arXiv subjects

Frits Veerman

Publications and source records attributed to Frits Veerman.

7 recordsLinked to original sources

arXiv2026

Constructing far-from-equilibrium patterns in a cross-diffusion vegetation-autotoxicity model

Using geometric singular perturbation theory, we construct stationary, periodic, front-type far-from-equilibrium patterns in a cross-diffusion model for plant growth under the influence of toxicity. We show how existing techniques for the analysis of far-from-equilibrium patterns in one spatial dimension can be extended to include cross-diffusion terms and prove the existence of a one-parameter family of these patterns. For a general cross-diffusion model class, we show when front-type patterns may appear depending on the shape of the critical manifold, which is determined by the specific properties of the reaction terms.

math.DS↗

arXiv2026

How inertia affects autotoxicity-mediated vegetation dynamics: from close-to to far-from-equilibrium patterns

In this work, the influence of inertial effects on the formation and evolution of vegetation patterns on sloped arid terrains is investigated from the onset of instability to far-from-equilibrium. Analyses are carried out in a hyperbolic extension of the one-dimensional Klausmeier model, where autotoxicity effects are also taken into account. As the system moves away from the wave bifurcation threshold, two classes of solutions arise: small-amplitude periodic migrating bands near onset and large-amplitude travelling pulses in far-from-equilibrium conditions. For the first class, results of LSA reveal that inertia has a twofold role at onset: it acts as a destabilising mechanism, thereby enlarging the parameter region in which uphill migrating vegetation bands can emerge, and it reduces the pattern migration speed. Its role also manifests itself close to onset, as proved by the Stuart-Landau equation for the pattern amplitude deduced via multiple-scale WNA. Indeed, it is shown that inertial effects may reverse the dynamical regime, from supercritical to subcritical, thus leading to hysteresis. For the second class of solutions, the travelling vegetation pulses are first captured via numerical simulations and then investigated via Geometric Singular Perturbation Theory (GSPT). In far-from-equilibrium conditions, inertia is shown to increase pulse speed while preserving the intrinsic multiscale structure of the solution, in full agreement with the numerical findings. Overall, the proposed combined analytical-numerical investigations have depicted several ecological scenarios as a function of the distance from the instability threshold, elucidating that inertia does not exclusively act as a time lag.

nlin.PS↗

arXiv2024

Far-from-equilibrium travelling pulses in sloped semi-arid environments driven by autotoxicity effects

In this work, an extension of the 1D Klausmeier model that accounts for the toxicity compounds is considered and the occurrence of travelling stripes is investigated. Numerical simulations are firstly conducted to capture the qualitative behaviours of the pulse-type solutions and, then, geometric singular perturbation theory is used to prove the existence of such travelling pulses by constructing the corresponding homoclinic orbits in the associated 4-dimensional system. A scaling analysis on the investigated model is performed to identify the asymptotic scaling regime in which travelling pulses can be constructed. Biological observations are extracted from the analytical results and the role of autotoxicity in travelling patterns is emphasized. Finally, the analytically constructed solutions are compared with the numerical ones, leading to a good agreement that confirms the validity of the conducted analysis. Numerical investigations are also carried out in order to gain additional information on vegetation dynamics.

math.DS↗

arXiv2023

Travelling pulses on three spatial scales in a Klausmeier-type vegetation-autotoxicity model

Reaction-diffusion models describing interactions between vegetation and water reveal the emergence of several types of patterns and travelling wave solutions corresponding to structures observed in real-life. Increasing their accuracy by also considering the ecological factor known as autotoxicity has lead to more involved models supporting the existence of complex dynamic patterns. In this work, we include an additional carrying capacity for the biomass in a Klausmeier-type vegetation-water-autotoxicity model, which induces the presence of two asymptotically small parameters: $\varepsilon$, representing the usual scale separation in vegetation-water models, and $\delta$, directly linked to autotoxicity. We construct three separate types of homoclinic travelling pulse solutions based on two different scaling regimes involving $\varepsilon$ and $\delta$, with and without a so-called superslow plateau. The relative ordering of the small parameters significantly influences the phase space geometry underlying the construction of the pulse solutions. We complement the analysis by numerical continuation of the constructed pulse solutions, and demonstrate their existence (and stability) by direct numerical simulation of the full PDE model.

math.DS↗

arXiv2023

Controlling pulse stability in singularly perturbed reaction-diffusion systems

The aim of this paper is to investigate the use of Pyragas control on the stability of stationary, localised coherent structures in a general class of two-component, singularly perturbed, reaction-diffusion systems. We use noninvasive Pyragas-like proportional feedback control to stabilise a singular pulse solution to a two-component, singularly perturbed reaction-diffusion system. We show that in a significant region of parameter space, the control can be adjusted to stabilise an otherwise unstable pulse.

math.DS↗

arXiv2019

Robust Stability of Multicomponent Membranes: the Role of Glycolipids

We present the multicomponent functionalized free energies that characterize the low-energy packings of amphiphilic molecules within a membrane through a correspondence to connecting orbits within a reduced dynamical system. To each connecting orbits we associate a manifold of low energy membrane-type configurations parameterized by a large class of admissible interfaces. The normal coercivity of the manifolds is established through criteria depending solely on the structure of the associated connecting orbit. We present a class of examples that arise naturally from geometric singular perturbation techniques, focusing on a model that characterizes the stabilizing role of cholesterol-like glycolipids within phospholipid membranes.

math.AP↗

arXiv2016

A predator-2 prey fast-slow dynamical system for rapid predator evolution

We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast--3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predator's diet choice. The two extremes at which the predator's diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast--slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.

math.DS↗