A non-local model for a swarm

 This paper describes continuum models for swarming behavior based on non-local interactions. The interactions are assumed to influence the velocity of the organisms. The model consists of integro-differential advection-diffusion equations, with convolution terms that describe long range attraction and repulsion. We find that if density dependence in the repulsion term is of a higher order than in the attraction term, then the swarm profile is realistic: i.e. the swarm has a constant interior density, with sharp edges, as observed in biological examples. This is our main result. Linear stability analysis, singular perturbation theory, and numerical experiments reveal that weak, density-independent diffusion leads to disintegration of the swarm, but only on an exponentially large time scale. When density dependence is put into the diffusion term, we find that true, locally stable traveling band solutions occur. We further explore the effects of local and non-local density dependent drift and unequal ranges of attraction and repulsion. We compare our results with results of some local models, and find that such models cannot account for cohesive, finite swarms with realistic density profiles. Received: 17 September 1997 / Revised version: 17 March 1998     

Swarming in homogeneous environments: A social interaction based framework

A great variety of biological groups form a self-organized swarming motion at some point during their life spans, which has two prominent collective features: common velocity and constant spacings among members. In this paper, we present a general individual-based motion framework to explain such collective motion of swarms in homogeneous environments. The motion framework utilizes the concept of social interactions that has been widely accepted throughout the literature. We assume that during the motion of the swarm, each member senses and interacts with its neighbors via virtual Attraction/Alignment/Repulsion (A/A/R) forces, while perceiving and following the gradient force of the environment. During the swarm's motion, the neighborhood and the interaction relations among members may dynamically change. To explicitly consider the effect of such dynamic change on the emergence of swarm's collective behavior, we use an algebraic graph to model the topology of the interaction and the neighborhood relations among the members.By using mathematical tools of nonsmooth analysis theory and Lyapunov stability theory, we analytically prove that if the A/A/R forces have limited ranges, and the attraction/repulsion forces are balanced at a certain range, the proposed framework leads to a parallel type of collective motion of the swarm. We mathematically show that the velocities of all swarm members asymptotically converge to a common value and the spacings among neighbors remain unchanging. In addition to the mathematical analysis, a few sets of simulation results are included to demonstrate the presented framework.The contributions of this paper are twofold: First, unlike most works in the literature that mainly use computer simulations to study the swarming phenomena, this paper provides an analytical methodology to investigate how the collective group behavior is self-organized by individual motions. Second, the presented motion framework works over a general range of A/A/R interactions. In other words, we analytically prove that the commonly used A/A/R model can lead to a collective motion of the swarm. In addition, we show that the alternative model in the literature that uses only attraction/repulsion (A/R) interactions is in fact a special case of the A/A/R model.     

How the Spatial Position of Individuals Affects Their Influence on Swarms: A Numerical Comparison of Two Popular Swarm Dynamics Models

Schools of fish and flocks of birds are examples of self-organized animal groups that arise through social interactions among individuals. We numerically study two individual-based models, which recent empirical studies have suggested to explain self-organized group animal behavior: (i) a zone-based model where the group communication topology is determined by finite interacting zones of repulsion, attraction, and orientation among individuals; and (ii) a model where the communication topology is described by Delaunay triangulation, which is defined by each individual''s Voronoi neighbors. The models include a tunable parameter that controls an individual''s relative weighting of attraction and alignment. We perform computational experiments to investigate how effectively simulated groups transfer information in the form of velocity when an individual is perturbed. A cross-correlation function is used to measure the sensitivity of groups to sudden perturbations in the heading of individual members. The results show how relative weighting of attraction and alignment, location of the perturbed individual, population size, and the communication topology affect group structure and response to perturbation. We find that in the Delaunay-based model an individual who is perturbed is capable of triggering a cascade of responses, ultimately leading to the group changing direction. This phenomenon has been seen in self-organized animal groups in both experiments and nature.     

Spatial scaling in a benthic population model with density-dependent disturbance.

This work investigates approaches to simplifying individual-based models in which the rate of disturbance depends on local densities. To this purpose, an individual-based model for a benthic population is developed that is both spatial and stochastic. With this model, three possible ways of approximating the dynamics of mean numbers are examined: a mean-field approximation that ignores space completely, a second-order approximation that represents spatial variation in terms of variances and covariances, and a patch-based approximation that retains information about the age structure of the patch population. Results show that space is important and that a temporal model relying on mean disturbance rates provides a poor approximation to the dynamics of mean numbers. It is possible, however, to represent relevant spatial variation with second-order moments, particularly when recruitment rates are low and/or when disturbances are large and weak. Even better approximations are obtained by retaining patch age information.     

Mutual interactions,potentials, and individual distance in a social aggregation

We formulate a Lagrangian (individual-based) model to investigate the spacing of individuals in a social aggregate (e.g., swarm, flock, school, or herd). Mutual interactions of swarm members have been expressed as the gradient of a potential function in previous theoretical studies. In this specific case, one can construct a Lyapunov function, whose minima correspond to stable stationary states of the system. The range of repulsion (r) and attraction (a) must satisfy r<a for cohesive groups (i.e., short range repulsion and long range attraction). We show quantitatively how repulsion must dominate attraction (Rr^d+1>cAa^d+1 where R, A are magnitudes, c is a constant of order 1, and d is the space dimension) to avoid collapse of the group to a tight cluster. We also verify the existence of a well-spaced locally stable state, having a characteristic individual distance. When the number of individuals in a group increases, a dichotomy occurs between swarms in which individual distance is preserved versus those in which the physical size of the group is maintained at the expense of greater crowding.     

Dynamics of prey-flock escaping behavior in response to predator's attack

The dynamic behavior of prey-flock in response to predator's attack was investigated by using molecular dynamics (MD) simulations in a two-dimensional (2D) continuum model. By locally applying interactive forces between prey individuals (e.g. attraction, repulsion, and alignment), a coherently moving state in the same direction was obtained among individuals in prey-flock. When a single predator was introduced to the prey population, the prey-flock was correspondingly deformed by the predator's continuous attacks towards the prey-flock's center. In response to the predator's attack, three regimes in the flock size (compression (Regime I), expansion (Regime II), compression (Regime III)) were revealed if the predator's attack speed (kappa) was comparatively low to the escape speed of prey-flock. If noise was added to the predator's attacking course, a higher degree of variation was observed in the patterns of compression and expansion in the prey-flock size. However, the scaling behavior in the changes in prey-flock size was present in different levels of noise with the increase in predation risk (R) when kappa takes an appropriately low value. During the procedure of escaping, order breaking in alignment (phi) of prey population was observed, while the degree of alignment was dependent upon the changes in parameters of kappa and R.     

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.     

From individuals to aggregations: the interplay between behavior and physics

This paper analyses the processes by which organisms form groups and how social forces interact with environmental variability and transport. For aquatic organisms, the latter is especially important-will sheared or turbulent flows disrupt organism groups? To analyse such problems, we use individual-based models to study the environmental and social forces leading to grouping. The models are then embedded in turbulent flow fields to gain an understanding of the interplay between the forces acting on the individuals and the transport induced by the fluid motion. Instead of disruption of groups, we find that flows often enhance grouping by increasing the encounter rate among groups and thereby promoting merger into larger groups; the effect breaks down for strong flows. We discuss the transformation of individual-based models into continuum models for the density of organisms. A number of subtle difficulties arise in this process; however, we find that a direct comparison between the individual model and the continuum model is quite favorable. Finally, we examine the dynamics of group statistics and give an example of building an equation for the spatial and temporal variations of the group-size distribution from the individual-based simulations. These studies lay the groundwork for incorporating the effects of grouping into models of the large scale distributions of organisms as well as for examining the evolutionary consequences of group formation.     

Dynamical modeling of collective behavior from pigeon flight data: flock cohesion and dispersion

Several models of flocking have been promoted based on simulations with qualitatively naturalistic behavior. In this paper we provide the first direct application of computational modeling methods to infer flocking behavior from experimental field data. We show that this approach is able to infer general rules for interaction, or lack of interaction, among members of a flock or, more generally, any community. Using experimental field measurements of homing pigeons in flight we demonstrate the existence of a basic distance dependent attraction/repulsion relationship and show that this rule is sufficient to explain collective behavior observed in nature. Positional data of individuals over time are used as input data to a computational algorithm capable of building complex nonlinear functions that can represent the system behavior. Topological nearest neighbor interactions are considered to characterize the components within this model. The efficacy of this method is demonstrated with simulated noisy data generated from the classical (two dimensional) Vicsek model. When applied to experimental data from homing pigeon flights we show that the more complex three dimensional models are capable of simulating trajectories, as well as exhibiting realistic collective dynamics. The simulations of the reconstructed models are used to extract properties of the collective behavior in pigeons, and how it is affected by changing the initial conditions of the system. Our results demonstrate that this approach may be applied to construct models capable of simulating trajectories and collective dynamics using experimental field measurements of herd movement. From these models, the behavior of the individual agents (animals) may be inferred.     

The Effect of Attractive Interactions and Macromolecular Crowding on Crystallins Association

In living systems proteins are typically found in crowded environments where their effective interactions strongly depend on the surrounding medium. Yet, their association and dissociation needs to be robustly controlled in order to enable biological function. Uncontrolled protein aggregation often causes disease. For instance, cataract is caused by the clustering of lens proteins, i.e., crystallins, resulting in enhanced light scattering and impaired vision or blindness. To investigate the molecular origins of cataract formation and to design efficient treatments, a better understanding of crystallin association in macromolecular crowded environment is needed. Here we present a theoretical study of simple coarse grained colloidal models to characterize the general features of how the association equilibrium of proteins depends on the magnitude of intermolecular attraction. By comparing the analytic results to the available experimental data on the osmotic pressure in crystallin solutions, we identify the effective parameters regimes applicable to crystallins. Moreover, the combination of two models allows us to predict that the number of binding sites on crystallin is small, i.e. one to three per protein, which is different from previous estimates. We further observe that the crowding factor is sensitive to the size asymmetry between the reactants and crowding agents, the shape of the protein clusters, and to small variations of intermolecular attraction. Our work may provide general guidelines on how to steer the protein interactions in order to control their association.     

Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations

In this paper we develop a general mathematical model describing the spatio-temporal dynamics of host-parasitoid systems with forced generational synchronisation, for example seasonally induced diapause. The model itself may be described as an individual-based stochastic model with the individual movement rules derived from an underlying continuum PDE model. This approach permits direct comparison between the discrete model and the continuum model. The model includes both within-generation and between-generation mechanisms for population regulation and focuses on the interactions between immobile juvenile hosts, adult hosts and adult parasitoids in a two-dimensional domain. These interactions are mediated, as they are in many such host-parasitoid systems, by the presence of a volatile semio-chemical (kairomone) emitted by the hosts or the hosts food plant. The model investigates the effects on population dynamics for different host versus parasitoid movement strategies as well as the transient dynamics leading to steady states. Despite some agreement between the individual and continuum models for certain motility parameter ranges, the model dynamics diverge when host and parasitoid motilities are unequal. The individual-based model maintains spatially heterogeneous oscillatory dynamics when the continuum model predicts a homogeneous steady state. We discuss the implications of these results for mechanistic models of phenotype evolution.P. Schofield gratefully acknowledges the financial support of the BBSRC and The Wellcome Trust.     

Travelling Waves in Hybrid Chemotaxis Models

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.     

Modeling Group Formation and Activity Patterns in Self-Organizing Collectives of Individuals

We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.     

Dual interaction producing both territorial and schooling behavior in fish

Fish schools are frequently observed without a leader and an explicit condition forming school. Several models of schooling have been proposed focusing on the influence of neighbors, and introducing probability distributions, while these models are based on the separation of schooling and territorial behavior. We frequently consider the duality of aggregation of animals, in which behavioral patterns involve both attraction and repulsion, antagonistic with each other. The idea of probability does not explain this duality that can provide both schooling and territorial behavior. From these biological facts, we have constructed a behavior model in which the influence of neighbors is formulated by the interface between the states of neighbors and a map of changes in these states. This interface uses a self-similar nowhere differentiable transition map which is temporally constructed, encompassing a crucial duality of repulsive and attractive forces hidden in the interaction among fishes. We tested it with computer simulations against the biological reality of schooling and territorial behavior. Due to the influence neighbors can have on duality, the same model can show both schooling behavior with a high degree of polarization and territorial behavior.     

Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion

In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell behaviour is driven by a molecular network describing the dynamics of E-cadherin and β-catenin. In this model, which we refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and computational implementations of the two approaches are very different, the results of the presented simulations are compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling artefacts specific to a given computational approach.     

About deterministic extinction in ratio-dependent predator-prey models

Ratio-dependent predator-prey models set up a challenging issue regarding their dynamics near the origin. This is due to the fact that such models are undefined at (0, 0). We study the analytical behavior at (0, 0) for a common ratio-dependent model and demonstrate that this equilibrium can be either a saddle point or an attractor for certain trajectories. This fact has important implications concerning the global behavior of the model, for example regarding the existence of stable limit cycles. Then, we prove formally, for a general class of ratio-dependent models, that (0, 0) has its own basin of attraction in phase space, even when there exists a non-trivial stable or unstable equilibrium. Therefore, these models have no pathological dynamics on the axes and at the origin, contrary to what has been stated by some authors. Finally, we relate these findings to some published empirical results.     

An ecological resilience perspective on cancer: Insights from a toy model

In this paper we propose an ecological resilience point of view on cancer. This view is based on the analysis of a simple ODE model for the interactions between cancer and normal cells. The model presents two regimes for tumor growth. In the first, cancer arises due to three reasons: a partial corruption of the functions that avoid the growth of mutated cells, an aggressive phenotype of tumor cells and exposure to external carcinogenic factors. In this case, treatments may be effective if they drive the system to the basin of attraction of the cancer cure state. In the second regime, cancer arises because the repair system is intrinsically corrupted. In this case, the complete cure is not possible since the cancer cure state is no more stable, but tumor recurrence may be delayed if treatment is prolongued. We review three indicators of the resilience of a stable equilibrium, related with size and shape of its basin of attraction: latitude, precariousness and resistance. A novel method to calculate these indicators is proposed. This method is simpler and more efficient than those currently used, and may be easily applied to other population dynamics models. We apply this method to the model and investigate how these indicators behave with parameters changes. Finally, we present some simulations to illustrate how the resilience analysis can be applied to validated models in order to obtain indicators for personalized cancer treatments.