Space-time fractional diffusion in cell movement models with delay

Estrada-Rodriguez, Gissell; Gimperlein, Heiko; Painter, Kevin J.; Stocek, Jakub

doi:10.1142/s0218202519500039

Cited by 30 publications

(31 citation statements)

References 35 publications

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“…Initially the robots are placed in the center of the arena with a distribution given Figure 2 shows the coverage as a function of time as the Lévy exponent α is varied for a fixed number of robots. This evidences improvements by long distance runs, compared with classical Brownian motion, similar to what is known for target search strategies [16,21].…”

Section: Macroscopic Transport Equation For Swarmingsupporting

confidence: 65%

“…A second quantity of interest is the mean first passage time for an unknown target. In this case [16,21] have shown analogous advantages for Lévy strategies in a system similar to Theorem 6.1, with delays between reorientations, but no alignment.…”

Section: Macroscopic Transport Equation For Swarmingmentioning

confidence: 68%

“…The numerical approximation of (65) by finite elements is described in [1,16]. From the solution of (65) we compute the area covered as a function of time, depending on the parameter α.…”

Section: Macroscopic Transport Equation For Swarmingmentioning

confidence: 99%

See 2 more Smart Citations

Interacting Particles with Lévy Strategies: Limits of Transport Equations for Swarm Robotic Systems

Estrada-Rodriguez¹,

Gimperlein²

2020

SIAM J. Appl. Math.

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Lévy robotic systems combine superdiffusive random movement with emergent collective behaviour from local communication and alignment in order to find rare targets or track objects. In this article we derive macroscopic fractional PDE descriptions from the movement strategies of the individual robots. Starting from a kinetic equation which describes the movement of robots based on alignment, collisions and occasional long distance runs according to a Lévy distribution, we obtain a system of evolution equations for the fractional diffusion for long times. We show that the system allows efficient parameter studies for a search problem, addressing basic questions like the optimal number of robots needed to cover an area in a certain time. For shorter times, in the hyperbolic limit of the kinetic equation, the PDE model is dominated by alignment, irrespective of the long range movement. This is in agreement with previous results in swarming of self-propelled particles. The article indicates the novel and quantitative modeling opportunities which swarm robotic systems provide for the study of both emergent collective behaviour and anomalous diffusion, on the respective time scales.

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Section: Macroscopic Transport Equation For Swarmingsupporting

confidence: 65%

Section: Macroscopic Transport Equation For Swarmingmentioning

confidence: 68%

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Interacting Particles with Lévy Strategies: Limits of Transport Equations for Swarm Robotic Systems

Estrada-Rodriguez¹,

Gimperlein²

2020

SIAM J. Appl. Math.

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“…The three operators A 2ρ , B 2σ , C 2τ are allowed to be a variation of fractional Laplacians, but also other elliptic operators, and may show different orders. Indeed, some components in tumor development, such as immune cells, exhibit an anomalous diffusion dynamics (as it observed in experiments [28]), but other components, like chemical potential and nutrient concentration are possibly governed by different fractional or non-fractional flows. However, taking all this into account, it is the case of pointing out that fractional operators are becoming more and more implemented in the field of biological applications: to this concern, a selection of notable and meaningful references is given by [1,7,28,29,45,48,50,51,54,62,64,70].…”

Section: Introductionmentioning

confidence: 81%

Asymptotic analysis of a tumor growth model with fractional operators

Colli

Gilardi

Sprekels

2020

ASY

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In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn-Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3-24). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn-Hilliard equation for the tumor cell fraction ϕ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic Colli -Gilardi -Sprekels analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

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“…The space-time adaptive methods developed in this paper will be of interest beyond variational inequalities. They will be of use, in particular, for the nonlinear systems of fractional diffusion equations arising in applications [30,31].…”

Section: Discussionmentioning

confidence: 99%

Space–time adaptive finite elements for nonlocal parabolic variational inequalities

Gimperlein

Stocek

2019

Computer Methods in Applied Mechanics and Engineering

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This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic variational inequalities associated to the fractional Laplacian we obtain a priori and a posteriori error estimates and study the resulting space-time adaptive mesh-refinement procedures. Particular emphasis is placed on mixed formulations, which include the contact forces as a Lagrange multiplier. Corresponding results are presented for elliptic problems. Our numerical experiments for 2dimensional model problems confirm the theoretical results: They indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space-time adaptive, as well as uniform and graded discretizations.

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Space-time fractional diffusion in cell movement models with delay

Cited by 30 publications

References 35 publications

Interacting Particles with Lévy Strategies: Limits of Transport Equations for Swarm Robotic Systems

Interacting Particles with Lévy Strategies: Limits of Transport Equations for Swarm Robotic Systems

Asymptotic analysis of a tumor growth model with fractional operators

Space–time adaptive finite elements for nonlocal parabolic variational inequalities

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