Diffusive instabilities in hyperbolic reaction-diffusion equations

Oscillatory periodic pattern dynamics in hyperbolic reaction-advection-diffusion models

In this work we consider a quite general class of two-species hyperbolic reaction-advection-diffusion system with the main aim of elucidating the role played by inertial effects in the dynamics of oscillatory periodic patterns. To this aim, first, we use linear stability analysis techniques to deduce the conditions under which wave (or oscillatory Turing) instability takes place. Then, we apply multiple-scale weakly nonlinear analysis to determine the equation which rules the spatiotemporal evolution of pattern amplitude close to criticality. This investigation leads to a cubic complex Ginzburg-Landau (CCGL) equation which, owing to the functional dependence of the coefficients here involved on the inertial times, reveals some intriguing consequences. To show in detail the richness of such a scenario, we present, as an illustrative example, the pattern dynamics occurring in the hyperbolic generalization of the extended Klausmeier model. This is a simple two-species model used to describe the migration of vegetation stripes along the hillslope of semiarid environments. By means of a thorough comparison between analytical predictions and numerical simulations, we show that inertia, apart from enlarging the region of the parameter plane where wave instability occurs, may also modulate the key features of the coherent structures, solution of the CCGL equation. In particular, it is proven that inertial effects play a role, not only during transient regime from the spatially-homogeneous steady state toward the patterned state, but also in altering the amplitude, the wavelength, the angular frequency, and even the stability of the phase-winding solutions.

Diffusive instability in hyperbolic reaction-diffusion equation with different inertia

This work considers a two-dimensional hyperbolic reaction-diffusion system with different inertia and explores criteria for various instabilities, like a wave, Turing, and Hopf, both theoretically and numerically. It is proven that wave instability may occur in a two-species hyperbolic reaction-diffusion system with identical inertia if the diffusion coefficients of the species are nonidentical but cannot occur if diffusion coefficients are identical. Wave instability may also arise in a two-dimensional hyperbolic reaction-diffusion system if the diffusivities of the species are equal, which is never possible in a parabolic reaction-diffusion system, provided the inertias are different. Interestingly, Turing instability is independent of inertia, but the stability of the corresponding local system depends on the inertia. Theoretical results are demonstrated with an example where the local interaction is represented by the Schnakenberg system.

Modern perspectives on near-equilibrium analysis of Turing systems

In the nearly seven decades since the publication of Alan Turing's work on morphogenesis, enormous progress has been made in understanding both the mathematical and biological aspects of his proposed reaction-diffusion theory. Some of these developments were nascent in Turing's paper, and others have been due to new insights from modern mathematical techniques, advances in numerical simulations and extensive biological experiments. Despite such progress, there are still important gaps between theory and experiment, with many examples of biological patterning where the underlying mechanisms are still unclear. Here, we review modern developments in the mathematical theory pioneered by Turing, showing how his approach has been generalized to a range of settings beyond the classical two-species reaction-diffusion framework, including evolving and complex manifolds, systems heterogeneous in space and time, and more general reaction-transport equations. While substantial progress has been made in understanding these more complicated models, there are many remaining challenges that we highlight throughout. We focus on the mathematical theory, and in particular linear stability analysis of 'trivial' base states. We emphasize important open questions in developing this theory further, and discuss obstacles in using these techniques to understand biological reality. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

Dynamics of a modified excitable neuron model: Diffusive instabilities and traveling wave solutions

We examine the dynamics of a spatially extended excitable neuron model between phase state and stable/unstable equilibrium point depending on the parameter regimes. The solitary wave profiles in the excitable medium are characterized by an improved Hindmarsh-Rose (H-R) spiking-bursting neuron model with an injected decaying current function. Linear stability and the nature of deterministic system dynamics are analyzed. Further investigation for the existence of wave using the reaction-diffusion H-R system and the criteria for diffusion-driven instabilities are performed. An approximation method is introduced to analyze traveling wave profiles for the oscillatory neuron model that allows the explicit analytical treatment of both the speed equations and shape of the traveling wave solution. The solitary wave profiles exhibited by the system are explored. The analytical expression for the solution scheme is validated with good accuracy in a wide range of the biophysical parameters of the system. The traveling wave fronts and speed equations control the variations of the information transmission, and the speed of signal transmission may be affected by the injection of certain drugs.

Diffusive instabilities in a hyperbolic activator-inhibitor system with superdiffusion

We investigate analytically and numerically the conditions for wave instabilities in a hyperbolic activator-inhibitor system with species undergoing anomalous superdiffusion. In the present work, anomalous superdiffusion is modeled using the two-dimensional Weyl fractional operator, with derivative orders α∈ [1,2]. We perform a linear stability analysis and derive the conditions for diffusion-driven wave instabilities. Emphasis is placed on the effect of the superdiffusion exponent α, the diffusion ratio d, and the inertial time τ. As the superdiffusive exponent increases, so does the wave number of the Turing instability. Opposite to the requirement for Turing instability, the activator needs to diffuse sufficiently faster than the inhibitor in order for the wave instability to occur. The critical wave number for wave instability decreases with the superdiffusive exponent and increases with the inertial time. The maximum value of the inertial time for a wave instability to occur in the system is τ_{max}=3.6. As one of the main results of this work, we conclude that both anomalous diffusion and inertial time influence strongly the conditions for wave instabilities in hyperbolic fractional reaction-diffusion systems. Some numerical simulations are conducted as evidence of the analytical predictions derived in this work.