8,425 research outputs found
Homogenization techniques for population dynamics in strongly heterogeneous landscapes
An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features
A fractional kinetic process describing the intermediate time behaviour of cellular flows
This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result for the associated advection-diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion
Edgeworth expansions for slow-fast systems with finite time scale separation
We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter ε. The corrections involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in ε and provide explicit expressions for the first two orders. At a formal level, the expressions derived are valid in the case when the fast dynamics is stochastic as well as when the fast dynamics is entirely deterministic. We corroborate our analytical results with numerical simulations and show that our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation
Interstellar Turbulence II: Implications and Effects
Interstellar turbulence has implications for the dispersal and mixing of the
elements, cloud chemistry, cosmic ray scattering, and radio wave propagation
through the ionized medium. This review discusses the observations and theory
of these effects. Metallicity fluctuations are summarized, and the theory of
turbulent transport of passive tracers is reviewed. Modeling methods, turbulent
concentration of dust grains, and the turbulent washout of radial abundance
gradients are discussed. Interstellar chemistry is affected by turbulent
transport of various species between environments with different physical
properties and by turbulent heating in shocks, vortical dissipation regions,
and local regions of enhanced ambipolar diffusion. Cosmic rays are scattered
and accelerated in turbulent magnetic waves and shocks, and they generate
turbulence on the scale of their gyroradii. Radio wave scintillation is an
important diagnostic for small scale turbulence in the ionized medium, giving
information about the power spectrum and amplitude of fluctuations. The theory
of diffraction and refraction is reviewed, as are the main observations and
scintillation regions.Comment: 46 pages, 2 figures, submitted to Annual Reviews of Astronomy and
Astrophysic
The cardiac bidomain model and homogenization
We provide a rather simple proof of a homogenization result for the bidomain
model of cardiac electrophysiology. Departing from a microscopic cellular
model, we apply the theory of two-scale convergence to derive the bidomain
model. To allow for some relevant nonlinear membrane models, we make essential
use of the boundary unfolding operator. There are several complications
preventing the application of standard homogenization results, including the
degenerate temporal structure of the bidomain equations and a nonlinear dynamic
boundary condition on an oscillating surface.Comment: To appear in Networks and Heterogeneous Media, Special Issue on
Mathematical Methods for Systems Biolog
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