114 research outputs found
Adiabatic stability under semi-strong interactions: The weakly damped regime
We rigorously derive multi-pulse interaction laws for the semi-strong
interactions in a family of singularly-perturbed and weakly-damped
reaction-diffusion systems in one space dimension. Most significantly, we show
the existence of a manifold of quasi-steady N-pulse solutions and identify a
"normal-hyperbolicity" condition which balances the asymptotic weakness of the
linear damping against the algebraic evolution rate of the multi-pulses. Our
main result is the adiabatic stability of the manifolds subject to this normal
hyperbolicity condition. More specifically, the spectrum of the linearization
about a fixed N-pulse configuration contains essential spectrum that is
asymptotically close to the origin as well as semi-strong eigenvalues which
move at leading order as the pulse positions evolve. We characterize the
semi-strong eigenvalues in terms of the spectrum of an explicit N by N matrix,
and rigorously bound the error between the N-pulse manifold and the evolution
of the full system, in a polynomially weighted space, so long as the
semi-strong spectrum remains strictly in the left-half complex plane, and the
essential spectrum is not too close to the origin
Large amplitude radially symmetric spots and gaps in a dryland ecosystem model
We construct far-from-onset radially symmetric spot and gap solutions in a
two-component dryland ecosystem model of vegetation pattern formation on flat
terrain, using spatial dynamics and geometric singular perturbation theory. We
draw connections between the geometry of the spot and gap solutions with that
of traveling and stationary front solutions in the same model. In particular,
we demonstrate the instability of spots of large radius by deriving an
asymptotic relationship between a critical eigenvalue associated with the spot
and a coefficient which encodes the sideband instability of a nearby stationary
front. Furthermore, we demonstrate that spots are unstable to a range of
perturbations of intermediate wavelength in the angular direction, provided the
spot radius is not too small. Our results are accompanied by numerical
simulations and spectral computations
Modelling honey bee colonies in winter using a Keller-Segel model with a sign-changing chemotactic coefficient
Thermoregulation in honey bee colonies during winter is thought to be
self-organised. We added mortality of individual honey bees to an existing
model of thermoregulation to account for elevated losses of bees that are
reported worldwide. The aim of analysis is to obtain a better fundamental
understanding of the consequences of individual mortality during winter. This
model resembles the well-known Keller-Segel model. In contrast to the often
studied Keller-Segel models, our model includes a chemotactic coefficient of
which the sign can change as honey bees have a preferred temperature: when the
local temperature is too low, they move towards higher temperatures, whereas
the opposite is true for too high temperatures. Our study shows that we can
distinguish two states of the colony: one in which the colony size is above a
certain critical number of bees in which the bees can keep the core temperature
of the colony above the threshold temperature, and one in which the core
temperature drops below the critical threshold and the mortality of the bees
increases dramatically, leading to a sudden death of the colony. This model
behaviour may explain the globally observed honey bee colony losses during
winter.Comment: 20 pages, 12 figure
Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model
We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the strongly localized activator pulses interact through the weakly localized inhibitor, the interaction is not tail-tail as in the weak interaction limit, and the pulses change amplitude and even stability as their separation distance evolves on algebraically slow time scales. The RG approach employed here validates the interaction laws of quasi-steady pulse patterns obtained formally in the literature, and establishes that the pulse dynamics reduce to a closed system of ordinary differential equations for the activator pulse locations. Moreover, we fully justify the reduction to the nonlocal eigenvalue problem (NLEP) showing the large difference between the quasi-steady NLEP operator and the operator arising from linearization about the pulse is controlled by the resolven
A geometric construction of traveling waves in a bioremediation.
Bioremediation is a promising technique for cleaning contaminated soil. We study an idealized bioremediation model involving a substrate (contaminant to be removed), electron acceptor (added nutrient), and microorganisms in a one-dimensional soil column. Using geometric singular perturbation theory, we construct traveling waves (TW) corresponding to motion of a biologically active zone, in which the microorganisms consume both substrate and acceptor. For certain values of the parameters, the traveling waves exist on a three-dimensional slow manifold within the five-dimensional phase space. We prove persistence of the slow manifold under perturbation by controlling the nonlinearity via a change of coordinates, and we construct the wave in the transverse intersection of appropriate stable and unstable manifolds in this slow manifold. We study how the TW depends on the half saturation constants and other parameters and investigate numerically a bifurcation in which the TW loses stability to a periodic wav
- …