In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances. In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.

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Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

A horizontal axis wind turbine blade in steady rotation endures cyclic transverse loading due to wind shear, tower shadowing and gravity, and a cyclic gravitational axial loading at the same fundamental frequency. These direct and parametric excitations motivate the consideration of a forced Mathieu equation with cubic nonlinearity to model its dynamic behavior. This equation is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. Order-two superharmonic resonance persists for the linear system. While the order-two subharmonic response level is dependent on the ratio of parametric excitation and damping, nonlinearity is essential for the order-two subharmonic resonance. Order-three resonances are present in the system as well and they are similar to those of the Duffing equation.

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Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades

In recent years, methods have been developed to study the existence, stability and bifurcations of pulses in singularly perturbed reaction–diffusion equations in one space dimension, in the context of a number of explicit model problems, such as the Gray–Scott and the Gierer–Meinhardt equations. Although these methods are in principle of a general nature, their applicability a priori relies on the characteristics of these models. For instance, the slow reduced spatial problem is linear in the models considered in the literature. Moreover, the nonlinearities in the fast reduced spatial problem are of a very specific, polynomial, nature. These properties are crucially used, especially in the stability and bifurcation analysis. In this paper, we present an explicit theory for pulses in two-component singularly perturbed reaction–diffusion equations that significantly extends and generalizes existing methods.

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An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations

Existence of pulses in excitable media with nonlocal coupling

The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures is presented. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm–Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.

https://iopscience.iop.org/article/10.1088/0951-7715/28/7/2211/pdf

Breathing pulses in singularly perturbed reaction-diffusion systems

In this paper, we study in detail the existence and stability of localized pulses in a Gierer-Meinhardt equation with an additional "slow" nonlinearity. This system is an explicit example of a general class of singularly perturbed, two component reaction-diffusion equations that goes significantly beyond well-studied model systems such as Gray-Scott and Gierer-Meinhardt. We investigate the existence of these pulses using the methods of geometric singular perturbation theory. The additional nonlinearity has a profound impact on both the stability analysis of the pulse—compared to Gray- Scott/Gierer-Meinhardt-type models a distinct extension of the Evans function approach has to be developed—and the stability properties of the pulse: several (de)stabilization mechanisms turn out to be possible. Moreover, it is shown by numerical simulations that, unlike the Gray-Scott/Gierer- Meinhardt-type models, the pulse solutions of the model exhibit a rich and complex behavior near the Hopf bifurcations.

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Pulses in a Gierer--Meinhardt Equation with a Slow Nonlinearity

Quasiperiodic phenomena in the Van der Pol-Mathieu equation

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 orb...

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Frits Veerman

Papers

An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations

Breathing pulses in singularly perturbed reaction-diffusion systems

Pulses in a Gierer--Meinhardt Equation with a Slow Nonlinearity

Quasiperiodic phenomena in the Van der Pol-Mathieu equation

A Predator--2 Prey Fast--Slow Dynamical System for Rapid Predator Evolution