arXiv subjects

Jan Haskovec

Publications and source records attributed to Jan Haskovec.

At least 37 records · Page 2Linked to original sources

arXiv2021

Cucker-Smale model with finite speed of information propagation: well-posedness, flocking and mean-field limit

We study a variant of the Cucker-Smale model where information between agents propagates with a finite speed $\mathfrak{c}>0$. This leads to a system of functional differential equations with state-dependent delay. We prove that, if initially the agents travel slower than $c$, then the discrete model admits unique global solutions. Moreover, under a generic assumption on the influence function, we show that there exists a critical information propagation speed $\mathfrak{c}^\ast>0$ such that if $\mathfrak{c}\geq\mathfrak{c}^\ast$, the system exhibits asymptotic flocking in the sense of the classical definition of Cucker and Smale. For constant initial datum the value of $\mathfrak{c}^\ast$ is explicitly calculable. Finally, we derive a mean-field limit of the discrete system, which is formulated in terms of probability measures on the space of time-dependent trajectories. We show global well-posedness of the mean-field problem and argue that it does not admit a description in terms of the classical Fokker-Planck equation.

math.AP↗

arXiv2021

Tensor PDE model of biological network formation

We study an elliptic-parabolic system of partial differential equations describing formation of biological network structures. The model takes into consideration the evolution of the permeability tensor under the influence of a diffusion term, representing randomness in the material structure, a decay term describing metabolic cost and a pressure force. A Darcy's law type equation describes the pressure field. In the spatially two-dimensional setting, we present a constructive, formal derivation of the PDE system from the discrete network formation model in the refinement limit of a sequence of unstructured triangulations. Moreover, we show that the PDE system is a formal $L^2$-gradient flow of an energy functional with biological interpretation, and study its convexity properties. For the case when the energy functional is convex, we construct unique global weak solutions of the PDE system. Finally, we construct steady state solutions in one- and multi-dimensional settings and discuss their stability properties.

math.AP↗

arXiv2021

Flocking in the Cucker-Smale model with self-delay and nonsymmetric interaction weights

We derive a sufficient condition for asymptotic flocking in the Cucker-Smale model with self-delay (also called reaction delay) and with non-symmetric interaction weights. The condition prescribes smallness of the delay length relative to the decay rate of the inter-agent communication weight. The proof is carried out by a bootstrapping argument combining a decay estimate for the group velocity diameter with a variant of the Gronwall-Halanay inequality.

math.AP↗

arXiv2021

Asymptotic behavior of the linear consensus model with delay and anticipation

We study asymptotic behavior of solutions of the first-order linear consensus model with delay and anticipation, which is a system of neutral delay differential equations. We consider both the transmission-type and reaction-type delay that are motivated by modeling inputs. Studying the simplified case of two agents, we show that, depending on the parameter regime, anticipation may have both a stabilizing and destabilizing effect on the solutions. In particular, we demonstrate numerically that moderate level of anticipation generically promotes convergence towards consensus, while too high level disturbs it. Motivated by this observation, we derive sufficient conditions for asymptotic consensus in the multiple-agent systems, which are explicit in the parameter values delay length and anticipation level, and independent of the number of agents. The proofs are based on construction of suitable Lyapunov-type functionals.

math.OC↗

arXiv2021

Asymptotic and exponential decay in mean square for delay geometric Brownian motion

We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions are written in terms of the parameters and are explicit for the case of asymptotic decay. For exponential decay, they are easily resolvable numerically. The analytical method is based on construction of a Lyapunov functional (asymptotic decay) and forward-backward estimate for the square mean (exponential decay).

math.PR↗

arXiv2021

Well posedness and asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation

We consider a variant of the Hegselmann-Krause model of consensus formation where information between agents propagates with a finite speed $\mathfrak{c}$. This leads to a system of ordinary differential equations (ODE) with state-dependent delay. Observing that the classical well-posedness theory for ODE systems does not apply, we provide a proof of global existence and uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in the spatially one-dimensional setting of the model, as long as agents travel slower than $\mathfrak{c}$. We also provide sufficient conditions for asymptotic consensus in the spatially multi-dimensional setting.

math.DS↗

arXiv2021

Direct proof of unconditional asymptotic consensus in the Hegselmann-Krause model with transmission-type delay

We present a direct proof of asymptotic consensus in the nonlinear Hegselmann-Krause model with transmission-type delay, where the communication weights depend on the particle distance in phase space. Our approach is based on an explicit estimate of the shrinkage of the group diameter on finite time intervals and avoids the usage of Lyapunov-type functionals or results from nonnegative matrix theory. It works for both the original formulation of the model with communication weights scaled by the number of agents, and the modification with weights normalized a'la Motsch-Tadmor. We pose only minimal assumptions on the model parameters. In particular, we only assume global positivity of the influence function, without imposing any conditions on its decay rate or monotonicity. Moreover, our result holds for any length of the delay.

math.DS↗

arXiv2020

A simple proof of asymptotic consensus in the Hegselmann-Krause and Cucker-Smale models with renormalization and delay

We present a simple proof of asymptotic consensus in the discrete Hegselmann-Krause model and flocking in the discrete Cucker-Smale model with renormalization and variable delay. It is based on convexity of the renormalized communication weights and a Gronwall-Halanay-type inequality. The main advantage of our method, compared to previous approaches to the delay Hegselmann-Krause model, is that it does not require any restriction on the maximal time delay, or the initial data, or decay rate of the influence function. From this point of view the result is optimal. For the Cucker-Smale model it provides an analogous result in the regime of unconditonal flocking with sufficiently slowly decaying communication rate, but still without any restriction on the length of the maximal time delay. Moreover, we demonstrate that the method can be easily extended to the mean-field limits of both the Hegselmann-Krause and Cucker-Smale systems, using appropriate stability results on the measure-valued solutions.

math.DS↗

arXiv2020

Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays

We study a variant of the Cucker-Smale system with distributed reaction delays. Using backward-forward and stability estimates on the quadratic velocity fluctuations we derive sufficient conditions for asymptotic flocking of the solutions. The conditions are formulated in terms of moments of the delay distribution and they guarantee exponential decay of velocity fluctuations towards zero for large times. We demonstrate the applicability of our theory to particular delay distributions - exponential, uniform and linear. For the exponential distribution, the flocking condition can be resolved analytically, leading to an explicit formula. For the other two distributions, the satisfiability of the assumptions is investigated numerically.

math.DS↗

arXiv2020

Exponential decay for negative feedback loop with distributed delay

We derive sufficient conditions for exponential decay of solutions of the delay negative feedback equation with distributed delay. The conditions are written in terms of exponential moments of the distribution. Our method only uses elementary tools of calculus and is robust towards possible extensions to more complex settings, in particular, systems of delay differential equations. We illustrate the applicability of the method to particular distributions - Dirac delta, Gamma distribution, uniform and truncated normal distributions.

math.DS↗

arXiv2020

Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime

We study a variant of the Cucker-Smale system with reaction-type delay. Using novel backward-forward and stability estimates on appropriate quantities we derive sufficient conditions for asymptotic flocking of the solutions. These conditions, although not explicit, relate the velocity fluctuation of the initial datum and the length of the delay. If satisfied, they guarantee monotone decay (i.e., non-oscillatory regime) of the velocity fluctuations towards zero for large times. For the simplified setting with only two agents and constant communication rate the Cucker-Smale system reduces to the delay negative feedback equation. We demonstrate that in this case our method provides the sharp condition for the size of the delay such that the solution be non-oscillatory. Moreover, we comment on the mathematical issues appearing in the formal macroscopic description of the reaction-type delay system.

math.CA↗

arXiv2019

Auxin transport model for leaf venation

The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organisation of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialised membrane-localised proteins. Many venation models have been based on polarly localised efflux-mediator proteins of the PIN family. Here, we investigate a modeling framework for auxin transport with a positive feedback between auxin fluxes and transport capacities that are not necessarily polar, i.e.\ directional across a cell wall. Our approach is derived from a discrete graph-based model for biological transportation networks, where cells are represented by graph nodes and intercellular membranes by edges. The edges are not a-priori oriented and the direction of auxin flow is determined by its concentration gradient along the edge. We prove global existence of solutions to the model and the validity of Murray's law for its steady states. Moreover, we demonstrate with numerical simulations that the model is able connect an auxin source-sink pair with a mid-vein and that it can also produce branching vein patterns. A significant innovative aspect of our approach is that it allows the passage to a formal macroscopic limit which can be extended to include network growth. We perform mathematical analysis of the macroscopic formulation, showing the global existence of weak solutions for an appropriate parameter range.

math.DS↗

arXiv2019

Murray's law for discrete and continuum models of biological networks

We demonstrate the validity of Murray's law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray's law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray's law for its linearly stable steady states.

math.AP↗

arXiv2019

Rigorous Continuum Limit for the Discrete Network Formation Problem

Motivated by recent physics 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 $Γ$-converge towards a continuum energy functional.

math.AP↗

arXiv2018

A mesoscopic model of biological transportation networks

We introduce a mesoscopic model for natural network formation processes, acting as a bridge between the discrete and continuous network approach proposed by Hu and Cai. The models are based on a common approach where the dynamics of the conductance network is subject to pressure force effects. We first study topological properties of the discrete model and we prove that if the metabolic energy consumption term is concave with respect to the conductivities, the optimal network structure is a tree (i.e., no loops are present). We then analyze various aspects of the mesoscopic modeling approach, in particular its relation to the discrete model and its stationary solutions, including discrete network solutions. Moreover, we present an alternative formulation of the mesoscopic model that avoids the explicit presence of the pressure in the energy functional.

math.AP↗

arXiv2018

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.

math.AP↗

arXiv2018

An Optimal Transport Approach for the Kinetic Bohmian Equation

We study the existence theory of solutions of the kinetic Bohmian equation, a nonlinear Vlasov-type equation proposed for the phase-space formulation of Bohmian mechanics. Our main idea is to interpret the kinetic Bohmian equation as a Hamiltonian system defined on an appropriate Poisson manifold built on a Wasserstein space. We start by presenting an existence theory for stationary solutions of the kinetic Bohmian equation. Afterwards, we develop an approximative version of our Hamiltonian system in order to study its associated flow. We then prove existence of solutions of our approximative version. Finally, we present some convergence results for the approximative system, the aim being to establish that, in the limit, the approximative solution satisfies the kinetic Bohmian equation in a weak sense.

math.AP↗

arXiv2017

Hydrodynamic Cucker-Smale model with normalized communication weights and time delay

We study a hydrodynamic Cucker-Smale-type model with time delay in communication and information processing, in which agents interact with each other through normalized communication weights. The model consists of a pressureless Euler system with time delayed non-local alignment forces. We resort to its Lagrangian formulation and prove the existence of its global in time classical solutions. Moreover, we derive a sufficient condition for the asymptotic flocking behavior of the solutions. Finally, we show the presence of a critical phenomenon for the Eulerian system posed in the spatially one-dimensional setting.

math.AP↗