Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model.

In this paper we analyze the effects of introducing the fractional-in-space operator into a Lotka-Volterra competitive model describing population super-diffusion. First, we study how cross super-diffusion influences the formation of spatial patterns: a linear stability analysis is carried out, showing that cross super-diffusion triggers Turing instabilities, whereas classical (self) super-diffusion does not. In addition we perform a weakly nonlinear analysis yielding a system of amplitude equations, whose study shows the stability of Turing steady states. A second goal of this contribution is to propose a fully adaptive multiresolution finite volume method that employs shifted Grünwald gradient approximations, and which is tailored for a larger class of systems involving fractional diffusion operators. The scheme is aimed at efficient dynamic mesh adaptation and substantial savings in computational burden. A numerical simulation of the model was performed near the instability boundaries, confirming the behavior predicted by our analysis.

Turing Pattern Formation in a Semiarid Vegetation Model with Fractional-in-Space Diffusion.

A fractional power of the Laplacian is introduced to a reaction-diffusion system to describe water's anomalous diffusion in a semiarid vegetation model. Our linear stability analysis shows that the wavenumber of Turing pattern increases with the superdiffusive exponent. A weakly nonlinear analysis yields a system of amplitude equations, and the analysis of these amplitude equations shows that the spatial patterns are asymptotic stable due to the supercritical Turing bifurcation. Numerical simulations exhibit a bistable regime composed of hexagons and stripes, which confirm our analytical results. Moreover, the characteristic length of the emergent spatial pattern is consistent with the scale of vegetation patterns observed in field studies.

Turing pattern dynamics in an activator-inhibitor system with superdiffusion.

The fractional operator is introduced to an activator-inhibitor system to describe species anomalous superdiffusion. The effects of the superdiffusive exponent on pattern formation and pattern selection are studied. Our linear stability analysis shows that the wave number of the Turing pattern increases with the superdiffusive exponent. A weakly nonlinear analysis yields a system of amplitude equations and the analysis of these amplitude equations predicts parameter regimes where hexagons, stripes, and their coexistence are expected. Numerical simulations of the activator-inhibitor model near the stability boundaries confirm our analytical results. Since diffusion rate manifests in both diffusion constant and diffusion exponent, we numerically explore their interactions on the emergence of Turing patterns. When the activator and inhibitor have different superdiffusive exponents, we find that the critical ratio of the diffusion rate of the inhibitor to the activator, required for the formation of the Turing pattern, increases monotonically with the superdiffusive exponent. We conclude that small ratio (than unity) of anomalous diffusion exponent between the inhibitor and activator is more likely to promote the emergence of the Turing pattern, relative to the normal diffusion.

Delay-driven irregular spatiotemporal patterns in a plankton system.

An inhomogeneous distribution of species density over physical space is a widely observed scenario in plankton systems. Understanding the mechanisms resulting in these spatial patterns is a central topic in plankton ecology. In this paper we explore the impact of time delay on spatiotemporal patterns in a prey-predator plankton system. We find that time delay can trigger the emergence of irregular spatial patterns via a Hopf bifurcation. Moreover, a phase transition from a regular spiral pattern to an irregular one was observed and the latter gradually replaced the former and persisted indefinitely. The characteristic length of the emergent spatial pattern is consistent with the scale of plankton patterns observed in field studies.

Delay-driven spatial patterns in a plankton allelopathic system.

Spatial patterns have received considerable attention in the physical, biological, and social sciences. Generally speaking, time delay is a prevailing phenomenon in aquatic environments, since the production of allelopathic substance by competitive species is not instantaneous, but mediated by some time lag required for maturity of species. A natural question is how delay affects the spatial patterns. Here, we consider a delayed plankton allelopathic system consisting of two competitive species and analytically investigate how the time delay affects the stability and spatial patterns. Based upon a stability analysis, we demonstrate that the delay can induce spatial patterns under some conditions. Moreover, by use of a series of numerical simulations performed with a finite difference scheme, we show that the delay plays an important role on pattern selection.

Traveling wave governs the stability of spatial pattern in a model of allelopathic competition interactions.

Inhomogenous distribution of populations across physical space is a widely observed scenario in nature and has been studied extensively. Mechanisms accounting for these observations are such as diffusion-driven instability and mechanochemical approach. While conditions have been derived from a variety of models in biological, physical, and chemical systems to trigger the emergence of spatial patterns, it remains poorly understood whether the spatial pattern possesses asymptotical stability. In a plankton allelopathic competitive system with distributed time delay, we found that spatial pattern arises as a result of Hopf bifurcation and, in the meantime, there exists a unique asymptotically stable traveling wave solution. The convergence of the traveling wave solution to the emergent pattern and its stability infer that the emergent spatial pattern is locally asymptotically stable.