We consider the discrete Couzin-Vicsek algorithm (CVA) [1,9,19,36], which describes the interactions of individuals among animal societies such as fish schools. In this article, we propose a kinetic (mean-field) version of the CVA model and provide its formal macroscopic limit. The final macroscopic model involves a conservation equation for the density of the individuals and a non conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.Acknowledgements: The first author wishes to thank E. Carlen and M. Carvalho for their interest in this work and their helpful suggestions.

We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or "agents", with the tendency to adjust to their 'environmental averages'. This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, rendezvous in mobile networks, etc. A natural question which arises in this context is to understand when and how clusters emerge through the self-alignment of agents, and what type of "rules of engagement" influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking of birds, fish or cells, rendezvous of mobile agents, and in general, concentration of other traits intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological and physical science assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, "Similarity breeds connection," reflects our intuition that increasing the intensity of alignment as the difference of positions decreases, is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents increase as a function of their difference in position. Heterophily -the tendency to bond more with those who are different rather than with those who are similar, plays a decisive rôle in the process of clustering. We point out that the number of clusters in heterophilious dynamics decreases as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus.Date: October 15, 2018. 1991 Mathematics Subject Classification. 92D25,74A25,76N10.

To Claude Bardos on his 70 th birthday, with friendship and admiration Abstract. We introduce a model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The proposed model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, our model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. The use of relative distances destroys the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems, which covers systems with non-symmetric influence matrices. In particular, we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with "leaders".The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our proposed model, and prove its unconditional flocking for slowly decaying influence functions.

The trajectories of Kuhlia mugil fish swimming freely in a tank are analyzed in order to develop a model of spontaneous fish movement. The data show that K. mugil displacement is best described by turning speed and its auto-correlation. The continuous-time process governing this new kind of displacement is modelled by a stochastic differential equation of Ornstein-Uhlenbeck family: the persistent turning walker. The associated diffusive dynamics are compared to the standard persistent random walker model and we show that the resulting diffusion coefficient scales non-linearly with linear swimming speed. In order to illustrate how interactions with other fish or the environment can be added to this spontaneous movement model we quantify the effect of tank walls on the turning speed and adequately reproduce the characteristics of the observed fish trajectories.

This paper considers a new model of individual displacement, based on fish motion, the so-called Persistent Turning Walker (PTW) model, which involves an Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The goal is to show that its large time and space scale dynamics is of diffusive type, and to provide an analytic expression of the diffusion coefficient. Two methods are investigated. In the first one, we compute the large time asymptotics of the variance of the individual stochastic trajectories. The second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories. The kinetic model is a Fokker-Planck type equation posed in an extended phase-space involving the curvature among the kinetic variables. We show that both methods lead to the same value of the diffusion constant. We present some numerical simulations to illustrate the theoretical results.Acknowledgements: The authors wish to thank Guy Théraulaz and Jacques Gautrais of the 'Centre de Recherches sur la Cognition Animale' in Toulouse, for introducing them to the model and for stimulating discussions.

As glioma cells infiltrate the brain they become associated with various microanatomic brain structures such as blood vessels, white matter tracts, and brain parenchyma. How these distinct invasion patterns coordinate tumor growth and influence clinical outcomes remain poorly understood. We have investigated how perivascular growth affects glioma growth patterning and response to antiangiogenic therapy within the highly vascularized brain. Orthotopically implanted rodent and human glioma cells are shown to commonly invade and proliferate within brain perivascular space. This form of brain tumor growth and invasion is also shown to characterize de novo generated endogenous mouse brain tumors, biopsies of primary human glioblastoma (GBM), and peripheral cancer metastasis to the human brain. Perivascularly invading brain tumors become vascularized by normal brain microvessels as individual glioma cells use perivascular space as a conduit for tumor invasion. Agent-based computational modeling recapitulated biological perivascular glioma growth without the need for neoangiogenesis. We tested the requirement for neoangiogenesis in perivascular glioma by treating animals with angiogenesis inhibitors bevacizumab and DC101. These inhibitors induced the expected vessel normalization, yet failed to reduce tumor growth or improve survival of mice bearing orthotopic or endogenous gliomas while exacerbating brain tumor invasion. Our results provide compelling experimental evidence in support of the recently described failure of clinically used antiangiogenics to extend the overall survival of human GBM patients.

This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attractionrepulsion. The starting point is the kinetic model considered in [10] with the addition of an attraction-repulsion interaction potential. Introducing different scalings than in [10], the non-local effects of the alignment and attraction-repulsion interactions can be kept in the hydrodynamic limit and result in extra pressure, viscosity terms and capillary force. The systems are shown to be symmetrizable hyperbolic systems with viscosity terms. A local-in-time existence result is proved in the 2D case for the viscous model and in the 3D case for the inviscid model. The proof relies on the energy method.Acknowledgements: The authors wish to acknowledge the hospitality

No abstract