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In this work an activator-depleted reaction–diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven instabilities. Quantitative relationships are found in the form of necessary conditions on the area of a disk-shape domain with respect to the diffusion and reaction rates for certain types of diffusion-driven instabilities to occur. Robust analytical methods are applied to find explicit expressions for the eigenvalues and eigenfunctions of the diffusion operator on a disk-shape domain with homogenous Neumann boundary conditions in polar coordinates. Spectral methods are applied using Chebyshev nonperiodic grid for the radial variable and Fourier periodic grid for the angular variable to verify the nodal lines and eigen-surfaces subject to the proposed analytical findings. The full classification of the parameter space in light of the bifurcation analysis is obtained and numerically verified by finding the solutions of the partitioning curves inducing such a classification. Spatiotemporal periodic behavior is demonstrated in the numerical solutions of the system for a proposed choice of parameters and a rigorous proof of the existence of infinitely many such points in the parameter plane is presented under a restriction on the area of the domain, with a lower bound in terms of reaction–diffusion rates. [ABSTRACT FROM AUTHOR]

We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many natural phenomena in areas such as developmental and cancer biology, cell motility and material science. In many of these applications, often one is interested in identifying parameters which will lead to a particular pattern for a given reaction-diffusion model. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally, we show that in some cases the inhomogeneous steady state can be a linear combination of eigenfunctions. Finally, we show an example suggesting that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary. [ABSTRACT FROM AUTHOR]

In this article we present robust, efficient and accurate fully implicit time-stepping schemes and nonlinear solvers for systems of reaction–diffusion equations. The applications of reaction–diffusion systems is abundant in the literature, from modelling pattern formation in developmental biology to cancer research, wound healing, tissue and bone regeneration and cell motility. Therefore, it is crucial that modellers, analysts and biologists are able to solve accurately and efficiently systems of highly nonlinear parabolic partial differential equations on complex stationary and sometimes continuously evolving domains and surfaces. The main contribution of our paper is the study of fully implicit schemes by use of the Newton method and the Picard iteration applied to the backward Euler, the Crank–Nicolson (and its modifications) and the fractional-step θ methods. Our results conclude that the fractional-step θ method coupled with a single Newton iteration at each timestep is as accurate as the fully adaptive Newton method; and both outperform the Picard iteration. In particular, the results strongly support the observation that a single Newton iteration is sufficient to yield as accurate results as those obtained by use of an adaptive Newton method. This is particularly advantageous when solving highly complex nonlinear partial differential equations on evolving domains and surfaces. To validate our theoretical results, various appropriate numerical experiments are exhibited on stationary planary domains and in the bulk of stationary surfaces. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]