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Each mini course will consists of 4 classes, with lectures and hands-on sessions. There will also be a 2-hour special lecture.
Prof. Laurent Desvillettes (Université Paris Cité, France)
Patterns appearing in the study of parabolic PDEs
Abstract: In this introductory course, we introduce the notion of patterns, starting with some classical example of destabilization of homogeneous steady solutions to parabolic systems of two equations (Turing instability). We show how patterns also appear in cases of cross diffusion systems coming out of population dynamics, and present some basic qualitative and quantitative properties of those patterns.
Prof. Thomas Hillen (University of Alberta, Canada)
Spatial Patterns in Biology
Abstract: Many spatial patterns in biology and medicine can be explained by partial differential equations. In these lectures I will choose a few specific biological examples to discuss the underlying models and their pattern formation properties. These include the mysterious striped patterns of tulips, merging-emerging chaotic patterns in chemotaxis, spatial patterns in non-local models, and the “monster on a leash” in go-or grow type models.
Prof. Ivonne Sgura (Università del Salento, Italy )
Prof. Anotida Madzvamuse (University of British Columbia, Canada)
Pattern formation on stationary, growing domains and evolving surfaces: modelling, numerical methods and applications
Abstract: In this Summer School, we will present recent advances in mathematical modelling of biophysical and energetic problems using systems of reaction-diffusion equations (RDEs) on stationary and evolving domains and surfaces. Our lectures will derive RDEs on domains that continuously deform in space and time (of which stationary domains are a special case). Turing diffusion-driven instability (TDDI) will be briefly analysed leading to mechanisms for pattern formation. We will show that the numerical approximation of the RDEs attains at the “steady state” solutions with different intriguing spatial morphologies, called Turing patterns like spots, holes, stripes and labyrinths. This is computationally challenging, because very fine meshes on large domains and integration for long times are needed to accurately describe the spatial structures of the patterns. For this reason, some lectures of the Course concern a selection of numerical methods, based on finite differences and finite elements in space, implicit-explicit (IMEX) schemes in time, for the approximation of RDEs. To speed up computations, in some cases, we will present the matrix-oriented approach recently developed.
In the second part of the Course, we extend the study of pattern formation to RDEs defined on (closed) surfaces in 3D and on evolving domains. In this case, we focus on: i) the new modeling framework; ii) evolving bulk-surface finite elements; and iii) related computer simulations for pattern formation. As an application, we present a Turing pattern formation morpho-chemical DIB RD-PDE system describing material localization in batteries. Participants will participate in interactive sessions of computer laboratory using MATLAB and FeNiCsX to obtain and discuss numerical simulations in realtime.
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