60 research outputs found
From individual to collective behaviour of coupled velocity jump processes: a locust example
A class of stochastic individual-based models, written in terms of coupled velocity jump processes, is presented and analysed. This modelling approach incorporates recent experimental findings on behaviour of locusts. It exhibits nontrivial dynamics with a “phase change” behaviour and recovers the observed group directional switching. Estimates of the expected switching times, in terms of number of individuals and values of the model coefficients, are obtained using the corresponding Fokker-Planck equation. In the limit of large populations, a system of two kinetic equations with nonlocal and nonlinear right hand side is derived and analyzed. The existence of its solutions is proven and the system’s long-time behaviour is investigated. Finally, a first step towards the mean field limit of topological interactions is made by studying the effect of shrinking the interaction radius in the individual-based model when the number of individuals grows
ODE-and PDE-based modeling of biological transportation networks
We study the global existence of solutions of a discrete (ODE based) model on
a graph describing the formation of biological transportation networks,
introduced by Hu and Cai. We propose an adaptation of this model so that a
macroscopic (PDE based) system can be obtained as its formal continuum limit.
We prove the global existence of weak solutions of the macroscopic PDE model.
Finally, we present results of numerical simulations of the discrete model,
illustrating the convergence to steady states, their non-uniqueness as well as
their dependence on initial data and model parameters
Rigorous continuum limit for the discrete network formation problem
Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their Γ-convergence towards a continuum energy functional.LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes)
The McKean-Vlasov Equation in Finite Volume
We study the McKean--Vlasov equation on the finite tori of length scale
in --dimensions. We derive the necessary and sufficient conditions for the
existence of a phase transition, which are based on the criteria first
uncovered in \cite{GP} and \cite{KM}. Therein and in subsequent works, one
finds indications pointing to critical transitions at a particular model
dependent value, of the interaction parameter. We show that
the uniform density (which may be interpreted as the liquid phase) is
dynamically stable for and prove, abstractly, that a
{\it critical} transition must occur at . However for
this system we show that under generic conditions -- large, and
isotropic interactions -- the phase transition is in fact discontinuous and
occurs at some \theta\t < \theta^{\sharp}. Finally, for H--stable, bounded
interactions with discontinuous transitions we show that, with suitable
scaling, the \theta\t(L) tend to a definitive non--trivial limit as
Singular Cucker-Smale Dynamics
The existing state of the art for singular models of flocking is overviewed,
starting from microscopic model of Cucker and Smale with singular communication
weight, through its mesoscopic mean-filed limit, up to the corresponding
macroscopic regime. For the microscopic Cucker-Smale (CS) model, the
collision-avoidance phenomenon is discussed, also in the presence of bonding
forces and the decentralized control. For the kinetic mean-field model, the
existence of global-in-time measure-valued solutions, with a special emphasis
on a weak atomic uniqueness of solutions is sketched. Ultimately, for the
macroscopic singular model, the summary of the existence results for the
Euler-type alignment system is provided, including existence of strong
solutions on one-dimensional torus, and the extension of this result to higher
dimensions upon restriction on the smallness of initial data. Additionally, the
pressureless Navier-Stokes-type system corresponding to particular choice of
alignment kernel is presented, and compared - analytically and numerically - to
the porous medium equation
On Cucker-Smale model with noise and delay
A generalization of the Cucker-Smale model for collective animal behaviour is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modelling: (i) stochasticity (imperfections) of individual behaviour; and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a byproduct, a new result regarding the asymptotic behaviour of delayed geometric Brownian motion is obtained. In the second part of the paper results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behaviour of the system - namely, that an introduction of intermediate time delay may facilitate flocking
Fluid dynamic description of flocking via Povzner–Boltzmann equation
We introduce and discuss the possible dynamics of groups of indistinguishable agents, which are interacting according to their relative positions, with the aim of deriving hydrodynamic equations. These models are developed to mimic the collective motion of groups of species such as bird flocks, fish schools, herds of quadrupeds or bacteria colonies. Our starting model for these interactions is the Povzner equation, which describes a dilute gas in which binary collisions of elastic spheres depend of their relative positions. Following the Cucker and Smale model, we will consider binary interactions between agents that are dissipative collisions in which the coefficient of restitution depends on their relative distance. Under the assumption of weak dissipation, it is shown that the Povzner equation is modified through a correction in the form of a nonlinear friction type operator. Using this correction we formally obtain from the Povzner equation in a direct way a fluid dynamic description of a system of weakly interacting agents interacting in a dissipative way, with a coefficient of restitution that depends on their relative distance
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