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- Matt Holzer, Arjen Doelman, Tasso J. Kaper
- J. Nonlinear Science
- 2013

In this article, we analyze traveling waves in a reaction–diffusionmechanics (RDM) system. The system consists of a modified FitzHugh–Nagumo equation, also known as the Aliev–Panfilov model, coupled bidirectionally with an elasticity equation for a deformable medium. In one direction, contraction and expansion of the elastic medium decreases and increases,… (More)

- T.W.N. Haine, Hai-lan Zhang, Darryn W. Waugh, Matt Holzer
- 2008

In a diffusive geophysical flow, there is not a single timescale or unique pathway for passive scalar transport from the reservoir’s surface into the interior because of irreversible diffusive mixing processes. Instead, there is a range of pathways and hence a transit-time distribution (TTD) since last surface contact. We explore the issues that arise when… (More)

We investigate the problem of attractor reconstruction from interspike times produced by an integrate-and-fire model of neuronal activity. Suzuki et. al. [14] found that the reconstruction of the Rössler attractor is incomplete if the integrate-and-fire model is used. We explain this failure using two observations. One is that the attractor reconstruction… (More)

- Tomás Gedeon, Matt Holzer
- Journal of mathematical biology
- 2004

It is known [8, 11, 16, 26] that phase locking can entrain frequency information when the leaky integrate-and-fire (IF) model of a neuron is forced by a periodic function. We show that this is still the case when the IF model is made more biologically realistic. We incorporate into our model spike dependent threshold modulation and refractory periods.… (More)

- Matt Holzer, Arnd Scheel
- 2012

We study invasion speeds in the Lotka-Volterra competition model when the rate of diffusion of one species is small. Our main result is the construction of the selected front and a rigorous asymptotic approximation of its propagation speed, valid to second order. We use techniques from geometric singular perturbation theory and geometric desingularization.… (More)

- Matt Holzer, Arnd Scheel
- J. Nonlinear Science
- 2014

This article is concerned with pointwise growth and spreading speeds in systems of parabolic partial differential equations. Several criteria exist for quantifying pointwise growth rates. These include the location in the complex plane of singularities of the pointwise Green’s function and pinched double roots of the dispersion relation. The primary aim of… (More)

The purpose of this paper is to explore spatio-temporal pattern formation via invasion fronts in the one and two dimensional Keller-Segel chemotaxis model. In the one-dimensional case, simulations show that solutions that begin near an unstable equilibrium evolve into periodic patterns. These in turn evolve into new patterns through a process known as… (More)

- Matt Holzer, Arnd Scheel
- SIAM J. Math. Analysis
- 2014

We study wavespeed selection in a staged invasion process. That is, we study a model in which an unstable homogeneous state is replaced via an invading front with a secondary state. This secondary state is also unstable and, in turn, replaced by a stable homogeneous state via a secondary invasion front. In particular, we are interested in the selected… (More)

- Matt Holzer
- 2015

This article is concerned with the rigorous validation of anomalous spreading speeds in a system of coupled Fisher-KPP equations of cooperative type. Anomalous spreading refers to a scenario wherein the coupling of two equations leads to faster spreading speeds in one of the components. The existence of these spreading speeds can be predicted from the… (More)

We identify a new mechanism for propagation into unstable states in spatially extended systems, that is based on resonant interaction in the leading edge of invasion fronts. Such resonant invasion speeds can be determined solely based on the complex linear dispersion relation at the unstable equilibrium, but rely on the presence of a nonlinear term that… (More)