, after the first set of paths are formed, an amount of virtual pheromone is placed on the edges based on the total time it takes for the ant to complete the path. If a path takes a long time to visit all nodes relatively little pheromone is placed, whereas if the path is relatively short a large amount of pheromone is placed. The next set of ants that are released have an increased chance of following a path which has a high level of pheromone on it. After many sets of ants are run on the network some of the paths will have a large amount of pheromone on and others will have relatively little, this indicates the potentially shorter routes through the network and so the parts which should be focused on as the search continues. Evaporation of pheromone from paths helps to prevent ants getting stuck in local optima, as paths that initially appear short and have large amounts of pheromone on them can eventually be avoided as the virtual pheromone evaporates. This basic algorithm with several modifications such as local search functions has proved to be one of the top performing algorithms on the travelling salesman problem. Variants of the ant colony optimisation algorithm have been used on several different problems such as graph colouring and vehicle routing [2]. They have been used very effectively on network routing problems [8]. In this case the algorithm is employed to find the quickest path between pairs of nodes in the network. It works in a similar way to the travelling salesman version of the algorithm. Ants are released to find routes between nodes, when they reach their destination they send a packet back along the outgoing path updating pheromone tables stored at every router based on the length of the path. This algorithm has several strong features. It is distributed so no single node has to store information about the whole network and send it to other nodes, which would create large amounts of traffic for maps that would be out of date by the time they arrived. It is adaptive; if the pheromone is made to evaporate quickly the algorithm will quickly find new and more efficient paths. It is tolerant to error, if a node fails 9

16 then the algorithm will automatically re-route packets and find the new optimal path without having to do a large amount of extra work. In addition to this it performs better than virtually all other routing algorithms which has led to it being adopted by BT for routing in their telecommunications network Collective sorting Sorting and clustering behaviour can be observed in many species of ants. In particular ants tend to sort their brood into concentric circles based on size and to group corpses together in cemeteries [2, 7]. This process is believed to be stigmergic. Ants pick up items, either brood or corpses from areas in which there are few things of a similar type and then take them to a place with similar items. This naturally leads to items being sorted into clusters. In the case of sorting by size the ant also considers the age of the larvae to determine its distance form the centre of the brood cluster. This behaviour has several applications. Firstly it may be used as a method for partitioning graphs into clusters [28]. The nodes of the graph are randomly distributed on a two dimensional space. Virtual ants are then released on this space and allowed to move randomly. If an ant encounters a node which is not heavily connected to those around it the node is picked up, the ant then continues walking. The ant may drop the node when it encounters an area to which the node it is carrying is closely connected. This results in the nodes of the graph being separated into clusters of tight local connectivity with few longrange links. A similar system may be applied to map multi-dimensional data onto a two dimensional graph. Initially the data is randomly located on the grid. Ants pick up data which is dissimilar to other points nearby and release it when it encounters data which is similar. This leads to the multi dimensional data being clustered in two dimensions. Clustering has also been implemented in robotic systems [2]. Many items are placed in an arena where robots are released. The robot are programmed to push a disc if there is no resistance, if there is resistance then the robot releases the disc, backs away and turns in a random direction before moving again. This system leads to clustering as initially robots will pick up free discs and move around with them. When a robot carrying a disc encounters another disc it will leave its disc in contact with the other disc and back away. This pair of discs now forms a larger target for other discs to be added to and at the same time is a more stable pattern because when a robot encounters the two discs at certain angles one disc will be pushed into the other and dropped immediately. Eventually large clusters of discs develop which become relatively stable. This model has been expanded to include sorting of red and blue discs [2]. Red discs are dealt with as before, blue discs are similar, except that when a blue disc is pushed against another the robot carries the blue disc back for a set distance before releasing it and moving off. This results in a central core of red discs surrounded by a ring of blue discs in a similar pattern to those achieved by ants brood sorting. This robotic example is promising though it is still a relatively simple task. 10

17 2.3.3 Division of labour Ant colonies automatically divide labour that must be carried out within the colonies amongst the workers [37]. Workers within these colonies are capable of completing many tasks and switching between tasks when necessary depending on the work which needs to be done. It is generally more efficient for specialised workers to execute separate tasks than for general workers to complete an equivalent sequence of tasks [2]. Ant colonies have developed a system to minimise the need to change jobs whilst ensuring that all work is performed. It is believed that workers have internal thresholds to govern when they need to do a job, for instance the removal of dead ants from the nest. If an ant encounters more than a given number of dead ants in the nest it will start to remove them. This threshold varies between different types of ant, for instance some worker ants have a relatively low threshold for tending the brood, however, their larger mandibled cousins have a higher threshold. In an ideal situation all of the brood tending work will be carried out by the smaller workers, because their threshold is lower there is a higher chance they will carry out the work allowing the larger mandibled workers to concentrate on defending the nest. If, however, a large number of the small workers die then the increasing stimulus to perform certain jobs will cause the larger mandibled workers to commence the jobs that are normally done by smaller mandibled workers. This method of internal thresholds allows the colony to assign the correct number of ants to achieve a given task whilst adapting to the changing numbers of individuals in the system and the tasks which need completing. By setting the worker ant threshold lower it allows specialisation and so higher efficiency in the execution of tasks. This system demonstrates stigmergy as the communication within the system is through the environment, by performing a task an ant changes the environment and makes it less likely that another ant will start to perform the task. This system of distributing labour has obvious applications in job shop problems [2]. These problems are characterised by a particular number of jobs having to be completed by a set number of machines whilst optimising some function such as time. Algorithms based on this system allow machines to bid for jobs based on their internal thresholds. If the machine has a short queue and so a relatively low threshold the machine is more likely to bid for a job, whereas a busy machine is less likely to bid for it. This system allows machines to automatically balance the number of jobs being performed between the machines present, it also allows the allocation to be adaptive so if a machine breaks or a job takes longer than usual the algorithm automatically balances the work between the machines. It is important to note that there is no central controller deciding which machine will get a job, each machine bids individually. This means that there is no central point of failure in the system. This system has so far had limited success in practical situations. There are instances of it being employed in factories [2], however, it has not gained wide spread usage. 11

18 2.3.4 Collective transport Individual ants are renowned for carrying many times their own weight, however, in some cases some objects are too large for a single ant to move. In these cases the ant recruits other ants to assist it in carrying the object back to the nest [2]. Initially an ant will try to pick up an object, if it is unsuccessful it may try again from other angles before deciding to recruit help. It does this in two ways [2]. Firstly it may recruit ants in the local vicinity by releasing pheromone from its location, this pheromone attracts ants to the object to help carry it. Alternatively the ant may lay a pheromone trail from the object to the nest which will cause other ants to exit the nest and assist in carrying the object. Once sufficient ants have been recruited they then work as a team to move the object. Initially the ants will position themselves around the object to be moved. The ants at the front will drag the object towards the nest; those behind will push it whilst those at the sides will angle themselves so that they too are pushing in the direction of motion. This system works well until an obstacle is reached, at which point the ants must somehow take the object around the obstacle. Once the ants find that the object has stopped moving they will attempt to realign and reposition. Individual ants may decide to change the direction they are pushing, when this occurs the ant will continue pushing in the same place, however, it will change the angle of its body. Often this is sufficient to dislodge an object which has become stuck. If this does not work some ants may reposition themselves at different points around the object being transported and push from their new location. This system of realigning and repositioning does not use any form of direct communication, all the communication is stigmergic i.e. through the environment, in particular the object being carried. It does, however, allow teams of ants to resolve deadlocks and to transport an object back to the nest. This system has particular applications in cooperative robotics. Cooperative robotics is a relatively new field that aims to use groups of robots to accomplish simple tasks. One of the key tests in this field is for a team of robots to push a box to the edge of an arena. The box itself is illuminated to make it easy to find, yet is too heavy for an individual robot to push itself. Solutions to this problem have previously been based on robots forming plans and communicating between each other their positions and orientations to coordinate the work. This system, however, is complex and is prone to deadlock. Kube and Zhang [27] have developed a system for this problem based on the method employed by ants. They have designed a subsumption architecture, where each robot has five behaviours. Find: the robot tries to locate the box. Follow: the robot will begin to follow another robot it detects. Slow: the robot slows down to avoid hitting the back of another robot. Goal: the robot heads towards the detected goal. Avoid: the robot avoids collisions with objects it encounters by turning away at a set angle. 12

19 Each of the behaviours will cause those above it in the list to become inactive so for example if the Avoid behaviour is active then the robot will ignore the Goal, Slow, Follow and Find behaviours. This architecture allows the group of robots to find each other and the goal in a similar way to the recruitment behaviour of ants, it does not use any direct communication. The problem of deadlock, however, still remains. In order to solve this Kube and Zhang used the same solution as ants i.e. to realign and reposition [27]. If a robot is in contact with the goal but not moving it will first try to turn to cause the block to move, if this is unsuccessful it will back off and reapproach. This method has been found to be a highly successful way of solving the box-pushing problem with simple robots. Although at the moment this system is still very simple it shows promise for future development. 2.4 Summary The models of swarm intelligence discussed show some of the range of behaviours that can be expected from self-organising systems. These systems complete tasks that appear relatively complex with no overall leader or blueprint, instead all of the communication is between agents and the environment. These properties were very important for the project model. If the model was to answer any questions it must have been based on local information and the concepts inherent in self-organisation. If it was not based on these properties then the results it obtained would have been of little scientific value. The applications of these behaviours show some of the advantages of swarm based systems. The advantages of these systems include error tolerance, flexibility, adaptability, distributed problem solving and efficiency. These factors are important in modern systems which are becoming increasingly complex and distributed so that a control point is incapable of controlling the system. There is, however, a lot of work that has to be done prior to these solutions being widely used. So far there has been little widespread application of swarm based solutions to problems. There are two reasons for this, firstly there is an element of social resistance, as Resnick [30] discusses, people do not trust distributed solutions to problems as they do not have a controller which can be said to be responsible. Secondly many of the ideas which are being researched are not yet sufficiently developed to provide efficient, reliable solutions to real world problems. For instance the research on collective transport has shown promising early results, however, this system is not yet capable of solving real problems. In spite of these problems swarm intelligence offers a new and promising method for solving these problems. 13

20 Chapter 3 Models Of Collective Construction 3.1 Introduction The study of collective construction is a relatively small field within swarm intelligence, when compared to work on ant foraging patterns there are relatively few significant models. The existing models of collective construction take a variety of forms which include models based on differential equations [9, 5], agent-based modes [16, 34, 3] and cellular automata [29]. In the next section the models which are most significant for this project will be examined. 3.2 Mathematical model of Construction in Termite Colonies Probably the first model of termite based collective construction was created by Deneubourg in 1977 [9]. This model gave a possible explanation for the formation of spatially separated pillars within the termite mound. The explanation for this was based on positive feedback; when material was placed in the world it gave out a cement pheromone. This cement pheromone attracted other termites towards the site and so increased the chance of more material being placed in the location of the first deposit. This in turn gave out more cement pheromone encouraging more building at that location, therefore a positive feedback cycle was formed. By attracting termites within a certain range of the building material the cement pheromone also acted to discourage the formation of pillars near to existing pillars. Workers were attracted to deposit material at an existing pillar rather than starting another pillar nearby due to the attractive effects of the cement pheromone being emitted from the established pillar. If two new pillars were started near each other then the pillars competed for material. As a pillar gained more material it became more attractive and so gained more material in the future. In the case of two new 14

21 pillars at some point one gained slightly more material than the other and so became more attractive which quickly led to it becoming dominant due to the positive feedback cycle. After this point the dominant pillar received all of the building work whereas the other pillar was ignored How it worked The model used a series of differential equations with one dimensional and two dimensional representations of the world to show how, over time, peaks of building material occurred which were spatial separated but temporally stable. In order to do this the quantities and interactions over space and time of three factors were simulated. These three factors were: the distribution of termites, the distribution of pheromone and the locations of placed building material which was actively giving out pheromone. The interactions of the three equations are summarised below, more detail may be seen in Deneubourg [9] and are reproduced by Bonabeau et al.[5]. The equations below show the change in pheromone distribution, termite distribution and active material distribution at every time step. δ t H k 2 P k 4 H D H 2 H (3.1) δ t C φ k 1 C D C 2 C γ C H (3.2) δ t P k 1 C k 2 P (3.3) Equation 3.1 modelled the concentration of pheromone at a specified time and location, H. k 4 H modelled the loss of pheromone at every location due to decay. k 2 P represented the amount of pheromone released at every location. D H 2 H modelled the diffusion of pheromone where D H was the constant diffusion parameter. Equation 3.2 modelled the change in the distribution of termites, C. There was assumed to be a spatially and temporally constant influx of termites to the system shown by φ. Termites were assumed to have a constant rate of material deposit shown by k 1 C. D C 2 C represented the random movement of termites modelled as a diffusive process, whereas γ C H gave the movement of termites due to the attractiveness of the cement pheromone. Equation 3.3 modelled the amount of material actively giving out pheromone, P. The first term (k 1 C) showed the amount of material deposited by termites, the second term ( k 2 P)the amount of material which had become inactive and so no longer gave out pheromone. These equations were evaluated over time using an Euler scheme Evaluation of the Model The primary strength of this model was its simplicity, it provided a minimum model of what was required in order to obtain the patterns seen. The model used relatively few effects to explain the construction of 15

22 the pillars meaning that the model could be analysed and understood in detail as was done in the original paper. The model also assumed very little about the termites. The termites were not assumed to be able to perform detailed calculations or to have memory in order to complete the task. They were also not required to communicate amongst themselves or respond to a leader in any way. All that was required was that the termites were slightly attracted towards the pheromone sources. There were several weaknesses in the way in which the world was modelled and in particular the effect of material in the world and the constraints placed on behaviour by it. Probably the most significant weakness of the model was that the effect of building material on the movement of termites and the diffusion of pheromone was ignored. The movement of termites was calculated as an average flow which ignored the presence of building material, therefore termites were effectively allowed to move through structures they had built. This was also the case with pheromone diffusion. Pheromone diffusion was again calculated as a flow which ignored the presence of building material, meaning that pheromone diffused through material and so sources of pheromone that were surrounded in material had an unrealistic effect. In reality if a source of pheromone was surrounded in building material that source would have had no further effect as its pheromone could not get to a location to influence termites, however, in this model the source continued to effect the future deposits and movements of termites. This effect may even have led to an increase in strength of the source as more termites were attracted to place material there. This effect was compounded by the model ignoring the presence of inactive material. Once material had stopped emitting pheromone it was removed from the simulation. As time progressed and more material was placed the inactive material would have become more influential on the termite behaviour, however, the model did not take this into account. The input of new termites to the system also suffered from a lack of physical plausibility. As can be seen from the first term of equation 3.2 there was a spatially and temporally uniform input of termites to the system. This was unrealistic in two ways, most importantly it was spatially unrealistic. Termites were allowed to enter the world at any point including in the middle of the world and in areas completely surrounded by building material. Obviously this was physically implausible and was a possible source of bias in the model, it was possible to imagine that if termites were forced to start from the edge of the world, in a more physically plausible manner, the structures produced would be different, for instance pillars near the periphery of the world would come to dominate due to their proximity to the source of building material. Less importantly the temporally uniform input of termites was unrealistic. It was known that termites use pheromone trails to recruit other termites to areas of building activity and so increase the number of termites on the building project [6]. This effect was completely ignored in this model, however, this was possibly not significant. In the early stages of construction there was relatively little recruitment due to pheromone trails not yet being established to the construction site, therefore, the omission of this effect did not prevent the model representing the early stages of construction. Although it had weaknesses this model was important for several reasons. Firstly it gave a new explanation for the construction of parts of the termite nest. At this time there was still a strong belief 16

23 that there was a leader in charge of the construction of the nest, the queen being the obvious candidate. This model showed that a self organising mechanism could be responsible for collective construction Extensions This model was later expanded by Bonabeau et al. [5] to try and explain other phenomenon observed in termite nests. The first modification was to add wind. This was manifest in the model by biasing the diffusion of pheromone in one particular direction. Equation 3.1 was modified to give equation 3.4 (notice the addition of an extra term). It was found that the presence of wind caused the structures formed to change, where pillars were observed before, at low wind strengths walls or pillared chambers were formed, at higher wind strengths no structures could be established as the pheromone templates created were not temporally stable. A similar effect was observed when a net movement of termites in a particular direction was introduced i.e. walls were formed in the direction of termite movement. δ t H k 2 P k 4 H D H 2 H u H (3.4) The second major addition was the introduction of a stable pheromone template caused by a stream of termites crossing the world. This was done by modifying equations 3.2 and 3.3 to give equations 3.5 and 3.6. Equation 3.5 contains an extra term v C T which corresponds to the emission of an attractive trail pheromone, the function T x y corresponds to the shape of the pheromone template. Notice also the addition of the function F x y, it was noted by Bruinsma [6] that termites do not build in the middle of trails, therefore, as well as attracting termites at high concentrations the trail pheromone must also inhibit building, therefore F x y 1 T x y. In this case for a single trail the template function is shown in equation 3.7 and for two crossing trails the template function is shown in equation 3.8. It was found that the trail pheromone emitted by termites crossing the world was sufficient to allow a covered tunnel to be introduced. When two paths crossed it was found that a chamber formed at the point of intersection. δ t C φ F x y k 1 C D C 2 C γ C H v C T (3.5) δ t P F x y k 1 C k 2 P (3.6) T x y e y y 0 λ y 2 (3.7) T x y min 1 e x x 0 λ x 2 e y y 0 λ y 2 (3.8) The final addition was the introduction of a queen termite which emitted a constant pheromone template. This was done by changing the template function 3.7 to 3.9 in order to better reflect the shape 17

24 of a queen. It was observed that the termites began to form a dome at a particular specified radius from the queen as observed in nature. T x y e x x 0 λ x 2 y y 0 λ y 2 (3.9) These modifications served as important proofs of concept for various structural ideas, however, the weaknesses in the original model were not solved. The problems associated with termites being able to move through walls and allowing pheromone to diffuse through them were still an issue. In addition new problems were caused by the two dimensional nature of this model. It was not possible in two dimensions to form enclosed volumes, so in the case of the queen chamber and covered passageways the model did not give decisive evidence that enclosed volumes would be formed. It was impossible to tell from the model if expanding it to three dimensions would alter the nature of these structures significantly. 3.3 Agent-based model of Wasp Building Behaviour In 1995 Theraulaz and Bonabeau [34, 4]published a paper describing the results of a model they had constructed on paper wasp nest construction. This model showed how complex three dimensional structures could be created by simple agents basing their decisions solely on stigmergy. The wasps in the simulation had no means of communicating and no plan or method of evaluating the structure as a whole. The only information available to them was the configuration of material in their local environment How it worked As opposed to the model proposed by Deneubourg [9] for termite nest construction the Theraulaz and Bonabeau model [34] was agent-based. In this model the behaviour of individual software agents was simulated over time according to rules specifying the agents behaviour. These rules used information from the agent s vicinity to determine its course of action. The model was based in a three dimensional lattice world [34] of a size specified at the start of the simulation. Within the world were a specified number of virtual wasps which each occupied a position in the lattice at any given time. These wasps had two distinct behaviours: movement and building. At each time step each wasp made one movement, this resulted in the wasp moving into one of the adjacent locations in the lattice chosen at random. At each time step each wasp could also have placed one piece of building material in its current location. The decision whether to place a piece of building material was determined by a set of rules (described below). This set of rules was identical for all the wasps in the simulation. The set of rules to determine if a piece of building material was placed corresponds to a set of yes/no decisions. Each rule gave a description of a particular local configuration of building material which would have resulted in new material being placed. The description contained the contents of all the adjacent locations in the lattice which would have resulted in the rule being satisfied i.e. a rule 18

25 specified that certain locations in the lattice must already contain material and specified that all the rest were empty. The rule could also have been rotated on the vertical axis so that the orientation in this direction was unimportant. It was important to note that each rule only specified the contents of the adjacent locations in the lattice and not any global information, therefore forcing the wasp to rely on local information and not a global plan. Each rule in a set described different situations in which material was placed, if one or more of these rules was satisfied by the wasps current location it placed material, whereas if none were satisfied the wasp moved in the next time step without placing material in its current location. When run with the appropriate rule sets it was found that this model could produce complex structures. The paper [34] demonstrated many examples of structures formed from this algorithm. Some of the structures resembled those that are found in paper wasps species in nature whereas others more closely resemble Human artefacts Evaluation of the Model This model had several strengths in its design. Most importantly it modelled individual wasps as agents rather than average quantities of wasps. This was important as it allowed small random fluctuations, starting from the movements of a single agent, to develop in the simulation. These small fluctuations may have been crucial especially if they occurred at the start of a simulation, a small chance movement may have led to one part of a structure coming to dominate over others. Relying on average quantities often leads to small fluctuations being smoothed out, resulting in the loss of some interesting effects. Importantly this model was also based in three dimensions. This allowed the process of construction to be accurately modelled from beginning to end. With a few notable exceptions [16, 17] it is usually the case that construction cannot be reduced from three dimensions to two dimensions without losing important details. Deneubourg used a two dimensional model [9] to simulate termite construction, however, as he noted this model could only accurately model the early stages of the construction process due to it being restricted to two dimensions. Modelling in three dimensions allowed structures such as enclosed spaces like domes or tunnels to be modelled, however, in two dimensions this was not possible and so detail was lost. Basing the model in three dimensions has allowed a better simulation of the complete construction process. Unlike the Deneubourg model this model introduced constraints on the movement of the agents. In this model agents were constrained not to move through the existing structure. Once building material was placed wasps were prevented from moving through that location. This was important as by constraining the movement of the agents the structure produced was also constrained. For instance if placing building material in a certain location prevented access to certain areas of a structure it was clear that the unreachable area of the structure would stop developing. The primary weakness of this model came in its interpretation. When viewed as a purely mathematical model of agent-based construction this model provided strong results for the effectiveness of stigmergy, however, as a model for construction in paper wasps it failed. There were too many large 19

26 inconsistencies when compared with construction in paper wasps that it could not be considered as an accurate model of the construction process. Firstly the material used in the model was cubic or cylindrical dependant on the lattice used which was a completely different substance to the paper used by paper wasps. Secondly the modelling of the wasps was physically unrealistic: wasps were assumed to have an unlimited supply of building material and had no long range sensory capabilities. Importantly the method used by the simulated wasps to decide whether to place material did not accurately model the process used by real wasps, for instance it is known that wasps use long range information [13, 14] as well as short range information to decide when to place building material. Despite the weaknesses in the modelling of the artificial wasps this model was important as it combined together an agent-based approach, three dimensional modelling and the use of local information to show that complex lifelike structures could be produced. Prior to this, models of decentralised construction had been primarily two dimensional based on differential equations similar to Deneubourg [9, 32]. This was the first attempt at modelling this process in an agent-based manner. This approach has subsequently been employed by others [17, 25]. In addition the importance of this model as a proof of concept for agent-based stigmergy cannot be ignored. This model provided a demonstration that stigmergic agents could build complex structures by following simple local rules Extensions This paper was later expanded by Bonabeau et al [3]. The work was expanded by using a genetic algorithm to discover interesting rule sets. In this work simulations were run with different rule sets, the structures were evaluated based on a fitness function. The fitter rule sets were then recombined to form a new generation of rule sets. This work produced several interesting structures, however, there were large weaknesses in the work. Most importantly the fitness function used in the genetic algorithm tried to define structure as an abstract concept without taking into account the function of the architecture created. Bonabeau et al [3] tried to quantify abstract structure by measuring parameters such as repeatability, rule use and regularity in the architecture created. This resulted in some fairly interesting structures and even some that looked vaguely like natural wasp nests, however, the architectures were effectively meaningless as they had no function. Eventually the fittest structures produced were cube like masses of material which were exceptionally regular and repeatable yet did not appear particularly structured as they lacked purpose. The application of a genetic algorithm to find interesting sets of rules was a good idea as it allowed interesting rule combinations to be combined to produce new rule sets and new architectures. The fitness function, however, was flawed as it tried to measure structure and not function. It is probable that the function of a virtual structure is harder to measure than qualities such as repeatability and regularity, however, a reasonable attempt at some measure of function, such as that used by Krink and Vollrath [26], would have produced much more relevant results than the measure of abstract structure which was used. 20

27 3.4 Summary The two models described above made important contributions to this project. The first model [9] is the currently accepted model of construction in termite colonies, however, as discussed above it has weaknesses. This project constructed a simulation along the lines of the second model discussed [34] above. This simulation attempted to replicate the same processes currently modelled by the Deneubourg mathematical model [9]. By creating an agent-based model in the style of that created by Theraulaz and Bonabeau [34] the intention was to resolve the problems present in the current mathematical model. Using an agent-based model allowed the accurate modelling of the constraints which the deposited material placed on the individual agents, therefore capturing behaviour that may have been missed by the original model in not considering these constraints. In addition to resolving this major weakness in the mathematical model it was possible to investigate the effect of inputting termites to the system in more plausible ways than was used by Deneubourg [9]. The different starting points for termites may have implications for the structure created. Whilst the agent-based model in this project was inspired by that of Theraulaz and Bonabeau [34] it was a much closer fit to the reality of termites than their model was to the reality of paper wasps. The cubic blocks were much nearer in shape to the pellets of material deposited by termites rather than the paper deposited by paper wasps. In addition it was much more accurate to assume that the termites only used local information to make decisions in the construction process. It is important to note that whilst the agent-based model was inspired by that of Theraulaz and Bonabeau it used a completely different mechanism for deciding when material was placed. Instead of using collections of rules, the termites used pheromone levels to make building decisions. In addition to the agent-based model the intention was to replicate the original mathematical model by Deneubourg [9] in order to compare and evaluate the results of the two models. 21

28 Chapter 4 Method 4.1 Introduction This chapter will first describe the methodology used to tackle this project. It will then describe in detail the workings of the agent-based simulation of collective construction. As was discussed in the last chapter this model shares strong features with the work of Theraulaz and Bonabeau [34, 3], however, it differs in the functions used by the agents to make decisions. 4.2 Methodology In this project a well known scientific methodology was followed, that of developing an alternative model of an already modelled system and then comparing the results in order to increase understanding of the constraints on the system. This methodology has already been used many times, for instance the many models of fish schooling [35, 22, 38] were all created to increase our understanding of the phenomenon. The model constructed was designed in a bottom up fashion. This meant that the behaviour of the individual agents in the system was designed to match that known in nature [15, 2, 7]. It was hoped that the behaviour of the system as a whole would match that of the real system. If it did then there was a strong indication that the behaviour of the agents was responsible for the patterns observed, however, if it did not then there is some important element of the system being missed. This bottom up approach was preferable to a top down approach as it was much easier to avoid adding bias to a bottom up model. A top down model is often based on assumptions about the behaviour of the system which should be observed rather than scientific observation of the behaviour of the subunits of the system. By 22

29 using a bottom up approach the intention was to construct a more accurate model. The methodology consisted of several stages. It closely followed the methodology set down by Camazine et al. [7](page 71) for the study of models of self-organising systems. The methodology was slightly changed in order to accommodate the implementation of the original model and to fit within the bounds of a computing final year project. In Camazine et al. s methodology four steps were defined, ((i) Observation of the system, (ii) Hypothecation of a mechanism, (iii) Experimental testing and solutions to the models or simulations, (iv) Comparison of results to see if the models agree). The first stage of the methodology was to undertake detailed reading on the subject in particular into the collective behaviour observed. This reading was vital as it provided the basis for the design of the model and allowed the design of relevant experiments and the interpretation of their results, the only deliverable from this stage was the background reading chapter of the final report. The second stage was to reimplement the original model. In this case there was an attempt to recreate the mathematical model and reproduce the results as described by Bonabeau et al. [5]. This stage formed the foundation of this project as it provided a model with which to compare the results of the later work. The obvious deliverable from this stage was the working mathematical model. The third stage of the methodology was to implement the new model. In this case the new model was an agent-based model of termite construction which took into account factors previously ignored by other models. The key deliverable from this stage was the working agent-based model. The fourth stage was to use the new model to reproduce the results from the original model. In this project the minimum aim was to replicate the conditions which led to the formation of pillars in the mathematical model. This allowed an understanding of the conditions which were important for pillars to form and the way in which the assumptions in the two models effected the construction process. The deliverable for this stage was the results of the simulations from both models. The fifth stage was the reproduction of other conditions which were believed to be important in the construction process. In this stage there was an attempt to replicate wind and termite trails. The results from the two models allowed further conclusions to be drawn about the system being examined from the assumptions present in both models, so moving nearer to solving the project problem. The deliverable from this stage was again the results from the two simulations. The sixth stage was to extend both models into previously un-modelled conditions. The exact conditions to be modelled were determined by the results form the previous stages. This work allowed new insights to be drawn about the construction process. The deliverable from this stage was the results from the two simulations. From the above it was possible to see the first stage of the methodology conformed to stage (i) of the methodology presented by Camazine et al. [7]. The second stage was an addition which allowed two different models to be dealt with simultaneously. The third stage was similar to the creation of a hypothesis i.e. stage (ii) of the methodology [7]. Stages four, five and six of the methodology in this project roughly conformed to stages (iii) and (iv) of the methodology [7]. These stages were altered slightly as a previous model was reimplemented and so at all points the results of the new model were 23

30 compared to those from the existing model. The following details how all stages of the methodology were completed. As was seen in the first chapter of this report, stage one was completed. The results section demonstrates that stages two and three were completed as without them no results would be available. The results and the discussion chapters show that stages four, five and six were completed and detailed. 4.3 Project Schedule Below is the initial project schedule. It sets down exactly what should have been done from the start of the project to the end of the project. This schedule included the deliverables which must be submitted prior to the final project hand in. It can be seen from this list that stages two and three of the methodology appear to have been mixed up. The reason for this is that it was not know how long stage three would take therefore it was imperative that it be completed as early as possible in the project. As can be seen from the schedule even though the majority of the work in stage three was done before stage two was started, stage two of the methodology was still completed prior to the results of the agent-based model being analysed i.e. prior to stage four. Implement world 20/10-2/11 Implement agents 3/11-9/11 Implement a system of diffusion 10/11-26/11 Complete simulation involving single pheromone 27/11-4/12 Write background reading chapter 5/12-12/12 Complete simulation involving multiple pheromones 15/12-22/12 Collect results from batch simulations of agent simulation 5/1-22/3 Implement Mathematical Model 24/1-8/2 Extend simulation to incorporate wind effects 9/2-28/2 Write Draft Chapter of final report 1/3-5/3 Gather results from mathematical model 6/3-21/3 Write draft of final report 22/3-18/4 Alterations to final report 19/4-27/4 24

31 This schedule was clearly based around the milestones which must be achieved to make the project a success. The completion of each item on the schedule led to the completion of a section of the project. Certain items on the schedule corresponded directly with minimum requirements such as Implement a system of diffusion. Other items on the schedule such as Gather results from mathematical model provided the information necessary to provide a solution to the problem tackled in this project. This plan was followed fairly closely. All of the items scheduled to be completed by the 22/12 were completed by that date. After this point, however, events occurred which prevented the plan being followed exactly. Implement Mathematical Model from the 24/1 until the 8/2, was stated on schedule, however, this proved impossible to compete on schedule. The construction of the mathematical model was started on the 24th January, however, due to technical difficulties and inaccuracies in the paper (to be discussed in later chapters) the model was not implemented correctly until the 12th March. It took a long time to implement the mathematical model, however, once implemented it was found that it was a very quick process to collect results therefore it was still possible to meet the deadline on the 21st March for collecting all the results from the mathematical model. The other major change to the schedule - Extend simulation to incorporate wind effects only took approximately half of the time allocated to it. This meant that extra time was available to further expand the project. With this time it was decided to implement pheromone trail based behaviour from the 20th February to the 28th February. It was, therefore, possible to further extend on the minimum aims. Throughout the project a diary of problems and events was kept, this is available at /home/cserv1 a/student/csz/cszdal/diary/diary.txt. 4.4 Design of the Agent-based Model The design of the agent-based model is detailed below. The important points in the model are discussed as they are encountered such as the implementation of physical constraints. The design paradigm i.e. using a three dimensional cubic lattice inhabited by agents and cubes of material can be seen to be heavily based on the work of Theraulaz and Bonabeau [34] World The world consisted of a discrete cubic grid typically of size This size was chosen as it was sufficient to keep edge effects on termite movement and pheromone diffusion to a minimum. The world was closed on the bottom (representing the ground) and open in all other directions. This allowed free movement of termites and pheromones out of the world in any direction except through the ground. The ground of the world was composed of a layer of building material. This material gave off no pheromone and occupied the bottom layer of the grid. In general building material was considered to be located at a point in the grid. One unit of building material was considered to fill one location in the cubic grid. The building material blocked termite movement, pheromone diffusion and prevented 25

32 any more material being placed in that location. The simulation used a fixed discrete time step. At each time step every termite was allowed to act and the pheromones were allowed to diffuse Pheromones The world was considered to contain three separate pheromone distributions. As was previously mentioned there were considered to be three main types of pheromone which influenced termite building behaviour; queen pheromone (given out by the queen), trail pheromone (given out by moving termites) and cement pheromone (given out by recently placed building material). Each pheromone distribution was represented by a discrete lattice of the same dimensions and granularity as the world. Each point on the lattice would therefore correspond to one location in the world and so represent the pheromone level at that location. The amount of each type of pheromone at each location in the lattice would change over time due to evaporation, diffusion and outside sources Evaporation Evaporation was simulated every time step by multiplying all locations in the lattice by a constant v such that 0 v 1, resulting in a loss of pheromone at a location proportional to the original concentration at that particular location Diffusion Diffusion was simulated at every time step using a finite volume method [21]. This method allowed each location in the lattice to exchange pheromone with the six neighbours with which it shared a face. The amount of pheromone transferred between a location and one of its neighbours was proportional to the pheromone gradient between the two locations. If the volume of pheromone at a particular location was x and the volume at one of its diffusion neighbours was y then the change of pheromone may be expressed as x 1 t α x y, where 0 α 6 α was constrained to be less than 1 6 to prevent the creation of new pheromone through the diffusion process. As each location exchanged pheromone with six other locations simultaneously if α was greater than 1 6 it was possible that more pheromone would be transferred out of the location than was 1 originally present. In this experiment it was decided to set α 7. This value meant that there would always be some pheromone left in the original location as the maximum which might have been transferred out (in the case where all the neighbours have no pheromone) was x 0 6x 7 where x 1 was the pheromone concentration. Also note that with α 7 no neighbour would have received more pheromone than was left in the original location after diffusion. As can be seen in the case with maximum diffusion, i.e. the case where none of the neighbours contained any pheromone, there was still x 7 left in the original location whereas the amount transferred was also x 7. In cases where there was pheromone diffusion in both directions the amount in the original location would, after diffusion, have been greater than x 7. 26

33 The finite volume method was implemented in an unusual way to allow restrictions on diffusion to be more easily implemented. With every time step, every location in the lattice shared its pheromone content between itself and its six neighbours. This was done by dividing the pheromone level into seven 1 equal parts (analogous to α 7 ), one part was kept in the original location the other six were transferred one each to the location s six neighbours. Every location in the distribution did this simultaneously resulting in pheromone being exchanged simultaneously between every pair of neighbours that shared a face in the grid. As the net transfer between a pair of neighbours was based on the difference of their pheromone levels this method equated to the finite volume method. It was easy to see that no pheromone was created or lost as a result of the diffusion process except at the edges of the world. Areas outside the world were assumed to contain no pheromone leading to pheromone being diffused out of the world at the edges but no pheromone diffusing back in. Although the finite volume method was relatively slow computationally when compared to a Gaussian blur it had one large advantage for the agent-based simulation, it allowed the process to take account of the configuration of the world by performing local updates. When designing the experiment it was important that realistic constraints be placed upon the behaviour of the world and the termites. One of the most important of these was to prevent the diffusion of pheromone through objects and material present in the world. This had important implications for the way in which the diffusion was implemented. A Gaussian blur would not have been able to take account of objects in the world as the spread of pheromone diffusion it calculates is fixed according to the Gaussian function. The method used, however, considered every location in the world individually so by checking a location s neighbours, when pheromone packets were passed out, the effects of material and objects could be considered. In the simulation it was necessary to prevent any pheromone diffusing into locations corresponding to places in the world occupied by material. This was implemented by checking if the location to which a particular volume of pheromone was being transferred contained a block. If it did not then it was treated as described above, if it did then the pheromone to be transferred was returned to its source location. In the finite volume method this was equivalent to setting the difference in pheromone levels between the two locations equal to zero leading, therefore, to no net flow. This meant that the pheromone to be transferred into the occupied location remained in the source location. In the next time step the total amount of pheromone was again divided into seven equal parts and the process repeated leading to an increase in pheromone flow in the non blocked directions. Note: pheromone was not allowed to diffuse into a location which contained building material, however, if there was any pheromone present in these locations then it could have diffused out Pheromone Sources In order for this simulation to catch the richness of behaviour seen in nature it was necessary to have pheromone sources which inputted pheromones to the distributions. All pheromone sources were considered to be point sources and so when pheromone was given out the level was increased in one particular location in the pheromone distribution corresponding to the 27

34 sources location. In addition all sources were considered to output only one type of pheromone so only one of the distributions had its pheromone level increased at that location. There were two types of pheromone source in this simulation, sources with a fixed output and sources with a fixed capacity. Sources with a fixed output added a specified amount to the pheromone level corresponding to their location in the world every time step. This specified amount remained the same throughout the simulation. The second type of source had a fixed capacity of pheromone which it could output during the life time of the simulation. When created a source of this type had a specified capacity and an output rate. The output rate was a value in the range 0 r 1 governing the proportion of pheromone outputted each time step. After pheromone was outputted the capacity of pheromone was reduced by the amount outputted leading to the amount outputted per time step decreasing over time for the source. The first type of source was normally used for the termite queen which was expected to maintain a constant template throughout the simulation. The second type of source was often associated with building material placed by the termites as it was expected that over time the material would become inactive and stop producing pheromone. It was possible that the location to which the pheromone was being outputted was occupied by building material, this was acceptable as the simulation only limited pheromone travelling into occupied locations and not out from them. The advantage of this was that it allowed an intuitive simulation of freshly placed building material giving off cement pheromone from their location in the world Termites The world contained a set number of termites throughout the simulation. Each termite was considered to be located in one square in the world grid. This assumption was unrealistic as it would have seemed to imply a termite was the same size as a block of building material. This discrepancy, however, could be reconciled by having the location correspond to the termites head i.e. the part of the body it used to lay down the building material. A termite could sense the levels of pheromone in its current location and in addition the pheromone gradients to the surrounding locations Movement Termites were constrained on the world grid. One movement took a termite from one location to potentially any one of the twentysix adjacent locations. There were constraints, however, on the sites which could be moved to. Firstly for a termite to move to a location, the new location must have been unoccupied by material and secondly one of the new location s neighbours must have been occupied by material or the ground. The first constraint prevented the termite moving through walls, the second constraint forced the termite to stick to surfaces and prevented it flying around the world. If some of the twentysix possible locations were outside the bounds of the world the termite may have moved into these locations and out of the world freely. If a termite moved out of the world a new termite was placed at a random location at ground level on the border of the world. 28

35 Without the influence of pheromones the termite movement was random. The termite was equally likely to move into any one of the feasible locations to which it was adjacent. For a termite to move the simulation first found all the locations adjacent to its current location to which it could move without breaking the constraints. One of these locations was then chosen at random and the termite was moved to it. If no movements were possible then the termite was removed and another was placed as if the termite had moved off the edge of the world. The presence of cement pheromone gradients in the world influenced the termites movement. Experimental evidence suggested that termites were likely to move up large pheromone gradients and might ignore small pheromone gradients. This could be seen in the pillar formation of termites [6] where the attractiveness of a potential pillar to a termite seemed to have been based on the distance the termite was from the pillar. As distance from the pillar increased the amount of pheromone present decreased and so the termites were more likely to ignore the gradient. The simulation attempted to capture the effect of pheromone on termites, therefore in this simulation weak pheromone gradients led to near random movement whereas strong pheromone gradients were more likely to lead to a termite moving up the gradient towards the source. Initially the simulation found the locations which could be moved to without breaking the movement constraints. Next pheromone gradients to each of these locations were calculated. One of the directions was then chosen based on the relative size of the gradient compared to the others present using Roulette wheel selection. This resulted in the termite having a tendency to follow the highest relative gradient. In addition to this it was necessary to make a termite more likely to follow a gradient if it had a large absolute value than if it had a small absolute value. In order to implement this there was a probability which was inversely proportional to the size of the gradient that the termite would make a random movement. A termite moved m times in this fashion every time step such that m 1. The reason for this was that in reality a termite moved faster than pheromone diffused. In all results in this paper m = Block Placement For material to be placed in a location, the location had to meet a number of conditions. The level of queen or trail pheromone had to be within a specified range ([0.01,0.5] for all results in this project]). All building material had to share a face with an already deposited piece of material or the ground. In addition there was a rough physics in the world which tried to prevent impossible structures from occurring. Therefore all blocks had to meet one of the following three conditions: 1: Either the location underneath or the location above the location to be filled had to be filled by material. 2: The location to be filled had to be horizontally next to a location which was filled and satisfied (1) 3: One face of the site had to neighbour three horizontal adjacent locations that each contained material. 29

36 The first constraint allowed vertical stacks to be built. The second allowed these stacks to be extended horizontally to a limited degree. The third constraint allowed the gradual construction of horizontal surfaces if sufficient support was available. If these conditions were met then there was a chance p that a block would be placed at the termites current location. If a block was placed the termite was removed and a new termite placed randomly on the edge of the world in contact with the ground. This corresponded to a termite leaving the area to find more building material. This was not modelled explicitly as it was not thought to be a vital part of the pattern formation process. Once a block was placed it became a fixed volume pheromone source for the cement pheromone with initial volume e. A fixed volume source gave a good representation of the real behaviour in that there was a high initial output which decreased over time Wind The effect of wind on pheromone distributions was modelled. The simulated wind was designed to take into account the effect of material present in the world and to change strength accordingly. Wind was simulated every time step and effected every location in the world during every time step. The wind came from one face of the world and travelled perpendicular to that face across the world. The strength of the wind, s, was specified to be in the range 0 to 1 and remained constant throughout the simulation. In order to take into account the locations of objects within the world the path of the wind every time step had to be traced. This was done by simultaneously following the path of wind across the world from all starting points and keeping a record of it to allow calculation. In order to implement this a three dimensional wind array was created of the same size and granularity as the world. The values in the two dimensional cross sectional array which correspond to the starting face of the world were initialised to the wind strength, this was called array a 1. All other points in the array were initialised to zero. The wind strength in the neighbouring cross section a 2 was then calculated from the strengths in a 1. This process was then repeated to calculate all values of wind strength in the three dimensional array. In general given the array a i it was possible to calculate the wind strength in array a i 1. For every point in a i all points in a i 1 which shared an edge corner or face with the point in a i were found. The number of these locations which were empty (i.e. they contained no material) were also found. For each of the empty locations the wind strength of the point in a i divided by the number of empty locations was added. This resulted in a three dimensional array giving the wind strength at all locations simultaneously taking into account the locations of obstructions within the world. In areas without any objects this resulted in the wind strength remaining constant as the strength in each location was spread between the same number of locations in the next array. In areas with obstructions this resulted in an increase of wind strength near the obstructions. The wind was funnelled around the obstruction with a wind shadow behind, since no wind passed through it. Over distance this shadow disappeared as the wind closed in again from the sides of the obstruction. When the wind strength had been calculated it was possible to then calculate the effect of the wind 30

37 on the pheromone levels at all locations. The effect of wind was performed as a simultaneous update of all locations. In doing this a steady stream of air movement throughout the simulation was effectively modelled. The wind could be said to have been moving at one world location per time step as pheromone from one location would only move to its neighbour in any one time step, although over several time steps pheromone would move several locations. It would also have been possible to have used a sequential update starting at the source of the wind. In this case the pheromone blown by the wind from one location to a neighbour would then have been used to calculate the amount of pheromone blown to the next neighbour and so on across the world. This was similar to simulating a gust of wind moving across the world within one time step. The amount of pheromone removed from a location was equal to the proportion s of the pheromone present. This amount was then added to the location which was one location downwind. This calculation was performed for all pheromone distributions separately. If the next location was filled then the pheromone was not moved, instead it was left in its current location. This method may have seemed strange initially as the wind only moved pheromone in one direction without spreading it, however, explicit spreading was not needed as the diffusion process resulted in pheromone spread Paths The effect of a flow of termites moving in a particular direction whilst emitting trail pheromone was investigated. The termites moving in this flow were considered separate from the population of termites which built the structures, they were considered to have different jobs. One set of termites was responsible for building structures, the other set was responsible for following a path. The two sets of termites, therefore, had different behavioural rules. The path following termites did not have any block placing behaviour only movement behaviour. The movement behaviour was based on the work by Deneubourg et al.[11]. A number of parameters specified the flow of termites across the world. A set of starting points was specified on an edge of the world at ground level. This set was a continuous group of points from which termites started their movement through the world. At each time step, and with probability, c, each of a fixed number of termites, t, might have entered the lattice at a randomly chosen entry point. Every time step every trail termite moved one location in the grid in the specified direction. The general direction of movement was fixed for every trail termite. This direction was decided by the start locations; the direction of movement was perpendicular to the edge on which the start locations lay and towards the centre of the world. For instance if the termite started from the far left of the world it would have moved right through the world. Every movement forced the termite to move one unit in the specified direction, however, the termite was not forced to move in a straight line, it could have moved to any location which shared a face, edge or corner with its current location as long as the new location was one unit further in the specified direction. The same restrictions on movement as applied to building termites still applied, however, so some locations may have been inaccessible. If no locations were consistent with the constraints then the 31

38 termite was removed. The movement each step was heavily based on the volume of trail pheromone. Trail termites were attracted to trail pheromone in the same way that builder termites were attracted to cement pheromone. Each time the termite moved it deposited an amount of trail pheromone q in the location from which it moved. This amount was the same for all trail termites. This deposited pheromone was free to diffuse and evaporate in the method described above. When a termite reached the end of the world it was removed. The building termites responded to the trail pheromone in a similar way to queen pheromone. If the level of trail pheromone at the current location fell within a range then there was a chance as described above of building material being placed. 4.5 Implementation of the Agent-based Model As well as the problems in the design of the model there were several problems which had to be tackled in its construction. The first of these problems was deciding how the model was going to be implemented, both the platform and the programming language had to be determined. It was decided to implement the model on the School s Linux platform as this platform had several advantages, most importantly the Linux machines in the school were faster than the Windows machines meaning the trials would take less time to run. In addition it was simpler to set trials running on multiple Linux machines through remote terminals than it was on the Windows machines. It was decided to implement the software in C++ as this language suited the project better than any of the other familiar languages (Java, C, SML, Prolog). It was decided not to use Java as it is an interpreted language and so is relatively slow. In this project speed of computation was significant as the large pheromone distributions and large number of agents required substantial computation. It was decided not to use SML and Prolog as their programming style did not really suit this task. The functional and logical programming styles have many important uses, however, the large amount of array processing meant that an Imperative language was better suited to the task. C++ was chosen over C even though both are imperative and relatively efficient, the object orientated paradigm offered by C++ is conceptually useful when dealing with agents. It would have been possible to learn a new language to implement this model, however, it was not believed that the time taken to learn a new language could be justified against the benefits it would provide. The remaining problems came in the implementation of the model. When the model was first implemented it was found that it was exceptionally slow. The reason for this was an attempt had been made to make the code easy to maintain in the future and unconsciously efficiency had been sacrificed. One example of this was that initially all placed material were stored as objects in a list object from the standard library. This method was chosen as it fitted with the object orientated paradigm and it was believed this would allow easy addition of material to the world. It was not considered that when all of the ground material was included in a world there would be objects at least in 32

39 the list. This created massive problems when the model had to check if there was material in a location. It was decided that in order to achieve the maximum efficiency coding niceties would be sacrificed, therefore the list was replaced with a binary array so the presence of material could be checked with a single lookup. Similar problems occurred in efficiently implementing the wind and agent movement. 4.6 Implementation of the Mathematical Model The mathematical model was implemented as described in Bonabeau et al. [5]. Initial this was done in the Matlab system running on Windows Matlab was chosen as the implementation environment for two reasons. Firstly it provided inbuilt methods for performing the Laplace and Nabla operations used by the model, which would make it relatively quick to develop a working model. Secondly Matlab has excellent visualisation features especially for visualising three dimensional data. The model deals with both one dimensional and two dimensional world representations. The three dimensional visualisation would therefore allow the one dimensional world to be visualised over time and for the two dimensional world data to be easily interpreted. There were, however, several problems in implementing the model on this system. The first problem encountered was with the paper itself. The paper [5] did not define the boundary conditions for any of the distributions in the model. In deciding the boundary conditions an attempt was made to choose those which would enable the model to produce similar results to those in the original paper and would be less likely to cause artefacts in the simulation. Careful examination of the results suggested that the distributions did not appear to have fixed boundary values therefore Dirichlet boundary conditions were rejected. As the boundaries did not appear to be fixed, boundary conditions were not added to the active material distribution and since it did not have a diffusion term boundary conditions were not strictly necessary. Both the termite and pheromone distributions, however, had diffusion terms and so needed boundary conditions. Neumann boundary conditions [21] with a gradient of zero were chosen for both distributions. These boundary conditions specify the gradient at the edge of the world and so specify the bias on moving in and out of the distribution. The choice of Neumann boundary conditions allowed the values at the edges of the world to change which was consistent with the observed results [5]. The gradient of zero meant that there was no bias for moving into or out of the world, this value was chosen since there was no other information on the conditions used and it was the value least likely to introduce artefacts to the simulation. There were many other possible boundary conditions which may have been physically plausible, however, this project was not concerned with investigating the effect of varying these conditions. It was also found that the initial conditions specified for the active material distribution did not give the range or distribution of the random values used. One graph hinted that the values may have been chosen from the range [3.3,3.4], therefore it was decided to draw values from a uniform distribution with this range to define the start levels of active material. Once the model was implemented trials were commenced, however, it was found that even though 33

40 the Windows machines were relatively new, due to problems with Matlab itself it was impossible to run trials of a similar order of magnitude to those run in the original paper. In order to obtain results within a reasonable amount of time (24 hours) it was found that the trial had to be approximately one hundred times smaller than the original trials run by Deneubourg [9] and Bonabeau et al [5]. This obviously suggested a problem with the implementation environment as the machine being used had a Pentium 4 processor and the original trials by Deneubourg had been performed in 1977 when computing power was significantly less advanced. When short trials (fewer time steps) were run with a world of the correct size it was found that the simulation did not behave in the expected manner, instead of oscillations starting to develop the system achieved a steady state relatively quickly. This was obviously a problem as without outside input the steady state would not be able to develop into the expected patterns. In order to remedy these problems it was decided to reimplement the model in C++ on a Linux system. In this case the only added difficulty was to explicitly define the Laplace and Nabla operations as no library routines were available. When run it was found that this simulation worked approximately one thousand times faster than the Matlab version. It was now possible to obtain results from a simulation of the same size as that used Bonabeau et al. [5] within three hours. In order to visualise the output it was decided to use Matlab. It was a relatively simple task to engineer the reimplementation to output results which could be read straight into Matlab and displayed as a mesh plot. There was, however, one final problem. When the results were examined it was noticed that there were approximately one hundred times as many oscillations as were expected in the simulation. After much analysis it was found that this result stemmed from a poorly labelled graph in the paper. The paper specified that a spatial step of 0.01 was used for all simulations and in the graphs displaying the results of the simulations one axis was labelled Space from 0 to 200. It has been assumed, therefore, that this meant that the world was of size two hundred and that because the spatial step was 0.01 there were twenty thousand spatial points. Close examination of the results revealed that the two hundred points on the space axis actually corresponded to two hundred spatial nodes each separated from each other by 0.01 causing the model world to be one hundred times too big and explaining why there were one hundred times too many oscillations. 34

41 Chapter 5 Results 5.1 Introduction This chapter will demonstrate the results obtained from the models. Firstly it will replicate and extend the work on the original mathematical model. Secondly it will demonstrate the results obtained from experiments with the agent-based model. 5.2 Results from the Mathematical Model This section will demonstrate that the Bonabeau et al. model [5] has been reimplemented and will then go on to demonstrate the extensions made to it Pillars The most important result of the original work with this mathematical model was the creation of spatially separate pillars from a random initial condition. The reimplementation of the mathematical model produced a similar result (see figures 5.1(left) and 5.1(right)). The first figure shows the formation of pillars in a one dimensional world over the length of the simulation. Initially a random distribution of material can be seen, this distribution quickly becomes almost uniform before small oscillations form and begin to increase. The second image shows the distribution of material at the end of the simulation in a two dimensional world. This distribution clearly shows peaks often joined by small ridges of material. The oscillations seen in the left hand image and the peaks seen in the right hand image were areas in which there are large concentrations of active material, therefore they may be considered to be pillars. These results are quantitatively similar at the finish time to those observed in the mathematical model. 35

42 t X 0 Figure 5.1: Left Temporal dynamics of P for a 1D system with k 1 k 2 k 4 0! 8888 D C 0! 01 D H 0! φ 3 γ 0! , temporal step is t 0! 0003, the spatial step is r 0! 01. Right Spatial dynamics for a 2D system same parameters as left at time = 100 There were small differences in the progress of the model to achieve these end states, the reasons for this will be discussed later in Chapter Wind t X t X 0 Figure 5.2: Temporal dynamics of a 2D system under the effect of wind at time = 100, same parameters as 5.1. Left u 0! 01 Right u 0! 1 The effect of wind on the formation of pillars is shown in figures 5.2(left), 5.2(right), 5.3(left) and 5.3(right). When a gentle wind was added to a one dimensional world (figure 5.2(left)) the temporally stable oscillations seen in figure 5.1(left) became unstable. Instead of remaining stationary the pattern moved slowly downwind over time. The application of a stronger wind resulted in no oscillations and, therefore, no structures being formed (figure 5.2(right)). 36

43 X Y X Y 100 Figure 5.3: Spatial dynamics of a 2D system under the effect of wind at time = 100, same parameters as 5.1. Left u 0! 01 Right u 0! 1 In two dimensions the application of light wind changed the nature of the structure produced. Without wind the simulation formed pillars (5.1(right)), however, under the effect of a light wind oscillations parallel to the direction of wind flow were observed (figure 5.3(left)). At higher wind strengths these oscillations were still observed, however, they were of greater amplitude (figure 5.3(right)). In both cases these oscillations may be interpreted as walls since they were stable in space and time. The results in one dimension closely matched the results obtained by Bonabeau et al. [5] in that they were quantitatively similar i.e. the heights and frequencies of the oscillations at the later time steps were identical. Again small differences were observed in the earlier time steps, these differences will also be discussed in the next chapter. The results for light winds in two dimensions were very similar to those of Bonabeau et al. [5], oscillations were identical in frequency but had a much higher amplitude. The results for heavy winds in two dimensions, however, appeared to differ quite substantially from what would have been expected. The results in one dimension suggested that as wind strength increased it became more difficult to build and the oscillations decreased in size. This may indicate that the model did not fully capture the dynamics of wind in this system. The paper [5] demonstrated the effect of heavy wind in one dimension, however, the case of heavy wind in two dimensions was not discussed Queen Figure 5.4 shows the effect of a pheromone template on the construction process. This template resembled the template produced by a queen termite in that it was an ellipsoid. It can be seen from the figures that the termites constructed a wall of material around the source of the pheromone following the contours of the template. This result was similar to that demonstrated by Bonabeau et al.[5]. Figure 5.4(left) shows the result of using the same parameters as Bonabeau et al. [5]. It can be seen that the walls were of a similar height, however, the diameter of the structure appeared to be much smaller. The 37

44 X 0 0 Y X Y 60 Figure 5.4: Spatial dynamics of a simulated pheromonal template created by the queen at time =100. Template function is T x y " e x x 0 λ x 2 y y 0 λ y 2, λ x 7, λ y 5, x 0 y Left λ x 7, λ y 5, Right λ x 17, λ y 17, All other parameters the same as 5.1 parameters were varied 5.4(right) and it was found that it was possible to obtain a structure of similar diameter to that of Bonabeau et al. [5]. The structure obtained, however, had much more material present in the centre of the dome than was present in the simulations run by Bonabeau et al.[5]. The most likely reason, in this case, may have been an inaccuracy in the paper, the function that Bonabeau et al. [5] said was used for the queen template trials did not appear to match the template function they actually used. Diagrams in the paper [5] showed that the template function used was 0 for a large oval area approximately units, whereas the function described in the paper (equation 3.9, see caption to figure 5.4) was clearly only 0 when x x 0 and y y 0 i.e. a single point, even when the parameters were changed this fact remained the same. Unfortunately there is no way of knowing what function was actually used in the original paper, however, experiments with different parameters show it was not the function they gave. This discrepancy in the function used was sufficient to explain why the radius of the structure created and the amount of material inside the structure in this model differs from that of the structures shown by Bonabeau et al. [5] Trails Figure 5.5(left) shows the effect of a trail shaped pheromone template. Originally demonstrated in Bonabeau et al. [5] and reproduced here using the same parameters, the result clearly shows that walls were built either side of the trail along its length. An extension to this, also shown by Bonabeau et al. [5], is shown in figure 5.5(right). In this case two trails were crossed and it can clearly be seen that a chamber formed at the intersection. In both cases the results were qualitatively similar i.e. similar locations of walls and chamber sizes, however, in this implementation the walls were much higher than were shown by Bonabeau et al. [5]. The reasons for this will be explained in Chapter 6. 38

45 X Y X Y 60 Figure 5.5: Spatial dynamics of a simulated pheromonal template created by termite trails at time = 100. Left template function T x y # e y y 0 λ y 2, λ y 5, y Right template function T x y min 1 e$ x x 0 λ x 2 e$ y y 0 λ y 2, λ x λ y 10, x 0 y 0 30.All other parameters the same as Extensions X 0 0 Y X Y 60 Figure 5.6: Spatial dynamics of a simulated pheromonal template created by trail templates at time = 100 where in both cases θ 12. Left template function T x y $ e y y 0 λ y 2, λ y 5, y Right template function T x y $ min 1 e% x x 0 λ x 2 e$ y y 0 λ y 2, λ x λ y 10, x 0 y 0 30.All other parameters the same as 5.1 The model of trails was extended to include the effect of constraining the entry of the construction termites to the edges of the world rather than all locations. This was achieved by replacing the φ term in equation 3.5 with m x y to give equation 5.1 shown below. δ t C m x y & F x y k 1 C D C 2 C γ C H ' v C T (5.1) The function m x y is 0 for all non edge locations and θ for locations adjacent to the edge. 39

46 X Y X Y 0 Figure 5.7: Spatial dynamics of a simulated pheromonal template created by trail templates at time = 100 where in both cases θ 30. Left template function T x y e y y 0 λ y 2, λ y 5, y Right template function T x y ( min 1 e$ x x 0 λ x 2 e y y 0 λ y 2, λ x λ y 10, x 0 y All other parameters the same as 5.1 This resulted in the structure shown in figure 5.6(left). Notice that the amount of material placed at the sides of the world is now much higher than in 5.5(left) as a larger proportion of termites entered the world near these locations. A similar result was observed in figure 5.6(right) for two crossing trails. Secondly, in addition to constraining entry to the edges of the world function m x y was changed to constrain termite entry to one end of the trail. Figure 5.7(left) shows the result of this. Notice in this case the building is much more concentrated on the trail, relatively less work is done at the sides of the world. Further, a similar situation is show in figure 5.7(right) where the entry of the termites was constrained to be from one end of each of the trails. It can be seen that the majority of the building work again occurred on the trails and particularly on the ends from which the termites started. The original model of pillar formation failed to demonstrate two properties. Firstly whether the pillars seen in the two dimensional model were temporally stable, secondly when the inactive material was considered were the oscillations seen really pillars? Figure 5.1(right) shows the simulation at time 100, figure 5.8 then shows the same run at time 180. It can clearly be observed that the pillars in both cases were in the same locations therefore the pillars were temporally stable. Figure 5.9 shows the total material deposited during the lifetime of the simulation. The oscillations seen previously in figure 5.1(right) are still present, however, careful observation of the scale produces some questions. Most obviously the oscillations vary between and This means that the oscillations were effectively only about 4% of the total amount of material present, therefore it would seem inaccurate to describe them as pillars. The amount of active material present and the small oscillations seen were important. In the original 40

47 Figure 5.8: Spatial dynamics of a 2D system, same parameters as 5.1 except time = 180 paper inactive material was not considered as it was assumed less important than the active material in the behaviour of the simulation. The amount of active material present here, however, suggests this assumption may have been invalid. In the quantities present the inactive material would have had a significant effect on the movement of termites and the diffusion of pheromone. More importantly, however, the inactive material corresponded to the actual structure being built, the lack of variation showed that this structure was effectively a solid mass of material which would serve no real function. By only considering active material in the original simulation Bonabeau et al. [5] missed much of the significant behaviour of this model. The results of this model seem to go against what should be expected. It would be expected that the small oscillations seen in the active material would be amplified, by a much greater degree over time, by positive feedback producing relatively large pillars. This model seems to show the opposite, in that over time the oscillations in the total material placed become relatively insignificant compared to the amount of material around them. The graph of total material placed (figure 5.9) shows that the final structure created did not resemble a group of pillars, instead it resembled a wall with a slightly bumpy top. This problem was hinted at by the original results. It can be seen from figure 5.1(right) that in the original results the variation in the amount of active material was relatively small compared to the amount of active material present. One explanation for this problem is that the model did not fully capture the dynamics of the positive feedback cycle. It has been shown that the size of the oscillations in the active material only increased at a slow rate(figures 5.1(right) and 5.8), even after large lengths of time the oscillations in the active material were still relatively small. This appears to be counterintuitive, if the positive feedback cycle was accurately modelled it would be expected that the oscillations in active material should grow rapidly once the pillars became established. As the oscillations in active 41

48 Figure 5.9: Spatial dynamics of a 2D system, same parameters as 5.1 graph, shows total material deposited, time = 100 material became more significant it would then be expected that the oscillations in the total material placed would become more significant and come to resemble pillars. An alternative explanation for the observed results is that the model may be accurate, however, the mechanism modelled is not sufficient to explain the occurrence of pillars in termite nests Summary The above results show that the model replicated the results obtained by Bonabeau et al. [5]. In all cases the results were qualitatively similar and in many cases quantitatively. It was shown that the model was sensitive to the way in which termites entered the world, and more seriously, the pillars observed were better understood as ripples on top of a wall. In the next section results are presented from the agentbased model which will further investigate some of the issues raised by the math model. In particular the agent-based model will attempt to replicate the experimental conditions investigated with the math model and examine the effect of physical constraints and an agent-based approach. 5.3 Results from the Agent-based Model In this section experiments were performed to investigate the effects of (i) a pheromone template caused by a queen termite (ii) wind (iii) a trail of termites crossing the world in one direction and (iv) two trails of termites crossing the world in orthogonal directions. Unless otherwise noted all pheromone distributions were initialised to be empty and the world initially contained no building material. 42

49 5.3.1 Queen The queen was placed at the centre of the world in contact with the ground. The queen was represented by a collection of blocks in a roughly half cylindrical configuration. Each block of the queen was a fixed output pheromone source for the queen pheromone. Throughout the simulation the queen remained stationary and was somewhat equivalent to building material in that she blocked the flow of pheromone through the area she occupied and allowed termites to climb over her but not to build on her. The simulation was initially run for 1000 time steps without the presence of termites to allow the pheromone template of the queen to stabilise. At this point a set number of termites were randomly distributed around the outside of the world in contact with the ground. From this time onward the simulation proceeded as described in the method. Previous models had predicted that a single source of queen pheromone would cause termites to construct a hollow chamber around the source. This prediction was verified by the agent-based simulation. The agent-based simulation produced royal chambers in all of the simulations of this type that were run. Figure 5.10 shows the stages of the dome constructed over the pheromone source and that this dome is hollow. Initially pillars form (figure 5.10(left)), these pillars are then joined by low walls (figure 5.10(centre)), the pillars and walls grow until a dome is formed (figure 5.10(right)). The prediction that the dome would follow the shape of the source was also demonstrated to be true. It was found that for small pheromone sources the granularity of the world made it difficult to tell if this was correct, therefore, figure 5.11 shows an unusually shaped large pheromone source. No claims are made that this is a realistic shape for a queen, however, it does demonstrate that the chamber formed around the pheromone source will follow the contours of the source. In order to investigate the effect of the number of termites in the model the time taken to construct the queen chamber with different numbers of termites was measured. These results shown in figure 5.12 clearly demonstrate that doubling the number of termites approximately halved the time taken for construction. Interestingly the relationship is not exact. Initially doubling the number of termites led to a better than doubling of construction speed, whereas doubling termites with a large number present less than doubled the construction speed. This can be explained by two factors. Initially doubling the number of termites led to more deposits and so more cement pheromone which helped to attract other termites more quickly. Later doubling the number of termites had less effect as there were only a limited number of places termites could build so the world started to become saturated Pillars Published research on termite nest building places a great deal of emphasis on the construction of pillars in the nest [9, 6, 5]. Bruinsma [6] showed that under certain conditions pillars and strips would be formed by the termites in the early periods of the construction process, therefore it was satisfying to see these features developing in the agent-based simulations. Figure 5.10 clearly shows the formation of pillars in the early stages of the construction of a queen chamber. These pillars were found to form 43

50 1 Figure 5.10: A royal chamber being constructed. Parameters: r 0! 5 α 7 v 0! 1 p 0! 1 n 300 m 5 e 10! 0 s 0! 0. Left image after 250 time steps, centre image after 500 time steps, right image after 5000 time steps Figure 5.11: A royal chamber being constructed with an alternative queen shape. Parameters as 5.10 after 2000 time steps automatically when cement pheromone was added to the simulation and unlike the mathematical model these pillars were physically present in the world and were not just an illusion caused by the active material. It is generally believed that the positive feedback cycle, discussed in Chapters 2 and 3, is responsible for the construction of the pillars, this was confirmed by the model. By performing simulations without the presence of the cement pheromone (the pheromone primarily responsible for the positive feedback effect) it was possible to observe that pillars were only formed when cement pheromone was present. Without the cement pheromone there was no force attracting termites towards already deposited material, therefore construction proceeded almost uniformly. In addition to the presence of the cement pheromone the number of termites was a key factor in the creation of a positive feedback cycle. Trials were run with varying numbers of termites and it was found that if there were insufficient termites pillars would not be formed and building would occur uniformly in a similar way to that observed when cement pheromone was not present. When building material was deposited the cement pheromone immediately started to dissipate. In systems where there were few termites it was found that the cement pheromone would completely dissipate prior to another termite encountering it and so prevented the positive feedback cycle forming. 44

51 2000 Material Placed Termites Figure 5.12: Graph of the total number of pieces of building material placed after 100 time steps by a specified number of termites. Error bars at 2 standard deviations. Bruinsma [6] performed many experiments on the effect of the number of termites present in the system and found that if there were too few termites no positive feedback cycle could be established. These results were confirmed with this model. It was found that when less than 80 termites were present no pillars were formed, whereas Bruinsma found that at least 40 [6] termites must be present for pillars to be observed. It is not surprising that these values differ. This simulation was based heavily on parameter values for which there was no good experimental data available, such as quantity of pheromone. In addition factors such as world size had a large effect on this result. It was also found that pillars were not created with larger numbers of termites which differed from the results observed from real termites. It is believed that this is an artefact of the model created. In order to make this model computationally tractable it was necessary to limit the world size to be In effect, because termites were constrained to only walk on the ground, this limited them to approximately locations at any time. When there were termites in the world a point was reached where termites were so densely distributed that building occurred uniformly. The formation of the positive feedback cycle relied on the initial deposits of material being relatively scarce, however, with so many termites there was no opportunity for the cement pheromone to diffuse and establish spatially separated building areas prior to more material being placed. With so much material being placed the cement pheromone had no opportunity to establish a template which would allow pillars to be formed and so building occurred uniformly Wind Initially one block was placed in the centre of the world. This block was a fixed output pheromone source of the queen pheromone. Wind blew into the world from the YZ plane and moved in the direction of 45

52 the positive X axis. After 1000 time steps termites were released from random locations around the outside of the world. This experiment was performed with a range of wind strengths from 0.0 to 0.95 in intervals of The experiment was repeated ten times for each wind strength. The graph of the results is shown below. Centre Offset Wind Strength Figure 5.13: Graph showing the relationship between the offset of the centre of the structure and the wind strength. Error bars at 2 standard deviations Back Gradient Front Gradient Gradient Wind Strength Figure 5.14: Graph showing the relationship between the gradients of the front and back slopes of the royal chamber and the wind strength Figure 5.13 shows the relationship between the offset of the centre of the structure and the pheromone source as wind was increased. The offset of the structure is defined as the distance between the pheromone source and the centre of the inside of the chamber (i.e. half the distance from the front wall to the back wall). At low values of wind it can be seen that there appears to be a linear relationship between centre offset and wind strength, however, at higher values of wind strength it appears that this 46

53 relationship ceases to hold. It appears that the wind did not stretch the structure as far as expected. The reason for this is quite simple, the wind strength is defined as the percentage of pheromone moved from one location to the neighbouring downwind location. As this value is a percentage it is capped at 100 (corresponding to 1.00), since it is impossible to move greater than one hundred percent of the pheromone into the neighbouring location. When the wind encountered obstacles it tried to flow around them, this was done by increasing the strength of the wind going around the sides of the obstacle. At high wind strengths this could mean that the wind strength of locations near an obstacle would be increased to greater than one hundred percent and so to avoid this some wind was lost. Therefore for relatively high wind strengths near obstacles, as were typically encountered near the ground, it would be expected that some wind strength would be lost. This effect explains why the relationship appears to be non linear at the higher wind strengths. As well as measuring the length of the structure the gradients of the front and back slopes were also measured. The gradient was measured on the line running through the pheromone source in the direction parallel to the wind. In order to measure the gradient two points were needed. In the case of the front gradient the first point corresponded to the piece of building material which was closest to the source of wind on the line defined. If multiple pieces were equally close the lowest, vertically, was taken. The second point corresponded to the piece of material which was vertically highest, in the event of a tie the closest to the source of wind was taken. In order to find the back gradient the same procedure was used except the algorithm looked for the blocks furthest from the source of wind. In many cases the dome had a large flat roof, therefore, by defining the gradient in this way the roof was ignored and only the gradient approaching it was measured. The centre offsets and gradients of structures were evaluated with a small Java program. This program took as input the output of the simulation program, i.e. a file describing the structure produced, it then looked at the structure and evaluated the desired statistics. This program saved a large amount of tedious work. In order to gain the statistics for the graphs two hundred trials were run, to evaluate these trials by hand would have taken weeks, whereas the program took less than a minute to gain all of the desired statistics. Figure 5.14 shows that as wind increased the front slope of the structure became steeper and the back slope shallower as would be expected in a structure affected by wind. The reason for this can be clearly seen when pheromone is shown together with the building material in figure It can be seen that as material was added to the front and back of the structure the pheromone template shortened, i.e. it was distorted less by the wind. This is because the material placed at the front of the structure helped to provide a wind break and the material at the back helped to prevent pheromone being blown away. This led to the distinctive shape observed and the highest point being relatively near the front of the structure. In addition to examining the wind in the abstract scenario previously discussed the wind was also examined in situations similar to those described for the queen pheromone experiments above. In these simulations the queen was set up as described before, wind again originated from the YZ plane and travelled in the positive X direction. The simulation was also run for 1000 time steps prior to termites 47

54 being introduced to allow the pheromone template to settle. Examples of the structures formed are shown below. Figure 5.15: A royal chamber being constructed under mildly windy conditions (wind emanates from the upper-left lattice edge). Parameters as 5.10, except: s 0! 15. Left image after 250 time steps, centre image after 500 time steps, right image after 5000 time steps Figure 5.16: A chamber being constructed under strongly windy conditions (wind emanated from the left lattice edge). Areas in which the trail pheromone s level was sufficient for building are shown in blue. Parameters as 5.10, except: s 0! 60 Left image prior to termites being released, centre image after 2000 time steps, right image after 5000 time steps. The pheromone source lies approximately one third of the distance from the left edge of the template These images (figure 5.15) show that under the effects of wind the queen chamber is lengthened in the downwind direction. This result shows that the behaviour of the simulation is as expected. It can be seen from figure 5.15 that the formation of the queen chamber proceeded as before for light winds. At higher wind strengths there was a change, pillars were no longer formed during the construction process. Instead construction proceeded in an almost uniform manner. At very high wind strengths the dome may have remained incomplete as the wind deformed the shape of the template sufficiently to prevent certain locations being filled Single Path Simulations of the building process were performed in the presence of a single trail of termites crossing the world. Under the majority of conditions used (see below) these simulations resulted in tunnels being formed. These tunnels stretched across the world from one edge to the opposite edge and were covered and clear of internal blockages (figure 5.17). This was a significant result as it showed that useable tunnels, i.e. paths that termites could move along and be sheltered by, could be formed under the correct 48

55 Figure 5.17: A covered walkway is constructed. Parameters as 5.10, save that no wind has been modelled, and a flow of trail termites has been introduced. Left q 10! 0 t 10 c 0! 5. the tunnel s interior is clear of obstructions with a regular cross-section, the tunnel is relative wide. Right q 10! 0 t 10 c 0! 05. the tunnel is cluttered with an irregular cross section, the tunnel is relativity narrow. Both images after 500 time step. Note that c is the chance of each of the t termites being introduced each time step. On average 5 termites were released every time step in the left simulation and on average 1 termite was released every other time step in the simulation on the right Figure 5.18: A covered walkway is constructed. Parameters as 5.10, save that no wind has been modelled, and a flow of trail termites has been introduced: q 10! 0 t 10 c 0! 1, blue areas are areas in which the level of trail pheromone within the range which allows building to occur Left The pheromone levels show that the path is straight. Centre, Right a secondary path is formed Left image after 1000 time steps, centre image after 2000 time steps, right image after 2010 time steps conditions. Two factors were of particular interest in the formation of enclosed paths. The first was the number of trail termites used to create the pheromone template, the second was the way in which the building termites were allowed to enter the world. It was found that the number of termites used to create the trail had an important impact on the characteristics of the cover created. If the number of trail termites released to create the trail was less than about one every other time step then no cover was created. Instead it was found that the entrance to the world used by the trail termites rapidly became blocked by building material. This was because after a termite had deposited trail pheromone it began to dissipate. The building termites were designed to place material in areas where the level of trail pheromone falls between two values, i.e. at a particular distance from the trail. It was found that for very low numbers of trail termites the trail would not be reinforced often enough. The level of trail pheromone at the entrance to the world would fall within the 49

56 range which encouraged builder termites to build and so the entrance would be blocked. For larger numbers of termites it was found that a covered trail would form. The most obvious effect of increasing the number of trail termites released each time step was an increase in the radius of the tunnel produced. As can be seen from the figures, the radius of the tunnel in figure 5.17(left) is much greater than that in figure 5.17(right). This effect is easily explained by the fact that more termites passing led to more pheromone being deposited and therefore the locations where the level of trail pheromone fell within the acceptable range for building were at a greater distance from the centre of the trail. The smoothness of both the inside and outside of the tunnel created was also looked at. With large numbers of trail termites it can be seen that both were relatively smooth and clear from obstruction, however, as the number of trail termites was decreased it can be seen that the inside of the tunnel became more cluttered (figure 5.17(right)) and the outside more rough. With larger numbers of termites the trail pheromone template created was more uniform as the random movement of termites tended to balance out, however, with fewer trail termites the random motion became more important. These small movements caused distortions in the trail pheromone template leading to areas inside the tunnel and on the outside surface where it was possible for a builder termite to build. This resulted in the small deposits observed on the inside and outside surface of the tunnel. An extension to this result was the presence of material located at a distance from either side of the tunnel. This material was only seen when there were fewer trail termites present. This was contrary to expectations since when there were less trail termites present it could be thought that the template they created would be smaller and so unable to account for these extra outcroppings. In order to explain this phenomenon it was necessary to look at the distributions of pheromone in the world. Figures 5.18(left) 5.18(centre) and 5.18(right) show the distribution of pheromone levels in the world which would cause a termite to build. The first image shows exactly what was expected, a thin path in the centre of the world, however the next two images show the formation of an alternative path. These extra paths were the cause of the random deposits of material far from the main path, the reason for these extra paths will be discussed in the next chapter. The presence of random deposits was also dependant on the method in which builder termites were allowed to enter the world. If builder termites were allowed to enter the world from any side then there were significantly more deposits than if they were forced to enter from the same locations as the trail termites Two Paths Similar results as those to single paths were observed for crossing paths but with a few additions. When two trails were crossed the simulation produced two crossing tunnels which were each clear of obstructions and covered along their length. As was predicted by previous models a small chamber was formed at the point where the two paths crossed. This chamber was not surprising due to the increased pheromone concentration in this area caused by twice as many trail termites passing through it. Inter- 50

57 Figure 5.19: Two examples of cross-road formation. Parameters again as 5.10, save that no wind was modelled, and two perpendicular streams of trail termites were introduced: t 1 10 t 2 10 c 0! 5. At each time step, between zero and 10 builder termites entered the lattice, with probability 0! 5 per termite. Left Builder termites entered from random locations along any edge of the ground plane. Some spreading of the down-stream sections of tunnel tends to occur. Right Builders entered from the same entry points used by trail termites. Spreading was suppressed. Both images after 5000 time steps estingly the source of the building termites was even more significant than in the case of single paths. When building termites were allowed to enter the world from any border was be seen that after the trails have crossed the tunnels appeared to fan out. Figure 5.19(left) shows this effect. Notice that the first half of each tunnel is relatively straight yet the second half spreads out considerably. Trials were run in which one of the paths was moved nearer or further from the starting point of the other and it was observed that the fanning always started after the two paths crossed. In the case where the builder termites were forced to enter the world at the same location as the trail termites this spreading was not observed (Figure 5.19(right)). 5.4 Summary In summary these results show that the work of Bonabeau et al [5]. has been replicated and some interesting extensions have been made to it concerning the level of material present and the effect of varying entry points of termites. The behaviour of the agent-based model and the results which it has produced has also been demonstrated. 51