Anotida Madzvamuse

A moving grid finite element method applied to a model biological pattern generator

Many problems in biology involve growth. In numerical simulations it can therefore be very convenient to employ a moving computational grid on a continuously deforming domain. In this paper we present a novel application of the moving grid finite element method to compute solutions of reaction-diffusion systems in two-dimensional continuously deforming Euclidean domains. A numerical software package has been developed as a result of this research that is capable of solving generalised Turing models for morphogenesis.

Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains

In this paper, we illustrate the application of time-stepping schemes to reaction-diffusion systems on fixed and continuously growing domains by use of finite element and moving grid finite element methods. We present two schemes for our studies, namely a first-order backward Euler finite differentiation formula coupled with a special form of linearisation of the nonlinear reaction terms (1-SBEM) and a second-order semi-implicit backward finite differentiation formula (2-SBDF) with no linearisation of the reaction terms. Our results conclude that for the type of reaction-diffusion systems considered in this paper, the 1-SBEM is more stable than the 2-SBDF scheme and that the 1-SBEM scheme has a larger region of stability (at least by a factor of 10) than that of the 2-SBDF scheme. As a result, the 1-SBEM scheme becomes a natural choice when solving reaction-diffusion problems on continuously deforming domains.

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

Numerical techniques for moving meshes are many and varied. In this paper we present a novel application of a moving grid finite element method applied to biological problems related to pattern formation where the mesh movement is prescribed through a specific definition to mimic the growth that is observed in nature. Through the use of a moving grid finite element technique, we present numerical computational results illustrating how period doubling behaviour occurs as the domain doubles in size.

A model for colour pattern formation in the butterfly wing of Papilio dardanus

The butterfly Papilio dardanus is well known for the spectacular phenotypic polymorphism in the female of the species. We show that numerical simulations of a reaction diffusion model on a geometrically accurate wing domain produce spatial patterns that are consistent with many of those observed on the butterfly. Our results suggest that the wing coloration is due to a simple underlying stripe–like pattern of some pigment–inducing morphogen. We focus on the effect of key factors such as parameter values for mode selection, threshold values which determine colour, wing shape and boundary conditions. The generality of our approach should allow us to investigate other butterfly species. The relationship between these key factors and gene activities is discussed in the context of recent biological advances.

The surface finite element method for pattern formation on evolving biological surfaces

In this article we propose models and a numerical method for pattern formation on evolving curved surfaces. We formulate reaction-diffusion equations on evolving surfaces using the material transport formula, surface gradients and diffusive conservation laws. The evolution of the surface is defined by a material surface velocity. The numerical method is based on the evolving surface finite element method. The key idea is based on the approximation of Γ by a triangulated surface Γh consisting of a union of triangles with vertices on Γ. A finite element space of functions is then defined by taking the continuous functions on Γh which are linear affine on each simplex of the polygonal surface. To demonstrate the capability, flexibility, versatility and generality of our methodology we present results for uniform isotropic growth as well as anisotropic growth of the evolution surfaces and growth coupled to the solution of the reaction-diffusion system. The surface finite element method provides a robust numerical method for solving partial differential systems on continuously evolving domains and surfaces with numerous applications in developmental biology, tumour growth and cell movement and deformation.

Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains

Reaction-diffusion systems have been widely studied in developmental biology, chemistry and more recently in financial mathematics. Most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions. It therefore becomes convenient to look for numerical solutions: finite difference, finite element, finite volume and spectral methods are typical examples of the numerical methods used. Most of these methods are locally based schemes. We examine the implications of mesh structure on numerically computed solutions of a well-studied reaction-diffusion model system on two-dimensional fixed and growing domains. The incorporation of domain growth creates an additional parameter - the grid-point velocity - and this greatly influences the selection of certain symmetric solutions for the ADI finite difference scheme when a uniform square mesh structure is used. Domain growth coupled with grid-point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure. We compare our results to those obtained by use of finite elements on unstructured triangular elements.

Stability analysis of non-autonomous reaction-diffusion systems: The effects of growing domains

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.

Implicit--Explicit Timestepping with Finite Element Approximation of Reaction--Diffusion Systems on Evolving Domains

We present and analyze an implicit--explicit timestepping procedure with finite element spatial approximation for semilinear reaction--diffusion systems on evolving domains arising from biological models, such as Schnakenbergs (1979). We employ a Lagrangian formulation of the model equations which permits the error analysis for parabolic equations on a fixed domain but introduces technical difficulties, foremost the space-time dependent conductivity and diffusion. We prove optimal-order error estimates in the

A mechanochemical model of striae distensae

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Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations

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