No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
From Individual to Collective Dynamics in Argentine Ants (Linepithema humile).
Abstract
In this paper we propose a model for the formation of paths in Argentine Ants when foraging in an empty arena. Based on experimental observations, we provide a distribution for the random change in direction that they approximately undergo while foraging as a mixture of a Gaussian and a Pareto distribution. By following the principles described in previous work [32], we consider persistence and reinforcement to create a model for the motion of ants in the plane. Numerical simulations based on this model lead to the formation of branched ant-trails analogous to those observed experimentally.
Copyright © 2015. Published by Elsevier Inc.
... Previous advancements in fully spatial models of ant behaviour, as demonstrated in works like Perna (2012); Boissard et al. (2013); Vela-Pérez et al. (2015); Amorim and Goudon (2021), involve exploring motion and trail formation in a free two-dimensional space without specifying the locations of the nest and food source. A different approach to the understanding of the dynamic response of ants to the chemical stimulus has been explored by Calenbuhr and Deneubourg (1992) where the trail is explicitly modeled by the stationary solution to an emission-diffusion problem and ants are characterized by the stimulus-response relationship. ...
... This scenario can lead to failures in trail establishment and subsequent trail following. Vela-Pérez et al. (2015) avoid this phenomenon (they refer to it as overcrowding) by incorporating an explicit mechanism to prevent it, which forces the ants to move away from the regions with pheromone concentrations larger than certain threshold. Alternatively, one may consider that ants deposit pheromone in a continuous trace. ...
... We note, however, that this behavior observed in our simulations was rare and temporary (simulations containing this behaviour were included in our results). Similar model behavior was mentioned in the work of Vela-Pérez et al. (2015), in which the authors incorporated a mechanism to avoid clustering in a small spatial region. ...
Colonies of ants can complete complex tasks without the need for centralised control as a result of interactions between individuals and their environment. Particularly remarkable is the process of path selection between the nest and food sources that is essential for successful foraging. We have designed a stochastic model of ant foraging in the absence of direct communication. The motion of ants is governed by two components - a random change in direction of motion that improves ability to explore the environment, and a non-random global indirect interaction component based on pheromone signalling. Our model couples individual-based off-lattice ant simulations with an on-lattice characterisation of the pheromone diffusion. Using numerical simulations we have tested three pheromone-based model alternatives: (1) a single pheromone laid on the way toward the food source and on the way back to the nest; (2) single pheromone laid on the way toward the food source and an internal imperfect compass to navigate toward the nest; (3) two different pheromones, each used for one direction. We have studied the model behaviour in different parameter regimes and tested the ability of our simulated ants to form trails and adapt to environmental changes. The simulated ants behaviour reproduced the behaviours observed experimentally. Furthermore we tested two biological hypotheses on the impact of the quality of the food source on the dynamics. We found that increasing pheromone deposition for the richer food sources has a larger impact on the dynamics than elevation of the ant recruitment level for the richer food sources.
... Therefore, numerous experimental works have been dedicated to studying and modeling how ants respond to pheromone at the individual level. Here, we name just a few from a vast literature [1,3,5,6,11,13,14,19,21,22,23,25,26,29,27,32] and concentrate on the important contributions in [4,17,16,24,27,30]. In [4], an individual-based model (IBM) is proposed, using directed pheromones to mediate the interaction between ants. ...
... In [27], an IBM model is presented, where formation of lanes is exhibited. Finally, in [16,24], valuable experimental results are reported, and corresponding individual-based models are studied. In particular, the authors show experimentally that individual ants' turning rate is governed by Weber's law: the turning rate is determined by the difference in pheromone on both sides of the ant, divided by the sum of pheromone on each side. ...
... However, until now, to the best of our knowledge (and despite advanced attempts such as [16,24,27]), the rigorous mathematical study of such individual models is very scarce: for instance, stability results ensuring that trail-following behavior is robust with respect to small perturbations are still lacking. Thus the main goals in this paper are: ...
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.
... 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.
... 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. ...
... This has the advantage of providing a bounded domain-so the agents cannot run off to infinity-while eliminating any artefacts produced by boundary conditions. Indeed, it is known [10,14] that ants may tend to aggregate on the edges of experimental domains for reasons that are not quite well understood. Also, it is not clear what boundary conditions the agents should verify at an individual level. ...
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.
... Table 1 Values of dimensional quantities used as baseline. The values of ω * were computed from the values of the turning angles of ants in the laboratory measured every 0.04 seg as reported in Table 1 of [37]. See also [14] for a useful compilation of parameter values of different species. ...
... for some ξ > 1 and a maximum value D max ϕ that, according to (22) and the laboratory results reported in [37], is of order t * × 10 −2 . Note that under this model, D ϕ (F N (r)) is a sigmoid function of r, and thus consistent with (27). Figure 4 compares paths of foraging ants for the case of ξ = 0 (Fickian diffusion) and ξ = 4. ...
We propose a Active Walker Model for the movement of individual under the feedback of a time-dependent pheromone field. It is assumed that the sole feedback mechanism is tropotaxis: ants can sense the gradient of the pheromone concentration field and adjust their orientation accordingly. We consider two types of pheromone fields: one emanating from the nest, and other actively produced by ants in their nest bound journey after finding a food source. We explicitly track the evolution of both fields in three dimensions. The model yields, under certain assumptions on the parameters, that ants can successfully recruit others to food sources, and that a trail network pattern will emerge.
... Ants communicate with each other through the use of pheromones to adjust their collective behaviour. [1][2][3] This mechanism often leads to intriguing self-organized patterns. For example, their foraging path can be understood as solving a certain optimization problem in terms of time and energy costs, [4][5][6][7][8][9] and the shape of the path is predictable by Fermat's principle of least time. ...
The Patlak-Keller-Segel equation is a canonical model of chemotaxis to describe self-organized aggregation of organisms interacting with chemical signals. We investigate a variant of this model, assuming that the organisms exert effective pressure proportional to the number density. From the resulting set of partial differential equations, we derive a Lyapunov functional that can also be regarded as the free energy of this model, and minimize it with a Monte Carlo method to detect the condition for self-organized aggregation. Focusing on radially symmetric solutions on a two-dimensional disc, we find that the chemical interaction competes with diffusion so that aggregation occurs when the relative interaction strength exceeds a certain threshold. Based on the analysis of the free-energy landscape, we argue that the transition from a homogeneous state to aggregation is abrupt yet continuous.
... 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.
... Ants communicate with each other through the use of pheromones to adjust their collective behaviour. [1][2][3] This mechanism often leads to intriguing self-organized patterns. For example, their foraging path can be understood as solving a certain optimization problem in terms of time and energy costs, [4][5][6][7][8][9] and the shape of the path is predictable by Fermat's principle of least time. ...
The Patlak-Keller-Segel equation is a canonical model of chemotaxis to describe self-organized aggregation of organisms interacting with chemical signals. We investigate a variant of this model, assuming that the organisms exert effective pressure proportional to the number density. From the resulting set of partial differential equations, we derive a Lyapunov functional that can also be regarded as the free energy of this model, and minimize it with a Monte Carlo method to detect the condition for self-organized aggregation. Focusing on radially symmetric solutions on a two-dimensional disc, we find that the chemical interaction competes with diffusion so that aggregation occurs when the relative interaction strength exceeds a certain threshold. Based on the analysis of the free-energy landscape, we argue that the transition from a homogeneous state to aggregation is abrupt yet continuous.
... Motion tracking and behavioral modeling of biological collectives, such as bird flocks, fish schools, insect swarms, and pedestrian crowds, have been the subject of active research in behavioral ecology [1], [2], [3], [4] and artificial life [5], [6], [7]. This is arguably one of the most challenging tasks in complex systems science, in which researchers need to elucidate unknown microscopic rules that are responsible for the observed or desired macroscopic emergent behaviors. ...
We propose a novel computational method to extract information about interactions among individuals with different behavioral states in a biological collective from ordinary video recordings. Assuming that individuals are acting as finite state machines, our method first detects discrete behavioral states of those individuals and then constructs a model of their state transitions, taking into account the positions and states of other individuals in the vicinity. We have tested the proposed method through applications to two real-world biological collectives, termites in an experimental setting and human pedestrians in an open space. For each application, a robust tracking system was developed in-house, utilizing interactive human intervention (for termite tracking) or online agent-based simulation (for pedestrian tracking). In both cases, significant interactions were detected between nearby individuals with different states, demonstrating the effectiveness of the proposed method.
We consider a non-local scalar conservation law in two space dimensions which arises as the formal hydrodynamic limit of a Fokker–Planck equation. This Fokker–Planck equation is, in turn, the kinetic description of an individual-based model describing the navigation of self-propelled particles in a pheromone landscape. The pheromone may be linked to the agent distribution itself, leading to a nonlinear, non-local scalar conservation law where the effective velocity vector depends on the pheromone field in a small region around each point, and thus, on the solution itself. After presenting and motivating the problem, we present some numerical simulations of a closely related problem, and then prove a well-posedness and stability result for the conservation law.
Individual fire ants are inherently active as they are living organisms that convert stored chemical energy into motion. However, each individual ant is not equally disposed to motion at any given time. In an active aggregation, most of the constituent ants are active, and vice versa for an inactive aggregation. Here we look at the role activity plays on the nonlinear mechanical behavior of the aggregation through large amplitude oscillatory shear measurements. We find that the level of viscous nonlinearity can be decreased by increasing the activity or by increasing the volume fraction. In contrast, the level of elastic nonlinearity is not affected by either activity or volume fraction. We interpret this in terms of a transient network with equal rates of linking and unlinking but with varying number of linking and unlinking events.
Significance
Bacterial chemotaxis is the process of bacterial migration in the direction of food and/or a chemoattractant and away from chemorepellents. Several models have been proposed for this process and these predict that their migration across obstacle-laden terrains would be slower when compared with terrains that do not contain obstacles. However, our experiments show that the presence of obstacles does not alter the average time it takes Escherichia coli cells to reach the food source. We show, both theoretically and experimentally, that a modified model that relies on adaptive movement following collisions with obstacles accurately explains this surprising result. These findings imply operational short-term memory of bacteria and motivate improved algorithms for self-autonomous robotic swarms.
Several models have been proposed to describe the behavior of ants when moving from nest to food sources. Most of these studies where based on numerical simulations with no mathematical justification. In this paper, we propose a mechanism for the formation of paths of minimal length between two points by a collection of individuals undergoing reinforced random walks taking into account not only the lengths of the paths but also the angles (connected to the preference of ants to move along straight lines). Our model involves reinforcement (pheromone accumulation), persistence (tendency to preferably follow straight directions in absence of any external effect) and takes into account the bifurcation angles of each edge (represented by a probability of willingness of choosing the path with the smallest angle). We describe analytically the results for 2 ants and different path lengths and numerical simulations for several ants.
During foraging, ant workers are known to make use of multiple information sources, such as private information (personal memory) and social information (trail pheromones). Environmental effects on foraging, and how these interact with other information sources, have, however, been little studied. One environmental effect is trail bifurcation asymmetry. Ants forage on branching trail networks and must often decide which branch to take at a junction (bifurcation). This is an important decision, as finding food sources relies on making the correct choices at bifurcations. Bifurcation angle may provide important information when making this choice. We used a Y-maze with a pivoting 90° bifurcation to study trail choice of Lasius niger foragers at varying branch asymmetries (0°, [both branches 45° from straight ahead], 30° [branches at 30° and 60° from straight ahead], 45°, 60° and 90° [one branch straight ahead, the other at 90°]). The experiment was carried out either with equal amounts of trail pheromone on both branches of the bifurcation or with pheromone present on only one branch. Our results show that with equal pheromone, trail asymmetry has a significant effect on trail choice. Ants preferentially follow the branch deviating least from straight, and this effect increases as asymmetry increases (47% at 0°, 54% at 30°, 57% at 45°, 66% at 60° and 73% at 90°). However, when pheromone is only present on one branch, the graded effect of asymmetry disappears. Overall, however, there is an effect of asymmetry as the preference of ants for the pheromone-marked branch over the unmarked branch is reduced from 65%, when it is the less deviating branch, to 53%, when it is the more deviating branch. These results demonstrate that trail asymmetry influences ant decision-making at bifurcations and that this information interacts with trail pheromone presence in a non-hierarchical manner.
Workers of the Argentine ant, Iridomyrmex humilis,start to explore a chemically unmarked territory randomly. As the exploratory front advances, other explorers are recruited and a trail extends from it to the nest. Whereas recruitment trails are generally constructed between two points, these exploratory trails have no fixed destination, and strongly resemble the foraging patterns of army ants. A minimal model shows how the exploratory pattern may be generated by the individual workers' simple trail-laying and -following behavior, illustrating how complex collective structures in insect colonies may be based on self-organization.
Biological systems that build transport networks, such as trail-laying ants and the slime mould Physarum, can be described in terms of reinforced random walks. In a reinforced random walk, the route taken by 'walking' particles depends on the previous routes of other particles. Here, we present a novel form of random walk in which the flow of particles provides this reinforcement. Starting from an analogy between electrical networks and random walks, we show how to include current reinforcement. We demonstrate that current-reinforcement results in particles converging on the optimal solution of shortest path transport problems, and avoids the self-reinforcing loops seen in standard density-based reinforcement models. We further develop a variant of the model that is biologically realistic, in the sense that the particles can be identified as ants and their measured density corresponds to those observed in maze-solving experiments on Argentine ants. For network formation, we identify the importance of nonlinear current reinforcement in producing networks that optimize both network maintenance and travel times. Other than ant trail formation, these random walks are also closely related to other biological systems, such as blood vessels and neuronal networks, which involve the transport of materials or information. We argue that current reinforcement is likely to be a common mechanism in a range of systems where network construction is observed.
We studied the formation of trail patterns by Argentine ants exploring an empty arena. Using a novel imaging and analysis technique we estimated pheromone concentrations at all spatial positions in the experimental arena and at different times. Then we derived the response function of individual ants to pheromone concentrations by looking at correlations between concentrations and changes in speed or direction of the ants. Ants were found to turn in response to local pheromone concentrations, while their speed was largely unaffected by these concentrations. Ants did not integrate pheromone concentrations over time, with the concentration of pheromone in a 1 cm radius in front of the ant determining the turning angle. The response to pheromone was found to follow a Weber's Law, such that the difference between quantities of pheromone on the two sides of the ant divided by their sum determines the magnitude of the turning angle. This proportional response is in apparent contradiction with the well-established non-linear choice function used in the literature to model the results of binary bridge experiments in ant colonies (Deneubourg et al. 1990). However, agent based simulations implementing the Weber's Law response function led to the formation of trails and reproduced results reported in the literature. We show analytically that a sigmoidal response, analogous to that in the classical Deneubourg model for collective decision making, can be derived from the individual Weber-type response to pheromone concentrations that we have established in our experiments when directional noise around the preferred direction of movement of the ants is assumed.
We measured the shape of the foraging trail networks of 11 colonies of the wood ant Formica aquilonia (Formica rufa group). We characterized these networks in terms of their degree of branching and the angles between branches, as well as
in terms of their efficiency. The measured networks were compared with idealized model networks built to optimize one of two
components of efficiency, total length (i.e., total amount of trail) and route factor (i.e., average distance between nest
and foraging site). The analysis shows that the networks built by the ants obtain a compromise between the two modes of efficiency.
These results are largely independent of the size of the network or colony size. The ants’ efficiency is comparable to that
of networks built by humans but achieved without the benefit of centralized control.
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 live in dynamically changing environments, where food sources become depleted and alternative sources appear. Yet most mathematical models of ant foraging assume that the ants' foraging environment is static. Here we describe a mathematical model of ant foraging in a dynamic environment. Our model attempts to explain recent empirical data on dynamic foraging in the Argentine ant Linepithema humile (Mayr). The ants are able to find the shortest path in a Towers of Hanoi maze, a complex network containing 32,768 alternative paths, even when the maze is altered dynamically. We modify existing models developed to explain ant foraging in static environments, to elucidate what possible mechanisms allow the ants to quickly adapt to changes in their foraging environment. Our results suggest that navigation of individual ants based on a combination of one pheromone deposited during foraging and directional information enables the ants to adapt their foraging trails and recreates the experimental results.
Although pedestrians have individual preferences, aims, and destinations, the dynamics of pedestrian crowds is surprisingly predictable. Pedestrians can move freely only at small pedestrian densities. Otherwise their motion is affected by repulsive interactions with other pedestrians, giving rise to self-organization phenomena. Examples of the resulting patterns of motion are separate lanes of uniform walking direction in crowds of oppositely moving pedestrians or oscillations of the passing direction at bottlenecks. If pedestrians leave footprints on deformable ground (for example, in green spaces such as public parks) this additionally causes attractive interactions which are mediated by modifications of their environment. In such cases, systems of pedestrian trails will evolve over time. The corresponding computer simulations are a valuable tool for developing optimized pedestrian facilities and way systems.
Trail traffic of the leaf-cutting ant Atta cephalotes involves intermingled flows of outbound and returning foragers. Head-on encounters between workers from the opposite flows are a common occurrence in this traffic. Each encounter momentarily delays the two ants involved, and these small delays might pose a significant cost to the colony's foraging performance when summed over thousands of workers along many metres of trail. We videotaped outbound and returning foragers over a 1 m course, and measured the encounter rates they experienced and their velocity. Our analysis indicates that locomotion speed is diminished by increasing encounter rate, but that the effect is small relative to the effects of ant body size and load mass. Head-on encounters allow exchange of information and leaf fragments between workers, and we consider how the benefits of such encounters may make this form of traffic organization superior to segregated outbound and returning lanes, despite the measurable c
ost of encounters in mixed traffic.
Foragers of the ant Lasius niger exploiting a 1 M sugar source were found to lay 43 % more trail marks than those exploiting a 0.05 or a 0.1 M source. The trail laying per forager decreased during the course of individual recruitment episodes, and the mean lifetime of the trail pheromone was estimated to be 47 min. A mathematical function describing the probability that a forager chooses one of two paths in relation to the amount of trail pheromone on them closely fitted experimental data. These results were incorporated into a model describing the recruitment dynamics of L. niger. Simulations of this model showed that the observed modulation of trail laying with respect to food source quality is sufficient in itself to account for the systematic selection of the richer source seen in the experiments.
Many biological systems use extensive networks for the transport of resources and information. Ants are no exception. How do biological systems achieve efficient transportation networks in the absence of centralized control and without global knowledge of the environment? Here, we address this question by studying the formation and properties of inter-nest transportation networks in the Argentine ant (Linepithema humile). We find that the formation of inter-nest networks depends on the number of ants involved in the construction process. When the number of ants is sufficient and networks do form, they tend to have short total length but a low level of robustness. These networks are topologically similar to either minimum spanning trees or Steiner networks. The process of network formation involves an initial construction of multiple links followed by a pruning process that reduces the number of trails. Our study thus illuminates the conditions under and the process by which minimal biological transport networks can be constructed.
The movement ecology framework depicts animal movement as the result of the combined effects of internal and external constraints on animal navigation and motion capacities. Nevertheless, there are still fundamental problems to understand how these modulations take place and how they might be translated into observed statistical properties of animal trajectories. Of particular interest, here, is the general idea of intermittence in animal movement. Intermittent locomotion assumes that animal movement is, in essence, discrete. The existence of abrupt interruptions in an otherwise continuous flow of movement allows for the possibility of reorientations, that is, to break down previous directional memories of the trajectory. In this study, we explore the potential links between intermittent locomotion, reorientation behavior, and search efficiency. By means of simulations we show that the incorporation of Lévy intermittence in an otherwise nonintermittent search strongly modifies encounter rates. The result is robust to different types of landscapes (i.e., target density and spatial distribution), and spatial dimensions (i.e., 2D, 3D). We propose that Lévy intermittence may come from reorientation mechanisms capable of organizing directional persistence on time (i.e., fractal reorientation clocks), and we rationalize that the explicit distinction between scanning and reorientation mechanisms is essential to make accurate statistical inferences from animal search behavior. Finally, we provide a statistical tool to judge the existence of episodic and strong reorientation behaviors capable of modifying relevant properties of stochastic searches, ultimately controlling the chances of finding unknown located items.
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 vertebrates, networks of capillary vessels supply tissues with nutrients. Capillary patterns are closely mimicked by endothelial cells cultured on basement membrane proteins that allow single randomly dispersed cells to self-organize into vascular networks. Here we provide a model including chemoattraction as the fundamental mechanism for cell-to-cell communication in order to identify key parameters in the complexity of the formation of vascular patterns. By flanking biological experiments, theoretical insights and numerical simulations, we provide strong evidence that endothelial cell number and the range of activity of a chemoattractant factor regulate vascular network formation and size. We propose a mechanism linking the scale of formed endothelial structures to the range of cell-to-cell interaction mediated by the release of chemoattractants.
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.
Active walker models have recently proved their great value for describing the formation of clusters, periodic patterns, and spiral waves as well as the development of rivers, dielectric breakdown patterns, and many other structures. It is shown that they also allow to simulate the formation of trail systems by pedestrians and ants, yielding a better understanding of human and animal behavior. A comparison with empirical material shows a good agreement between model and reality. Our trail formation model includes an equation of motion, an equation for environmental changes, and an orientation relation. It contains some model functions, which are specified according to the characteristics of the considered animals or pedestrians. Not only the kind of environmental changes differs: Whereas pedestrians leave footprints on the ground, ants produce chemical markings for their orientation. Nevertheless, it is more important that pedestrians steer towards a certain destination, while ants usually find their food sources by chance, i.e. they reach their destination in a stochastic way. As a consequence, the typical structure of the evolving trail systems depends on the respective species. Some ant species produce a dendritic trail system, whereas pedestrians generate a minimal detour system. The trail formation model can be used as a tool for the optimization of pedestrian facilities: It allows urban planners to design convenient way systems which actually meet the route choice habits of pedestrians. Comment: For related work see http://www.theo2.physik.uni-stuttgart.de/helbing.html and http://summa.physik.hu-berlin.de:80/~frank/index.html and http://galaxy.cau.edu/molnar
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.
We propose and numerically analyze a PDE model of ant foraging behavior. Ant
foraging is a prime example of individuals following simple behavioral rules
based on local information producing complex, organized and ``intelligent''
strategies at the population level. One of its main aspects is the widespread
use of pheromones, which are chemical compounds laid by the ants used to
attract other ants to a food source. In this work, we consider a continuous
description of a population of ants and simulate numerically the foraging
behavior using a system of PDEs of chemotaxis type. We show that, numerically,
this system accurately reproduces observed foraging behavior, such as trail
formation and efficient removal of food sources.
Particle tracking is a widely used and promising technique for elucidating complex dynamics of the living cell. The cytoplasm is an active material, in which the kinetics of intracellular structures are highly heterogeneous. Tracer particles typically undergo a combination of random motion and various types of directed motion caused by the activity of molecular motors and other non-equilibrium processes. Random switching between more and less directional persistence of motion generally occurs. We present a method for identifying states of motion with different directional persistence in individual particle trajectories. Our analysis is based on a multi-scale turning angle model to characterize motion locally, together with a Hidden Markov Model with two states representing different directional persistence. We define one of the states by the motion of particles in a reference data set where some active processes have been inhibited. We illustrate the usefulness of the method by studying transport of vesicles along microtubules and transport of nanospheres activated by myosin. We study the results using mean square displacements, durations, and particle speeds within each state. We conclude that the method provides accurate identification of states of motion with different directional persistence, with very good agreement in terms of mean-squared displacement between the reference data set and one of the states in the two-state model.
Behavioral evidence indicates that (Z)-9-hexadecenal (Z9-16∶ALD) is a trail pheromone component ofIridomyrmex humilis, and that the true trail pheromone may be multicomponent. Trail-following responses ofI. humilis workers to several concentrations of syntheticZ9-16∶ALD, a constituent of the Pavan's gland, were found to be comparable to responses to gaster extract trails containing ca. 100 times lessZ9-16∶ALD. Of the five aldehyde analogs tested, only (Z)-7-hexadecenal (Z7-16∶ALD) elicited significant trail-following. However, following responses to severalZ9-16∶ALD-Z7-16∶ALD combinations were lower than responses toZ9-16∶ALD alone. Trails on filter paper of biologically relevant concentrations ofZ9-16∶ALD lose activity within 2 hr in the laboratory. The release rate ofZ9-16∶ALD measured from filter paper trails was 0.25 ± 0.10 pg/cm-sec. This was used to estimate the trail-following threshold for this compound of Argentine ant workers.
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.
Many ants use pheromone trails to organize collective foraging. This study investigated the rate at which a well-established Pharaoh's ant, Monomorium pharaonis (L.), trail breaks down on two substrates (polycarbonate plastic, newspaper). Workers were allowed to feed on sucrose solution from a feeder 30 cm from the nest. Between the nest and the feeder, the trail had a Y-shaped bifurcation. Initially, while recruiting to and exploiting the feeder, workers could only deposit pheromone on the branch leading to the feeder. Once the trail was established (by approximately 60 ants per min for 20 min), the ants were not allowed to reinforce the trail and were given a choice between the marked and unmarked branches. The numbers of ants choosing each branch were counted for 30 min. Initially, most went to the side on which pheromone had been deposited (80% and 70% on the plastic and paper substrates, respectively). However, this decayed to 50% within 25 min for plastic and 8 min for paper. From these data, the half-life times of the pheromone are estimated as approximately 9 min and 3 min on plastic and paper, respectively. The results show that, for M. pharaonis, trail decay is rapid and is affected strongly by trail substrate.
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.
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.
Heterogeneous, "aggregated" patterns in the spatial distributions of individuals are almost universal across living organisms, from bacteria to higher vertebrates. Whereas specific features of aggregations are often visually striking to human eyes, a heuristic analysis based on human vision is usually not sufficient to answer fundamental questions about how and why organisms aggregate. What are the individual-level behavioral traits that give rise to these features? When qualitatively similar spatial patterns arise from purely physical mechanisms, are these patterns in organisms biologically significant, or are they simply epiphenomena that are likely characteristics of any set of interacting autonomous individuals? If specific features of spatial aggregations do confer advantages or disadvantages in the fitness of group members, how has evolution operated to shape individual behavior in balancing costs and benefits at the individual and group levels? Mathematical models of social behaviors such as schooling in fishes provide a promising avenue to address some of these questions. However, the literature on schooling models has lacked a common framework to objectively and quantitatively characterize relationships between individual-level behaviors and group-level patterns. In this paper, we briefly survey similarities and differences in behavioral algorithms and aggregation statistics among existing schooling models. We present preliminary results of our efforts to develop a modeling framework that synthesizes much of this previous work, and to identify relationships between behavioral parameters and group-level statistics.
In this paper we present an individual-based model describing the foraging behavior of ants moving in an artificial network of tunnels in which several interconnected paths can be used to reach a single food source. Ants lay a trail pheromone while moving in the network and this pheromone acts as a system of mass recruitment that attracts other ants in the network. The rules implemented in the model are based on measures of the decisions taken by ants at tunnel bifurcations during real experiments. The collective choice of the ants is estimated by measuring their probability to take a given path in the network. Overall, we found a good agreement between the results of the simulations and those of the experiments, showing that simple behavioral rules can lead ants to find the shortest paths in the network. The match between the experiments and the model, however, was better for nestbound than for outbound ants. A sensitivity study of the model suggests that the bias observed in the choice of the ants at asymmetrical bifurcations is a key behavior to reproduce the collective choice observed in the experiments.
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.
- S Camazine
- H L Deneubourg
- N R Franks
- J Sneyd
- G Theraulaz
- E Bonaneau
S. Camazine, H. L. Deneubourg,N. R. Franks, J. Sneyd, G. Theraulaz and E. Bonaneau " Self-Organization in Biological Systems, "
Princeton University Press, 2003.
A continuous model of ant foraging with pheromones and trail formation
- P Amorin
P. Amorin, A continuous model of ant foraging with pheromones and
trail formation, ariv:1402.5611 (2014).