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Browsing IPS - ESTB - DMI by Author "Barreira, Raquel"
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- Cross-Diffusion in Reaction-Diffusion Models:Analysis, Numerics, and ApplicationsPublication . Madzvamuse, Anotida; Barreira, Raquel; Gerisch, Alf
- Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulationsPublication . Madzvamuse, Anotida; Ndakwo, Hussaini; Barreira, RaquelBy introducing linear cross-diffusion for a two-component reaction-diffusion system withactivator-depletedreaction kinetics (Gierer and Meinhardt,Kybernetik 12:30–39,1972;PrigogineandLefever,JChemPhys48:1695–1700,1968;Schnakenberg, J Theor Biol 81:389–400,1979), we derivecross-diffusion-driveninstability conditions and show that they are a generalisation of the classical diffusion-driveninstabilityconditionsintheabsenceofcross-diffusion.Ourmostrevealingresultis that, in contrast to the classical reaction-diffusion systems without cross-diffusion,it is no longer necessary to enforce that one of the species diffuse much faster than theother.Furthermore,it is no longer necessary to have an activator–inhibitor mecha-nism as premises for pattern formation, activator–activator,inhibitor–inhibitorreac-tion kinetics as well asshort-range inhibitionandlong-range activationall have thepotential of giving rise to cross-diffusion-driven instability. To support our theoreti-cal findings, we compute cross-diffusion induced parameter spaces and demonstratesimilarities and differences to those obtained using standard reaction-diffusion theory.Finite element numerical simulations on planary square domains are presented to back-up theoretical predictions. For the numerical simulations presented, we choose parameter values from and outside the classical Turing diffusively-driven instability space;outside, these are chosen to belong to cross-diffusively-driven instability parameterspaces. Our numerical experiments validate our theoretical predictions that parameterspaces induced by cross-diffusion in both theuandvcomponents of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spacesof the cross-diffusion induced parameter spaces. Our results allow experimentalists tohave a wider range of parameter spaces from which to select reaction kinetic parametervalues that will give rise to spatial patterning in the presence of cross-diffusion.
- Encontro Nacional da Sociedade Portuguesa de Matemática 2016: livro de resumosPublication . Guerra, Telma; Marques, Anabela; Carlos, Clara; Salvador, Dina; Dias, Mariana; Dias, Otília; Barreira, Raquel; Silva, Vanda Rosado
- Exhibiting cross-diffusion-induced patterns for reaction-diffusionsystems on evolving domains and surfacesPublication . Madzvamuse, Anotida; Barreira, RaquelThe aim of this manuscript is to present for the first time the application of the finite element method for solvingreaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore wepresent pattern formation generated by the reaction-diffusion system with cross-diffusion on evolving domains andsurfaces. A two-component reaction-diffusion system with linear cross-diffusion in bothuandvis presented. Thefinite element method is based on the approximation of the domain or surface by a triangulated domain or surfaceconsisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. Afinite element space of functions is then defined by taking the continuous functions which are linear affine on eachsimplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of patternformation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusionparameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems.Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; themethodology can deal with complicated evolution laws of the domain and surface, and these include uniformisotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing inthe domain or on the surface
- Numerical modelling of operational risks for the banking industryPublication . Barreira, Raquel; Pryer, Tristan; Tang, Q.
- A practical approach to model banking risks using Loss Distribution Approach (LDA) in Basel II frameworkPublication . Barreira, Raquel; Pryer, Tristan; Tang, QiIn Basel II Capital Accord, the Advanced Measurement Approaches (AMA) is stated as one of the pillar stone methods for calculating corporate risk reserves. One of the common yet cumbersome methods is the one known as loss distribution approach (cf. [3]). In this article, we present an easy to implement scheme through electronic means and discuss some of the mathematical problems we encountered in the process together with proposed solution methods and further sought on the issues.
- Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusionPublication . Madzvamuse, Anotida; Ndakwo, Hussaini; Barreira, RaquelThis article presents stability analytical results of a two compo-nent reaction-diffusion system with linear cross-diffusion posed on continuouslyevolving domains. First the model system is mapped from a continuously evolv-ing domain to a reference stationary frame resulting in a system of partialdifferential equations with time-dependent coefficients. Second, by employingappropriately asymptotic theory, we derive and prove cross-diffusion-driven in-stability conditions for the model system for the case of slow, isotropic domaingrowth. Our analytical results reveal that unlike the restrictive diffusion-driveninstability conditions on stationary domains, in the presence of cross-diffusioncoupled with domain evolution, it is no longer necessary to enforce cross norpure kinetic conditions. The restriction toactivator-inhibitorkinetics to inducepattern formation on a growing biological system is no longer a requirement.Reaction-cross-diffusion models with equal diffusion coefficients in the principalcomponents as well as those of theshort-range inhibition, long-range activa-tionandactivator-activatorform can generate patterns only in the presence ofcross-diffusion coupled with domain evolution. To confirm our theoretical find-ings, detailed parameter spaces are exhibited for the special cases of isotropicexponential, linear and logistic growth profiles. In support of our theoreticalpredictions, we present evolving or moving finite element solutions exhibitingpatterns generated by ashort-range inhibition, long-range activationreaction-diffusion model with linear cross-diffusion in the presence of domain evolution.
- The surface finite element method for pattern formation on evolving biological surfacesPublication . Barreira, Raquel; Elliott, C. M.; Madzvamuse, AnotidaIn this article we propose models and a numerical method for patternformation on evolving curved surfaces. We formulate reaction-diffusion equations onevolving surfaces using the material transport formula, surface gradients and diffusiveconservation laws. The evolution of the surface is defined by a material surface veloc-ity. The numerical method is based on the evolving surface finite element method. Thekey idea is based on the approximation of by a triangulated surface hconsistingof a union of triangles with vertices on . A finite element space of functions is thendefined by taking the continuous functions on hwhich are linear affine on each sim-plex of the polygonal surface. To demonstrate the capability, flexibility, versatility andgenerality of our methodology we present results for uniform isotropic growth as wellas anisotropic growth of the evolution surfaces and growth coupled to the solutionof the reaction-diffusion system. The surface finite element method provides a robustnumerical method for solving partial differential systems on continuously evolvingdomains and surfaces with numerous applications in developmental biology, tumourgrowth and cell movement and deformation.
- XXII Jornadas de Classificação e Análise de Dados (JOCLAD 2015): Livro de ResumosPublication . Ferreira, Ana Sousa; Marques, Anabela; Carlos, Clara; Dias, Otília; Barreira, Raquel; Guerra, Telma