1. Stability bounds of a delay visco-elastic rheological model with substrate friction
- Author
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Malik A. Dawi, Jose J. Muñoz, Universitat Politècnica de Catalunya. Doctorat en Enginyeria del Terreny, and Universitat Politècnica de Catalunya. Departament de Matemàtiques
- Subjects
Differential equations ,Delay differential equations ,Biologia ,Oscillations ,Friction ,Cells ,Matemàtiques i estadística::Matemàtica aplicada a les ciències [Àrees temàtiques de la UPC] ,74 Mechanics of deformable solids::74C Plastic materials, materials of stress-rate and internal-variable type [Classificació AMS] ,Dynamical Systems (math.DS) ,Matemàtiques i estadística::Equacions diferencials i integrals::Equacions diferencials ordinàries [Àrees temàtiques de la UPC] ,Quantitative Biology::Cell Behavior ,Matemàtiques i estadística::Equacions diferencials i integrals [Àrees temàtiques de la UPC] ,Matemàtiques i estadística::Anàlisi numèrica [Àrees temàtiques de la UPC] ,Equacions diferencials funcionals ,65 Numerical analysis::65P Numerical problems in dynamical systems [Classificació AMS] ,74 Mechanics of deformable solids::74D Materials of strain-rate type and history type, other materials with memory [Classificació AMS] ,FOS: Mathematics ,Differentiable dynamical systems ,Visco-elasticity ,Mathematics - Dynamical Systems ,Strength of materials ,Tissues and Organs (q-bio.TO) ,Biology ,Resistència de materials ,92 Biology and other natural sciences::92C Physiological, cellular and medical topics [Classificació AMS] ,Anàlisi numèrica ,Applied Mathematics ,Quantitative Biology - Tissues and Organs ,Sistemes dinàmics diferenciables ,Agricultural and Biological Sciences (miscellaneous) ,FOS: Biological sciences ,Modeling and Simulation ,34 Ordinary differential equations::34k Functional-differential and differential-difference equations [Classificació AMS] ,37 Dynamical systems and ergodic theory::37N Applications [Classificació AMS] ,Rheology ,Plastics ,Stability ,Numerical analysis - Abstract
The version of record is available online at: http://dx.doi.org/10.1007/s00285-021-01699-8 Cells and tissues exhibit sustained oscillatory deformations during remodelling, migration or embryogenesis. Although it has been shown that these oscillations correlate with intracellular biochemical signalling, the role of these oscillations is as yet unclear, and whether they may trigger drastic cell reorganisation events or instabilities remains unknown. Here, we present a rheological model that incorporates elastic, viscous and frictional components, and that is able to generate oscillatory response through a delay adaptive process of the rest-length. We analyse its stability as a function of the model parameters and deduce analytical bounds of the stable domain. While increasing values of the delay and remodelling rate render the model unstable, we also show that increasing friction with the substrate destabilises the oscillatory response. This fact was unexpected and still needs to be verified experimentally. Furthermore, we numerically verify that the extension of the model with non-linear deformation measures is able to generate sustained oscillations converging towards a limit cycle. We interpret this sustained regime in terms of non-linear time varying stiffness parameters that alternate between stable and unstable regions of the linear model. We also note that this limit cycle is not present in the linear model. We study the phase diagram and the bifurcations of the non-linear model, based on our conclusions on the linear one. Such dynamic analysis of the delay visco-elastic model in the presence of friction is absent in the literature for both linear and non-linear rheologies. Our work also shows how increasing values of some parameters such as delay and friction decrease its stability, while other parameters such as stiffness stabilise the oscillatory response. JJM and MD have been financially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) with grant DPI2016-74929-R and by the local government Generalitat de Catalunya with grant 2017 SGR 1278.
- Published
- 2021