293 results found
Degond P, Hecht S, Vauchelet N, 2020, Incompressible limit of a continuum model of tissue growth for two cell populations, Networks and Heterogeneous Media, Vol: 15, Pages: 57-85, ISSN: 1556-1801
This paper investigates the incompressible limit of a system modelling the growth oftwo cells population. The model describes the dynamics of cell densities, driven by pressureexclusion and cell proliferation. It has been shown that solutions to this system of partialdifferential equations have the segregation property, meaning that two population initiallysegregated remain segregated. This work is devoted to the incompressible limit of suchsystem towards a free boundary Hele Shaw type model for two cell populations.
Degond P, Ferreira MA, Merino-Aceituno S, et al., 2020, A New Continuum Theory for Incompressible Swelling Materials, Publisher: Society for Industrial & Applied Mathematics (SIAM)
Barré J, Degond P, Peurichard D, et al., Modelling pattern formation through diﬀerential repulsion, Networks and Heterogeneous Media, ISSN: 1556-1801
Motivated by experiments on cell segregation, we present a twospeciesmodel of interacting particles, aiming at a quantitative description ofthis phenomenon. Under precise scaling hypothesis, we derive from the microscopicmodel a macroscopic one and we analyze it. In particular, we determinethe range of parameters for which segregation is expected. We compare ouranalytical results and numerical simulations of the macroscopic model to directsimulations of the particles, and comment on possible links with experiments.
Degond P, Ferreira M, Merino-Aceituno S, et al., A new continuum theory for incompressible swelling materials, SIAM: Multiscale Modeling and Simulation, ISSN: 1540-3459
Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well-known. Here, we take an opposite route and consider a simple packing heuristics, i.e. the particles can’t overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can’t swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets while in the parallel direction, it is determined by a Laplace-Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.
Ferreira M, Despin-Guitard E, Duarte F, et al., Interkinetic nuclear movements promote apical expansion in pseudostratified epithelia at the expense of apicobasal elongation, PLoS Computational Biology, ISSN: 1553-734X
Pseudostratified epithelia (PSE) are a common type of columnar epithelia found in a wealth of embryonic and adult tissues such as ectodermal placodes, the trachea, the ureter, the gut and the neuroepithelium. PSE are characterized by the choreographed displacement of cells’ nuclei along the apicobasal axis according to phases of their cell cycle. Such movements, called interkinetic movements (INM) have been proposed to influence tissue expansionand shape and suggested as culprit in several congenital diseases such as CAKUT(Congenital anomalies of kidney and urinary tract)and esophageal atresia. INM rely on cytoskeleton dynamics just as adhesion, contractility and mitosis do. Therefore, long term impairment of INM without affecting proliferation and adhesion is currently technically unachievable. Here we bypassed this hurdle by generating a 2D agent-based model of a proliferating PSE and compared its output to the growth of the chick neuroepitheliumto assess the interplay between INM and these other importantcell processes during growth of a PSE. We found that INM directly generates apical expansion and apical nuclear crowding. In addition, our data strongly suggest that apicobasal elongation of cells is not an emerging property of a proliferative PSE but rather requires a specific elongation program. We then discuss how such program might functionally link INM, tissue growth and differentiation.
Degond P, Merino-Aceituno S, Nematic alignment of self-propelled particles: from particle to macroscopic dynamics, Mathematical Models and Methods in Applied Sciences (M3AS), ISSN: 0218-2025
Starting from a particle model describing self-propelled particles interacting throughnematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics.After diffusive rescaling of the kinetic equation, we formally show that the distributionfunction converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system isconsistent with symmetries typical of a nematic material. The derivation is carried overby means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introducedto overcome the lack of momentum conservation of the system, plays a central role. Thiscross diffusion system poses many new challenging questions.
Degond P, Engel M, Liu J-G, et al., A Markov jump process modelling animal group size statistics, Communications in Mathematical Sciences, ISSN: 1539-6746
We translate a coagulation-framentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the nonlinear evolution equation witha Markov jump process of a type introduced in classic work of H. McKean. In particular this formalizesa model suggested by H.-S. Niwa [J. Theo. Biol. 224 (2003)] with simple coagulation and fragmentationrates. Based on the jump process, we develop a numerical scheme that allows us to approximate the equilibrium for the Niwa model, validated by comparison to analytical results by Degond et al. [J. NonlinearSci. 27 (2017)], and study the population and size distributions for more complicated rates. Furthermore,the simulations are used to describe statistical properties of the underlying jump process. We additionally discuss the relation of the jump process to models expressed in stochastic differential equations anddemonstrate that such a connection is justified in the case of nearest-neighbour interactions, as opposedto global interactions as in the Niwa model.
Aceves Sanchez P, Bostan M, Carrillo de la Plata JA, et al., 2019, Hydrodynamic limits for kinetic ﬂocking models of Cucker-Smale type, Mathematical Biosciences and Engineering, Vol: 16, Pages: 7883-7910, ISSN: 1547-1063
We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in , is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.
Degond P, Pulvirenti M, 2019, Propagation of chaos for topological interactions, Annals of Applied Probability, Vol: 29, Pages: 2594-2612, ISSN: 1050-5164
We consider aN-particle model describing an alignment mechanism due to a topolog-ical interaction among the agents. We show that the kinetic equation, expected to holdin the mean-field limitN→∞, as following from the previous analysis in Ref.  can berigorously derived. This means that the statistical independence (propagation of chaos)is indeed recovered in the limit, provided it is assumed at time zero.
Chertock A, Degond P, Hecht S, et al., 2019, Incompressible limit of a continuum model of tissue growth with segregation for two cell populations, Mathematical Biosciences and Engineering, Vol: 16, Pages: 5804-5835, ISSN: 1547-1063
This paper proposes a model for the growth of two interacting populations of cells that do not mix. The dynamics is driven by pressure and cohesion forces on the one hand and proliferation on the other hand. Contrasting with earlier works which assume that the two populations are initially segregated, our model can deal with initially mixed populations as it explicitly incorporates a repul-sion force that enforces segregation. To balance segregation instabilities potentially triggered by the repulsion force, our model also incorporates a fourth order diffusion. In this paper, we study the influ-ence of the model parameters thanks to one-dimensional simulations using a finite-volume method for a relaxation approximation of the fourth order diffusion. Then, following earlier works on the single population case, we provide formal arguments that the model approximates a free boundary Hele Shaw type model that we characterise using both analytical and numerical arguments.
Peurichard D, Ousset M, Paupert J, et al., 2019, Extra-cellular matrix rigidity may dictate the fate of injury outcome, Journal of Theoretical Biology, Vol: 469, Pages: 127-136, ISSN: 0022-5193
After injury, while regeneration can be observed in hydra, planaria and some vertebrates, regeneration is rare in mammals and particularly in humans. In this paper, we investigate the mechanisms by which biological tissues recover after injury. We explore this question on adipose tissue, using the mathematical framework recently developed in Peurichard et al., J. Theoret. Biol. 429 (2017), pp. 61-81. Our assumption is that simple mechanical cuesbetween the Extra-Cellular Matrix (ECM) and differentiated cells can explain adipose tissue morphogenesis and that regeneration requires after injury the same mechanisms. We validate this hypothesis by means of a two-dimensional Individual Based Model (IBM) of interacting adipocytes and ECM fiber elements. The model successfully generates regeneration or scar formation as functions of few key parameters, and seems to indicate that the fate of injury outcome could be mainly due to ECM rigidity
Degond P, Merino Aceituno S, Vergnet F, et al., 2019, Coupled Self-Organized Hydrodynamics and Stokes models for suspensions of active particles, Journal of Mathematical Fluid Mechanics, Vol: 21, ISSN: 1422-6928
We derive macroscopic dynamics for self-propelled particles in a fluid. The starting point is a coupled Vicsek–Stokes system. The Vicsek model describes self-propelled agents interacting through alignment. It provides a phenomenological description of hydrodynamic interactions between agents at high density. Stokes equations describe a low Reynolds number fluid. These two dynamics are coupled by the interaction between the agents and the fluid. The fluid contributes to rotating the particles through Jeffery’s equation. Particle self-propulsion induces a force dipole on the fluid. After coarse-graining we obtain a coupled Self-Organised Hydrodynamics–Stokes system. We perform a linear stability analysis for this system which shows that both pullers and pushers have unstable modes. We conclude by providing extensions of the Vicsek–Stokes model including short-distance repulsion, finite particle inertia and finite Reynolds number fluid regime.
Singh K, Muljadi B, Raeini AQ, et al., 2019, The architectural design of smart ventilation and drainage systems in termite nests, Science Advances, Vol: 5, ISSN: 2375-2548
Termite nests have been widely studied as effective examples for ventilation and thermoregulation;however, the mechanisms by which the nest properties are controlled by the micro-structure of the outer walls remain unclear. Here, we combine multi-scale X-ray imaging with three-dimensional flow field simulations to investigate the impact of the architectural design of nest walls on CO2and heat transport as well as on water drainage. We show that termites construct an outer wall that contains both small and percolating large pores at the micro-scale. The network of larger micro-scale pores in the outer wall provides a permeability that is 1-2 orders of magnitude greater than that of the smaller pores, andaCO2diffusivitythat is a factor of up to eight times larger. The largerpores and resultant high porosity also reduce the solid mass required for nest construction by ~11-14%. This is energetically favorable and reduces the overall weight of the nest, thus lowering the risk of collapse. In addition, the pore network offers enhanced thermal insulation to the inner parts of the nest and allows quick drainage ofrainwater thereby restoring the ventilation and providing structural stability to the wet nest.
Bailo R, Carrillo JA, Degond P, 2019, Pedestrian models based on rational behaviour, Crowd Dynamics, Volume 1, Editors: Bellomo, Gibelli, Publisher: Springer Nature, Pages: 259-292
Following the paradigm set by attraction-repulsion-alignment schemes, a myriad of individual-based models have been proposed to calculate the evolution of abstract agents. While the emergent features of many agent systems have been described astonishingly well with force-based models, this is not the case for pedestrians. Many of the classical schemes have failed to capture the fine detail of crowd dynamics, and it is unlikely that a purely mechanical model will succeed. As a response to the mechanistic literature, we will consider a model for pedestrian dynamics that attempts to reproduce the rational behaviour of individual agents through the means of anticipation. Each pedestrian undergoes a two-step time evolution based on a perception stage and a decision stage. We will discuss the validity of this game theoretical-based model in regimes with varying degrees of congestion, ultimately presenting a correction to the mechanistic model in order to achieve realistic high-density dynamics.
Degond P, Frouvelle A, Merino Aceituno S, et al., Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations, Stochastic dynamics out of equilibrium, Publisher: Springer Nature, ISSN: 2194-1017
The goal of these lecture notes is to present in a unified wayvarious models for the dynamics of aligning self-propelled rigid bodiesat different scales and the links between them. The models and methodsare inspired from [17,18], but, in addition, we introduce a new modeland apply on it the same methods. While the new model has its owninterest, our aim is also to emphasize the methods by demonstratingtheir adaptability and by presenting them in a unified and simplifiedway. Furthermore, from the various microscopic models we derive thesame macroscopic model, which is a good indicator of its universality.
Aceves-Sanchez P, Bostan M, Carrillo J-A, et al., 2019, Hydrodynamic limits for kinetic flocking models of Cucker-Smale type, Publisher: AMER INST MATHEMATICAL SCIENCES-AIMS
Motsch S, Moussaid M, Guillot E, et al., 2018, Modeling crowd dynamics through coarse-grained data analysis, Mathematical Biosciences and Engineering, Vol: 15, Pages: 1271-1290, ISSN: 1547-1063
Understanding and predicting the collective behaviour of crowdsis essential to improve the efficiency of pedestrian flows in urban areas andminimize the risks of accidents at mass events. We advocate for the develop-ment of crowd traffic management systems, whereby observations of crowdscan be coupled to fast and reliable models to produce rapid predictions of thecrowd movement and eventually help crowd managers choose between tailoredoptimization strategies. Here, we propose a Bi-directional Macroscopic (BM)model as the core of such a system. Its key input is the fundamental diagramfor bi-directional flows, i.e. the relation between the pedestrian fluxes anddensities. We design and run a laboratory experiments involving a total of119 participants walking in opposite directions in a circular corridor and showthat the model is able to accurately capture the experimental data in a typicalcrowd forecasting situation. Finally, we propose a simple segregation strat-egy for enhancing the traffic efficiency, and use the BM model to determinethe conditions under which this strategy would be beneficial. The BM model,therefore, could serve as a building block to develop on the fly prediction ofcrowd movements and help deploying real-time crowd optimization strategies.
Degond PAA, Minakowski P, Zatorska E, 2018, Transport of congestion in two-phase compressible/incompressible flows, Nonlinear Analysis: Real World Applications, Vol: 42, Pages: 485-510, ISSN: 1468-1218
We study the existence of weak solutions to the two-phase fluid model with congestion constraint. The model encompasses the flow in the uncongested regime (compressible) and the congested one (incompressible) with the free boundary separating the two phases. The congested regime appears when the density in the uncongested regime achieves a threshold value that describes the comfort zone of individuals. This quantity is prescribed initially and transported along with the flow. We prove that this system can be approximated by the fully compressible Navier–Stokes system with a singular pressure, supplemented with transport equation for the congestion density. We also present the application of this approximation for the purposes of numerical simulations in the one-dimensional domain.
Aoki K, Degond P, Yang T, 2018, Special issue celebrating the 10th anniversary of KRM PREFACE, KINETIC AND RELATED MODELS, Vol: 11, Pages: I-I, ISSN: 1937-5093
Degond PAA, Manhart A, Yu H, 2018, An age-structured continuum model for myxobacteria, Mathematical Models and Methods in Applied Sciences, Vol: 28, Pages: 1737-1770, ISSN: 1793-6314
Myxobacteria are social bacteria, that can glide in 2D and form counter-propagating,interacting waves. Here we present a novel age-structured, continuous macroscopic modelfor the movement of myxobacteria. The derivation is based on microscopic interactionrules that can be formulated as a particle-based model and set within the SOH (Self-Organized Hydrodynamics) framework. The strength of this combined approach is thatmicroscopic knowledge or data can be incorporated easily into the particle model, whilstthe continuous model allows for easy numerical analysis of the different effects. Howeverwe found that the derived macroscopic model lacks a diffusion term in the density equa-tions, which is necessary to control the number of waves, indicating that a higher orderapproximation during the derivation is crucial. Uponad-hocaddition of the diffusionterm, we found very good agreement between the age-structured model and the biology.In particular we analyzed the influence of a refractory (insensitivity) period following areversal of movement. Our analysis reveals that the refractory period is not necessaryfor wave formation, but essential to wave synchronization, indicating separate molecularmechanisms.
Degond PAA, Minakowski P, Navoret L, et al., 2018, Finite volume approximations of the Euler system with variable congestion, Computers and Fluids, Vol: 169, Pages: 23-39, ISSN: 0045-7930
We are interested in the numerical simulations of the Euler system with variable congestion encoded by a singular pressure (Degond et al., 2016). This model describes for instance the macroscopic motion of a crowd with individual congestion preferences. We propose an asymptotic preserving (AP) scheme based on a conservative formulation of the system in terms of density, momentum and density fraction. A second order accuracy version of the scheme is also presented. We validate the scheme on one-dimensionnal test-cases and compare it with a scheme previously proposed in Degond et al. (2016) and extended here to higher order accuracy. We finally carry out two dimensional numerical simulations and show that the model exhibit typical crowd dynamics.
David I, Kohnke P, Fehrenbach J, et al., 2018, New objective measurements of semen wave motion are associated with fertility in sheep, Reproduction, Fertility and Development, Vol: 30, Pages: 889-896, ISSN: 1031-3613
In sheep, wave motion in semen is currently used by AI centres to select ejaculates for insemination. Despite its low cost, convenience and established ability to predict fertility, the subjectivity of this assessment is a limiting factor for its applicability. The aims of the present study were to establish an objective method for the analysis of wave motion and to assess the associations of objective parameters with fertility after cervical insemination. Collective sperm motion in undiluted semen was observed by phase contrast microscopy at low magnification in a 100-µm deep glass chamber. Images of moving dark waves over a grey background were recorded and analysed by the optic flow method, producing several velocity-related parameters. Turbulence was assessed from the motion of fluorescent polystyrene beads. Among objective parameters, optical flow entropy and the average speed of beads were both able to discriminate ejaculates suitable for insemination. Two synthetic variables of optic flow and bead motion and a global objective variable were computed from linear combinations of individual parameters and compared with the subjective motion score for their predictive value. These were as efficient as the wave motion score for assessing fertility and can be proposed for the assessment of ram semen in routine AI procedures.
Degond PAA, Frouvelle A, Merino Aceituno S, et al., 2018, Quaternions in collective dynamics, Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Vol: 16, Pages: 28-77, ISSN: 1540-3459
We introduce a model of multiagent dynamics for self-organized motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbors. The body attitudes are represented through unitary quaternions. We prove the correspondence with the model presented in [P. Degond, A. Frouvelle, and S. Merino-Aceituno, Math. Models Methods Appl. Sci., 27 (2017), pp. 1005--1049], where the body attitudes are represented by rotation matrices. Differently from this previous work, the individual-based model introduced here is based on nematic (rather than polar) alignment. From the individual-based model, the kinetic and macroscopic equations are derived. The benefit of this approach, in contrast to that of the previous one, is twofold: first, it allows for a better understanding of the macroscopic equations obtained and, second, these equations are prone to numerical studies, which is key for applications.
Blanchet A, Degond PAA, 2017, Kinetic models for topological nearest-neighbor interactions, Journal of Statistical Physics, Vol: 169, Pages: 929-950, ISSN: 1572-9613
We consider systems of agents interacting through topological interactions. Thesehave been shown to play an important part in animal and human behavior. Precisely, thesystem consists of a finite number of particles characterized by their positions and velocities.At random times a randomly chosen particle, the follower, adopts the velocity of its closestneighbor, the leader. We study the limit of a system size going to infinity and, under theassumption of propagation of chaos, show that the limit kinetic equation is a non-standardspatial diffusion equation for the particle distribution function. We also study the case whereinthe particles interact with their K closest neighbors and show that the corresponding kineticequation is the same. Finally, we prove that these models can be seen as a singular limitof the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys163:41–60, 2016). The proofs are based on a combinatorial interpretation of the rank as wellas some concentration of measure arguments.
Barré J, Degond PAA, Zatorska E, 2017, Kinetic theory of particle interactions mediated by dynamical networks, Multiscale Modeling & Simulation, Vol: 15, Pages: 1294-1323, ISSN: 1540-3467
We provide a detailed multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we formally derive coupled kinetic equations for the particle and link densities, following the approach of [Degond et al., M3AS, 2016]. Assuming that the process of remodelling the network is very fast,we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneous distribution of particles for a general potential. For the Hookean potential we obtain a precise condition for the phase transition, and, using the central manifold reduction, we characterize the type of bifurcation at the instability onset.
Leroy-Lerêtre M, Dimarco G, Cazalès M, et al., 2017, Are tumor cell lineages solely shaped by mechanical forces?, Bulletin of Mathematical Biology, Vol: 79, Pages: 2356-2393, ISSN: 1522-9602
This paper investigates cell proliferation dynamics in small tumor cell aggregates using an individual-based model (IBM). The simulation model is designed to study the morphology of the cell population and of the cell lineages as well as the impact of the orientation of the division plane on this morphology. Our IBM model is based on the hypothesis that cells are incompressible objects that grow in size and divide once a threshold size is reached, and that newly born cell adhere to the existing cell cluster. We performed comparisons between the simulation model and experimental data by using several statistical indicators. The results suggest that the emergence of particular morphologies can be explained by simple mechanical interactions.
Barré J, Carrillo de la plata J, Degond PAA, et al., 2017, Particle interactions mediated by dynamical networks: assessment ofmacroscopic descriptions, Journal of Nonlinear Science, Vol: 28, Pages: 235-268, ISSN: 0938-8974
We provide a numerical study of the macroscopic model of Barré et al.(Multiscale Model Simul, 2017, to appear) derived from an agent-based model for asystem of particles interacting through a dynamical network of links. Assuming thatthe network remodeling process is very fast, the macroscopic model takes the formof a single aggregation–diffusion equation for the density of particles. The theoreticalstudy of the macroscopic model gives precise criteria for the phase transitions ofthe steady states, and in the one-dimensional case, we show numerically that thestationary solutions of the microscopic model undergo the same phase transitions andbifurcation types as the macroscopic model. In the two-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement withthe predicted theoretical values. This study provides a partial validation of the formalderivation of the macroscopic model from a microscopic formulation and shows thatthe former is a consistent approximation of an underlying particle dynamics, makingit a powerful tool for the modeling of dynamical networks at a large scale.
Peurichard D, Delebecque F, Lorsignol A, et al., 2017, Simple mechanical cues could explain adipose tissue morphology, Journal of Theoretical Biology, Vol: 429, Pages: 61-81, ISSN: 1095-8541
The mechanisms by which organs acquire their functional structure and realize its maintenance (or homeostasis) over time are still largely unknown. In this paper, we investigate this question on adipose tissue. Adipose tissue can represent 20 to 50% of the body weight. Its investigation is key to overcome a large array of metabolic disorders that heavily strike populations worldwide. Adipose tissue consists of lobular clusters of adipocytes surrounded by an organized collagen fiber network. By supplying substrates needed for adipogenesis, vasculature was believed to induce the regroupment of adipocytes near capillary extremities. This paper shows that the emergence of these structures could be explained by simple mechanical interactions between the adipocytes and the collagen fibers. Our assumption is that the fiber network resists the pressure induced by the growing adipocytes and forces them to regroup into clusters. Reciprocally, cell clusters force the fibers to merge into a well-organized network. We validate this hypothesis by means of a two-dimensional Individual Based Model (IBM) of interacting adipocytes and extra-cellular-matrix fiber elements. The model produces structures that compare quantitatively well to the experimental observations. Our model seems to indicate that cell clusters could spontaneously emerge as a result of simple mechanical interactions between cells and fibers and surprisingly, vasculature is not directly needed for these structures to emerge.
Degond PAA, Frouvelle A, Merino-Aceituno S, 2017, A new flocking model through body attitude coordination, Mathematical Models & Methods in Applied Sciences, Vol: 27, ISSN: 1793-6314
We present a new model for multi-agent dynamics where each agent is described byits position and body attitude: agents travel at a constant speed in a given directionand their body can rotate around it adopting di erent con gurations. In this manner,the body attitude is described by three orthonormal axes giving an element inSO(3)(rotation matrix). Agents try to coordinate their body attitudes with the ones of theirneighbours. In the present paper, we give the Individual Based Model (particle model) for this dynamics and derive its corresponding kinetic and macroscopic equations.
Degond PAA, Engel M, 2017, Numerical approximation of a coagulation-fragmentation model for animal group size statistics, Networks and Heterogeneous Media, Vol: 12, Pages: 217-243, ISSN: 1556-181X
We study numerically a coagulation-fragmentation model derivedby Niwa  and further elaborated by Degond et al. . In  a unique equi-librium distribution of group sizes is shown to exist in both cases of continuousand discrete group size distributions. We provide a numerical investigation ofthese equilibria using three different methods to approximate the equilibrium:a recursive algorithm based on the work of Ma et. al. , a Newton methodand the resolution of the time-dependent problem. All three schemes are val-idated by showing that they approximate the predicted small and large sizeasymptotic behaviour of the equilibrium accurately. The recursive algorithm isused to investigate the transition from discrete to continuous size distributionsand the time evolution scheme is exploited to show uniform convergence toequilibrium in time and to determine convergence rates.
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