Active random walkers simulate trunk trail formation by ants

Interaction of red crabs with yellow crazy ants during migration on Christmas Island

Invasive species have had a profound impact on ecosystems all over the world. Their presence can lead to fundamental changes in the biodiversity of a given ecosystem as well as the extinction of native species. In particular, this work looks at the effect on the Gecarcoidea natalis (Red Crab) population on Christmas Island due to the presence of vast arrays of supercolonies containing Anoplolepis gracilipes (Yellow Crazy Ant). We primarily study the inter-species interaction occurring during the crab migration to the island coast. We propose a microscopic model for the dynamics of the crabs and ants with the goal of increasing crab survival. Through analysis of the model, we investigate a range of potential preventative measures that could be taken to preserve the native crab population dependent on their locations. The main result of this work is that by considering the locations of ant supercolonies incorporated into Monte Carlo simulations of the model, we can identify the order that the supercolonies need to be removed to provide the greatest chance at survival for the crabs per migration cycle.

Modeling ant foraging: A chemotaxis approach with pheromones and trail formation

We consider a continuous mathematical description of a population of ants and simulate numerically their foraging behavior using a system of partial differential equations of chemotaxis type. We show that this system accurately reproduces observed foraging behavior, especially spontaneous trail formation and efficient removal of food sources. We show through numerical experiments that trail formation is correlated with efficient food removal. Our results illustrate the emergence of trail formation from simple modeling principles.

A model for collective dynamics in ant raids

Ant raiding, the process of identifying and returning food to the nest or bivouac, is a fascinating example of collective motion in nature. During such raids ants lay pheromones to form trails for others to find a food source. In this work a coupled PDE/ODE model is introduced to study ant dynamics and pheromone concentration. The key idea is the introduction of two forms of ant dynamics: foraging and returning, each governed by different environmental and social cues. The model accounts for all aspects of the raiding cycle including local collisional interactions, the laying of pheromone along a trail, and the transition from one class of ants to another. Through analysis of an order parameter measuring the orientational order in the system, the model shows self-organization into a collective state consisting of lanes of ants moving in opposite directions as well as the transition back to the individual state once the food source is depleted matching prior experimental results. This indicates that in the absence of direct communication ants naturally form an efficient method for transporting food to the nest/bivouac. The model exhibits a continuous kinetic phase transition in the order parameter as a function of certain system parameters. The associated critical exponents are found, shedding light on the behavior of the system near the transition.

Tumor invasion optimization by mesenchymal-amoeboid heterogeneity

Metastasizing tumor cells migrate through the surrounding tissue and extracellular matrix toward the blood vessels, in order to colonize distant organs. They typically move in a dense environment, filled with other cells. In this work we study cooperative effects between neighboring cells of different types, migrating in a maze-like environment with directional cue. Using a computerized model, we measure the percentage of cells that arrive to the defined target, for different mesenchymal/amoeboid ratios. Wall degradation of mesenchymal cells, as well as motility of both types of cells, are coupled to metabolic energy-like resource level. We find that indirect cooperation emerges in mid-level energy, as mesenchymal cells create paths that are used by amoeboids. Therefore, we expect to see a small population of mesenchymals kept in a mostly-amoeboid population. We also study different forms of direct interaction between the cells, and show that energy-dependent interaction strength is optimal for the migration of both mesenchymals and amoeboids. The obtained characteristics of cellular cluster size are in agreement with experimental results. We therefore predict that hybrid states, e.g. epithelial-mesenchymal, should be utilized as a stress-response mechanism.

Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality

Animal spacing has important implications for population abundance, species demography and the environment. Mechanisms underlying spatial segregation have their roots in the characteristics of the animals, their mutual interaction and their response, collective as well as individual, to environmental variables. This review describes how the combination of these factors shapes the patterns we observe and presents a practical, usable framework for the analysis of movement data in confined spaces. The basis of the framework is the theory of interacting random walks and the mathematical description of out-of-equilibrium systems. Although our focus is on modelling and interpreting animal home ranges and territories in vertebrates, we believe further studies on invertebrates may also help to answer questions and resolve unanswered puzzles that are still inaccessible to experimental investigation in vertebrate species.

Current-reinforced random walks for constructing transport networks

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.

How random is social behaviour? Disentangling social complexity through the study of a wild house mouse population

Out of all the complex phenomena displayed in the behaviour of animal groups, many are thought to be emergent properties of rather simple decisions at the individual level. Some of these phenomena may also be explained by random processes only. Here we investigate to what extent the interaction dynamics of a population of wild house mice (Mus domesticus) in their natural environment can be explained by a simple stochastic model. We first introduce the notion of perceptual landscape, a novel tool used here to describe the utilisation of space by the mouse colony based on the sampling of individuals in discrete locations. We then implement the behavioural assumptions of the perceptual landscape in a multi-agent simulation to verify their accuracy in the reproduction of observed social patterns. We find that many high-level features--with the exception of territoriality--of our behavioural dataset can be accounted for at the population level through the use of this simplified representation. Our findings underline the potential importance of random factors in the apparent complexity of the mice's social structure. These results resonate in the general context of adaptive behaviour versus elementary environmental interactions.

Individual rules for trail pattern formation in Argentine ants (Linepithema humile)

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.

Many ant species produce large dendritic networks of trails around their nest. These networks result from self-organized feedback mechanisms: ants leave small amounts of a chemical -a pheromone- as they move across space. In turn, they are attracted by this same pheromone so that eventually a trail is formed. In our study, we introduce a new image analysis technique to estimate the concentrations of pheromone directly on the trails. In this way, we can characterise the ingredients of the feedback loop that ultimately leads to the formation of trails. We show that the response to pheromone concentrations is linear: an ant will turn to the left with frequency proportional to the difference between the pheromone concentrations on its left and right sides. Such a linear individual response was rejected by previous literature, as it would be incompatible with the results of a large number of experiments: trails can only be reinforced if the ants have a disproportionally higher probability to select the trail with higher pheromone concentration. However, we show that the required non-linearity does not reside in the perceptual response of the ants, but in the noise associated with their movement.

Trail formation based on directed pheromone deposition

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.

Model of fasciculation and sorting in mixed populations of axons

We extend a recently proposed model [Chaudhuri et al., Europhys. Lett. 87, 20003 (2009)] aiming to describe the formation of fascicles of axons during neural development. The growing axons are represented as paths of interacting directed random walkers in two spatial dimensions. To mimic turnover of axons, whole paths are removed and new walkers are injected with specified rates. In the simplest version of the model, we use strongly adhesive short-range inter-axon interactions that are identical for all pairs of axons. We generalize the model to adhesive interactions of finite strengths and to multiple types of axons with type-specific interactions. The dynamic steady state is characterized by the position-dependent distribution of fascicle size and fascicle composition. With distance in the direction of axon growth, the mean fascicle size and emergent time scales grow monotonically, while the degree of sorting of fascicles by axon type has a maximum at a finite distance. To understand the emergence of slow time scales, we develop an analytical framework to analyze the interaction between neighboring fascicles.

Structure and formation of ant transportation networks

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.

Modeling vortex swarming in Daphnia

Based on experimental observations in Daphnia, we introduce an agent-based model for the motion of single and swarms of animals. Each agent is described by a stochastic equation that also considers the conditions for active biological motion. An environmental potential further reflects local conditions for Daphnia, such as attraction to light sources. This model is sufficient to describe the observed cycling behavior of single Daphnia. To simulate vortex swarming of many Daphnia, i.e. the collective rotation of the swarm in one direction, we extend the model by considering avoidance of collisions. Two different ansatzes to model such a behavior are developed and compared. By means of computer simulations of a multi-agent system we show that local avoidance - as a special form of asymmetric repulsion between animals - leads to the emergence of a vortex swarm. The transition from uncorrelated rotation of single agents to the vortex swarming as a function of the swarm size is investigated. Eventually, some evidence of avoidance behavior in Daphnia is provided by comparing experimental and simulation results for two animals.

Phase transitions and superuniversality in the dynamics of a self-driven particle

We study an active random walker model in which a particle's motion is determined by a self-generated field. The field encodes information about the particle's path history. This leads to either self-attractive or self-repelling behavior. For self-repelling behavior, we find a phase transition in the dynamics: when the coupling between the field and the walker exceeds a critical value, the particle's behavior changes from renormalized diffusion to one characterized by a diverging diffusion coefficient. The dynamical behavior for all cases is surprisingly independent of dimension and of the noise amplitude.

Coupled computational simulation and empirical research into the foraging system of Pharaoh's ant (Monomorium pharaonis)

The Pharaoh's ant (Monomorium pharaonis), a significant pest in many human environments, is phenomenally successful at locating and exploiting available food resources. Several pheromones are utilized in the self-organized foraging of this ant but most aspects of the overall system are poorly characterised. Agent-based modelling of ants as individual complex X-machines facilitates study of the mechanisms underlying the emergence of trails and aids understanding of the process. Conducting simultaneous modelling, and simulation, alongside empirical biological studies is shown to drive the research by formulating hypotheses that must be tested before the model can be verified and extended. Integration of newly characterised behavioural processes into the overall model will enable testing of general theories giving insight into division of labour within insect societies. This study aims to establish a new paradigm in computational modelling applicable to all types of multi-agent biological systems, from tissues to animal societies, as a powerful tool to accelerate basic research.

The effect of a non-uniform turning kernel on ant trail morphology

An ordinary differential equation model is constructed for the formation of pheromone trails by ants on a pre-determined network. At each junction of the trails the probability that an ant will turn through any particular angle is given by a turning kernel. We prove analytically using analogies with thermodynamics that turning behaviour determines trail morphology when the turning kernel is steep. We conjecture that this is also true in general for non-uniform turning kernels and present numerical simulations as evidence. Using this conjecture we show the existence of three types of collective foraging: individuals exploring without the use of a trail network, and two distinct types of trail networks; one that consists of low pheromone concentration trails that bend, branch and dissipate and one that consists of high pheromone concentration, straight, unbranched trails. We show that the form of the pheromone response function is crucial in determining the existence and stability of the steady states corresponding to these three foraging strategies, and examine the bifurcations between different trail morphologies as a function of turning kernel steepness for a particular response function.

Self-assembling of networks in an agent-based model

We propose a model to show the self-assembling of networklike structures between a set of nodes without using preexisting positional information or long-range attraction of the nodes. The model is based on Brownian agents that are capable of producing different local (chemical) information and respond to it in a nonlinear manner. They solve two tasks in parallel: (i) the detection of the appropriate nodes, and (ii) the establishment of stable links between them. We present results of computer simulations that demonstrate the emergence of robust network structures and investigate the connectivity of the network by means of both analytical estimations and computer simulations.

Statistical mechanics of canonical-dissipative systems and applications to swarm dynamics

We develop the theory of canonical-dissipative systems, based on the assumption that both the conservative and the dissipative elements of the dynamics are determined by invariants of motion. In this case, known solutions for conservative systems can be used for an extension of the dynamics, which also includes elements such as the takeup/dissipation of energy. This way, a rather complex dynamics can be mapped to an analytically tractable model, while still covering important features of nonequilibrium systems. In our paper, this approach is used to derive a rather general swarm model that considers (a) the energetic conditions of swarming, i.e., for active motion, and (b) interactions between the particles based on global couplings. We derive analytical expressions for the nonequilibrium velocity distribution and the mean squared displacement of the swarm. Further, we investigate the influence of different global couplings on the overall behavior of the swarm by means of particle-based computer simulations and compare them with the analytical estimations.

Freezing by heating in a driven mesoscopic system

We investigate a simple model corresponding to particles driven in opposite directions and interacting via a repulsive potential. The particles move off-lattice on a periodic strip and are subject to random forces as well. We show that this model-which can be considered as a continuum version of some driven diffusive systems-exhibits a paradoxical, new kind of transition called here "freezing by heating." One interesting feature of this transition is that a crystallized state with a higher total energy is obtained from a fluid state by increasing the amount of fluctuations.

Active Brownian particles with energy depots modeling animal mobility

In the model of active motion studied here, Brownian particles have the ability to take up energy from the environment to store it in an internal depot and to convert internal energy into kinetic energy. Considering also internal dissipation, we derive a simplified model of active biological motion. For the take-up of energy two different examples are discussed: (i) a spatially homogeneous supply of energy, and (ii) the supply of energy at spatially localized sources (food centers). The motion of the particles is described by a Langevin equation which includes an acceleration term resulting from the conversion of energy. Dependent on the energy sources, we found different forms of periodic motion (limit cycles), i.e. periodic motion between 'nest' and 'food'. An analytic approximation allows the description of the stationary motion and the calculation of critical parameters for the take-up of energy. Finally, we derive an analytic expression for the efficiency ratio of energy conversion, which considers the take-up of energy, compared to (internal and external) dissipation.