Rui Dilão

Modelling butterfly wing eyespot patterns

Eyespots are concentric motifs with contrasting colours on butterfly wings. Eyespots have intra– and interspecific 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 organization of eyespots, their phenotypic plasticity and seasonal variability, and predicts effects from microsurgical manipulations on pupal wings as reported in the literature.

Turing instabilities and patterns near a Hopf bifurcation

We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from vector fields with one fixed point and a supercritical Hopf bifurcation. For the Brusselator and the Ginzburg-Landau reaction-diffusion equations, we obtain the bifurcation diagrams associated with the transition between time periodic solutions and asymptotically stable solutions (Turing patterns). In two-component systems of reaction-diffusion equations, we show that the existence of Turing instabilities is neither necessary nor sufficient for the existence of Turing pattern type solutions. Turing patterns can exist on both sides of the Hopf bifurcation associated with the local vector field, and, depending on the initial conditions, time periodic and stable solutions can coexist.

A general approach to the modelling of trophic chains

Abstract Based on the law of mass action (and its microscopic foundation) and mass conservation, we present here a method to derive consistent dynamic models for the time evolution of systems with an arbitrary number of species. Equations are derived through a mechanistic description, ensuring that all parameters have ecological meaning. After discussing the biological mechanisms associated to the logistic and Lotka–Volterra equations, we show how to derive general models for trophic chains, including the effects of internal states at fast time scales. We show that conformity with the mass action law leads to different functional forms for the Lotka–Volterra and trophic chain models. We use mass conservation to recover the concept of carrying capacity for an arbitrary food chain.

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 γN=DΔt/(Δx)2 assumes a fixed constant value, where N is an odd positive integer parametrizing the algorithm. The error between the solutions of the discrete and the continuous equations goes to zero as (Δx)2(N+2) and the values of γN 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 traveling 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.

Nonlinear Dynamics in Particle Accelerators

Introduction to acclerator physics introduction to the qualitative theory of nonlinear differential equations simple synchrotron accelerators the dynamics and stability of guiding and focusing nonlinear magnets motion in the transverse plane acceleration of charged particles radio-frequency time series analysis beam-beam interaction.

Validation of a Morphogenesis Model of Drosophila Early Development by a Multi-objective Evolutionary Optimization Algorithm

We apply evolutionary computation to calibrate the parameters of a morphogenesis model of Drosophila early development. The model aims to describe the establishment of the steady gradients of Bicoid and Caudal proteins along the antero-posterior axis of the embryo of Drosophila . The model equations consist of a system of non-linear parabolic partial differential equations with initial and zero flux boundary conditions. We compare the results of single- and multi-objective variants of the CMA-ES algorithm for the model the calibration with the experimental data. Whereas the multi-objective algorithm computes a full approximation of the Pareto front, repeated runs of the single-objective algorithm give solutions that dominate (in the Pareto sense) the results of the multi-objective approach. We retain as best solutions those found by the latter technique. From the biological point of view, all such solutions are all equally acceptable, and for our test cases, the relative error between the experimental data and validated model solutions on the Pareto front are in the range 3% *** 6%. This technique is general and can be used as a generic tool for parameter calibration problems.

Topological entropy and approaches to chaos in dynamics of the interval

Abstract A method for computing the growth number and the topological entropy in maps of the interval, given the kneading sequence, is presented. This technique is applied to the study of the parameter dependence and universal properties of the topological entropy in the approach of aperiodic regimes. In particular we show that period doubling does not change entropy.

Stable knotted strings

Abstract We solve the Cauchy problem for the relativistic closed string in Minkowski space M 3+1 , including the cases where the initial data has a knot like topology. We give the general conditions for the world sheet of a closed knotted string to be a time periodic surface. In the particular case of zero initial string velocity the period of the world sheet is proportional to half the length ( l ) of the initial string and a knotted string always collapses to a link for t = l /4. Relativistic closed strings are dynamically evolving or pulsating structures in spacetime, and knotted or unknotted like structures remain stable over time. The generation of arbitray n -fold knots, starting with an initial simple link configuration with non zero velocity is possible.

COMPUTING THE TOPOLOGICAL ENTROPY OF UNIMODAL MAPS

We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy of maps with direct and reverse bifurcations.

Computing the Topological Entropy of Multimodal Maps via Min-Max Sequences

We derive an algorithm to recursively determine the lap number (minimal number of monotonicity segments) of the iterates of twice differentiable l-modal map, enabling to numerically calculate the topological entropy of these maps. The algorithm is obtained by the min-max sequences—symbolic sequences that encode qualitative information about all the local extrema of iterated maps.