Pattern formation in systems with nonlocal interactions

Belintsev, B. N.; Livshits, M. A.; Volkenstein, M.V.

doi:10.1007/bf01294173

Cited by 15 publications

(10 citation statements)

References 2 publications

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“…The inclusion of such terms 4 makes the theory successively more non-local. The physical origins of effectively non-local interactions can be manifold [52][53][54], ranging from global packing constraints to hydrodynamic and chemical interactions in biological systems. In microbial suspensions, such non-localities may arise naturally from active stresses that are generated by the swimming strokes of the organisms and transported through the fluid.…”

Section: Introductionmentioning

confidence: 99%

Minimal continuum theories of structure formation in dense active fluids

Dunkel¹,

Heidenreich²,

Bär³

et al. 2013

New J. Phys.

107

145

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Self-sustained dynamical phases of living matter can exhibit remarkable similarities over a wide range of scales, from mesoscopic vortex structures in microbial suspensions and motility assays of biopolymers to turbulent largescale instabilities in flocks of birds or schools of fish. Here, we argue that, in many cases, the phenomenology of such active states can be efficiently described in terms of fourth-and higher-order partial differential equations. Structural transitions in these models can be interpreted as Landau-type kinematic transitions in Fourier (wavenumber) space, suggesting that microscopically different biological systems can share universal long-wavelength features. This general idea is illustrated through numerical simulations for two classes of continuum models for incompressible active fluids: a Swift-Hohenberg-type scalar field theory, and a minimal vector model that extends the classical Toner-Tu theory and appears to be a promising candidate for the quantitative description of dense bacterial suspensions. We discuss how microscopic symmetry-breaking mechanisms can enter macroscopic continuum descriptions of collective microbial motion near surfaces, and conclude by outlining future applications.

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Section: Introductionmentioning

confidence: 99%

Minimal continuum theories of structure formation in dense active fluids

Dunkel¹,

Heidenreich²,

Bär³

et al. 2013

New J. Phys.

107

145

View full text Add to dashboard Cite

show abstract

“…Such pure mathematical approach is presented in the paper (Sattinger,D.H. 1979) and later used to analyze more concrete formal models in (Belintsev,B.N. et al, 1981).…”

Section: Sot-based Computation Of Symmetry Breaking Instability Of Onmentioning

confidence: 99%

“…1979). These conditions in more concrete form are substantially considered in (Belintsev,B.N. et al, 1981) and applied to a formal model of a dissipative system with counteraction of local activation and an effective nonlocal inhibition.…”

Section: Sot-based Computation Of Symmetry Breaking Instability Of Onmentioning

confidence: 99%

Integrating Mechanical Control Theory Into Models of Biological Development — Analytical Review

Petrov

Aegerter

2011

J. Mech. Med. Biol.

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This paper presents both a general review on developmental biomechanics and a concrete proposition for the computation of a symmetry breaking instability of a model of biological development in terms of self-organization theory. The necessary biological and physical facts taken from the literature are described and discussed in the context of a unified statement of the problems for mathematical modeling of pattern formation. This is then applied to planar cell polarization (PCP) of the Drosophila wing. In this way, the process is modeled by an elastopolarization equation. In terms of this statement, the mechanical specificity (interaction with basal plate) of wing PCP is characterized. Some aspects of modeling somite formation as well as other developmental processes are also concerned.

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“…Nonlocal coupling schemes that do not change the homogeneous steady states can nevertheless generate patterns [58] or control propagation of fronts and pulses [59][60][61]. Nonlocal coupling also plays an essential role in the formation of chimera states in discrete arrays of oscillators [62][63][64][65][66][67].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection

et al. 2014

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A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.

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Pattern formation in systems with nonlocal interactions

Cited by 15 publications

References 2 publications

Minimal continuum theories of structure formation in dense active fluids

Minimal continuum theories of structure formation in dense active fluids

Integrating Mechanical Control Theory Into Models of Biological Development — Analytical Review

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection

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