We present an efficient and accurate numerical method for solving a ratio-dependent predatorprey model with a Turing instability. The system is discretized by a finite difference method with a semi-implicit scheme which allows much larger time step sizes than those required by a standard explicit scheme. A proof is given for the positivity and boundedness of the numerical solutions depending on...

The purpose of this work is to find the region necessary and sufficient conditions for diffusion instability on parameter plane (τ, d) Gierer–Meinhardt system, where τ relaxation parameter, d dimensionless coefficient; derive analytically dependence critical wave number characteristic size spatial region; obtain explicit representations secondary spatially distributed structures, formed as a re...

The population model by Busenberg and Travis is a paradigmatic in ecology tumor modeling because its ability to capture interesting phenomena such as segregation of populations. Its singular mathematical structure enforces the consideration regularized problems deduce properties fundamental existence solutions. In this article we perform weakly nonlinear stability analysis general class study c...

We present a theoretical and experimental study of the sideband instabilities in Turing patterns of stripes. We compare numerical computations of the Brusselator model with experiments in a chlorine dioxide-iodine-malonic acid (CDIMA) reaction in a thin gel layer reactor in contact with a continuously refreshed reservoir of reagents. Spontaneously evolving Turing structures in both systems typi...

This paper proves that contractive ordinary differential equation systems remain contractive when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems. An important biochemical system is shown to satisfy the required conditions.

In this work, we analyse a pair of two-dimensional coupled reaction-diusion equations known as the Gray-Scott model, in which spot patterns have been observed. We focus on stationary patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. A complete bifurcation study of these solutions is presented. The main mathematic...