2,311 research outputs found
Continuation for thin film hydrodynamics and related scalar problems
This chapter illustrates how to apply continuation techniques in the analysis
of a particular class of nonlinear kinetic equations that describe the time
evolution through transport equations for a single scalar field like a
densities or interface profiles of various types. We first systematically
introduce these equations as gradient dynamics combining mass-conserving and
nonmass-conserving fluxes followed by a discussion of nonvariational amendmends
and a brief introduction to their analysis by numerical continuation. The
approach is first applied to a number of common examples of variational
equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including
certain thin-film equations for partially wetting liquids on homogeneous and
heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal
equations. Second we consider nonvariational examples as the
Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard
equations and thin-film equations describing stationary sliding drops and a
transversal front instability in a dip-coating. Through the different examples
we illustrate how to employ the numerical tools provided by the packages
auto07p and pde2path to determine steady, stationary and time-periodic
solutions in one and two dimensions and the resulting bifurcation diagrams. The
incorporation of boundary conditions and integral side conditions is also
discussed as well as problem-specific implementation issues
Formation of localized states in dryland vegetation: Bifurcation structure and stability
In this paper, we study theoretically the emergence of localized states of
vegetation close to the onset of desertification. These states are formed
through the locking of vegetation fronts, connecting a uniform vegetation state
with a bare soil state, which occurs nearby the Maxwell point of the system. To
study these structures we consider a universal model of vegetation dynamics in
drylands, which has been obtained as the normal form for different vegetation
models. Close to the Maxwell point localized gaps and spots of vegetation exist
and undergo collapsed snaking. The presence of gaps strongly suggest that the
ecosystem may undergo a recovering process. In contrast, the presence of spots
may indicate that the ecosystem is close to desertification
Integrated Environmental Modelling Framework for Cumulative Effects Assessment
Global warming and population growth have resulted in an increase in the intensity of natural and anthropogenic stressors. Investigating the complex nature of environmental problems requires the integration of different environmental processes across major components of the environment, including water, climate, ecology, air, and land. Cumulative effects assessment (CEA) not only includes analyzing and modeling environmental changes, but also supports planning alternatives that promote environmental monitoring and management. Disjointed and narrowly focused environmental management approaches have proved dissatisfactory. The adoption of integrated modelling approaches has sparked interests in the development of frameworks which may be used to investigate the processes of individual environmental component and the ways they interact with each other. Integrated modelling systems and frameworks are often the only way to take into account the important environmental processes and interactions, relevant spatial and temporal scales, and feedback mechanisms of complex systems for CEA. This book examines the ways in which interactions and relationships between environmental components are understood, paying special attention to climate, land, water quantity and quality, and both anthropogenic and natural stressors. It reviews modelling approaches for each component and reviews existing integrated modelling systems for CEA. Finally, it proposes an integrated modelling framework and provides perspectives on future research avenues for cumulative effects assessment
Tropical tree cover in a heterogeneous environment: a reaction-diffusion model
This is the final version. Available from Public Library of Science via the DOI in this record.Observed bimodal tree cover distribution sat 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 for one set of parameters. Here, we perform ananalytical 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.EPSR
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.Comment: 28 pages, 6 figures, 2 tables, supplementary info include
On the development and analysis of coupled surface-subsurface models of catchments. Part 2. A three-dimensional benchmark model and its properties
The objective of this three-part work is the formulation and rigorous
analysis of a number of reduced mathematical models that are nevertheless
capable of describing the hydrology at the scale of a river basin (i.e.
catchment). Coupled effects of surface and subsurface flows are considered.
In this second part, we construct a benchmark catchment scenario and
investigate the effects of parameters within their typical ranges. Previous
research on coupled surface-subsurface models have focused on numerical
simulations of site-specific catchments; here our focus is broader and
emphasises the study of general solutions to the mathematical models, and their
dependencies on dimensionless parameters. This study provides a foundation
based on the examination of a geometrically simple three-dimensional benchmark
scenario. We develop a nondimensional coupled surface-subsurface model, and
extract the key dimensionless parameters. We then apply asymptotic methods in
order to discuss some potential simplifications, including the reduction of the
geometry to a two-dimensional form, where the principal groundwater and
overland flows occur in the hillslope direction. Numerical solutions
demonstrate the effects of model parameters and provide guidance on the
validity of the dimensional reductions
- …