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Knobloch E, Yochelis A. Instability mechanisms of repelling peak solutions in a multi-variable activator-inhibitor system. CHAOS (WOODBURY, N.Y.) 2022; 32:123129. [PMID: 36587350 DOI: 10.1063/5.0125535] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2022] [Accepted: 11/15/2022] [Indexed: 06/17/2023]
Abstract
We study the linear stability properties of spatially localized single- and multi-peak states generated in a subcritical Turing bifurcation in the Meinhardt model of branching. In one spatial dimension, these states are organized in a foliated snaking structure owing to peak-peak repulsion but are shown to be all linearly unstable, with the number of unstable modes increasing with the number of peaks present. Despite this, in two spatial dimensions, direct numerical simulations reveal the presence of stable single- and multi-spot states whose properties depend on the repulsion from nearby spots as well as the shape of the domain and the boundary conditions imposed thereon. Front propagation is shown to trigger the growth of new spots while destabilizing others. The results indicate that multi-variable models may support new types of behavior that are absent from typical two-variable models.
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Affiliation(s)
- Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel
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Al Saadi F, Champneys A. Unified framework for localized patterns in reaction-diffusion systems; the Gray-Scott and Gierer-Meinhardt cases. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200277. [PMID: 34743600 DOI: 10.1098/rsta.2020.0277] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
A recent study of canonical activator-inhibitor Schnakenberg-like models posed on an infinite line is extended to include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model to the Gray-Scott model. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localized pattern parameter regime as the homotopy parameter varies. Several distinct codimension-two bifurcations are discovered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other pattern-formation systems arising as models of natural phenomena. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
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Affiliation(s)
- Fahad Al Saadi
- Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK
- Department of Systems Engineering, Military Technological College, Muscat, Oman
| | - Alan Champneys
- Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK
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Mangioni SE, Deza RR. Stochastic dissipative solitons. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032116. [PMID: 26465435 DOI: 10.1103/physreve.92.032116] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/02/2013] [Indexed: 06/05/2023]
Abstract
By the effect of aggregating currents, some systems display an effective diffusion coefficient that becomes negative in a range of the order parameter, giving rise to bistability among homogeneous states (HSs). By applying a proper multiplicative noise, localized (pinning) states are shown to become stable at the expense of one of the HSs. They are, however, not static, but their location fluctuates with a variance that increases with the noise intensity. The numerical results are supported by an analytical estimate in the spirit of the so-called solvability condition.
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Affiliation(s)
- Sergio E Mangioni
- Instituto de Física de Mar del Plata, Universidad Nacional de Mar del Plata, and CONICET, Deán Funes 3350, B7602AYL Mar del Plata, Buenos Aires, Argentina
| | - Roberto R Deza
- Instituto de Física de Mar del Plata, Universidad Nacional de Mar del Plata, and CONICET, Deán Funes 3350, B7602AYL Mar del Plata, Buenos Aires, Argentina
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Yochelis A, Knobloch E, Köpf MH. Origin of finite pulse trains: Homoclinic snaking in excitable media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032924. [PMID: 25871189 DOI: 10.1103/physreve.91.032924] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2013] [Indexed: 06/04/2023]
Abstract
Many physical, chemical, and biological systems exhibit traveling waves as a result of either an oscillatory instability or excitability. In the latter case a large multiplicity of stable spatially localized wavetrains consisting of different numbers of traveling pulses may be present. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customized initial stimuli, thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.
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Affiliation(s)
- Arik Yochelis
- Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Jacob Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990, Israel
| | - Edgar Knobloch
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Michael H Köpf
- Département de Physique, École Normale Supérieure, 24 rue Lhomond, 75005 Paris, France
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Breña-Medina V, Champneys A. Subcritical Turing bifurcation and the morphogenesis of localized patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032923. [PMID: 25314520 DOI: 10.1103/physreve.90.032923] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2014] [Indexed: 05/03/2023]
Abstract
Subcritical Turing bifurcations of reaction-diffusion systems in large domains lead to spontaneous onset of well-developed localized patterns via the homoclinic snaking mechanism. This phenomenon is shown to occur naturally when balancing source and loss effects are included in a typical reaction-diffusion system, leading to a super- to subcritical transition. Implications are discussed [corrected]for a range of physical problems, arguing that subcriticality leads to naturally robust phase transitions to localized patterns.
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Affiliation(s)
- Víctor Breña-Medina
- Departamento de Nanotecnología, Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Juriquilla No. 3001, Querétaro 76230, Mexico and Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, United Kingdom
| | - Alan Champneys
- Department of Engineering Mathematics, University of Bristol, Queen's Building, University Walk, Bristol BS8 1TR, United Kingdom
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Ben Isaac E, Manor U, Kachar B, Yochelis A, Gov NS. Linking actin networks and cell membrane via a reaction-diffusion-elastic description of nonlinear filopodia initiation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022718. [PMID: 24032875 DOI: 10.1103/physreve.88.022718] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/24/2012] [Revised: 07/08/2013] [Indexed: 06/02/2023]
Abstract
Reaction-diffusion models have been used to describe pattern formation on the cellular scale, and traditionally do not include feedback between cellular shape changes and biochemical reactions. We introduce here a distinct reaction-diffusion-elasticity approach: The reaction-diffusion part describes bistability between two actin orientations, coupled to the elastic energy of the cell membrane deformations. This coupling supports spatially localized patterns, even when such solutions do not exist in the uncoupled self-inhibited reaction-diffusion system. We apply this concept to describe the nonlinear (threshold driven) initiation mechanism of actin-based cellular protrusions and provide support by several experimental observations.
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Affiliation(s)
- Eyal Ben Isaac
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
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Kao HC, Knobloch E. Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:026211. [PMID: 22463303 DOI: 10.1103/physreve.85.026211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/23/2011] [Indexed: 05/31/2023]
Abstract
The transition from subcritical to supercritical stationary periodic patterns is described by the one-dimensional cubic-quintic Ginzburg-Landau equation A(t) = μA + A(xx) + i(a(1)|A|(2)A(x) + a(2)A(2)A(x)*) + b|A|(2)|A - |A|(4)A, where A(x,t) represents the pattern amplitude and the coefficients μ, a(1), a(2), and b are real. The conditions for Eckhaus instability of periodic solutions are determined, and the resulting spatially modulated states are computed. Some of these evolve into spatially localized structures in the vicinity of a Maxwell point, while others resemble defect states. The results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during the transition from subcriticality to supercriticality.
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Affiliation(s)
- Hsien-Ching Kao
- Department of Physics, University of California, Berkeley, California 94720, USA.
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Yochelis A, Sheintuch M. Principal bifurcations and symmetries in the emergence of reaction-diffusion-advection patterns on finite domains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:056201. [PMID: 20365054 DOI: 10.1103/physreve.80.056201] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2008] [Revised: 07/12/2009] [Indexed: 05/29/2023]
Abstract
Pattern formation mechanisms of a reaction-diffusion-advection system, with one diffusivity, differential advection, and (Robin) boundary conditions of Danckwerts type, are being studied. Pattern selection requires mapping the domains of coexistence and stability of propagating or stationary nonuniform solutions, which for the general case of far from instability onsets, is conducted using spatial dynamics and numerical continuations. The selection is determined by the boundary conditions which either preserve or destroy the translational symmetry of the model. Accordingly, we explain the criterion and the properties of stationary periodic states if the system is bounded and show that propagation of nonlinear waves (including solitary) against the advective flow corresponds to coexisting family that emerges nonlinearly from a distinct oscillatory Hopf instability. Consequently, the resulting pattern selection is qualitatively different from the symmetric finite wavenumber Turing or Hopf instabilities.
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Affiliation(s)
- Arik Yochelis
- Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
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Yochelis A, Sheintuch M. Towards nonlinear selection of reaction-diffusion patterns in presence of advection: a spatial dynamics approach. Phys Chem Chem Phys 2009; 11:9210-23. [PMID: 19812842 DOI: 10.1039/b903266e] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We present a theoretical study of nonlinear pattern selection mechanisms in a model of bounded reaction-diffusion-advection system. The model describes the activator-inhibitor type dynamics of a membrane reactor characterized by a differential advection and a single diffusion; the latter excludes any finite wave number instability in the absence of advection. The focus is on three types of different behaviors, and the respective sensitivity to boundary and initial conditions: traveling waves, stationary periodic states, and excitable pulses. The theoretical methodology centers on the spatial dynamics approach, i.e. bifurcation theory of nonuniform solutions. These solutions coexist in overlapping parameter regimes, and multiple solutions of each type may be simultaneously stable. The results provide an efficient understanding of the pattern selection mechanisms that operate under realistic boundary conditions, such as Danckwerts type. The applicability of the results to broader reaction-diffusion-advection contexts is also discussed.
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Affiliation(s)
- Arik Yochelis
- Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
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