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A PDE model for the dynamics of trail formation by ants
Abstract
Armies of ants are known to move in trails. These trails are formed by a chemotactic force induced by pheromone secreted by the ants. In this paper we develop a mathematical model consisting of two partial differential equations, which explain when and how these trails are formed. The first equation, for the ants, includes the chemotaxis effect of pheromone and the dispersion caused by overcrowding. The second equation is a reaction–diffusion equation for the pheromone concentration. The strength of the chemotactic force, χ, plays a critical role in the analysis. We prove that trails cannot be formed if χ is small, while many trails exist if χ is large.
... In the following, we validate our model against the one presented in Ryan (2016) and then show that our model using parameters in Table 1 can produce more sophisticated results that capture foraging ant behavior seen in experiment or nature. The beauty of our simple model is that it coarse-grains over microscopic underpinnings described in other models to capture the macroscopic picture but nevertheless captures details described in more specific models (see (Amorim et al. 2019;Malíčková et al. 2015;Fontelos and Friedman 2015;Vela-Pérez et al. 2015), for example). We validate against (Ryan 2016) due to the direct relation between our model and the one presented there. ...
... This is in contrast to our modeling paradigm, where an ant moves towards the direction of higher pheromone concentration (position-jump process). Indeed, such models result in more robust, biological realistically results than position-jump models (Amorim et al. 2019;Malíčková et al. 2015;Ramirez et al. 2018;Mokhtari et al. 2022;Vela-Pérez et al. 2015;Fontelos and Friedman 2015). For example, in Amorim et al. (2019) a velocity-jump model was used to show that the positioning of ant antennae at the front of their bodies is necessary for ants to have the ability to follow a pheromone signal. ...
Foraging for resources is an essential process for the daily life of an ant colony. What makes this process so fascinating is the self-organization of ants into trails using chemical pheromone in the absence of direct communication. Here we present a stochastic lattice model that captures essential features of foraging ant dynamics inspired by recent agent-based models while forgoing more detailed interactions that may not be essential to trail formation. Nevertheless, our model’s results coincide with those presented in more sophisticated theoretical models and experiments. Furthermore, it captures the phenomenon of multiple trail formation in environments with multiple food sources. This latter phenomenon is not described well by other more detailed models. We complement the stochastic lattice model by describing a macroscopic PDE which captures the basic structure of lattice model. The PDE provides a continuum framework for the first-principle interactions described in the stochastic lattice model and is amenable to analysis. Linear stability analysis of this PDE facilitates a computational study of the impact various parameters impart on trail formation. We also highlight universal features of the modeling framework that may allow this simple formation to be used to study complex systems beyond ants.
... These studies emphasize the types of scalar fields, derived from the chemical concentration, that are necessary for the steering mechanism of a particle to follow a trail of chemoattractant. Fontelos and Friedman [22] proposed a PDE model and proved the existence of trails if the interaction intensity with the field is sufficiently large. In [23], the authors derived a model where each particle senses the gradient of the concentration field ahead of its position. ...
In this paper, we propose a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic local PDE system, for the population density and the chemical field. The main novelty lies in the transport term of the population density, which depends on the second-order derivatives of the chemical field. This term is derived as an anticipation-reaction steering mechanism of an infinitesimally small ant as its size approaches zero. We establish global-in-time existence and uniqueness for the model, and the propagation of regularity from the initial data. Then, we build a numerical scheme and present various examples that provide hints of trail formation.
... From it, they obtain an analogous Keller-Segel instability criterion for an active chemotaxis system. In [36], they derive a PDE system from the AAA model phenomenologically and discuss the emergence of trails using a linear stability analysis. Like the AAA model, the model we study is similar to the Keller-Segel model (1.1). ...
We investigate an interacting particle model to simulate a foraging colony of ants, where each ant is represented as an active Brownian particle. The interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of the pheromone field. Unlike conventional models, our study introduces a parameter that enables the reproduction of two distinctive behaviors: the well-known Keller-Segel collapse and the formation of traveling clusters, without relying on external constraints such as food sources or nests. We consider the associated mean-field limit partial differential equation (PDE) of this system and establish the analytical and numerical foundations for understanding these particle behaviors. Remarkably, the mean-field PDE not only supports Keller-Segel collapse and lane formation but also unveils a bistable region where these two behaviors compete. The patterns associated with these phenomena are elucidated by the shape of the growing eigenfunctions derived from linear stability analysis. This study not only contributes to our understanding of complex ant colony dynamics but also introduces a novel parameter-dependent perspective on pattern formation in collective systems. 1. Introduction. Collective behaviours of animals can persist over large distances and for long times. One example can be found in the self-organisation of ants. Amongst the different known species of ants (Formicidae), ranging in the 20.000s, there are many ways in which they use chemicals to organise collectively. Using chemical (olfactory) cues, they organise themselves into trails connecting their nests and food sources or into ant armies to capture moving prey [4, 40]. The foraging trails can last for months and extend over vast distances. It is common to see ants move bidirectionally along the trails, split into incoming and outgoing groups, either bringing food back to the nest or going out to get new food [37, 48]. Ant colonies have long amazed researchers with their ability to regulate traffic and prevent jams in crowded conditions, as may occur in trails along confined spaces. In fact, ants appear to do better than humans at traffic regulation at high densities [32]. In lab experiments using narrow bridges connecting the nest and a food source, researchers found that ants can sustain a constant flow at high densities as opposed to the typical reduction in flow due to congestion in traffic models [63]. To understand such behaviour, one may turn to microscopic or individual-based models that track every single ant in the colony and their interactions. In the case of ants, as customary in active matter systems, one typically accounts for the time evolution of the position and the orientation of each ant. Microscopic models of active matter have been used to study a vast array of different complex behaviours such as lane formation [5, 21, 67], bird flocking [24, 27, 30, 50], biological flows [59, 65] and clustering [17, 22, 25, 51]. Ants utilise different pheromone molecules for various tasks such as trail formation, navigation , bridge building, nestmate recognition and alarming in cases of danger; see, for example, [10, 28, 29] and references therein. Our starting point is an individual-based model proposed in [31], following the experimental data of [63]. It consists of a set of stochastic differential equations (SDEs) for the position and orientation of each ant and a partial differential equation (PDE) describing the evolution of the pheromone concentration. The SDE model represents each ant as an active Brownian particle; that is, the position evolves according to a Brownian motion with a bias in the direction of the orientation, and the orientation changes gradually by a periodic Brownian motion. Ants change their orientations to align with the upward gradient of a pheromone field they lay. This mechanism is akin to that of autophoretic colloids studied in [53, 62], leading to a so-called Active Attractive Alignment (AAA) model in [52]. These active colloids, which are synthetically manufactured and used as micro-engines and cargo carriers, are seen in experiments to undergo dynamic clustering even at slow densities of less than 10% [52]. The ant model in [31] differs from the AAA model in that the chemical sensing is not at the particle' centre but at a look-ahead distance that represents the location of the ants' antennas. We will show that this difference is
... It is made experiment or nature. The beauty of our simple model is that it coarse-grains over microscopic underpinnings described in other models to capture the macroscopic picture but nevertheless captures details described in more specific models (see [4,35,47,73], for example). We validate against [57] due to the direct relation between our model and the one presented there. ...
Foraging for resources is an essential process for the daily life of an ant colony. What makes this process so fascinating is the self-organization of ants into trails using chemical pheromone in the absence of direct communication. Here we present a stochastic lattice model that captures essential features of foraging ant dynamics inspired by recent agent-based models while forgoing more detailed interactions that may not be essential to trail formation. Nevertheless, our model's results coincide with those presented in more sophisticated theoretical models and experiment. Furthermore, it captures the phenomenon of multiple trail formation in environments with multiple food sources. This latter phenomenon is not described well by other more detailed models. An additional feature of this approach is the ability to derive a corresponding macroscopic PDE from the stochastic lattice model which can be described via first principle interactions and is amenable to analysis. Linear stability analysis of this PDE reveals the key biophysical parameters that give rise to trail formation. We also highlight universal features of the modeling framework that this simple formation may allow it to be used to study complex systems beyond ants.
... Mathematical modeling is intended to shed some light on the emergence of such collective behavior, based on limited exchanges of information and simple individual rules; as initiated in the seminal work [20,21]. In the specific case of ants, it is worth mentioning the works of [5,6,10,11,14,16] which offer a large variety of approaches by using individual-based models or more macroscopic PDEs systems. ...
We establish well-posedness for a model of self-propelled agents interacting through pheromone which they themselves produce. The model consists of an arbitrary number of agents modeled by a system of ordinary differential equations, for which the acceleration term includes the influence of a chemical signal, or pheromone, which induces a turning-like behaviour. The signal is produced by the agents themselves and obeys a diffusion equation. We prove that the resulting system, which is non-local in both time and space, enjoys well-posedness properties, using a fixed point method, and show some numerical results.
... with ρ → 0 when t → −τ , being τ some characteristic time for the duration of the pheromones. Let us mention the pioneer work of Fontelos and Friedman [12], where the behavior of ants was modeled using a system of two partial differential equations, one governing the distribution of ants, and the other one devoted to the generation and decay of the pheromones, see also [5]. Since new pheromones appear in the path each ant travelled, it is enough to know their paths and weighting them with a function of t in order to model the decay of the chemical trail. ...
In this work we study a kinetic model of active particles with delayed dynamics, and its limit when the number of particles goes to infinity. This limit turns out to be related to delayed differential equations with random initial conditions. We analyze two different dynamics, one based on the full knowledge of the individual trajectories of each particle, and another one based only on the trace of the particle cloud, loosing track of the individual trajectories. Notice that in the first dynamic the state of a particles is its path, whereas it is simply a point in R d in the second case. We analyse in both cases the corresponding mean-field dynamic obtaining an equation for the time evolution of the distribution of the particles states. Well-posedness of the equation is proved by a fixed-point argument. We conclude the paper with some possible future research directions and modelling applications.
... The authors of that work argue convincingly about why such a result apparently contradicts earlier experiments which appeared to show that a different, nonlinear (in L and R) response law holds for ant movement (see Camazine et al. (2001) and the discussion in Perna et al. (2012)). Also, the model in Fontelos and Friedman (2015) can be seen as a version of Weber's law. Here, we support and extend the conclusions in Perna et al. (2012) related to the applicability of Weber's law to ant movement. ...
We analyze an ant navigation model based on Weber’s law, where the ants move across a pheromone landscape sensing the area using two antennae. The key parameter of the model is the angle 2 β representing the span of the ant’s sensing area. We show that when β< π/ 2 ants are able to follow (straight) pheromone trails proving that for initial conditions close to the trail, there exists a Lyapunov function that ensures ant trajectories converge on and follow the pheromone trail, with these solutions being locally asymptotically stable. Furthermore, we indicate that the features of the ant trajectories such as convergence speed or oscillation wave length are controlled by the angle β. For β> π/ 2 , we present numerical evidence that indicates that ants are unable to follow pheromone trails. We also assess our model by comparing it to previous experimental results, showing that the solutions’ behavior falls into biologically meaningful ranges. Our work provides solid mathematical support for experimental studies where it was found that ant perception follows a Weber’s law, by proving that such models lead to the desired robust and stable trail following.
... However, the discussion is limited to only a single robot. Ant trail formation based on a reaction-diffusion type model is presented in [23]. Robot area coverage that considers different types of stochastic behavior is presented in [24][25][26][27]. ...
Design of robot swarms inspired by self-organization in social insect groups is currently an active research area with a diverse portfolio of potential applications. In this work, the authors propose a control law for efficient area coverage by a robot swarm in a 2D spatial domain, inspired by the unique dynamical characteristics of ant foraging. The novel idea pursued in the effort is that dynamic, adaptive switching between Brownian motion and Lévy flight in the stochastic component of the search increases the efficiency of the search. Influence of different pheromone (the virtual chemotactic agent that drives the foraging) threshold values for switching between Lévy flights and Brownian motion is studied using two performance metrics — area coverage and visit entropy. The results highlight the advantages of the switching strategy for the control framework, particularly in cases when the object of the search is scarce in quantity or getting depleted in real-time.
... The sensitivity of the system to the chemoattractant has been related to the system mass, which in turn influences the stability of the system as represented by a typical Keller-Segel reaction diffusion system (Blanchet et al. 2006). Furthermore, chemotactic sensitivity and its effect on the formation (or lack thereof) of pheromone trails has also been studied (Fontelos and Friedman 2015). There have also been interesting efforts to correlate pheromone concentration with the magnitude of ant turning angle which was shown to follow Weber's law (Perna et al. 2012). ...
This work proposes a control law for efficient area coverage and pop-up threat detection by a robot swarm inspired by the dynamical behavior of ant colonies foraging for food. In the first part, performance metrics that evaluate area coverage in terms of characteristics such as rate, completeness and frequency of coverage are developed. Next, the Keller–Segel model for chemotaxis is adapted to develop a virtual-pheromone-based method of area coverage. Sensitivity analyses with respect to the model parameters such as rate of pheromone diffusion, rate of pheromone evaporation, and white noise intensity then identify and establish noise intensity as the most influential parameter in the context of efficient area coverage and establish trends between these different parameters which can be generalized to other pheromone-based systems. In addition, the analyses yield optimal values for the model parameters with respect to the proposed performance metrics. A finite resolution of model parameter values were tested to determine the optimal one. In the second part of the work, the control framework is expanded to investigate the efficacy of non-Brownian search strategies characterized by Lévy flight, a non-Brownian stochastic process which takes variable path lengths from a power-law distribution. It is shown that a control law that incorporates a combination of gradient following and Lévy flight provides superior area coverage and pop-up threat detection by the swarm. The results highlight both the potential benefits of robot swarm design inspired by social insect behavior as well as the interesting possibilities suggested by considerations of non-Brownian noise.
In this work we study a kinetic model of active particles with delayed dynamics, and its limit when the number of particles goes to infinity. This limit turns out to be related to delayed differential equations with random initial conditions. We analyze two different dynamics, one based on the full knowledge of the individual trajectories of each particle, and another one based only on the trace of the particle cloud, loosing track of the individual trajectories. Notice that in the first dynamic the state of a particles is its path, whereas it is simply a point in Rd in the second case. We analyze in both cases the corresponding mean-field dynamic obtaining an equation for the time evolution of the distribution of the particles states. Well-posedness of the equation is proved by a fixed-point argument. We conclude the paper with some possible future research directions and modeling applications.
In this paper we analyze a system of PDEs recently introduced in [P. Amorim,
{\it Modeling ant foraging: a {chemotaxis} approach with pheromones and trail
formation}], in order to describe the dynamics of ant foraging. The system is
made of convection-diffusion-reaction equations, and the coupling is driven by
chemotaxis mechanisms. We establish the well-posedness for the model, and
investigate the regularity issue for a large class of integrable data. Our main
focus is on the (physically relevant) two-dimensional case with boundary
conditions, where we prove that the solutions remain bounded for all times. The
proof involves a series of fine \emph{a priori} estimates in Lebesgue spaces.
We experimentally investigated both individual and collective behavior of the Argentine ant Linepithema humile as they crossed symmetrical and asymmetrical bifurcations in gallery networks. Ants preferentially followed the branch that
deviated the least from their current direction and their probability to perform a U-turn after a bifurcation increased with
the turning angle at the bifurcation. At the collective level, colonies were better able to find the shortest path that linked
the nest to a food source in a polarized network where bifurcations were symmetrical from one direction and asymmetrical from
the other than in a network where all bifurcations were symmetrical. We constructed a model of individual behavior and showed
that an individual’s preference for the least deviating path will be amplified via the ants’ mass recruitment mechanism thus
explaining the difference found between polarized and non-polarized networks. The foraging efficiency measured in the simulations
was three times higher in polarized than in non-polarized networks after only 15min. We conclude that measures of transport
network efficiency must incorporate both the structural properties of the network and the behavior of the network users.
Ants are known to be able to find paths of minimal length between the nest
and food sources. The deposit of pheromones while they search for food and
their chemotactical response to them has been proposed as a crucial element in
the mechanism for finding minimal paths. We investigate both individual and
collective behavior of ants in some simple networks representing basic mazes.
The character of the graphs considered is such that it allows a fully rigorous
mathematical treatment via analysis of some markovian processes in terms of
which the evolution can be represented. Our analytical and computational
results show that in order for the ants to follow shortest paths between nest
and food, it is necessary to superimpose to the ants' random walk the
chemotactic reinforcement. It is also needed a certain degree of persistence so
that ants tend to move preferably without changing their direction much. It is
also important the number of ants, since we will show that the speed for
finding minimal paths increases very fast with it.
Many human social phenomena, such as cooperation, the growth of settlements, traffic dynamics and pedestrian movement, appear to be accessible to mathematical descriptions that invoke self-organization. Here we develop a model of pedestrian motion to explore the evolution of trails in urban green spaces such as parks. Our aim is to address such questions as what the topological structures of these trail systems are, and whether optimal path systems can be predicted for urban planning. We use an 'active walker' model that takes into account pedestrian motion and orientation and the concomitant feedbacks with the surrounding environment. Such models have previously been applied to the study of complex structure formation in physical, chemical and biological systems. We find that our model is able to reproduce many of the observed large-scale spatial features of trail systems.
We show how the movement rules of individual ants on trails can lead to a collective choice of direction and the formation of distinct traffic lanes that minimize congestion. We develop and evaluate the results of a new model with a quantitative study of the behaviour of the army ant Eciton burchelli. Colonies of this species have up to 200 000 foragers and transport more than 3000 prey items per hour over raiding columns that exceed 100 m. It is an ideal species in which to test the predictions of our model because it forms pheromone trails that are densely populated with very swift ants. The model explores the influences of turning rates and local perception on traffic flow. The behaviour of real army ants is such that they occupy the specific region of parameter space in which lanes form and traffic flow is maximized.
In many biological systems, movement of an organism occurs in response to a di#usible or otherwise transported signal, and in its simplest form this can be modeled by di#usion equations with advection terms of the form first derived by Patlak [Bull. of Math. Biophys.,15 (1953), pp. 311--338]. However, other systems are more accurately modeled by random walkers that deposit a nondi#usible signal that modifies the local environment for succeeding passages. In these systems, one example of which is the myxobacteria, the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates. Davis [Probab. Theory Related Fields, 84 (1990), pp. 203--229] has studied this question for a certain class of random walks, and here we extend this analysis to the continuum limit of such walks. We first derive several general classes of partial di#erential equations that depend on how the movement rules are a#ected by the local modulator concentration. We then show that a variety of dynamics is possible, which we classify as aggregation, blowup, or collapse, depending on whether the dynamics admit stable bounded peaks, whether solutions blow up in finite time, or whether a suitable spatial norm of the density function is asymptotically less than its initial value.
By exploring the way in which certain animal groups coordinate among themselves,Collective Animal Behavioroffers a great deal of insight for managers seeking to better understand how collective behavior takes shape within a company. Using concrete examples, Sumpter . . . offers a clear account whose scope extends well beyond the natural sciences.
In this article we propose a mechanism for the formation of paths of minimal length between two points (trails) by a collection of individuals undergoing reinforced random walks. This is the case, for instance, of ant colonies in search for food and the development of ant trails connecting nest and food source. Our mechanism involves two main ingredients: (1) the reinforcement due to the gradients in the concentration of some substance (pheromones in the case of ants) and (2) the persistence understood as the tendency to preferably follow straight directions in absence of any external effect. Our study involves the formulation and analysis of suitable Markov chains for the motion in simple labyrinths, that will be understood as graphs, and numerical computations in more complex graphs reproducing experiments performed in the past with ants.
Leta
i,iX®\overrightarrow X
=X
0,X
1, ... on the integers as follows. Initially the weight of each interval (i, i+1), i an integer, equals 1. If at timen an interval (i, i+1) has been crossed exactlyk times by the motion, its weight is
1 + åj = 1k aj1 + \sum\limits_{j = 1}^k {a_j }
. Given (X
0,X
1, ...,X
n)=(i0, i1, ..., in), the probability thatX
n+1 isi
n–1 ori
n+1 is proportional to the weights at timen of the intervals (i
n–1,i
n) and (i
n,iin+1). We prove that
X®\overrightarrow X
either visits all integers infinitely often a.s. or visits a finite number of integers, eventually oscillating between two adjacent integers, a.s., and that
limn ® ¥\mathop {\lim }\limits_{n \to \infty }
X
n
/n=0 a.s. For much more general reinforcement schemes we proveP (
X®\overrightarrow X
visits all integers infinitely often)+P (
X®\overrightarrow X
has finite range)=1.
We propose an Individual-Based Model of ant-trail formation. The ants are modeled as self-propelled particles which deposit directed pheromone particles and interact with them through alignment interaction. The directed pheromone particles intend to model pieces of trails, while the alignment interaction translates the tendency for an ant to follow a trail when it meets it. Thanks to adequate quantitative descriptors of the trail patterns, the existence of a phase transition as the ant-pheromone interaction frequency is increased can be evidenced. We propose both kinetic and fluid descriptions of this model and analyze the capabilities of the fluid model to develop trail patterns. We observe that the development of patterns by fluid models require extra trail amplification mechanisms that are not needed at the Individual-Based Model level.
Social insects--ants, bees, termites, and wasps--can be viewed as powerful problem-solving systems with sophisticated collective intelligence. Composed of simple interacting agents, this intelligence lies in the networks of interactions among individuals and between individuals and the environment. A fascinating subject, social insects are also a powerful metaphor for artificial intelligence, and the problems they solve--finding food, dividing labor among nestmates, building nests, responding to external challenges--have important counterparts in engineering and computer science. This book provides a detailed look at models of social insect behavior and how to apply these models in the design of complex systems. The book shows how these models replace an emphasis on control, preprogramming, and centralization with designs featuring autonomy, emergence, and distributed functioning. These designs are proving immensely flexible and robust, able to adapt quickly to changing environments and to continue functioning even when individual elements fail. In particular, these designs are an exciting approach to the tremendous growth of complexity in software and information. Swarm Intelligence draws on up-to-date research from biology, neuroscience, artificial intelligence, robotics, operations research, and computer graphics, and each chapter is organized around a particular biological example, which is then used to develop an algorithm, a multiagent system, or a group of robots. The book will be an invaluable resource for a broad range of disciplines.
Stochastic models of biased random walk are discussed, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor. In particular the turning frequency and turn angle distribution are derived from certain biological hypotheses on the background of related experimental observations. Under suitable assumptions it is shown that solutions of the underlying differential-integral equation approximately satisfy the well-known Patlak-Keller-Segel diffusion equation, whose coefficients can be expressed in terms of the microscopic parameters. By an appropriate energy functional a precise error estimation of the diffusion approximation is given within the framework of singular perturbation theory.
Vertex-Reinforced Random Walk (VRRW), defined by Pemantle (1988a), is a random process in a continuously changing environment which is more likely to visit states it has visited before. We consider VRRW on arbitrary graphs and show that on almost all of them, VRRW visits only finitely many vertices with a positive probability. We conjecture that on all graphs of bounded degree, this happens a.s., and provide a proof only for trees of this type. We distinguish between several different patterns of localization and explicitly describe the long-run structure of VRRW, which depends on whether a graph contains triangles or not. While the results of this paper generalize those obtained by Pemantle and Volkov (1998) for Z,ideas of proofs are different and typically based on a large deviation principle rather than a martingale approach.
The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.