arXiv subjects

Eric Siero

Publications and source records attributed to Eric Siero.

6 recordsLinked to original sources

arXiv2026

Busse balloon deformation and splitting by non-local interaction: the influence of grazing on a Klausmeier vegetation model

Large areas on all continents except Antarctica are covered by dryland vegetation patterns, with wavelengths typically in the range of tens of meters. In models, many wavelengths are simultaneously stable, and we argue that this multi-stability also holds for real world patterns. We then study the shape of the Busse balloon representation of multi-stability for a previously introduced Klausmeier model with non-local grazing. For this we first extend application of singular perturbation to Klausmeier/Gray-Scott models to include non-local interaction, providing analytical control near the homoclinic limit, and then use numerical continuation to demonstrate deformation of Busse balloons away from the typical banana shape, in four non-local grazing regimes. Since the Busse balloon has been invoked to support "evasion of tipping", we underscore the importance of the shape of the Busse balloon when inferring ecosystem response to, e.g., climate change.

math.DS↗

arXiv2024

Blurring the Busse balloon: Patterns in a stochastic Klausmeier model

We investigate (in)stabilities of periodic patterns under stochastic forcing in reaction-diffusion equations exhibiting a so-called Busse balloon. Specifically, we used a one-dimensional Klausmeier model for dryland vegetation patterns. Using numerical methods, we can accurately describe the transient dynamics of the stochastic solutions and compare several notions of stability. In particular, we show that stochastic stability heavily depends on the model parameters, the intensity of the noise and the location of the wavenumber of the periodic pattern within the deterministic Busse balloon. Furthermore, the boundary of the Busse balloon becomes blurred under the stochastic perturbations.

math.AP↗

arXiv2023

Stability and Fluctuations in Complex Ecological Systems

From 08-12 August, 2022, 32 individuals participated in a workshop, Stability and Fluctuations in Complex Ecological Systems, at the Lorentz Center, located in Leiden, The Netherlands. An interdisciplinary dialogue between ecologists, mathematicians, and physicists provided a foundation of important problems to consider over the next 5-10 years. This paper outlines eight areas including (1) improving our understanding of the effect of scale, both temporal and spatial, for both deterministic and stochastic problems; (2) clarifying the different terminologies and definitions used in different scientific fields; (3) developing a comprehensive set of data analysis techniques arising from different fields but which can be used together to improve our understanding of existing data sets; (4) having theoreticians/computational scientists collaborate closely with empirical ecologists to determine what new data should be collected; (5) improving our knowledge of how to protect and/or restore ecosystems; (6) incorporating socio-economic effects into models of ecosystems; (7) improving our understanding of the role of deterministic and stochastic fluctuations; (8) studying the current state of biodiversity at the functional level, taxa level and genome level.

q-bio.PE↗

arXiv2020

The impact of advection on the stability of stripes on lattices near planar Turing instabilities

Striped patterns are known to bifurcate in reaction-diffusion systems with differential isotropic diffusions at a supercritical Turing instability. In this paper we study the impact of weak anisotropy by directional advection on the stability of stripes with respect to various lattice modes, and the role of quadratic terms therein. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using centre manifold reduction we derive a rigorous parameter expansion for the critical eigenvalues for lattice mode perturbations, specifically nearly square and nearly hexagonal ones. This provides detailed formulae for the loci of stability boundaries under the influence of the advection and quadratic terms. In particular, the well known destabilising effect of quadratic terms can be counterbalanced by advection, which leads to intriguing arrangements of stability boundaries. We illustrate these results numerically by a specific example. Finally, we show numerical computations of these stability boundaries in the extended Klausmeier model for vegetation patterns and show stripes bifurcate stably in the presence of sufficiently strong advection.

math.AP↗

arXiv2019

The impact of advection on large-wavelength stability of stripes near planar Turing instabilities

It is well known that for reaction-diffusion systems with differential isotropic diffusions, a Turing instability yields striped solutions. In this paper we study the impact of weak anisotropy by directional advection on such solutions, and the role of quadratic terms. We focus on the generic form of planar reaction-diffusion systems with two components near such a bifurcation. Using Lyapunov-Schmidt reduction and Floquet-Bloch decomposition we derive a rigorous parameter expansion for existence and stability against large wavelength perturbations. This provides detailed formulae for the loci of bifurcations and so-called Eckhaus as well as zigzag stability boundaries under the influence of the advection and quadratic terms. In particular, while destabilisation of the background state is through modes perpendicular to the advection (Squire-theorem), we show that stripes can bifurcate zigzag unstably. We illustrate these results numerically by an example. Finally, we show numerical computations of these stability boundaries in the extended Klausmeier model for vegetation patterns and show stripes bifurcate stably in the presence of advection.

math.AP↗

arXiv2013

Quasilinear parabolic reaction-diffusion systems: user's guide to well-posedness, spectra and stability of travelling waves

This paper is concerned with quasilinear parabolic reaction-diffusion-advection systems on extended domains. Frameworks for well-posedness in Hilbert spaces and spaces of continuous functions are presented, based on known results using maximal regularity. It is shown that spectra of travelling waves on the line are meaningfully given by the familiar tools for semilinear equations, such as dispersion relations, and basic connections of spectra to stability and instability are considered. In particular, a principle of linearized orbital instability for manifolds of equilibria is proven. Our goal is to provide easy access for applicants to these rigorous aspects. As a guiding example the Gray-Scott-Klausmeier model for vegetation-water interaction is considered in detail.

math.DS↗