arXiv subjects

Matt Holzer

Publications and source records attributed to Matt Holzer.

At least 19 recordsLinked to original sources

arXiv2026

Predicting the occurrence of Braess paradox in the synchronization threshold of coupled oscillator systems

We study Braess-type paradoxes in coupled oscillator networks where the addition of an edge to the network leads to an increase in the critical coupling required for the existence of a stable phase-locked synchronized solution. By introducing a continuous edge-weight parameter and constructing an augmented system that allows application of the Implicit Function Theorem, we derive an explicit first-order formula for the sensitivity of the critical coupling, $K'_c(0)$, to small changes in the network structure. This formula reveals that an edge is locally Braess if the ordering of the oscillators being coupled is opposite the ordering of their components in the critical eigenvector at the saddle-node bifurcation, providing a simple criterion for Braess paradox to occur. We then employ this local approximation as a predictor for the occurrence of Braess paradox when the edge is fully incorporated in the network. This prediction is tested numerically in several classes of random networks, demonstrating both high predictive fidelity and the nonlinear limitations of the local approximation. We also examine several explicit examples, highlighting possible motifs by which Braess-type paradoxes occur. Together, these results establish a predictive framework for understanding how incremental changes in network topology can produce counterintuitive synchronization outcomes.

math.DS↗

arXiv2026

Rate-Induced Tipping in a Non-Uniformly Moving Habitat and Determination of the Critical Rate

A habitat that is moving due to environmental change may result in tipping to extinction if the rate at which it moves is too great. We use a scalar reaction-diffusion equation with a non-autonomous reaction term, representing a spatially localized habitat moving from one asymptotic location to another, as a context for studying this phenomenon. The movement is characterized by displacement $d$ and rate parameter $r$. The system admits three steady states in both asymptotic habitat locations: a stable extinction state $u_0^*=0$, an unstable pulse (so-called edge state) $u_1^*(x)>0$, which gives rise to the Allee effect, and a stable pulse (populated base state) $u_2^*(x)>u_1^*(x)$, which corresponds to a thriving population at its carrying capacity. Numerical simulations for a specific model identify a critical displacement $d^*$ and, for $d > d^*$, demonstrate the existence of a \textit{critical rate} $r_c(d)$ at which rate-induced tipping occurs: for $r> r_c$ an initially thriving population becomes extinct due to habitat movement being too rapid. We provide analytical results for two limiting cases. For $r\ll 1$, solutions track the moving base state with error $O(r)$. For $r\gg 1$, solutions converge to the extinction state provided $d$ is sufficiently large. For $d$ too small, no tipping occurs regardless of $r$. Numerical simulations complement and extend these analytical results. At the critical rate $r=r_c(d)$, we identify a pulse-to-pulse heteroclinic connection between the base state at the past asymptotic location and the edge state at the future asymptotic location of the habitat. We also establish the uniqueness of this critical rate and non-degeneracy of the heteroclinic connection as $r$ varies.

math.AP↗

arXiv2025

Selection mechanisms in front invasion

We review progress on questions related to front propagation into unstable states and point out open problems in the area. We strive to highlight different theoretical perspectives and challenges while also addressing more practical questions with examples and guides to computational methods. Throughout we take a dynamical systems point of view that focuses on the ability of invasion processes to act as a selection mechanism in complex systems.

math.AP↗

arXiv2025

Pulled fronts are not (just) pulled

Front propagation into unstable states is often determined by the linearization, that is, propagation speeds agree with predictions from the linearized equation at the unstable state. The leading edge behavior is then a Gaussian tail propagating with the linear spreading speed. Fronts following this leading edge are commonly referred to as pulled fronts, alluding to the idea that they are ``pulled'' by this leading-edge Gaussian tail. We describe here a class of examples that exhibits how these leading-order effects do not completely describe the dynamics in the wake of the front. In fact, leading edge behavior predicts at most two possible invasion scenarios, associated with positive and negative amplitudes of the Gaussian tail, but our examples exhibit three or more invasion fronts with different states in the wake. The resulting invasion process therefore leaves behind a state that is not solely determined by the leading edge, and thus not just pulled by the Gaussian tail.

math.AP↗

arXiv2024

Persistence of steady-states for dynamical systems on large networks

The goal of this work is to identify steady-state solutions to dynamical systems defined on large, random families of networks. We do so by passing to a continuum limit where the adjacency matrix is replaced by a non-local operator with kernel called a graphon. This graphon equation is often more amenable to analysis and provides a single equation to study instead of the infinitely many variations of networks that lead to the limit. Our work establishes a rigorous connection between steady-states of the continuum and network systems. Precisely, we show that if the graphon equation has a steady-state solution whose linearization is invertible, there exists related steady-state solutions to the finite-dimensional networked dynamical system over all sufficiently large graphs converging to the graphon. The proof involves setting up a Newton--Kantorovich type iteration scheme which is shown to be a contraction on a suitable metric space. Interestingly, we show that the first iterate of our defined operator in general fails to be a contraction mapping, but the second iterate is proven to contract on the space. We extend our results to show that linear stability properties further carry over from the graphon system to the graph dynamical system. Our results are applied to twisted states in a Kuramoto model of coupled oscillators, steady-states in a model of neuronal network activity, and a Lotka--Volterra model of ecological interaction.

math.DS↗

arXiv2023

Pushed and pulled fronts in a logistic Keller-Segel model with chemorepulsion

We analyze spatial spreading in a population model with logistic growth and chemorepulsion. In a parameter range of short-range chemo-diffusion, we use geometric singular perturbation theory and functional-analytic farfield-core decompositions to identify spreading speeds with marginally stable front profiles. In particular, we identify a sharp boundary between between linearly determined, pulled propagation, and nonlinearly determined, pushed propagation, induced by the chemorepulsion. The results are motivated by recent work on singular limits in this regime using PDE methods.

math.AP↗

arXiv2022

Pushed fronts in a Fisher-KPP-Burgers system using geometric desingularization

We study traveling fronts in a system of one dimensional reaction-diffusion-advection equations motivated by problems in reactive flows. In the limit as a parameter tends to infinity, we construct the approximate front profile and determine the leading order expansion for the selected wavespeed. Such fronts are often constructed as transverse intersections of stable and unstable manifolds of the traveling wave differential equation. However, a re-scaling of the dependent variable leads to a lack of hyperbolicity for one of the end states making the definition of one such manifold unclear. We use geometric blow-up techniques to recover hyperbolicity and following an analysis of the blown up vector field are able to show the existence of a traveling front with a leading order expansion of its speed.

math.AP↗

arXiv2022

Pushed-to-pulled front transitions: continuation, speed scalings, and hidden monotonicity

We analyze the transition between pulled and pushed fronts both analytically and numerically from a model-independent perspective. Based on minimal conceptual assumptions, we show that pushed fronts bifurcate from a branch of pulled fronts with an effective speed correction that scales quadratically in the bifurcation parameter. Strikingly, we find that in this general context without assumptions on comparison principles, the pulled front loses stability and gives way to a pushed front when monotonicity in the leading edge is lost. Our methods rely on far-field core decompositions that identify explicitly asymptotics in the leading edge of the front. We show how the theoretical construction can be directly implemented to yield effective algorithms that determine spreading speeds and bifurcation points with exponentially small error in the domain size. Example applications considered here include an extended Fisher-KPP equation, a Fisher-Burgers equation, negative taxis in combination with logistic population growth, an autocatalytic reaction, and a Lotka-Volterra model.

math.AP↗

arXiv2021

Pattern Formation in Random Networks Using Graphons

We study Turing bifurcations on one-dimensional random ring networks where the probability of a connection between two nodes depends on the distance between the two nodes. Our approach uses the theory of graphons to approximate the graph Laplacian in the limit as the number of nodes tends to infinity by a nonlocal operator -- the graphon Laplacian. For the ring networks considered here, we employ center manifold theory to characterize Turing bifurcations in the continuum limit in a manner similar to the classical partial differential equation case and classify these bifurcations as sub/super/trans-critical. We derive estimates that relate the eigenvalues and eigenvectors of the finite graph Laplacian to those of the graphon Laplacian. We are then able to show that, for a sufficiently large realization of the network, with high probability the bifurcations that occur in the finite graph are well approximated by those in the graphon limit. The number of nodes required depends on the spectral gap between the critical eigenvalue and the remaining ones, with the smaller this gap the more nodes that are required to guarantee that the graphon and graph bifurcations are similar. We demonstrate that if this condition is not satisfied then the bifurcations that occur in the finite network can differ significantly from those in the graphon limit.

math.DS↗

arXiv2021

Epidemic spreading on complex networks as front propagation into an unstable state

We study epidemic arrival times in meta-population disease models through the lens of front propagation into unstable states. We demonstrate that several features of invasion fronts in the PDE context are also relevant to the network case. We show that the susceptible-infected-recovered model on a network is linearly determined in the sense that the arrival times in the nonlinear system are approximated by the arrival times of the instability in the system linearized near the disease free state. Arrival time predictions are extended to an susceptible-exposed-infected-recovered model. We then study a recent model of social epidemics where high order interactions of individuals lead to faster invasion speeds. For these pushed fronts we compute corrections to the estimated arrival time in this case. Finally, we show how inhomogeneities in local infection rates lead to faster average arrival times.

q-bio.PE↗

arXiv2021

Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotony of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems. Most notably, when the profile of the coefficient diffusion is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the coefficient diffusion is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.

math.AP↗

arXiv2020

Locked fronts in a discrete time discrete space population model

A model of population growth and dispersal is considered where the spatial habitat is a lattice and reproduction occurs generationally. The resulting discrete dynamical systems exhibits velocity locking, where rational speed invasion fronts are observed to persist as parameters are varied. In this article, we construct locked fronts for a particular piecewise linear reproduction function. These fronts are shown to be linear combinations of exponentially decaying solutions to the linear system near the unstable state. Based upon these front solutions, we then derive expressions for the boundary of locking regions in parameter space. We obtain leading order expansions for the locking regions in the limit as the migration parameter tends to zero. Strict spectral stability in exponentially weighted spaces is also established.

math.DS↗

arXiv2020

Invasion into remnant instability: a case study of front dynamics

We study the invasion of an unstable state by a propagating front in a peculiar but generic situation where the invasion process exhibits a remnant instability. Here, remnant instability refers to the fact that the spatially constant invaded state is linearly unstable in any exponentially weighted space in a frame moving with the linear invasion speed. Our main result is the nonlinear asymptotic stability of the selected invasion front for a prototypical model coupling spatio-temporal oscillations and monotone dynamics. We establish stability through a decomposition of the perturbation into two pieces: one that is bounded in the weighted space and a second that is unbounded in the weighted space but which converges uniformly to zero in the unweighted space at an exponential rate. Interestingly, long-time numerical simulations reveal an apparent instability in some cases. We exhibit how this instability is caused by round-off errors that introduce linear resonant coupling of otherwise non-resonant linear modes, and we determine the accelerated invasion speed.

math.AP↗

arXiv2019

Asymptotic stability of the critical pulled front in a Lotka-Volterra competition model

We prove that the critical pulled front of Lotka-Volterra competition systems is nonlinearly asymptotically stable. More precisely, we show that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof relies on pointwise semigroup methods and utilizes in a crucial way that the faster decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.

math.AP↗

arXiv2018

Asymptotic stability of the critical Fisher-KPP front using pointwise estimates

We propose a simple alternative proof of a famous result of Gallay regarding the nonlinear asymptotic stability of the critical front of the Fisher-KPP equation which shows that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof is based on pointwise semigroup methods and the key remark that the faster algebraic decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.

math.AP↗

arXiv2018

Invasion fronts on graphs: the Fisher-KPP equation on homogeneous trees and Erdős-Réyni graphs

We study the dynamics of the Fisher-KPP equation on the infinite homogeneous tree and Erdős-Réyni random graphs. We assume initial data that is zero everywhere except at a single node. For the case of the homogeneous tree, the solution will either form a traveling front or converge pointwise to zero. This dichotomy is determined by the linear spreading speed and we compute critical values of the diffusion parameter for which the spreading speed is zero and maximal and prove that the system is linearly determined. We also study the growth of the total population in the network and identify the exponential growth rate as a function of the diffusion coefficient, $α$. Finally, we make predictions for the Fisher-KPP equation on Erdős-Rényi random graphs based upon the results on the homogeneous tree. When $α$ is small we observe via numerical simulations that mean arrival times are linearly related to distance from the initial node and the speed of invasion is well approximated by the linear spreading speed on the tree. Furthermore, we observe that exponential growth rates of the total population on the random network can be bounded by growth rates on the homogeneous tree and provide an explanation for the sub-linear exponential growth rates that occur for small diffusion.

nlin.PS↗

arXiv2018

Bifurcation to locked fronts in two component reaction-diffusion systems

We study invasion fronts and spreading speeds in two component reaction-diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the spreading speed of the PDE system does not depend continuously on the coefficient of diffusion.

nlin.PS↗

arXiv2017

Estimating epidemic arrival times using linear spreading theory

We study the dynamics of a spatially structured model of worldwide epidemics and formulate predictions for arrival times of the disease at any city in the network. The model is comprised of a system of ordinary differential equations describing a meta-population SIR compartmental model defined on a network where each node represents a city and edges represent flight paths connecting cities. Making use of the linear determinacy of the system, we consider spreading speeds and arrival times in the system linearized about the unstable disease free state and compare these to arrival times in the nonlinear system. Two predictions are presented. The first is based upon expansion of the heat kernel for the linearized system. The second assumes that the dominant transmission pathway between any two cities can be approximated by a one dimensional lattice or homogeneous tree and gives a uniform prediction for arrival times independent of specific network features. We test these predictions on a real network describing worldwide airline traffic.

nlin.PS↗