arXiv subjects

Hwai-Ray Tung

Publications and source records attributed to Hwai-Ray Tung.

8 recordsLinked to original sources

arXiv2026

Beyond species area curves: a theoretical approach to the relationship between diversity and area

Species area curves, which describe the number of species present as a function of area, have long been used to understand biodiversity and inform conservation efforts. While understanding the number of species is important, it leaves out information about the population sizes of each species. In this work, we examine the relationship between the effective number of species from different diversity indices, like Simpson's index and the Shannon index, and area. These effective number of species are also referred to as Hill numbers. Using a spatial and neutral model that has previously been used to understand species area curves, we show through a combination of theory and simulations that the relationship between the effective number of species and area is linear when the area is sufficiently large and resembles a power law for smaller areas.

q-bio.PE↗

arXiv2026

Search at bounded speed with immigration

Many biophysical search processes employ searchers which enter or "immigrate" into the domain progressively over time. Existing search time estimates can become unphysical for fast immigration since they imply that searchers move at infinite speed. In this paper, we investigate search times of immigrating searchers that move at bounded speed. In the fast immigration limit, we determine the full probability distribution and all the moments of the $k$th search time in terms of the early time probability distribution of a single searcher. We apply these rigorous mathematical results to several canonical models of stochastic search. We further analyze different models of "diffusion" and use these results to investigate when and how the minutiae of searcher dynamics affect search times. We compare our theory to numerical simulations.

math.PR↗

arXiv2026

Stochastic search with space-dependent diffusivity

The canonical model of stochastic search tracks a randomly diffusing "searcher" until it finds a "target." Owing to its many applications across science and engineering, this perennially popular problem has been thoroughly investigated in a variety of models. However, aside from some exactly solvable one-dimensional examples, very little is known if the searcher diffusivity varies in space. For such space-dependent or "heterogeneous" diffusion, one must specify the interpretation of the multiplicative noise, which is termed the Itô-Stratonovich dilemma. In this paper, we investigate how stochastic search with space-dependent diffusivity depends on this interpretation. We obtain general formulas for the probability distribution and all the moments of the stochastic search time and the so-called splitting probabilities assuming that the targets are small or weakly reactive. These asymptotic results are valid for general space-dependent diffusivities in general domains in any space dimension with targets of general shape which may be in the interior or on the boundary of the domain. We illustrate our theory with stochastic simulations. Our analysis predicts that stochastic search can depend strongly and counterintuitively on the multiplicative noise interpretation.

cond-mat.stat-mech↗

arXiv2025

Escape from heterogeneous diffusion

Many physical processes depend on the time it takes a diffusing particle to find a target. Though this classical quantity is now well-understood in various scenarios, little is known if the diffusivity depends on the location of the particle. For such heterogeneous diffusion, an ambiguity arises in interpreting the stochastic process, which reflects the well-known Itô versus Stratonovich controversy. Here we analytically determine the mean escape time and splitting probabilities for an arbitrary heterogeneous diffusion in an arbitrary three-dimensional domain with small targets that can be perfectly or imperfectly absorbing. Our analysis reveals general principles for how search depends on heterogeneous diffusion and its interpretation (e.g. Itô, Stratonovich, or kinetic). An intricate picture emerges in which, for instance, increasing the diffusivity can decrease, not affect, or even increase the escape time. Our results could be used to determine the appropriate interpretation for specific physical systems.

cond-mat.stat-mech↗

arXiv2025

Passage times of fast inhomogeneous immigration processes

In many biophysical systems, key events are triggered when the fastest of many random searchers find a target. Most mathematical models of such systems assume that all searchers are initially present in the search domain, which permits the use of classical extreme value theory. In this paper, we explore $k$th passage times of inhomogeneous immigration processes where searchers are added to the domain over time either through time inhomogeneous rates or a Yule (pure birth) process. We rigorously prove convergence in distribution and convergence of moments of the $k$th passage times for both processes as immigration rates grow. In particular, we relate immigration with time inhomogeneous rates to previous work where all searchers are initially present through a coupling argument and demonstrate how immigration through a Yule process can be viewed as a time inhomogeneous immigration process with a random time shift. For Yule immigration, we find that the extreme distributions depart from the classical family of Frechet, Gumbel, and Weibull, and we compare our results to classical theorems on branching Brownian motion. This work offers one of the few examples where extreme value distributions can be obtained exactly for random variables which are neither independent nor identically distributed.

math.PR↗

arXiv2025

First passage times with fast immigration

Many scientific questions can be framed as asking for a first passage time (FPT), which generically describes the time it takes a random "searcher" to find a "target." The important timescale in a variety of biophysical systems is the time it takes the fastest searcher(s) to find a target out of many searchers. Previous work on such fastest FPTs assumes that all searchers are initially present in the domain, which makes the problem amenable to extreme value theory. In this paper, we consider an alternative model in which searchers progressively enter the domain at a constant "immigration" rate. In the fast immigration rate limit, we determine the probability distribution and moments of the $k$-th fastest FPT. Our rigorous theory applies to many models of stochastic motion, including random walks on discrete networks and diffusion on continuous state spaces. Mathematically, our analysis involves studying the extrema of an infinite sequence of random variables which are both not independent and not identically distributed. Our results constitute a rare instance in which extreme value statistics can be determined exactly for strongly correlated random variables.

math.PR↗

arXiv2022

Competitive exclusion in a model with seasonality: three species cannot coexist in an ecosystem with two seasons

Chan, Durrett, and Lanchier introduced a multitype contact process with temporal heterogeneity involving two species competing for space on the d-dimensional integer lattice. Time is divided into two seasons. They proved that there is an open set of the parameters for which both species can coexist when their dispersal range is sufficiently large. Numerical simulations suggested that three species can coexist in the presence of two seasons. The main point of this paper is to prove that this conjecture is incorrect. To do this we prove results for a more general ODE model and contrast its behavior with other related systems that have been studied in order to understand the competitive exclusion principle.

q-bio.PE↗

arXiv2020

Controlling the spread of COVID-19 on college campuses

This research was done during the DOMath program at Duke University from May 18 to July 10, 2020. At the time, Duke and other universities across the country were wrestling with the question of how to safely welcome students back to campus in the Fall. Because of this, our project focused on using mathematical models to evaluate strategies to suppress the spread of the virus on campus, specifically in dorms and in classrooms. For dorms, we show that giving students single rooms rather than double rooms can substantially reduce virus spread. For classrooms, we show that moving classes with size above some cutoff online can make the basic reproduction number $R_0<1$, preventing a wide spread epidemic. The cutoff will depend on the contagiousness of the disease in classrooms.

math.PR↗