arXiv subjects

Mary Lou Zeeman

Publications and source records attributed to Mary Lou Zeeman.

3 recordsLinked to original sources

arXiv2026

A Radial and Tangential Framework for Studying Transient Reactivity in Two-Dimensional Systems

Even if a linear system of ordinary differential equations has a globally attracting equilibrium at the origin, small disturbances from the equilibrium may lead to large transient excursions before the system stabilizes. This counter-intuitive phenomenon of transient amplification is called reactivity and is often associated with systems that are non-normal. Here, we establish a new framework for analyzing reactivity and transient dynamics in two-dimensional linear ODEs. Our work is facilitated by decomposing the corresponding vector field into sinusoidal radial and tangential components. Using this decomposition, we introduce a structure of orthovectors and orthovalues as dual to the eigenstructure. Since diagonalization masks transient reactivity, we combine the eigenstructure and the orthostructure to propose alternative matrix forms which capture both transient and asymptotic behavior and which highlight reactivity features more directly. Leveraging these matrix forms, we analytically quantify the maximal amplification in globally attracting systems, and we provide new insight into how a nonautonomous linear system can be unstable, even when all the frozen-time systems are stable.

math.DS↗

arXiv2022

Nitrogen-induced hysteresis in grassland biodiversity: a theoretical test of litter-mediated mechanisms

The global rise in anthropogenic reactive nitrogen (N) and the negative impacts of N deposition on terrestrial plant diversity are well-documented. The R* theory of resource competition predicts reversible decreases in plant diversity in response to N loading. However, empirical evidence for the reversibility of N-induced biodiversity loss is mixed. In a long-term N-enrichment experiment in Minnesota, a low-diversity state that emerged during N addition has persisted for decades after additions ceased. Hypothesized mechanisms preventing recovery of biodiversity include nutrient recycling, insufficient external seed supply, and litter inhibition of plant growth. Here we present an ODE model that unifies these mechanisms, produces bistability at intermediate N inputs, and qualitatively matches the observed hysteresis at Cedar Creek. Key features of the model, including native species' growth advantage in low-N conditions and limitation by litter accumulation, generalize from Cedar Creek to North American grasslands. Our results suggest that effective biodiversity restoration in these systems may require management beyond reducing N inputs, such as burning, grazing, haying, and seed additions. By coupling resource competition with an additional inter-specific inhibitory process, the model also illustrates a general mechanism for bistability and hysteresis that may occur in multiple ecosystem types.

q-bio.PE↗

arXiv2018

Quantifying resilience to recurrent ecosystem disturbances using flow-kick dynamics

Shifting ecosystem disturbance patterns due to climate change (e.g. storms, droughts, wildfires) or direct human interference (e.g. harvests, nutrient loading) highlight the importance of quantifying and strengthening the resilience of desired ecological regimes. Although existing metrics capture resilience to isolated shocks, gradual parameter changes, and continuous noise, quantifying resilience to repeated, discrete disturbances requires novel analytical tools. Here we introduce a flow-kick framework that quantifies resilience to disturbances explicitly in terms of their magnitude and frequency. We present a resilience boundary between disturbances that cause either escape from a basin of attraction or stabilization within it, and use the resilience boundary to build resilience metrics tailored to repeated, discrete disturbances. The flow-kick model suggests that the distance-to-threshold resilience metric overestimates resilience in the context of repeated disturbances. It also reveals counterintuitive triggers for regime shifts, such as increasing recovery times between disturbances, or increasing disturbance magnitude and recovery times proportionately.

q-bio.PE↗