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Modelling the swarming behaviour of army ants Watmough, James 1992

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MODELLING THE SWARMING BEHAVIOUR OF ARMY ANTS By James Watmough B.A.Sc (Engineering Physics) University of British Columbia  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS AND INSTITUTE OF APPLIED MATHEMATICS  We accept this thesis as conforming to the required standard  £  uUMA  November 1992  ©  James Watmough, 1992  In presenting this thesis  in  partial  fulfilment  degree at the University of British Columbia,  of the  requirements  for an  advanced  I agree that the Library shall make it  freely available for reference and study. I further agree that permission for extensive copying  of this thesis for scholarly purposes may be granted by the head of my  department  or  by  his  or  her  representatives.  It  is  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of British Columbia Vancouver, Canada  Date OQ-’r.  DE-6 (2/88)  C5  abstract  A model of the collective motion of a group of social insects interacting through a trail network is developed around a set of simple rules. The model assumes that individuals interact with the existing trail network both by following the trails, and by adding to and reenforcing trails. The main hypothesis of the model is that the mechanics of the motion of the individuals are sufficient to determine the collective motion of the group, or equivalently, that the parameters governing the behaviour of the individuals are sufficient to determine the patterns of the trail networks produced. Both a simulation and a mathematical model are derived from this same set of rules. The mathematical model is derived for a single spatial dimension to study the propagation of the raiding column. The results are compared both to the swarming behaviour of army ants observed in the field, and to the results of simulations of other authors which were based on different assumptions. We conclude that it is possible that many aspects of the swarming pattern are controlled by the parameters governing the behaviour of the individual.  11  Table of Contents  abstract  ii  List of Tables  v  List of Figures  vi  1  ft.  2  3  Introduction  1  1.1  The Swarm  1  1.2  Organization in Social Insects  5  1.3  Communication Through a Trail Network  5  1.4  Using Simulations to Model Trail Formation  7  1.5  Overview  8  Mechanics of Motion 2 1  Rules of Motion  2.2  Results of the Computer Simulation  9  9 11  A Model in One Spatial Dimension  18  3.1  Derivation of the Model  18  3.2  Reduction to the Travelling Wave Form  23  3.3  Analysis of the State Space  28  3.3.1  Local Analysis of the Steady States  28  3.3.2  Global Analysis  30  3.4  Stability of the Travelling Wave as a Solution to the PDEs  37  3.5  Implications and Generalizations  37  ‘U  Discussion  43  Bibliography  48  A Dimensional Analysis  50  B Elgenvalues in 3 Dimensions  52  C The Location of D Details of the Computer Simulation  58  List of Notations  60  iv  List of Tables  B.1 Real eigenvalues of a 3x3 matrix  54  B.2 Imaginary eigenvalues of a 3x3 matrix  54  V  List of Figures  1.1  Development of the swarm  3  1.2  Typical raiding patterns of army ants  4  1.3  Mechanism of trail detection  7  2.1  Comparison of simulation to field observations  13  2.2  Dependence of trail network on forward persistence  14  2.3  Dependence of trail network on the affinity of the ants to the trails  15  2.4  Dependence of trail network on the rate of trail deposition  16  2.5  The formation of a trunk trail  17  3.1  The State Space  32  3.2  Intersection of W with T°  34  3.3  A schematic showing the boundaries of A— and their intersections with W?  3.4  A section showing the intersection of W and W? with a plane of constant T  36  3.5  Stable travelling wave solution:  38  3.6  Stable wave resulting from oscillating input  39  3.7  Simulations of Deneubotirg et al  46  E  .1, c  C.1 Locating the eigenspace of u 0  =  .5, and -y  =  1  .  35  57  vi  Chapter 1  Introduction  1.1  The Swarm  Aspects of trail following and trail formation are utilized to varying degrees by many social organisms, from the repeated use of the same trail by an individual, to the formation of vast trail networks by large numbers of insects. Many means of communication and interaction within the trail network are used. Individuals may use familiar landmarks as roadsigns; or they may orient themselves with respect to the sun, moon, stars, or even the earth’s magnetic field. Able [1] gives many good examples. One of the most striking examples of trail formation are the raiding columns formed by army ants. Here, up to 20 million’ individuals will march as a single entity. Although these ant colonies display an extensive caste system, it is known [21] that the networks form without the use of any preset pattern, and without direction from any ant, or group of ants, which may be considered an authority. It is possible that the ants may take some clues from their surroundings [22]; however, in this paper we study the formation of a trail network independent of these effects. To model the collective motion in the raiding column we take the hypothesis that there is no preset or ‘inherited’ pattern followed by the ants, and further, that there are no ‘leader’ ants to be given the task of deciding the direction of the day’s pillage. In fact, an amusing characteristic of the ‘front line’ of the raiding column is that any ant who discovers herself to be on unexplored territory will immediately ‘about face’ and attempt to reenter the swarm. The column itself is maintained not by any central intelligence on behalf of the upper caste, but by the determination of individuals to push forward and follow their reluctant leaders [21]. ‘The number of ants in a. swarm is species dependent, this figure is for Dorylus (Anomrna) wilverthis. (see Schneirla [29]) 1  A typical scenario has the ants forming a bivouac at night, linking hundreds of thousands of workers around the queen and her brood. The bivouac begins to break apart towards morning as the light levels increase [21]. Pressure from this dissolving mass of ants then forces the outer ants away from the nest site. Observations of Schneirla [29] on the species Eciton burchelli, a new world ant, suggest that these ants begin a radial expansion which grows fastest in the direction offering the least resistant terrain. After about 30 minutes a trunk trail will have developed as the ants attempt to remain together (see figure 1.1). Here we have the first indication that the motion is not simply due to diffusion, since the regions in which travel is difficult, or prey is scarce, are abandoned entirely. If the motion were simply diffusive, these areas would be explored, albeit on a slower time scale. After an hour a typical swarm pattern has formed (figure 1.2), and a few of the ants begin to return along the newly formed trails laden with pieces of their favourite prey. The fully developed E. burchelli swarm is composed of a loose network of trails 10 to 15 meters wide, and about one meter deep. It will progress at a fairly steady rate of 10 to 12 meters per hour. Behind the swarm, the trail network gradually converges to a single trail trunk. The swarm front expands quickest into areas which are rich in booty, and those which present level, unobstructed terrain. Thus, the swarm does not simply sweep forward, but will veer left and right into these preferred areas as it proceeds. As pressure from the unfolding bivouac increases, the swarm front will sprout subswarms from the flanks. Some of these will grow, and some will perish, depending on the terrain and the abundance of prey. For the species Eciton hcimaturn, the swarm front will be several orders of magnitude smaller, and the raid will propagate as a narrow column rather than the broad fan of Eciton burchelli [29]. Ants moving along the trail trunk have been clocked at speeds varying erratically between 3 and 5 cm/s while heading away from the nest. Those returning to the nest with food proceed at a steadier pace usually in the range of 4 to 5 cm/s. Note that this is an order of magnitude larger than the aforementioned column speed (10 meters/hour  2  3 mm/s).  Ps I 2  3  4  Figure 1.1: The development of the swarm: (a) initial radial expansion as the ants begin to leave the bivouac; (b) the formation of a truck trail; (c) the fully developed swarm. Taken from Schneirla [29].  3  Ec/fon homo turn  Ec/fol?  Column Raid  burche///  Swarm  Raid  raid front ‘  ..:  .  swarm  c  S  F /  1 1  booty cache  (  7’  booty cache  (..r column  b ivouoc\ 5 meters  bose column  5melerg  bivouac  Figure 1.2: Typical raiding patterns of army ants: (left) E. hamaturn during the nomadic phase; (right) E. burchelli. Taken from Rettenmeyer [28].  4  1.2  Organization in Social Insects  Camazine [7] and Deneubourg et al. [10] have both presented arguments suggesting that organi zational phenomena can be broken into two broad classes. As one extreme the society employs a central scrutinizer to oversee the entire organizational process, and at the other they rely on a large number of simple automata in a parallel arrangement. In the first case, commonly referred to as the Blueprint Hypothesis , there is a requirement for a complex algorithm, and 2 a higher level of information processing. Unfortunately, such systems are in general not robust to changes in the environment, and are known to be inflexible to changes in the needs of the society. The second case has long been known a.s Self-Organization [7, 8, 11].  Under the self  organization hypothesis there is no need to relay all information to a central processing unit for decision making. This in turn leads to a system that is more robust to changes in the infor mation supplied to the system. Since all the processing is done on the level of the individual, and in a parallel arrangement, the algorithms ca.n be far simpler. in addition the information required by the system can be both localized and unrefined. If any mistakes occur in the algo rithm of a single individual, this will have little effect on the operation of the entire system. In fact, if there is a. small amount of randomness in the behaviour of the individuals, such systems will lead to an optimal solution even in a changing environment [10]. Several examples of the possible use of this system by ants are given in [2, 5, 8, 9, 10, 11], and for honey bees in [7].  1.3  Communication Through a Trail Network  The dominant form of communication and orientation along the trails is through the use of chemicals. The chemicals used, and the complexity of the communication varies a great deal between species.  No less tha.n ten anatomical structures have been noted as being used in  0n a philosophical note, it is possible that the term ‘blueprint’ does not convey the correct meaning. We 2 are not begging the question of what gives rise to the pattern, we are distinguishing between two distinct levels of intelligence possible in the system. In the blueprint hypothesis, decisions are made by a well defined central intelligence; whereas iii the self-organizational hypothesis, the apparent intelligent behaviour of the system has no tangible source, but arises out of the coordinated efforts of a large number of individuals.  5  depositing trail indicators. In addition, as many 14 chemicals can be used as indicators in a single species [21]. Fortunately, in the large colonies typical of the army ants the means of communication appear to be simpler. The species Eciton hamatum use only excretions from the hindgut [21]. Although many species of ants will deposit pheromone only as a periodic trail marker, using several different combinations in accordance with some measure of worth of the booty they have fouiid along a particular trail, army ants are known to lay pheromone in a continuous track both when foraging and when returning to the nest [3, 12]. Although it has also been noted that many differences in the collective motion of a group can be attributed to its size [14], as well as the type of food foraged [12], we will concern ourselves herein only with simple communication through the use of chemical trail markers. In this manner we attempt to determine the dependence of the collective motion on the parameters describing this communication. According to Calenbuhr [5], ants use a bi-receptor mechanism to determine their motion relative to the pheromone trail. This allows them to detect not only the local concentrations of pheromone, but to determine the local gradients in this concentration as well. As a result, the ants move in a continual weaving fashion in and out of the trail (see figure 1.3). There is evidence that the resolution degrades at higher concentrations due to saturation of the antennae [6]. This means that two trails, each marked with a large level of pheromone, will appear to the ant to be equal in concentration; whereas, if the two trails are marked with pheromone in the same relative proportions but at lower concentrations, the ant will be able to distinguish between them. The ability of the ants to detect and follow trails is not perfect; thus, they will follow a given trail for a finite distance, and this distance will be chosen from a random distribution with a certain mean and variance. It has been noted by several authors (see the discussion in Calenbuhr [5]) that the mean distance that an ant will follow a trail increases with the concentration of pheromone on that trail.  6  1’il Figure 1.3: Adapted from Schneirla [29]. The ant will use it’s antenna to detect the stronger of two trails. As the ant attempts to follow a single trail it will weave in and out of the pheromone track. 1.4  Using Simulations to Model Trail Formation  With problems involving simple interactions between many similar units, it is useful to study the formation of the pattern using a simulation. In our case we are dealing with the interactions of large numbers of ants through a chemical trail network. Although a mathematical model of the motion has been derived in two dimensions, it is difficult to analyze except for special cases. Using a simulation gives us the advantage of being able to explore the development of the network in two dimensions without being forced to make specific assumptions. Furthermore, the results of the simulation are displayed in ‘real time’ on the computer screen. These visual images provide a excellent means to both study the problem, and convey the results to a wider audience. We may expand on the initial hypothesis of the model and view the results of various assumptions on the formation of the trail network. This can sometimes be done more easily, 7  and often far faster by using a simulation than by using a more rigorous mathematical analysis. The insight provided by this experimentation can also be used to direct a further, more detailed, analysis of the problem. Such an analysis will still be necessary as it is not possible, using the simulation, to place definite bounds on region of parameter space in which different results occur.  1.5  Overview  In section 2.1 we construct a set of simple rules governing the response of individuals both to each other, and to the trail network. In section 2.2, these rules are incorporated into a computer simulation. In section 3.1 the rules mentioned above are used to derive a system of partial differential equations (P DEs). However, due to the complexity of the analysis of a model with two spatial dimensions, we limit our analysis to a model with only one spatial dimension. This model represents the propagation of the leading edge of the trail network.  In section 3.2, using  various techniques of applied analysis, we simplify the PDEs to a system of ordinary differential equations (ODEs) whose solutions represent travelling wave solutions of the original PDEs. In section 3.3, these ODEs are then studied analytically, and we are able to show how the speed of propagation of the raiding column depends on the parameters of the model. Section 3.5 deals with generalizations on this model which provide us with a method for relating the motion resulting from our rules to similar motions caused by other mechanisms.  8  Chapter 2  Mechanics of Motion  2.1  Rules of Motion  To model the behaviour of the swarm using a system of automatons, we develop a set of  rules or algorithms to be followed by each individual. As noted in section 1.2, one advantage of self-organization is that it is flexible, and requires little processing on the part of each individual. Hence, our rules are simple, and yet applicable over a wide range of settings. Also, although it is clear that external conditions such as food distribution and terrain are important in determining the pattern of the swarm, we focus on the dependence of the pattern on the behaviour of the individual ants. Hence, our rules will depend not on external constraints, but on the parameters governing the behaviour of the individuals. The computer simulation is an ideal environment for this type of experimentation. In any real system, it is generally difficult to separate the subject from its environment. Many physiological experiments can be performed on an individual; however, it becomes increasingly difficult to design experiments if  one’s interest is in the interactions between many individuals. With the above points in mind, and using the observations of section 1.3 as a guide, we propose the following rules of motion: 1. The automatons (ants) move at a fixed speed. 2. As each ant moves it deposits a trail pheromone in its path. This deposition will occur regardless of whether the ant was following a trail, or searching randomly. 3. The trail pheromone evaporates at a steady rate.  4. If an ant is following a trail there is a probability that it will lose the trail. When the ant loses the trail it becomes ‘lost’ (see rule 7). 9  5. If an ant following a trail comes to a point at which two or more trails intersect, there is a probability that it will turn and follow the new trail. This will be modelled as follows: i) If there is a trail pheromone directly in front of the ant, it will continue moving hi its current direction. ii) If there is no trail pheromone ahead of the ant, it will move either to the left or to the right depending on which side has the stronger trail. iii) If both the above tests fail, or if the trails to the left and right are of equal strength, then the ant will move a.s if it were lost (rule 7). 6. If a lost ant encounters a trail, there is a probability that it will turn and follow it. This probability will depend on the density of pheromone on the trail, and the angle the trail makes with the ant’s current direction of motioii. If the ant does not follow the trail it will remain ‘lost’ (see rule 7). 7. If the ant is lost, it will move randomly. However, this random motion will have a certain directional persistence (see below). These rules are meant only as a. starting point for the simulations. They are flexible, and can be generalized to include many types of interactions. For example, the rate of trail deposition need not be constant. Also, the probability that a lost ant turns to follow a trail may depend not only on the concentration of the trail pheromone, but also on the density of ants following the trail. We have purposefully kept the behaviour of the ants simple in order to minimize the parameters involved in the initial model. As the model is studied, further parameters may be added, and a more general model developed. The above rules were used to design an algorithm for the computer simulation. The simu lation begins by releasing ants one at a time from a central nest location. Each ant is given a position and a velocity. This velocity is composed of a speed, which is assumed to be constant, and a direction. At the beginning of each time interval each ant takes a survey of its local envi ronment, and chooses a new direction depending on the local concentration and concentration 10  gradient (computed by discrete differences) of trail pheromone. If there are no trails at the ant’s position, or if the ant fails to detect a trail, then it will choose this new direction randomly. However, as mentioned in rule 7, the random turn will have a certain directional persistence. Each ant then deposits a pheromone droplet at its current position, moves one step in its new direction, and assumes a new position. At the end of each time interval the concentration of trail pheromone at each position is decremented by one unit. The behaviour of each individual is captured by the following parameters: the affinity of the ants to the trails, the directional persistence of the random turning, and the rate of trail deposition. If the ants have a higher affinity to the trails, they will tend to remain on a trail longer. A higher directional persistence indicates that the ant will tend to continue in a single direction for a longer period of time. Likewise, a lower persistence indicates that turns will occur more frequently. These parameters are explained in more detail in appendix D. Several other parameters have been included in the simulation, but have not as yet been fully explored. We will discuss these in chapter 3.5  2.2  Results of the Computer Simulation  Several runs of the simulation were performed for various values of the parameters. The key results of the runs can be summarized as follows: • If there was a strong directional persistence to the movement of the lost ants, then a tendency for a few dominant trails to form was observed. If, on the other hand, the ants had lower directional persistence, then a denser network of trails generally formed (see figure 2.2). • The ratio of followers to lost ants did not depend on the directional persistence of the lost ants. • The ratio of followers to lost ants decreased as the affinity of the ants to the trails de creased. 11  • There was a slight increase in the number of trails in the network, and a drop in the strength of the trails formed as the affinity of the ants to the trails was decreased. This led to a denser network of weaker trails (see figure 2.3). • The density of the network increased as the rate of trail deposition decreased (see fig ure 2.4). Each of the runs of figures 2.2 to 2.4 was of the same length. In figure 2.1 we compare the results of two runs with the raiding patterns of figure 1.2. This shows that by varying the three parameters in concert, we can shift from the observed raiding pattern of E. burcheili to that of F. hamaturn. That is, the pattern shifts from a highly branched, area.  covering fan to a  more tree like structure. It is possible, then, to achieve the  range of patterns observed on the field by varying the parameters governing the behaviour of the individual, without explicitly including a dependence of the behaviour on precise locations of food, or other spatial information.  Deneubourg [12] suggests that variations in the fowi  distribution might be responsible for differences in the raiding pattern. Our results do not in fact disagree with this proposal. We will discuss in the chapter 3.5 how variations in the surroundings ma affect the parameters governing the motion of the individuals. For example, a homogeneous food distribution may lead to a high degree of random motion, whereas a patchy distribution may lead to a more directed searching behavior. Figure 2.5 shows the development of a stable raiding column. This pattern was observed occasionally for a high affinity, a strong directional persistence, and a narrow range of deposition rates. In general, the results of the simulation show that a network of trails can form for any set of parameter values. By varying individual parameters we observe a shift in the resulting pattern from trails which are strongly marked and carry a high proportion of traffic, to weaker trails, and from a dense trail network to a loose one.  12  (a)  (b)  -3  (c)  (d)  Figure 2.1: A comparison of the results of the simulation to field observations: (a) observations of an E. burchelli swarm; (b) results of a simulation with ‘ = 255, and r = 12; (c) d = .749, 4 observations of a E. harnatum swarm; (d) results of a simulation with = .925, = 255, and r = 6. The pattern has shifted from a highly branched network in which the central trails lead to a large area (b), to a network with fewer dominant trails which concentrate activity over several smaller areas (d). 13  .  .  Figure 2.2: The dependence of the pattern of the trail network on the forward persistence of the lost ants: (top) the results of a simulation with a high forward persistence (d = .925, = 255, and T = 12); (bottom) the results of a simulation with a low forward persistence (d = .749, = 255, and T = 12). An increase in the density of the network is observed as the forward persistence is decreased. 14  a  4  ..  I  S  m  SI  s.  *  Figure 2.3: The dependence of trail network on the affinity of the ants to the trails. (top) the results of a simulation with a high affinity (Wd = .749, = 255, and r = 12); (bottom) the results of a simulation with a low affinity ( = 245, and r = 12). d = .749, 1  15  (a)  (b)  a  a a  (c)  Figure 2.4: The dependence of trail network on the rate of trail deposition: a) the results of a simulation with a low rate of deposition (d = .749, = 255, and r = 3; b) the results of a simulation with a medium rate of deposition (‘I’d = .749, = 255, and r = 6); c) the results of a simulation with a high rate of deposition (‘I’d = .749, = 250, and T = 12).  16  4.  4,v  I  S  4  S  S  •  44 4.  V time  248  -  4•  4.  S  4.  S 4.  4.S  S  I  S  time  328  4.  V 4.  —4 4.  4.  S  S  *  I  time  395  %  4.  I  Figure 2.5: Simulation results showing the formation of a trunk trail: .238, = 250, ‘Pd and r = 3. The positions of the ants have been included for clarity. Such trunk trails occur occasionally in a. small region of parameter space. In a. larger region of parameter space, a trunk trail may form, but not persist.  17  Chapter 3  A Model in One Spatial Dimension  To gain a more complete understanding of the effects of the behaviour of the individual on the resulting pattern, we will derive and study a simple mathematical model of the group’s motion. The derivation is based on the velocity jump process of Othmer et al. and  [25]. In sections 3.2  3.3, using various techniques of applied analysis we examine special solutions of this  model known as travelling waves. These can be studied qualitatively using the state space of an associated dynamical system. The stability of the travelling wave solution is examined numerically in section 3.4. Finally in section 3.5 we discuss possible extensions to the model, and examine -the general class of equations to which our model belongs. 3.1  Derivation of the Model  Consider again the trails formed in figure 1.2, and their development in figure Li. We may develop equations for the full model in two spatial dimensions; however, this proves to be a difficult model to analyze. We instead consider how the parameters of the model may effect the propagation of the trail network in a single dimension. Let x E (0, oc) represent a distance from the nest, measured along the dominant direction of the motion of the swarm. Note that this is not equivalent to assuming the motion is one dimensional, but merely integrates the variations in density across the dominant direction of motion. We then study the advance of the swarm into an unexplored region. The ants are divided into two groups. Let F(x, t) represent the total density of ants that are currently following a trail, and let L(x, t) represent the total density of ants that are lost and laying a new trail. Let T(x, L) dx be the total length of trails between x and x+dx. Although the  18  trails are not oriented, the ants may move along them in either direction. Assume that the ants move at a constant speed; we denote this speed s for the lost ants, and v for the followers. To distinguish between the two directions of motion, let F+(x, i) and L+(x, t) denote the densities ants moving away from the nest (with velocities +s, and +v), and F(x,t),L(x,t) denote the densities of ants moving towards the nest (with velocities —s, and —v). Pfistner [26] uses a similar basis for a model of swarming in myxobacteria. As stated in rule 2 of section 2.2, each of these ants continuously secretes trail pheromone at a constant rate. Since this rate could be different for followers and lost ants, we define the deposition rate for the followers, and  TI  Tf  as  as that for the lost ants. This deposition produces  a trail which evaporates at a constant rate -y. We may write the governing equation for the evolution of the trails as  a  jT(x t)  =  — T 7 (x, t) + TfF(x, t) + r L(x, t). 1  (3.1)  Since the ants are not perfect followers, they have some probability per unit time of hecoming lost ants (see rule 4 of section 2.1). To model this we introduce the parameter  E  as the rate  at which followers become lost. Those ants which remain following the trails may reverse their direction. Let p+ be the rate at which followers moving away from the nest turn towards the nest, and let p be the rate at which followers moving towards the nest turn and begin moving away. The rate at which lost ants find trails (see rule 6 of section 2.1) will increase with the local density of the trail network. Let c be the rate at which lost ants encountering trails turn and follow these trails. As an ant encountering a trail can follow it in either direction, let  be  the probability that an ant ‘choosing’ to follow a trail reverses its direction. The rate of change of the density of followers at any given point (x, t) is then governed by the equations OF  F+  8F-  =  —cF + c(1  =  —cF  —  /3)LT + a/3LT  —  pF + pF  (3.2)  0F —  + u/3LT + c(1  —  /3)LT + pF  —  pF  (3.3)  If we assume that the ants following a trail have some sense of which direction leads to the 19  nest and that they prefer to travel away from the nest, then we can assume that p is much larger than the other parameters in the system. Equation 3.3 then implies that F(x,t) decays exponentially on the order of F(x,0)e’. The assumption may not be valid in the later stages of the swarm; however, for the initia.l stages, when there is a large outward flux of ants from the unfolding bivouac, it. will hold. Adding equations (3.2) and(3.3) produces 8F  a +  F)  -  to be exponentially small, then F+  If we assume F  aF -—-  at  8F + v—  ax  =  =  —  -cF + aLT F+  F  (3.4) F, and equation 3.4 becomes  -eF + aLT.  (3.5)  To model the motion of the lost ants, assume that the they reverse their direction at random times chosen from  a Poisson distribution with mean \.  That is, their movements will be  composea of runs at a constant spend either towards,. or away from the nest, se-arate-d at random intervals by 180° turns. As with the lost ants which become followers, those followers which become lost may either continue moving in the same direction, or reverse their direction when they lose the trail  .  Assume that the same constant  governs this probability. We can  now write the following equations for the mean rate of change of the lost ants with time: a a L(x,t) + s—L(x,t)  e(l  =  —  a —  a s—L(x,t)  —  /3)F(x,t) + e/3F(x,t)  aLT  —  AL(x,t) + AL(x,t)  c/3F(x,t) + e(1  =  —  (3.6)  —  /3)F(x,t)  aLT + AL(x,t)  —  (3.7)  L(x,t)  Adding equations (3.6) and (3.7) we obtain a  a + s—(L  —  L)  20  =  eF  —  aLT,  (3.8)  and subtracting them, 0 —  8 L) + sj_L  =  —  2i3)(F  —  F-)  —  —  LiT  —  2A(L  —  Li.  (3.9)  To eliminate the terms involving L± and F± in the last two equations we first differentiate (3.8) with respect to t, and then multiply (3.9) by s and differentiate the result with respect to x. Taking the difference of the resulting equations yields L 2 0  L 2 9 —  8 = —  (F  —  nLT)  0 + sb— [E(1  —  —  8 2As—(L  2/3)(F  —  Although this equation is hyperbolic, if we assume that  F-) s2  —  —  >>  L)  a(L  (3.10)  —  L)T].  1 and that A >> 1 then the  equation can be approximated by the parabolic equation (32/2x)  L 2 0 =  0 s—(L  —  17).  (3.11)  This is equivalent to observing the motion of the ants on a time scale which is larger than the rate of reversal of the lost ants, and on a spatial scale which is much smaller than the distance a lost ant would have travelled if it had moved at a constant speed s without reversing. This is known as the diffusion limit and is discussed in more detail in Othmer et al. [25]. Finally, using (3.8) we replace the flux term on the r.h.s of the above equation by quantities involving only the total densities F,L, and T. This yields the following interaction-diffusion equation for L: OL  2  L 2 0  =  -  eF + aLT.  (3.12)  Collectively, equations (3.1),(3.5), and (3.12) represent the evolution of the trail network in the case where the frequency of the random turns of the lost ants (A), and the rate at which ants following trails towards the nest turn away from the nest (pj are large relative to the remaining parameters of the system.  After deriving these equations from a simple 21  random process (microscopic), it is useful to discuss the physical and biological significance (macroscopic) of each term. For convenience we repeat the complete system below.  IJt  =  L 1 T  +  Tf F  (i) = --  DL =  (3.13)  —  (ii)  a  —----(vF) —€F + aLT} (iv) (ni) L 2 6 ---+eF-aLT.  (3.14)  (3.15)  (v) T represents the total length of trails per unit length of the strip (0, oo), and F and L represent the total densities of ants moving left (towards the nest) and right (away from the nest) respectively. The parameter  j =  is known as a motility coefficient, and represents  the diffusivity of the lost ants due to random motion. The terms over brace (i) depict the reinforcement of trails by each ant, and term (ii) the decay due to evaporation of the trails. Term (iii) of equation (3.14) represents the directed motion, or convection of followers along the trails at the velocity v. The terms grouped under (iv) which appear in equations (3.14) and (3.15) represent the rate,  E,  at which followers lose the trail, and the rate, a, per unit length of  trail, at which the lost ants begin to follow an existing trail. Finally, the exploratory motion of the lost ants is represented by the diffusion term (v) with a motility coefficient ii. A further discussion of the general form of these equations appears in section 3.5 For the remainder of this chapter we will be interested in finding solutions to equations (3.133.15) which represent the propagation of the swarm. One distinguishing trait of any swarm is that it maintains a semi-rigid profile as it moves. Okubo [24] gives an excellent account of the differences between coherent swarming motion and simple diffusion, and describes the forces that are necessary to maintain the swarm shape. Solutions representing the movement of a 22  fixed profile are known as travelling waves. For examples of such waves and their applications to biology and chemistry see [15, 23].  3.2  Reduction to the Travelling Wave Form  In the following analysis we assume that each of the coefficients in the system (3.13  -  3.15) is  a constant. Deviations from these assumptions will be discussed at the end of this chapter. A first step in the analysis is a reduction of the equations to a dimensionless form. The details of this are left to appendix A. Here we merely note that upon introducing the rescalings:  v =  =  T  2  au —T  —  -  =  —  ‘  —t, II  F*  EIL  —i-F, -‘IV  =  V —x, /1  —,  2 V Tf  aILTj  L*  —L, 2 -yv  =  (3.16)  r”  =  —.  Ti  and dropping the *‘s, we obtain the dimensionless equations: T  =  -y(L + TF  F  =  —F—EF+LT,  (3.18)  =  L + eF  (3.19)  —  —  T).  (3.17)  LT.  We are interested in examining solutions to this system which represent waves of a popula tion propagating into an unexplored area. These solutions are characterized by a fixed profile moving at a constant speed  c.  Such solutions are known as travelling waves, and are studied  by transforming the system to the moving coordinates  z =  x  —  ct,  and t’  =  t. The steady state  solutions of the PDEs in these new coordinates, found by setting the derivatives of T, F, and L with respect to the new time t’ to zero, correspond to waves with a fixed profile moving at a  constant speed c in the original coordinate system. These steady state solutions will satisfy an autonomous system of ODEs in the variable  z.  23  To represent the system (3.13-3.15) in the moving coordinates we use the chain rule of calculus:  OF  OFOz  OFOt’  OF 9z  OF at’  —  (3.20)  OF =  Oz  at’ and  are found by again applying the chain rule to the transformations z(x, t) ax Ox and t’(x,t) = t: —  dz  =  dt’  =  —dx ax at’ -—--dx ax  +  +  az —dt at at’ —dt at  az —=1, ax at’ = 0, ax  dx—cdt dt  =  —  =  az —=—c; at at’ = 1. at  x  —  et  (3.21)  —  If we substitute these results back into (3.20) we find that:  OF a2 OF  OF ax ‘ OF  —  (3.22)  OF  Using these results we transform (3.17-3.19) to the moving coordinates, and set  aF  aT  -b—-  =  OL =  0, and  =  0. This produces the following system of ordinary differential equations:  (1  Here, the  ‘  —  —cT’  =  -y(L + rF  c)F’  =  —cF + LT;  —cL’  =  L” + F  —  —  T);  (3.23) (3.24)  LT.  (3.25)  indicates differentiation with respect to the wave variable z.  These equations can be further simplified by the following operations. First, add equations (3.24) and (3.25) to produce the equation (1  —  c)F’  —  24  cL’  =  L”.  (3.26)  This equation can be integrated once from —oo to z to give (1— c)F—cL  =  (3.27)  L’+k.  The constant of integration, k, is given by k  =  0 cL  —  (1  —  c)Fo + (L’)o,  where the subscripts’ indicate evaluation at the point z  =  (3.28)  —.  Replacing equation 3.25 with equation 3.27 leaves us with an autonomous system of first order ODEs. Such a system is commonly referred to as a dynamical system. The analysis of these systems is introduced for applications to travelling waves in [15, 23], and in a mathematical setting in [20, 31]. Since the system of ODEs is autonomous, its solutions can be represented as curves in the three dimensional state space (T, F, L). In addition, at each point in this state space there is an associated vector (T’, F’, L’) which is tangent to the solution curves passing though that point. A fixed point is defined to be a point where the tangent vector (T’, F’, L’) vanishes. A homogeneous spatial distribution which is a steady state solution of the equations in the original coordinate system, (3.13-3.15), appears in the state space as a single fixed point. We can study the shape of solutions to the PDEs by examining the trajectories of the solution curves in the T-F-L state space. If these trajectories are to represent a wave profile traveffing through a population, then they must satisfy the following criteria: • Populations must remain bounded. Hence, the trajectories are restricted to homodinic orbits (closed curves passing through a single fixed point), heterodinic orbits (curves connecting two fixed points), or limit cycles (closed curves which do not pass through any fixed points). • A population density must remain either positive or zero. Hence the trajectories must be contained in the positive octant of the state space. The use of zero as a subscript will become apparent soon. 1  25  • The waves represent a swarm propagating into an empty region of (physical) space. Hence the trajectory must end at the origin (T, F, L)  0.  =  Further, the origin represents a  homogeneous spatial distribution (no ants, and no trails) and must therefore be a fixed point. With regards to the variable z, the solution will asymptotically approach (0, 0, 0) as z  cc. This fixed point will be referred to as the trivial steady state, and denoted by  U’.  • Finally, we expect that some fixed density is established behind the wave; hence the trajectory must originate at a fixed point in the state space. With regards to the variable z, the solution vi1l asymptoticajly approach this fixed point as z  —  —cc. This fixed point  will he referred to as the populated steady state, and denoted by u 0 Thus the travelling wave of interest will he represented in the state space by a curve connecting a point in the first octant to the origin. An inspection of (3.27) shows that the origin u , will be a fixed point if and only if k 1 In addition, if reduces to cL 0  0 it  is a fixed point then by definition (L’)o (1  —  =  =  0.  0. With these simplifications (3.28)  )T, there will be a 0 c)Fo. Thus, given any initial steady state (To, F ,L 0  unique wave speed 0 F =  ) 0 (L + F  (3.29)  for which the final state will be unexplored territory. We have now reduced the original system of PDEs to the following system of first order ODEs:  TI  F’ L’  =  7 —[T  =  (l_C)[+LT];  (3.31)  =  (1  (3.32)  —  —  L  c)F  —  —  TF];  cL.  The solution of interest satisfies the boundary conditions  26  (3.30)  T  0 T  F  =uo,  0 F  =  L  0 L (3.33)  T  1 T  F  0  1 F  =  L  0  =  1 L  +  and the restriction that T, F, L he positive for  ti,  0 <  —00  z <  +00.  Any solution to this system will  be a solution to the original system of PDEs which represents a travelling wave propagating from the populated steady state u 0 into the trivial steady state u 1 with a velocity c. Setting tile i.h.s of equations (3.30-3.32) to zero yields the system of equations for tile fixed point s: T =  (1  —  c)F  L+TF;  (3.34)  LT;  (3.35)  cL.  (3.36)  There are only two points satisfying these constraints. One of these is the origin, and the other,  which must be u 0 if the travelling wave is to exist, is given by (C  1—c (C  =  (1  —  c)(i  —  (337)  c + rc)  (C  1 c + rc Note that since u 0 lies in the positive octant of tile state space, —  we mllst restrict tile values of  c so that 0 < c < 1. Recalling that we are dealing with a dimensionless system, tills  15  simply  a statement that the wave propagate forward, and that tile speed of propagation of the swarm front be 110  faster that the velocity v  of the followers in the original coordinates.  27  3.3  Analysis of the State Space  The problem of finding travel]ing wave solutions to the initial system of PDEs has now been reduced to that of finding bounded, positive solutions to the system of three first order ODEs. To show that such solutions exist will require a further examination of the state space of the system. First, we perform a local analysis of the fixed points u 0 and u. This will provide us with information on the behaviour of the dynamical system near these two points. Next, we perform a global analysis to determine the nature of the solutions in regions containing both the fixed points. In this manner, we establish conditions for the existence of solution curves connecting these two points.  3.3.1  Local Analysis of the Steady States  Linearizing equations (3.30) thru (3.32) about a point  (T,F,T) yields the system:  L  T-T  F’  (i-c)  (1-c)  =  (3.38)  (1-c)  1—c  0 Substituting (T, F, L)  F—F  —  —c  L—L  (0, 0,0), it follows that at the fixed point u , the linearized system has 1  the eigenvalues:  3 A  =  7/c;  =  —E/(1 ,  28  >0 —  c);  < 0  <0  (3.39)  and eigenvectors: (1—c) + 2 r(c(1—c)—€)  1  7 —  i—C+Ec/7  =  0  C  C  —  (3.40)  0  i—c 0  1—c  —+c C  Since 7, c, c > 0, we have that A indicates that  u  >  ,A 2 3 < 0 over the entire parameter space. This 0, and A  is the intersection of a two dimensional stable manifold Wj , and a one 9  dimensional unstable manifold 1’V. At the populated steady state u 0 the eigenvalues of 3.38 are roots of the cubic equation  2 + 3 A + BA+C=O, AA  (3.41)  where  A  =  c+  1  —  7/c,  c  I’  B  =  —  I  (3.42)  €  I +  c(1  —  C  +  TC)  I  (3.43)  (7 =  (3.44)  i—c  Following the technique of Derrick [13] outlined in appendix B, we can use the fact that B < 0, and C > 0, for all values of the parameters to show that two of the eigenvalues will have real parts greater than zero, and the third will be real and negative. Thus, although it is not a simple task to determine the exact elgenvalues and eigenvectors of the populated state for general parameter values, we can assert that there will always be a two dimensional unstable manifold W’ and a one dimensional stable manifold W intersecting at  . 0 u  Further, a detailed  analysis (see Appendix C) of the local dynamics near this fixed point shows that both of the unstable eigenvectors must point into the region of phase space where each of the derivatives T’, F’, and L’ are negative, and that each of their components must be negative. 29  We now know the behaviour of the vector field, and thus the dynamics of (3.30-3.32) near the fixed points of the system We also know that the system has only two such fixed points If there is to be a trajectoiy tiavelling fiom u 0 to u 1 it must be contained in both the unstable manifold of Uo, and the stable manifold of tt. To show that such a trajectory exists will require a knowledge of the global behaviour of the dynamical system. 3.3.2  Global Analysis  For systems of two ODEs, a qualitative analysis of the global structure of the state space can often be performed quickly and easily. In contrast, our system consists of three ODEs, and such analysis proves more difficult.  By a careful examination of the geometry of the state  space and the flow of the associated vector field, and by using some numerical experiments we are able determine conditions under which the existence of the desired solution is guaranteed. Furthermore, we can determine the shape of the wave, and the effect of the parameter values on that shape.  -  To begin, we note that the heterodinic orbit must be contained in both Wj , the stable 9 , and T’V’, the unstable manifold of ‘u 1 manifold of u . Conversely, since there is a unique 0  solution passing through each point in state space which is not a fixed point, if W intersects with W at any point then this is sufficient to guarantee the existence of the heterodinic orbit. We show that W will fail to intersect W only if there is a separatrix 2 that bounds W. However, numerical experiments gave convincing evidence of a structurally stable heterodinic orbit for each set of parameter values tested, and so we suspect that such a separatrix does not arise in this system. Let T°, F°, and L° denote the nuhisurfaces of the vector space. These are the surfaces where the derivatives of the state variables vanish, and are given by the following equations: T°  =  { (T, F, L) I T  —  L  —  TF  =  0 };  (3.45)  A solution that is topologically different from neighbouring solutions. In JR 2 3 this may be a family of such solutions (see [4, 20])  30  F°  =  { (T, F, L) I  L°  =  {(T,F,L) (1— c)F— cL  LT  —  eF  These nulisurfaces are shown in figure 3.1. Let T+ and T  =  0 }; =  (3.46) 0}.  (3.47)  be the regions in which T’ is positive  and negative respectively. These are the sets defined by: T  =  {(T,F,L)IT—L—rF>0};  T  =  {(T,F,L)IT—L—TF<0}.  The sets F±, L± are defined in a similar fashion. Using the directions ‘left’, ‘right’, ‘up’, ‘down’, ‘ahead’, and ‘behind’ defined in figure 3.1, T  lies above T°, F— lies below F°, and L  lies to  the right of L°. Finally, we define the following sets (as illustrated in figures 3.1 and 3.2): Q1  =  {(T,F,L)IT,F,L>0};  111  =  {(T,F,L)IF=0, T,L>0};  112  =  {(T,F,L)IT=0, F,L>0};  =  F+flL;  (3.48)  FflL; B  T°flA;  E  T°flW.  The region Qi is the first octant of the state space. 111 is the back wall of the first octant, and 112 the floor.  is the region in which both F and L are increasing, and is located above  F° and to the left of L°. A  is the region in which both F and L are decreasing. It is located  below F° and to the right of L°. The  regions  B and E lie in the plane T° as shown in figure 3.2.  To analyze the flow of the vector field in the state space, the following observations are necessary: 1. The flow through the surface F°flT flL  (the portion of F° to the right of L° and  below T°), is oriented from F+ into F—. That is, it passes through F° from above. 31  back  left  T-axs forward  right  F° T0  down  —  L-axis  F-axis  I  —  nulisurface intersecUons F-L .:.:::.:...  T-F  T-L (hidden)  Figure 3.1: The State Space (Octant I). showing the nulisurfaces T°, F°, and L°, and the regions A±, 1 and 2•  32  2. The flow through 11’ is into Qi (forward), since F’> 0 on 11’. 3. The flow through 112 is out of  Q  (down), since T’ > 0 on 112.  4. The flow through L° is from L into L  (left to right) below the F° surface, and from  L— into L+ (right to left) above it. To see this note that, since L° is a vertical plane, the direction of flow across this plane will be determined by the sign of F’. From these points we can conclude that any trajectory initiated in A— flQi flT—, that is, the portion of the first octant below T°, below F°, and to the right of L°, must exit that region through either B, or 112. this follows since the flow on all other boundaries of this region is directed into this region. Further, a trajectory cannot remain in this region since at each point, the flow velocity is bounded away from zero, and each component of the flow is decreasing. Thus, not oniy are there no fixed points, but there are no closed curves (or limit sets) contained entirely in this region. The unstable manifold W’ must pass through u 0 tangent to the unstable eigenspace of the linearized system near u 0 (see Wiggins [31]). By the analysis of Appendix C, the unstable eigenvectors of this system are directed into the sets A— and A+. If we consider the local unstable manifold as a disc tangent to W’ spanned by these unstable eigenvectors, then this disc must be imbedded in the state space near u , and intersect both A+ and A—. This immediately 0 shows that the intersection of T’V’ with T°, which is the intersection E, will be non-empty. Figure 3.2 shows the intersection of T° with  Q  as viewed from below. The intersection E  will be a continuous curve on this plane which must originate at u . By the observations of 0 the preceding paragraph, E must also originate in B. Also, a trajectory which intersects A cannot exit this region through F° unless it first passes through B. Hence the curve E cannot  W which pass through from U cross the line F° fl T°. It can also be shown that trajectories in 0 below must do so from within A—. Thus it follows that E does not cross the line L°  fl T°.  This  argument involves first showing that the portion of W’ forward of u 0 must also lie to the right of u , and then that a trajectory in this region must pass through L° into A— before passing 0  33  T° ntersect L 0  0 intersect F° T  1 U  Figure 3.2: A view of T° from below, showing the sets B, E, and the intersection of W with T°. through T°. These restrictions on the curve E imply that it is contained entirely in B. Thus, it appears that unless l’V’ is bounded in some manner, the curve E will extend to u. This would imply the existence of a heteroclinic orbit connecting  UQ  to u . 1  Now, we examine the stable manifold 1’V. From the eigenvectors given by 3.40, it follows that l47j intersects A— for all parameter values. Also, a similar analysis to that used to examine W in the above paragraph, it follows that WjS intersects F°, and L°, but cannot intersect 112 or B. Further, the intersections of T47j with L° and F° cannot rise above T°, nor can they fall below 112 (see figure 3.3). Since the flow in A— below T° is bounded and oriented downwards, we can conclude that Wj 5 must either approach arbitrarily close to u , or pass below it. 0 If W? passes below u , and does not intersect W’, then there must be a third manifold 0 positioned between u 0 and l1 . This manifold will intersect B, and prevent E from reaching u. 9 If fVj does not intersect u, it is still possible that E extends to u . In this case the heterocinic 1  will exist, and may not be unique. 1-lowever, it will not be a structurally stable feature of the vector field. Finally, if W 9 approaches arbitrarily close to u 1 , then the previous observations 0 about the location of l’V imply tl1at the two manifolds must intersect. This is an intersection of two two-dimensional manifolds, and will be structurally stable to small perturbations of the vector field. This means that such an orbit occurs at a single point in parameter space, it must  34  0 intersect T  0 intersect F° T  0 U  F°  Intersections of the stable manifold of u 1 with F . and L 0 . 0  Figure 3.3: A schematic showing the boundaries of A— and their intersections with Wj. The F° surface has been distorted for clarity. also occur at all parameter values in a neighbourhood of that point. We can now use numerical techniques to show the existence of the homoclinic orbit for specific parameter values. To do this we show that the intersection of 14’ and Wj 9 contains at least one point. As mentioned above, since the tangent vector at each point is unique, there can be oniy one solution passing through each point in the state space. By definition, the solution passing through any point in the unstable manifold of the fixed point 0 u tends to u 0 as z  —  —cc. Also, the solution of any point in the stable manifold of the fixed point u 1 tends  to u as z  —  cc. Thus, if a point x lies in both of these manifolds, then there must be a  unique solution passing through x which connects u . Numerically we can compute the 1 0 to u local stable and unstable manifolds of u 1 and u 0 respectively (that is, the eigenspaces of the linearized system), and solve the system forwards in time from u , and backwards in time from 0 . To determine if the two manifolds intersect, we compute their intersection with a plane of 1 u constant T. This intersection is shown in figure 3.4. The points represent the intersections of  35  the manifolds with a plane of constant T. The fact that these intersections cross implies that  S S —  .  .  S. • 4 ’b  • S  Figure 3.4: A section showing the intersection of W ’ and W? with a plane of constant T: 0 c = 1,7 = 4, c = .25, T = .17. The crossing of the intersections implies that W and Wj have a common point, and thus a common trajectory. The figure was created using PhasePlane [18] there is a point on the plane which is common to both W’ and Wf. In this manner we can determine the existence of a heteroclinic orbit for any choice of parameter values. Further, the intersection will persist for small perturbations from these values, and by the above arguments, the heterodinic orbit will persist as well. That is, the heterodinic orbIt is stable at least to small perturbations of parameter space. These experiments were performed on various points in parameter space, with similar results in each case. Thus we provide a strong case for the  36  existence of the travelling wave solution, if not for all parameter values, then over a large portion of the parameter space.  3.4  Stability of the Travelling Wave as a Solution to the PDEs.  Note that c is not a parameter of the original system of PDEs. This implies that there is a redundancy in the solutions to the ODEs. That is, we only require the existence of the hete rocinic orbit in the dynamical system for a single speed c given any point in 7, E, T-space. To determine the stability of the travelling wave as a solution to the original system of PDEs, we solve the full system of PDEs (equations 3.13 to 3.15) for various initial conditions. The solu tions were found using the method of lines with a fourth order implicit Runge-Ktitta algorithm. This method consists of discretizing the spatial domain of the PDEs, yielding a coupled system of ODEs which can then be solved using the Runge-Kutta algorithm. An upwind scheme was used for the discretization of the parabolic equation for F. The travelling wave was found to be stable for each case tested. Further, an oscillating wave was never observed. The results are shown in the figures 3.5 and 3.6.  3.5  Implications and Generalizations  In summary, although we have been unable to show the existence of the travelling wave solution for every parameter value, we have shown that the wave does exist at certain parameter values. Also, we have shown that it persists for small perturbations from these parameter values. Thus, the heteroclinic orbit which represents the travelling wave in the (T, F, L)-state space is structurally stable to small perturbations of the parameters. Using a qualitative global analysis of the dynamical system, we were also able to determine the shape of the travelling wave, and the effect of the parameters on that shape. These effects can be summarized as follows: 1. We noted that the speed of the wave would depend on the ratio of the density of the followers to the total density of ants at the nest. That is, a higher proportion of followers to lost ants at the nest would result in a faster propagation of the trail network. 37  SLOPED !WP1J  0.15  -  N  \  \  \.  \  \  \  \.  \,  0.10  “. — . —. —  \  C  0  Legend Followers time = 100 -time= 80 -time= 60 -time= 40 Lostants -time= 100 80 60 40 Trails time = 100 -time= 80 -time= 60 -time= 40 Lost ants initial data Followers initial data  ‘.  •\.  0  0.  -  -  0.05  -  0 0  20  40  60  80  100  distance from nest  Figure 3.5: Stable travelling wave solution:  38  =  .1, c  =  .5, and  =  1  OSC!LLA11WG !t4P’J1’  -  ?NtT1AW( LOS1  0.6  Legend followers time = 20 time = 40 -time =60 time = 80 -time = 100 lost ants -  -  -  0.4 C 0  lost ants • initial data  D 0. 0  -  0.2  0 0  20  100  80  40 distance from nest  Figure 3.6: Stable wave resulting from oscillating input: e were initially lost.  39  =  .1, c  =  .5, and y  =  1. All ants  2. The travelling wave solution represents a realistic profile of a population density; partic ularly in that it is bounded, and everywhere positive. This follows from the fact that if the heteroclinic orbit exists, it must lie in the first octant of the state space. 3. If the eigenvalues of the linearized system near u 0 are real, then the solution will be monotonic decreasing in each of the variables T, F, and L. If they are complex, then the solution will oscillate as it approaches u . The oscillating wave form was not observed as 0 a stable solution of the PDEs. 4. The monotonic travelling wave was observed to be a stable solution to the original system of PDEs. The model also shows that the density of the lost ants attains its largest value close to the nest. That is, L(z)  <  0 at any point z along the wave. It would be desirable to develop a model L  iii  which the lost ants obtain a maximum near the front of the wave. To examine this possibility further we study the class of equations intO which our model falls. The equations are part of a larger class given by the following system: =  F  G(T, F, L)  = _-  =  (vF)  —  —  T;  (3.49)  R(T,F,L);  (3.50)  L 2 0 ii----+R(T,F,L).  (3.51)  Equation 3.51 represents the growth and decay of the network. The network decays expo nentially, and its growth is governed by the two interacting species F and L according to the function G. The first species, F, is transported along the network at a velocity v; whereas the second, L, diffuses with a motility coefficient of  t.  The the interactions of the two species are  represented in the function R. Thus, the behaviour of the lost individuals is analogous to that of a diffusing substance, and the behaviour of the followers is similar to that of a substance which is being either transported or convected through a network. These two substaices can  40  be considered to interact both through and with the trail network. The network itself can be compared to a reactive medium. Using tile techniques of section 3.2 we may determine the associated dynamical system for the travelling waves admitted by the PDEs (3.49-3.51): —cT’  =  =  L’  =  G  (3.52)  —  (vF)’  —  (v  —  (3.53)  —  c)F  —  cL.  (3.54) (3.55)  From this we see that the shape of the nullsurface L° is determined by the parameter v, and that changes in the reaction term R do not affect the shape of this uullsiirface. In fact the equation R(T, F, L)  =  0 determines the nuilsurface F°.  One method of generalizing the original model is to allow the parameters to be functions of the position in the state space. In fact, the experiments cited by Calenhuhr [6] indicate that €  is an increasing function of the trail strength. Further experimentation via both numerical  analysis, and the simulation, is ullder way to determine the effects of altering the parameters in this fashion. The results of this study will be presented in a later paper. Note that the only way of forcillg the density of the lost ants to attain its maximum nearer the wave front is to change the nullsurface L°. From an examination of the generalized equations, we see that this can be accomplished only by introducing a dependence of v on the state variables T, F, and L. The 2 2 + v*F T dependarice was introduced in a numerical analysis of the PDEs; however, no = 2 + F T 2 significant difference was noted in the results. The shape of the stable wave solution remained unchanged. When generalizing equations in this fashion, one must be careful with the interpretation of the results since we have  not  maintained a rigorous connection between the behaviour of the  individuals and the parameters in the model. However, we may now note the similarities of the form of these equations with other models; two interesting examples are the motion of gliding 41  myxobacteria [26], and the transport of chemicals within cells [27].  42  Discussion  The main goals of this thesis were to study the collective behaviour of a group of interacting individuals, and to determine how the patterns of motion of the group are affected by variations in the parameters  governing  the behaviour of the individuals.  We focused on the specific  example of the propagation of a swarm of army ants. This problem was modeled by a system of automata interacting with a trail network. The behaviour of the each automaton was governed by a set of rules which were dependent only on the local density of the trail network. These rules did not explicitly refer to any external variables such as the distribution of food sources or variations in the terrain.  Rather we suggest that such variations affect the parameters  governing the behaviour of the individuals and do not directly influence the collective motion of the population. The model was studied using both a computer simulation, and a mathematical model. To visualize the motion of the swarm in two dimensions, a computer simulation of a system of automata. was developed. Each automaton was equipped with a set of simple rules which governed its behaviour, and its interactions with the network of trails.  These interactions  were represented by the following parameters: directional persistence of the ants not following a trail, affinity of ants to the trails, and the rate of deposition of new trail material. The patterns were studied to qualitatively determine the density of the trail network, the tortuosity of the pattern, and the distribution of lost ants relative to followers. At first, each parameter was varied individually to determine its effect on the resulting patteri. By varying all three parameters in concert, we were able to obtain patterns characteristic of the swarms observed in nature. Hence, we have shown that the parameters governing the behaviour of the individual are sufficient to model the problem. External influences may then be responsible for reinforcing these behavioural responses, and maintaining a stable pattern.  43  In addition to the simulation, a mathematical model was derived from the rules governing the behaviour of the individual automata. To simplify the analysis, the mathematical model examines the behaviour of the swarm in only a single spatial dimension. That is, we examine the density of both the ants and the trail network as a function of the distance from the nest. The mathematical model provides a means of determining the relationships between certain aspects of the swarm pattern and the parameters of the model. Specifically, we have shown that the swarm will propagate as a travelling wave for any of the parameter values, and that the speed of this wave will increase as the affinity of the ants for the trail increases. Through a further qualitative analysis we have also determined the shape of the travelling wave for any parameter value. Numerical experiments show the wave to be stable. This model was also generalized to consider the class of equations that could be used to model similar behaviour. In this way, it is possible to link the phenomenon in question with other problems. The process used to generate the equations for the one dimensional model can also be extended to derive the equation of motion in two dimensions. We have extended the analysis of Othmer et al.  [25] to model the motion of automata interacting with a trail network in  two dimensions. Although the full system in two dimensions is difficult to analyze, several simplifying assumptions can be made. For example, Edelstein-Keshet [16] has developed a model with no spatial dependence to examine the orientational aspects of the pattern, and to determine the conditions necessary for a transition from a loose network of trails to a network with a strong directional order. By deriving this and other simple models from the more general model in two dimensions it is possible to accurately integrate the results obtained with each of these models. Having developed both the mathematical model and the simulation, we may now use them to study the effects of slight alterations in the behaviour of the individual on the formation and stability of the trail network. It is obvious tl1at the full behaviour of the swarm cannot be captured in any simple mathematical model. However, by using simple models we may study the dependance of the resulting patterns on the behaviour of the individuals. Once these simple  44  models have been analyzed, they may be extended to include a more complex behaviour. For example, Calenbuhr [61 has shown that the mean distance an ant follows a trail will increase as the strength of the trail increases. This implies that the probability that a follower loses the trail at any point will decrease as the trail strength increases. This has been implemented in recent versions of the simulation, but studies of the model are not as yet complete. Another possibility is that the followers may reverse their direction of travel along the trails if they do not encounter other ants following the same trail. Gordon [19] has recently experimented with such behaviour. We are not the first to use a simulation to model the swarming behaviour of army ants. Deneubourg et al [12] have used a simulation to study the motion of ants swarming into a region covered with various food distributions. His automata (ants) initially swarmed forward from a nest leaving a small amount of trail pheromone in their path. At each step forward, each ant’s choice of a new direction depended on the relative distribution of pheromone ahead left and ahead right of its current position. If an ant found food, it returned to the nest leaving a larger pheromone deposition in its wake. Deneubourg observed that as the swarm progressed forward a network of trails formed immediately behind the front, and a single trail dominated nearer the nest. By varying the food distribution, various patterns of trail networks arose. Three possible patterns are shown in figure 3.7. Case (a) shows a raiding pattern typical of E. harnatum, obtained for a patchy food distribution. Case (c) was obtained using a homogeneous food distribution, and is more characteristic of the broad swarm of E. burchelli. To examine the formation and stability of these trail networks Ermentrout and Edelstein Keshet [17] studied a system of automatons (ants) interacting on a torus through the use of trail pheromones. They began with an initially random distribution of ants and found that three distinct final patterns arose: random milling; the formation of strong trails, but no stable pattern; and the formation of a small number of very strong, long lived trails. In addition to the simulation, they also analyzed the formation and stability of strong and weak trails using a mathematical model. Their results indicated that no stable trails would form if the density  45  E. burchelli  E. hamatum  Step 900  E. raPax  Step 1100  1  Step 1300  Figure 3.7: Simulations of Deneubourg et al. [12] showing the dependence of the raiding pattern on the food distribution. The inserts are sketches of actual swarms taken from Schneirla [28]. a mathematical model. Their results indicated that no stable trails would form if the density of ants was below a certain threshold; strong stable trails would develop and persist at larger densities; and if the density was increased still further, both weak and strong trails would persist. The work of Deneubourg et. al .suggests that there is a significant dependence of the final pattern on the distribution of food in the environment. They concluded tha.t a single mechanism is implemented by different species, and that the environment is responsible for any differences in behaviour. We have shown tha.t it is possible to obtain a wide range of patterns by varying the parameters governing the behaviour of the individuals without referring explicitly to changes in the environment. This is not in contradiction with the results of the previous simulations and experiments of Deneubourg and Franks. We suggest that variations in the environment to  46  not directly influence the motion of the ants, but rather influence tile parameters governing the behaviour of the ants. That is, a change in the environment effects a change in the collective motion of tile swarm through the chain : environment 3 motion  —  —  individual behaviour  —  individual  collective motion. For example, an increase in the local food density may lead to a  decrease in the directional persistence of the individuals, which then leads to a collective motion which will cover a given area more  Our simulation then, studies the effect of the  effectively.  behaviour of the individual on the collective motion of the group. We have developed a model for tile collective is based  on parameters governing  motion of army ant  raiding swarms. This model  the motion of individual ants. We have shown, using both a  simulation and a mathematical analysis, that the model exhibits similar swarming behaviour both to previous simulations based on different assumptions, and to patterns observed in nature.  1t is interesting to note that this is not a deterministic chain of cause and effect; there is a possibility of 3 random fluctuations in the effects at each link in the chain.  47  Bibliography  [1] K. Able. Mechanisms of orientation, navigation, and homing. In S. Gauthreaux, editor, Animal Migration, Orientation, and Navigation, pages 283—373. Academic Press, N.Y., 1980. [2] S. Aron, J. M. Pasteels, and J. L. Deiioubourg. Trail-laying behavior during exploratory recruitment in the argentine ant, iridomyrmex humilis (Mayr). Biology of Behaviour, 14(3):207—2i7, 1989. [3] R. Beckers, J. L. Deneubourg, S. 4oss, and J. M. Pasteels. Collective decision making through food recruitment. Insectes ,Sociaux (Paris), 37(3):258—267, 1990. [4] G. W. Bluman and S. Kumei. Symmetries and Differential Equations. Applied Mathe matical Sciences 81. Springer-Verlag, Berlin, 1989. [5] V. Calenbuhr and J. Deneubourg. A model for trail following in ants: individual and collective behaviour. In W. Alt and G. Hoffmann, editors, Biological Motion, Proceed ings, Königswinter, 1989), Lecture iVotes in Biomathernatics, 89, pages 453—469. Springer Verlag, Berlin, 1989. -  [6] V. Calenbuhr and J. L. Deneubourg. A model for osmotropotactic orientation. Journal of Theoretical Biology, submitted. [7] S. Carnazine. Self-organizing pattern formation on the combs of honeybee colonies. Be havioral Ecology and Sociobiology, 28:61—76, 1991.  [8] J. Deneubourg, S. Aron, S. Goss, and J. Pasteels. The self-organizing exploratory pattern of the Argentine ant .Jo’urna.i of Insect Behavior, 3(2):159—168, 1990. [9] J. Denetibourg, S. Coss, Sendva-Franks, C. Detrain, and L. Crétian. The dynamics of collective sorting; robot-like ants and ant-like robots. In Simulation of Animal Behaviour; From Animals to Animuts. MIT Press, Cambridge. [10] J. L. Deneubourg, Goss, Beckers, and Sandini. Collectively self-solving problems. In Babloyantz, editor, Self organization, Emergent Properties, and Learning. Plenum, 1991. [11] J. L. Deneubourg and S. Goss. Collective patterns and decision making. Ethology, Ecology and Evolution, 1, 1989. [12] J. L. Deneuhourg, S. Goss, Franks, and J. M. Pasteels. The blind leading the blind: Chem ically mediated morphogenesis and army ant raid patterns. Journal of Insect Behavior, 2:719—725, 1989.  48  [13] B. Derrick, personal communication. [14] C. Detrain, J. Deneuborg, S. Goss, and Y. Quinet. Dynamics of collective exploration in the ant Pheidole pallidula. Psyche, in press. [15] L. Edelstein-Keshet. Mathematical Models in Biology. Random House, 1988. [16] L. Edelstein-Keshet. Trail Following as an Adaptable Mechanism for Population Be haviour. preprint, 1992. [17] L. E.delstein-Keshet and B. Ermentrout. Trail following in social insects: Patterns of motion and behaviour of individuals. preprint, 1991. [18] B. Ermentrout. PhasePlane. 1990. [191 D. Gordon. personal communication. [20] J. Gnckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bi furcations of Vector Fields. Applied Mathematical Sciences 42. Springer-Verlag, Berlin, 1983. [21] B. Hölldobler aud E. 0. Wilson. The Ant.s. Harvard University Press, Cambridge, 1990. [22] K. Jaffe, C. R.amos, C. Lagalla, and L. Parra. Orientation cues used by ants. Insectes Sociaux (Paris), 37(2):101—1L5, 1990. -  [23] J. Murray. Mathematical Biology. Springer Verlag, Berlin, 1989. [24] A. Okubo. Diffusion and Ecological Problems: Mathematical Models. Springer Verlag, Berlin, 1986. [25] H. G. Othmer, S. Dunbar, and W. Alt. Models of dispersal in biological systems. Journal of Mathematical Biology, 26:263—298, 1988. [26] B. Pfistner. A one-dimensional model of the swarming behavior of myxobacteria. In W. Alt aud G. Hoffmann, editors, Biological Motion, Proceedings, Königswinter, (1989), Lecture Notes in Biomathematics, 89, pages 556—565. Springer Verlag, Berlin, 1989. [27] M. Reed and J. Blum. A model for fast axonal transport. Journal of Cell Motility, 5:507— 527, 1985. [28] C. W. Rettenmeyer. Behavioral studies of army ants. Bulletin, 44:281—465, 1963.  University of Kansas Scientific  [29] T. C. Schneirla. Army Ants, a study in social organization. W. H. Freeman & Co., 1971. [30] G. Strang. Linear Algebra and its Applications. Academic Press, INC., Orlando, Florida, 1976. [31] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer— Verlag, Berlin, 1990.  49  Appendix A  Dimensional Analysis  The ideas of dimensional analysis were introduced by the engineering physicist Buckingham, and are used extensively in physics to reduce the number of independent parameters appearing in a problem. These ideas are part of a larger theory of using symmetries in the reduction of differential equatiois (see Bluman and Kurnei [4]). We present a simple method of using an arbitrary scaling of a differential equation to reduce the number of independent parameters. Equations (3.13  -  3.15) have two independent, and three dependent variables, each of these  variables can be rescaled independently without changing the nature of the solutions. To rescale the system we introduce the new coordinates t*,x*,T*,F*,L* EJ defined by t  x  T  =  TT,  F  FF*,  =  L  =  LL*,  Le B?.  Substituting these values into the PDE system yields. TaT* =  rlLL*+r/.F*_ T 7 T*,  -  (v7)  :  -  jJ =  2 8(xj  EF* + cJJTL*T*,  + EF*  —  aJJTL*T*.  These equations may be reorganized to the form  1OT*  L* +  F*  -  T*,  \.yTJ DF =  1v’\  —  I  ÔF*  I  50  EtF* + oLTT L*T*, F -  —  —  (A.1)  /  /  Dt  2r*  —\  ‘-‘-  t—l \2)  =  D(x*)2  —  Ft +=_F*_o1TL*T*. L  Since all solutions to the original equations remain as solutions to the rescaled equations re gardless of the scalings  we may choose  so as to eliminate as many  coefficients as possible. For example, choosing:  v  a’p. T-—-, 2 v  2  =  =  (A.2)  V  x  =  —,  cx  IL  TI  7V 2’  leaves us with the system: i DT  L+?*F*_T*,  —-b--— DES  DF  =  OL*  Dt*  L* 2 L1  +c*F*_L*T*  — —  This rescaled system contains only the three parameters: 7*  =  ES  =  ([/ V Tf  T  *  =  —.  Ti  The Buckingham Pi theorem provides a more rigorous method to ensure that the largest possible reduction in the number of parameters has been achieved. This will be the case provided the new parameters and coordinates are all dimensionless.  51  Appendix B  Eigenvalues in 3 Dimensions  In this appendix we develop the relations of Derrick [13] between the properties of a 3x3 matrix, and its eigenvalues. Consider the system  =  Ax,  (BJ)  near the fixed point x=O. We will examine the case where A is a 3x3 matrix with real elements. The eigenvalues of the system are given by the roots of the equation + A + B\ + C  =  0,  (B.2)  where  A  =  —trace A,  (B.3)  B  =  A 1 1 + A 22 + , 33 A  (B.4)  —det A.  (B.5)  C  are the cofactors of A [29]. For the case where each of these roots are real, we will name them a,,8. and  ‘,  such that  <  ‘y. We  iow  have that  A  =  —a—/3—-y,  (B.6)  B  =  cq3+/3’y+-ya’,  (B.7)  C’  =  —a,  (B.8)  and we can methodically construct table B.1. Tins table shows the relationship between the properties of the matrix A, and its possible real eigenvalues.  52  Similarly, for the case of one real root and two complex roots, let the roots be given by a, ± iw. We then have that  A  =  —(a+23),  B  =  2a/9 +  C  =  +w 2 —n’(f3 ) ,  32  , 2 +w  (B.9) (B.lo) (B.11)  and we construct table B.2. One immediate consequence is that a necessary condition for a single eigenvalue to pass through zero (a bifurcation) is that C  =  0. Also, necessary conditions for a Hopf bifurcation  (pure imaginary eigenvalue pair) are:  B  >  0,  (B.12)  .4B  =  C.  (B.13)  53  sgnA +  sgnB +  sgnC +  sgna’ —  -  +  + + —  -  + —  —  + +  —  + —  + -  sgn/3  — —  + —  —  —  —  +  + —  + +  — —  — —  —  —  + —  —  sgw’  +  — —  + + + + + + + + +  Conditions none  /37>IaI(/3+7), IaH/3+7. a/3>(a+/3)7,  (a+/3)>7.  Iaj(/3+y)>/37, I>a/ 7k+/ , 3  IaI>!3+7. Ic+/3I>7.  /3+7>Ial,  /37>ja/3+7).  none  7>Ia+/31, a!3>71a+/31. Iaj(/3+7)>/37, IaI>/3+7. 7Ia+/3a/3, 7>Ia+/3l.  Table B.1: Real eigenvalues of a 3x3 matrix, a <  sgnA  sgnB +  sgnC +  sgna —  sgn/3 +  — —  + + + —  —  + —  —  + +  — —  —  —  + —  + —  +  —  —  + —  + —  + + —  +  —  + —  + + —  + —  /3  < 7,  Conditions IaI>2/3, /3 >2Ia/3 + 2  none 2I/3 > a  /32  w > 2a1/31. +2  IaI>2/3, . +w 2 21a1/3>/3 2j/3>a, 2 +w 2a1/31>/3 . 2/3>IaI, /3 >2IaI/3. + 2 none a>21/31, 2/3>  + 2 /3 > 2a/3I. ’ > /32 2, +  al, 21a1/3  a>21/31,  +w 2 2a1/31>/3 .  Table B.2: Imaginary eigenvalues of a 3x3 matrix. A  54  a,,B,7 E JR  =  a, A 3 , 2  =  /3 +  icy’  Appendix C  The Location of WU o,loc  Herin we show that the eigenvectors of the linearized system near u 0 must be directed into the negative octant of state space relative to u , and into the region where all three components of 0 the tangent vectors are negative. To analyze the eigenvectors near the fixed point u 0 we divide the local state space into eight regions using the nuliplanes of T, F, and L.  Over these regions we superimpose the  eight octants of the local state space. The positive octant being the region where all three components of u 0  —  (T, F, L) are greater than zero, and the negative octant the region where  all three components of u 0  —  (T F, L) are less than zero. We will take the convention that ‘up’  and ‘down’ refer to the positive and negative T directions respectively; ‘left’/’right’ to the L directions; and ‘forward’/’back’ to the F directions (see figure 3.1). Figure C.1 shows the intersections of the F and L nullsurfaces with planes of constant T. If the eigenvectors are to point into the positive octant relative to u , then they must also point 0 into the region where T’, F’, and L’ are positive. To do otherwise would lead immediately to a contradiction. Similar arguments lead to restrictions on the possibility of the eigenvectors pointing into the other local octants. Imposing these restrictions on the L components of the eigenvectors, it follows that they must point into the shaded regions of figure C.1 (b). Figure C.1 (a) shows that F’ > 0 immediately above u , and thus the eigenvectors cannot point into the 0 shaded region behind and above u . Thus, this region is no longer shaded in figure C.1 (a). 0 Similarly, from figure C.1 (c) it follows that the eigenvectors cannot point ahead and below u. Thus this region is no longer shaded in figure C.1 (c). If the eigenvectors are to point into a region above u , then they must also point into T+. Thus the eigeiivectors cannot point into 0  55  the region ahead of u 0 and below T°. Similarly, they cannot point into the region behind u 0 and above T°. This restricts the eigenvectors to point into the negative octant of state space relative to u 0 intersected with the region where all three components of the tangent vectors are negative.  56  L  L  F°  F  F (a) T  0 L  L0FO  0 F  >  0 T  F (b) T=T 0  (C)  T <T 0  Figure C.1: Sections of constant T showing the location of the eigenspace of u . The sections 0 are for values of T where: a) T > T , b) T = T 0 , and c) T < T 0 .The shaded regions indicate 0 the possible locations of the eigenvectors uo.  57  Appendix D  Details of the Computer Simulation  The computer simulation was developed to simulate the motion of the ants on a rectangular grid of arbitrary size. Data stacks were used to keep track of the positions and velocities of the ants, and of the positions and strengths of the trails. The velocity of the ants is composed of a constant speed, and a direction which is an integer multiple of  450•  The program begins at the  bottom of the ant stack and performs the following algorithm for each ant: 1. The affinity’ is computed according to the following formula: STRENGTH * SATURATION  low+  (D.1)  where STRENGTH is the strength of the trail at the ant’s location, and SATURATION, low’ and  are the parameters governing the detection mechanism.  2. A random number in the range [0,255] is generated and compared to the parameter If the random number is larger than  ,  .  then the ant is assumed to be lost (step 4).  Otherwise, the ant attempts to follow the trail (step 3). 3. The point directly ahea.d of the ant is tested for the presence of trail pheromone. If the test is successful, the ant will move to that location. If the test is negative (no trail) the ant will check for trails to the left and right, and either i) turn (either incrementing of decrementing 2 the velocity by one) and move to the stronger trail if the strengths differ, ii)  or, move  randomly (step 4) if the strengths are equal.  ‘An integer in the range [0,255] 2 T hese calculations are performed modulo eight  58  4. A random number is generated, and compared to the TURNING KERNEL entered at the start of the run. This will yield a random velocity which is added to that of the ants (addition is performed modulo eight). The ant then moves one step in this direction and its new position and velocity are stored in the ant stack. 5. After the ant is moved, the trail stack is searched for the new position. If there is a trail there, the pheromone level is increased by r, otherwise, a new trial is added to the stack, and given a strength  T.  After processing all the ants in the ant stack, the program decrements the strength of each trail in the trail stack by one,  removing  any trails of zero strength.  The turning kernel governs the relative bias that the lost ant shows towards turning a certain angle. The kernel is assumed to have no external bias, hence it depends oniy on the differences between the ants current direction and the new direction. Further, the kernel is assumed to be symmetric with respect to left and right  turns.  The forward  persistence,  is an  indkatiom of  the forward bias of the kernel, and is computed by =  jcosB()d.  B(i,b) is the relative probability of turning through an angle B(—’b)  59  (D.2) ‘,  and is subject to the constraints  =  B(b),  (D.3)  =  1.  (D.4)  List of Notations  The rate at which lost ants follow trails per encounter  19  Eigenvalue of A  50  /3  The probability of an ant reversing when it moves on or off of a trail  /3  Eigenvalue of A  50  7  The rate at which the trail pheremone decays  19  nondimensionalized version  19  23  Eigenvalue of A  50  The rate at which followers lose trails  19  nondimensionalized version A  ...  The mean rate of reversal of the lost ants  23 -  .  20  Eigenvalues of the fixed point u 1  29  The region of state space where F’ < 0, and L’ < 0  31  The region of state space where F’ > 0, and L’ > 0  31  The motility coefficient of the lost ants  22  111  The ‘floor’ of the first octant of state space  31  112  The ‘back wall’ of the first octant of state space  31  p  The rate at which followers turn away from the nest  19  The rate at which followers turn towards the nest  19  r  The ratio of follower deposition rate to that of the lost ants  23  Tj  The rate at which followers deposit trails  19  Ti  The rate at which lost ants deposit trails  19  The affinity of the ants to the trails  58  Lower bound of  58  A+  low  60  Turn angle of lost ant  58  Forward persistence of the lost ants  58  Eigenvectors of the fixed point u 1  29  w  Complex portion of eigenvalue of A  50  A  3x3 matrix  50  Cofactors of A  50  A  —trace A  50  B  Intersection of A  B  ZA,  50  B(I’)  The turning kernel governing the random motion of the lost ants  58  C  —detA  50  c  The speed of the travelling wave  23  E  The intersection of W’ with B  31  F(x,t)  Total density of followers  18  F(x,t)  Density of followers moving towards the nest  18  F+(x, t)  Density of followers moving away from the nest  18  F°  F’ nulisurface  30  F  Region of state space where F’ < 0  30  Region of state space where F’ > 0  30  G(T, F, L)  Growth function of the generalized system  40  L(x, t)  Total density of lost ants  18  L(x,t)  Density of lost ants moving towards the nest  18  L+(x, t)  Density of lost ants moving away from the nest  18  rO  L / nuilsurface  30  L  Region of state space where L’ < 0  30  L+  Region of state space where L’> 0  30  The first octant of state space  31  with T°  31  61  R(T, F, L)  Reaction function of the generalized system  40  s  Speed of lost ants  18  T(x, t)  Total length of trails per unit length of network  18  time  18  time in the moving coordinate system  23  T’ nulisurface  30  T  Region of state space where T’ < 0  30  T+  Region of state space where T’ > 0  30  uo  The populated steady state  26  The trivial steady state  26  v  Speed of followers  18  ’ 5 W  The stable manifold of the populated steady state  29  ” 0 W  The unstable manifold of the populated steady state  29  9 Wj  The stable manifold of the trivial steady state  29  W  The unstable manifold of the trivial steady state  29  x  Distance away from the nest  18  z  Distance from the wavefront  23  62  


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