scholarly journals Mean-field models for segregation dynamics

2020 ◽  
pp. 1-22
Author(s):  
MARTIN BURGER ◽  
JAN-FREDERIK PIETSCHMANN ◽  
HELENE RANETBAUER ◽  
CHRISTIAN SCHMEISER ◽  
MARIE-THERESE WOLFRAM
Keyword(s):  
Random Walk ◽  
Mean Field ◽  
Single Species ◽  
Random Motion ◽  
Local Effects ◽  
Mean Field Models ◽  
Existence And Stability ◽  
Non Local ◽  
The Rich ◽  
Multiple Species

In this paper, we derive and analyse mean-field models for the dynamics of groups of individuals undergoing a random walk. The random motion of individuals is only influenced by the perceived densities of the different groups present as well as the available space. All individuals have the tendency to stay within their own group and avoid the others. These interactions lead to the formation of aggregates in case of a single species and to segregation in the case of multiple species. We derive two different mean-field models, which are based on these interactions and weigh local and non-local effects differently. We discuss existence and stability properties of solutions for both models and illustrate the rich dynamics with numerical simulations.

scholarly journals Non-local effects in the mean-field disc dynamo: II – numerical and asymptotic solutions

2004 ◽  
Vol 98 (4) ◽  
pp. 345-363 ◽  
Author(s):  
Ashley P. Willis ◽  
Anvar Shukurov ◽  
Andrew M. Soward ◽  
Dmitry Sokoloff
Keyword(s):  
Mean Field ◽  
Asymptotic Solutions ◽  
Local Effects ◽  
Non Local ◽  
The Mean

scholarly journals Derivation and analysis of continuum models for crossing pedestrian traffic

2017 ◽  
Vol 27 (07) ◽  
pp. 1301-1325 ◽  
Author(s):  
Sabine Hittmeir ◽  
Helene Ranetbauer ◽  
Christian Schmeiser ◽  
Marie-Therese Wolfram
Keyword(s):  
Nonlinear Partial Differential Equation ◽  
Mean Field ◽  
Transition Rates ◽  
Dimensional Model ◽  
Global Weak Solutions ◽  
One Dimensional ◽  
Mean Field Models ◽  
Differential Equation Models ◽  
Pedestrian Traffic ◽  
The Rich

In this paper, we study hyperbolic and parabolic nonlinear partial differential equation models, which describe the evolution of two intersecting pedestrian flows. We assume that individuals avoid collisions by sidestepping, which is encoded in the transition rates of the microscopic 2D model. We formally derive the corresponding mean-field models and prove existence of global weak solutions for the parabolic model. Moreover we discuss stability of stationary states for the corresponding one-dimensional model. Furthermore we illustrate the rich dynamics of both systems with numerical simulations.

scholarly journals A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis

2016 ◽  
Author(s):  
Andreas Buttenschön ◽  
Thomas Hillen ◽  
Alf Gerisch ◽  
Kevin J. Painter
Keyword(s):  
Random Walk ◽  
Mean Field ◽  
Polarization Vector ◽  
Jump Process ◽  
Biological Properties ◽  
Cellular Adhesion ◽  
Cell Polarization ◽  
Local Models ◽  
A Cell ◽  
Non Local

AbstractCellular adhesion provides one of the fundamental forms of biological interaction between cells and their surroundings, yet the continuum modelling of cellular adhesion has remained mathematically challenging. In 2006, Armstrong et al. proposed a mathematical model in the form of an integro-partial differential equation. Although successful in applications, a derivation from an underlying stochastic random walk has remained elusive. In this work we develop a framework by which non-local models can be derived from a space-jump process. We show how the notions of motility and a cell polarization vector can be naturally included. With this derivation we are able to include microscopic biological properties into the model. We show that particular choices yield the original Armstrong model, while others lead to more general models, including a doubly non-local adhesion model and non-local chemotaxis models. Finally, we use random walk simulations to confirm that the corresponding continuum model represents the mean field behaviour of the stochastic random walk.

scholarly journals A new and accurate continuum description of moving fronts

2017 ◽  
Author(s):  
S T Johnston ◽  
R E Baker ◽  
M J Simpson
Keyword(s):  
Random Walk ◽  
Cell Biology ◽  
Spatial Clustering ◽  
Mean Field ◽  
Mean Field Models ◽  
Alternative Description ◽  
Continuum Description ◽  
Computationally Expensive ◽  
Random Fluctuations ◽  
Moving Front

AbstractProcesses that involve moving fronts of populations are prevalent in ecology and cell biology. A common approach to describe these processes is a lattice-based random walk model, which can include mechanisms such as crowding, birth, death, movement and agent-agent adhesion. However, these models are generally analytically intractable and it is computationally expensive to perform sufficiently many realisations of the model to obtain an estimate of average behaviour that is not dominated by random fluctuations. To avoid these issues, both mean-field and corrected mean-field continuum descriptions of random walk models have been proposed. However, both continuum descriptions are inaccurate outside of limited parameter regimes, and corrected mean-field descriptions cannot be employed to describe moving fronts. Here we present an alternative description in terms of the dynamics of groups of contiguous occupied lattice sites and contiguous vacant lattice sites. Our description provides an accurate prediction of the average random walk behaviour in all parameter regimes. Critically, our description accurately predicts the persistence or extinction of the population in situations where previous continuum descriptions predict the opposite outcome. Furthermore, unlike traditional mean-field models, our approach provides information about the spatial clustering within the population and, subsequently, the moving front.

scholarly journals Non-local effects in the mean-field disc dynamo I. An asymptotic expansion

2000 ◽  
Vol 93 (1-2) ◽  
pp. 97-114 ◽  
Author(s):  
Vladimir Priklonsky ◽  
Anvar Shukurov ◽  
Dmitry Sokoloff ◽  
Andrew Soward
Keyword(s):  
Asymptotic Expansion ◽  
Mean Field ◽  
Local Effects ◽  
Non Local ◽  
The Mean

scholarly journals Light bullets in a time-delay model of a wide-aperture mode-locked semiconductor laser

2018 ◽  
Vol 376 (2124) ◽  
pp. 20170372 ◽  
Author(s):  
A. Pimenov ◽  
J. Javaloyes ◽  
S. V. Gurevich ◽  
A. G. Vladimirov
Keyword(s):  
Mean Field ◽  
Dissipative Structures ◽  
Type Model ◽  
Delay Model ◽  
Wide Aperture ◽  
Existence And Stability ◽  
Non Local ◽  
Light Bullets ◽  
Delay Differential ◽  
Mode Locked Lasers

Recently, a mechanism of formation of light bullets (LBs) in wide-aperture passively mode-locked lasers was proposed. The conditions for existence and stability of these bullets, found in the long cavity limit, were studied theoretically under the mean field (MF) approximation using a Haus-type model equation. In this paper, we relax the MF approximation and study LB formation in a model of a wide-aperture three section laser with a long diffractive section and short absorber and gain sections. To this end, we derive a non-local delay-differential equation (NDDE) model and demonstrate by means of numerical simulations that this model supports stable LBs. We observe that the predictions about the regions of existence and stability of the LBs made previously using MF laser models agree well with the results obtained using the NDDE model. Moreover, we demonstrate that the general conclusions based upon the Haus model that regard the robustness of the LBs remain true in the NDDE model valid beyond the MF approximation, when the gain, losses and diffraction per cavity round trip are not small perturbations anymore. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)’.

Ion Pairing and Dielectric Decrement in Glycosaminoglycan Brushes

2020 ◽  
Author(s):  
James Sterling ◽  
Wenjuan Jiang ◽  
Wesley M. Botello-Smith ◽  
Yun L. Luo
Keyword(s):  
Molecular Dynamics ◽  
Ion Pairs ◽  
Mean Field ◽  
Salt Bridge ◽  
Ion Pairing ◽  
Donnan Potential ◽  
Mean Field Models ◽  
Ion Exclusion ◽  
Dynamics Simulations ◽  
Contact Ion Pairs

Molecular dynamics simulations of hyaluronic acid and heparin brushes are presented that show important effects of ion-pairing, water dielectric decrease, and co-ion exclusion. Results show equilibria with electroneutrality attained through screening and pairing of brush anionic charges by cations. Most surprising is the reversal of the Donnan potential that would be expected based on electrostatic Boltzmann partitioning alone. Water dielectric decrement within the brush domain is also associated with Born hydration-driven cation exclusion from the brush. We observe that the primary partition energy attracting cations to attain brush electroneutrality is the ion-pairing or salt-bridge energy associated with cation-sulfate and cation-carboxylate solvent-separated and contact ion pairs. Potassium and sodium pairing to glycosaminoglycan carboxylates and sulfates consistently show similar abundance of contact-pairing and solvent-separated pairing. In these crowded macromolecular brushes, ion-pairing, Born-hydration, and electrostatic potential energies all contribute to attain electroneutrality and should therefore contribute in mean-field models to accurately represent brush electrostatics.

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