Existence of Pulses for the System of Competition of Species

Marion, Martine; Volpert, Vitaly

doi:10.1007/s10884-017-9582-6

Cited by 8 publications

(6 citation statements)

References 13 publications

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“…that is a positive stationary solution w 0 (x) with the zero limits at infinity [94], [95]. Moreover, such solution exists if and only if the wave speed is positive.…”

Section: Clot Growth As a Reaction-diffusion Wavementioning

confidence: 99%

Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones

Belyaev

Dunster

Gibbins

et al. 2018

Physics of Life Reviews

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Hemostasis is a complex physiological mechanism that functions to maintain vascular integrity under any conditions. Its primary components are blood platelets and a coagulation network that interact to form the hemostatic plug, a combination of cell aggregate and gelatinous fibrin clot that stops bleeding upon vascular injury. Disorders of hemostasis result in bleeding or thrombosis, and are the major immediate cause of mortality and morbidity in the world. Regulation of hemostasis and thrombosis is immensely complex, as it depends on blood cell adhesion and mechanics, hydrodynamics and mass transport of various species, huge signal transduction networks in platelets, as well as spatiotemporal regulation of the blood coagulation network. Mathematical and computational modeling has been increasingly used to gain insight into this complexity over the last 30 years, but the limitations of the existing models remain profound. Here we review state-of-the-art-methods for computational modeling of thrombosis with the specific focus on the analysis of unresolved challenges. They include: a) fundamental issues related to physics of platelet aggregates and fibrin gels; b) computational challenges and limitations for solution of the models that combine cell adhesion, hydrodynamics and chemistry; c) biological mysteries and unknown parameters of processes; d) biophysical complexities of the spatiotemporal networks' regulation. Both relatively classical approaches and innovative computational techniques for their solution are considered; the subjects discussed with relation to thrombosis modeling include coarse-graining, continuum versus particle-based modeling, multiscale models, hybrid models, parameter estimation and others. Fundamental understanding gained from theoretical models are highlighted and a description of future prospects in the field and the nearest possible aims are given.

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“…that is a positive stationary solution w 0 (x) with the zero limits at infinity [94], [95]. Moreover, such solution exists if and only if the wave speed is positive.…”

Section: Clot Growth As a Reaction-diffusion Wavementioning

confidence: 99%

Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones

Belyaev

Dunster

Gibbins

et al. 2018

Physics of Life Reviews

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“…The model system and the homotopy can be found in [58,62]. Theorem 4.2 on the existence of pulses is generalized for monotone systems of two equations [35,36]: Theorem 4.5. Let the system…”

Section: Monotone and Locally Monotone Systemsmentioning

confidence: 99%

Method of Monotone Solutions for Reaction-Diffusion Equations

Volpert

Vougalter

2021

J Math Sci

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Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reaction-diffusion systems for which there exist two sub-classes of solutions separated in the function space, monotone and non-monotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

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“…The first one is that the wave speed should be positive. This condition is equivalent to the existence of stationary solutions of system (1.1), (1.2) in the form of pulses [65,115]. The second conditions concerns the initial perturbation.…”

Section: Modelmentioning

confidence: 99%

Mathematical modelling of atherosclerosis

Khatib¹,

Kafi²,

Sequeira³

et al. 2019

Math. Model. Nat. Phenom.

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The review presents the state of the art in the atherosclerosis modelling. It begins with the biological introduction describing the mechanisms of chronic inflammation of artery walls characterizing the development of atherosclerosis. In particular, we present in more detail models describing this chronic inflammation as a reaction-diffusion wave with regimes of propagation depending on the level of cholesterol (LDL) and models of rolling monocytes initializing the inflammation. Further development of this disease results in the formation of atherosclerotic plaque, vessel remodelling and possible plaque rupture due its interaction with blood flow. We review plaque-flow interaction models as well as reduced models (0D and 1D) of blood flow in atherosclerotic vasculature.

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Existence of Pulses for the System of Competition of Species

Cited by 8 publications

References 13 publications

Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones

Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones

Method of Monotone Solutions for Reaction-Diffusion Equations

Mathematical modelling of atherosclerosis

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