One of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack... Show moreOne of the effects of climate change is the phenomenon of desertification, a process that occurs in semi-arid and arid areas and causes land degradation as well as vegetation loss. Due to the lack of resources, vegetation self-organizes to sustain itself by forming large-scale spatial patterns. In this thesis, the underlying mathematical structure of these observed vegetation patterns is studied using partial differential equations models. The vegetation patterns are analyzed using techniques from geometrical singular perturbation theory and numerical simulations. Additionally, novel multi-front patterns are constructed that arise within one of the models studied. This interdisciplinary research allows for cross-fertilization of both mathematics and ecology. Show less
Jaïbi O., Doelman A., Chirilus-Bruckner M., Meron E. 2020
In this paper we consider the 2-component reaction–diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics (Gilad... Show moreIn this paper we consider the 2-component reaction–diffusion model that was recently obtained by a systematic reduction of the 3-component Gilad et al. model for dryland ecosystem dynamics (Gilad et al., 2004). The nonlinear structure of this model is more involved than other more conceptual models, such as the extended Klausmeier model, and the analysis a priori is more complicated. However, the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations. Moreover, we find that the model exhibits a richness of analytically tractable patterns that exceeds that of Klausmeier-type models. Our study focuses on the 4-dimensional dynamical system associated with the reaction–diffusion model by considering traveling waves in 1 spatial dimension. We use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic/homoclinic/periodic orbits that ‘jump’ between (normally hyperbolic) slow manifolds, representing various kinds of localized vegetation patterns. The basic 1-front invasion patterns and 2-front spot/gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model. By exploiting the rich nonlinear structure of the model, we construct many multi-front patterns that are novel, both from the ecological and the mathematical point of view. In fact, we argue that these orbits/patterns are not specific for the model considered here, but will also occur in a much more general (singularly perturbed reaction–diffusion) setting. We conclude with a discussion of the ecological and mathematical implications of our findings. Show less
Vast, often populated, areas in dryland ecosystems face the dangers of desertification. Loosely speaking, desertification is the process in which a relatively dry region loses its vegetation -... Show moreVast, often populated, areas in dryland ecosystems face the dangers of desertification. Loosely speaking, desertification is the process in which a relatively dry region loses its vegetation - typically as an effect of climate change. As an important step in this process, the lack of resources forces the vegetation in these semi-arid areas to organise itself into large-scale spatial patterns. In this thesis, these patterns are studied using conceptual mathematical models, in which vegetation patterns present themselves as localised structures (for example pulses or fronts). These are analysed using mathematical techniques from (geometric singular) perturbation theory and via numerous numerical simulations. The study of these ecosystem models leads to new advances in both mathematics and ecology. Show less
In this paper we develop an extended center manifold reduction method: a methodology to analyze the formation and bifurcations of small-amplitude patterns in certain classes of multi-component,... Show moreIn this paper we develop an extended center manifold reduction method: a methodology to analyze the formation and bifurcations of small-amplitude patterns in certain classes of multi-component, singularly perturbed systems of partial differential equations. We specifically consider systems with a spatially homogeneous state whose stability spectrum partitions into eigenvalue groups with distinct asymptotic properties. One group of successive eigenvalues in the bifurcating group are widely interspaced, while the eigenvalues in the other are stable and cluster asymptotically close to the origin along the stable semi-axis. The classical center manifold reduction provides a rigorous framework to analyze destabilizations of the trivial state, as long as there is a spectral gap of sufficient width. When the bifurcating eigenvalue becomes commensurate to the stable eigenvalues clustering close to the origin, the center manifold reduction breaks down. Moreover, it cannot capture subsequent bifurcations of the bifurcating pattern. Through our methodology, we formally derive expressions for low-dimensional manifolds exponentially attracting the full flow for parameter combinations that go beyond those allowed for the (classical) center manifold reduction, i.e. to cases in which the spectral gap condition no longer can be satisfied. Our method also provides an explicit description of the flow on these manifolds and thus provides an analytical tool to study subsequent bifurcations. Our analysis centers around primary bifurcations of transcritical type–that can be either of co-dimension 1 or 2–in two- and three-component PDE systems. We employ our method to study bifurcation scenarios of small-amplitude patterns and the possible appearance of low-dimensional spatio-temporal chaos. We also exemplify our analysis by a number of characteristic reaction–diffusion systems with disparate diffusivities. Show less
In this thesis, the existence and stability of pulse solutions in two-component, singularly perturbed reaction-diffusion systems is analysed using dynamical systems techniques. New phenomena in... Show moreIn this thesis, the existence and stability of pulse solutions in two-component, singularly perturbed reaction-diffusion systems is analysed using dynamical systems techniques. New phenomena in very general types of systems emerge when geometrical techniques are applied. Show less
In this thesis I considered the dynamics of self-propelling particles (SPP). Flocking of living organisms like birds, fishes, ants, bacteria etc. is an area where the theory of the collective... Show moreIn this thesis I considered the dynamics of self-propelling particles (SPP). Flocking of living organisms like birds, fishes, ants, bacteria etc. is an area where the theory of the collective behaviour of SPP can be applied. One can often see how these animals develop coherent motion, amazing the observer by the diversity of its forms and shapes. In this thesis a hydrodynamic model with so-called kinematic constraints, which are imposed on the orientations of the velocities of the particles, is proposed. The tendency of the particles to adjust their velocities to the ones of the neighbours leads to the emergence of a coherent motion. In our model two types of stationary flows are obtained: linear and vortical hydrodynamic flows. A remarkable property of the vortical flow is that it has finite flocking behaviour, where the density and the velocity fields are coupled. From the physical point of view these flows are of interest because of their realization in nature. The stability properties of the stationary flows are determined. Further a hydrodynamic model is derived from the discrete description using the averaging procedure. The connection between the discrete and continuous approaches is analysed. Show less
