Joaquim Sainhas | Searcharxiv

arXiv2005

Modelling butterfly wing eyespot patterns

Eyespots are concentric motifs with contrasting colours on butterfly wings. Eyespots have intra- and inter-specific visual signalling functions with adaptive and selective roles. We propose a reaction-diffusion model that accounts for eyespot development. The model considers two diffusive morphogens and three non-diffusive pigment precursors. The first morphogen is produced in the focus and determines the differentiation of the first eyespot ring. A second morphogen is then produced, modifying the chromatic properties of the wing background pigment precursor, inducing the differentiation of a second ring. The model simulates the general structural organisation of eyespots, their phenotypic plasticity and seasonal variability, and predicts effects from microsurgical manipulations on pupal wings as reported in the literature.

q-bio.TO↗

arXiv1997

Validation and Calibration of Models for Reaction-Diffusion Systems

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps ($Δx$ and $Δt$) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction-diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite $Δx$ and $Δt$, if the parameter $γ_N=D Δt/(Δx)^2$ assumes a fixed constant value, where $N$ is an odd positive integer parametrizing the alghorithm. The error between the solutions of the discrete and the continuous equations goes to zero as $(Δx)^{2(N+2)}$ and the values of $γ_N$ are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction-diffusion equations. Comparison between numerical and analytical solutions of reaction-diffusion equations give global discretization errors of the order of $10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of $10^{-3}$.

patt-sol↗