arXiv subjects

Nicolas Verschueren

Publications and source records attributed to Nicolas Verschueren.

4 recordsLinked to original sources

arXiv2021

Tropical Tree Cover in a Heterogeneous Environment: a Reaction-diffusion Model

Observed bimodal tree cover distributions at particular environmental conditions and theoretical models indicate that some areas in the tropics can be in either of the alternative stable vegetation states forest or savanna. However, when including spatial interaction in nonspatial differential equation models of a bistable quantity, only the state with the lowest potential energy remains stable. Our recent reaction-diffusion model of Amazonian tree cover confirmed this and was able to reproduce the observed spatial distribution of forest versus savanna satisfactorily when forced by heterogeneous environmental and anthropogenic variables, even though bistability was underestimated. These conclusions were solely based on simulation results. Here, we perform an analytical and numerical analysis of the model. We derive the Maxwell point (MP) of the homogeneous reaction-diffusion equation without savanna trees as a function of rainfall and human impact and show that the front between forest and nonforest settles at this point as long as savanna tree cover near the front remains sufficiently low. For parameters resulting in higher savanna tree cover near the front, we also find irregular forest-savanna cycles and woodland-savanna bistability, which can both explain the remaining observed bimodality.

q-bio.PE↗

arXiv2021

Localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk

Axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation posed on a disk with Neumann boundary conditions are studied via numerical continuation and bifurcation analysis. Axisymmetric localized solutions in the form of spots and rings known from earlier studies persist and snake in the usual fashion until they begin to interact with the boundary. Depending on parameters, including the disk radius, these states may or may not connect to the branch of domain-filling target states. Secondary instabilities of localized axisymmetric states may create multi-arm localized structures that grow and interact with the boundary before broadening into domain filling states. High azimuthal wavenumber wall states referred to as daisy states are also found. Secondary bifurcations from these states include localized daisies, i.e., wall states localized in both radius and angle. Depending on parameters, these states may snake much as in the one-dimensional Swift-Hohenberg equation, or invade the interior of the domain, yielding states referred to as worms, or domain-filling stripes.

nlin.PS↗

arXiv2020

Dissecting the snake: the transition from localized patterns to isolated spikes in pattern formation systems

An investigation is undertaken of coupled reaction-diffusion systems in one spatial dimension that are able to support, in different regions of their parameter space, either an isolated spike solution, or stable localized patterns with an arbitrary number of peaks. The distinction between the two cases is drawn through the behavior of the far field, where there is either an oscillatory or a monotonic decay. Several examples are studied, including the Lugiato-Lefever model, a generalized Schakenberg system that arises in cellular-level morphogensis and a continuum model of urban crime spread. In each, it is found that localized patterns connected via a so-called homoclinic snaking curve in parameter space transition into a single spike solution as a second parameter is varied, via a change in topology of the snake into a series of disconnected branches. In each case, the transition is caused by a so-called Belyakov-Devaney transition between complex and real spatial eigenvalues of the fair field of the primary pulse. A codimension-two problem is studied in detail where a non-transverse homoclinic orbit undergoes this transition. A Shilnikov-style analysis is undertaken which reveals the asymptotics of how the infinite family of folds of multi-pulse orbits are all destroyed at the same parameter value. The results are shown to be consistent with numerical experiments on two of the examples.

nlin.PS↗

arXiv2016

Effects of source and loss terms on the wave-pinning description of cell polarisation

A system of two Schnakenberg-like reaction-diffusion equations is investigated analytically and numerically. The system has previously been used as a minimal model for concentrations of GTPases involved in the process of cell polarisation. Source and loss terms are added, breaking the mass conservation, which was shown previously to be responsible for the generation of stable fronts via a so-called wave-pinning mechanism. The extended model gives rise to a unique homogeneous equilibrium in the parameter region of interest, which loses stability via a pattern formation, or Turing bifurcation. The bistable character of the reaction terms ensures that this bifurcation is subcrtical for sufficiently small values of the driving parameter multiplying the nonlinear kinetics. This subcriticality leads to the onset of a multitude of localised solutions, through the homoclinic snaking mechanism. As the driving parameter is further decreased, the multitude of solutions transforms into a single pulse through a Belyakov-Devaney transition in which there is the loss of a precursive pattern. An asymptotic analysis is used to probe the conservative limit in which the source and loss terms vanish. Matched asymptotic analysis shows that on an infinite domain the pulse solution transitions into a pair of fronts, with an additional weak quadratic core and exponential tails. On a finite domain, the core and tails disappear, leading to the mere wave-pinning front and its mirror image.

nlin.PS↗