arXiv subjects

Dang H. Nguyen

Publications and source records attributed to Dang H. Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

Stochastic Extinction with Relaxed Boundedness Conditions

We study stochastic extinction for a class of Markov processes motivated by models in ecology and epidemiology. Extinction is often characterized by a boundedness condition and a condition on boundary Lyapunov exponents (invasion rates). While the latter is typically sharp, the former is often restrictive and can be improved. Building on the ideas initiated in \cite{benaim2018stochastic}, we develop a streamlined approach that relaxes this boundedness condition and yields concise and accessible criteria for extinction. In particular, we establish extinction criteria in two settings: with and without a linearly bounded quadratic variation condition. In the first case, our result is comparable to, and slightly improves upon, the main results in \cite{foldes2024stochastic}. In the second case, where the quadratic variation is not linearly bounded, we obtain new extinction results that fall outside the scope of existing frameworks. Several examples are provided to illustrate the applicability of our results and to highlight situations where previous conditions are not practically verifiable.

math.PR↗

arXiv2025

Dynamics of stochastic microorganism flocculation models

In this paper we study the dynamics of stochastic microorganism flocculation models. Given the strong influence of environmental and seasonal fluctuations that are present in these models, we propose a stochastic model that includes multiple layers of stochasticity, from small Brownian fluctuations, to possibly large changes due to environmental `shifts'. We are able to give a full classification of the asymptotic behavior of these models. New techniques had to be developed to prove the persistence and extinction of the process as the system is not in Kolmogorov form and, as a result, the analysis is significantly more involved.

math.PR↗

arXiv2025

VinDr-CXR-VQA: A Visual Question Answering Dataset for Explainable Chest X-Ray Analysis with Multi-Task Learning

We present VinDr-CXR-VQA, a large-scale chest X-ray dataset for explainable Medical Visual Question Answering (Med-VQA) with spatial grounding. The dataset contains 17,597 question-answer pairs across 4,394 images, each annotated with radiologist-verified bounding boxes and clinical reasoning explanations. Our question taxonomy spans six diagnostic types-Where, What, Is there, How many, Which, and Yes/No-capturing diverse clinical intents. To improve reliability, we construct a balanced distribution of 41.7% positive and 58.3% negative samples, mitigating hallucinations in normal cases. Benchmarking with MedGemma-4B-it demonstrates improved performance (F1 = 0.624, +11.8% over baseline) while enabling lesion localization. VinDr-CXR-VQA aims to advance reproducible and clinically grounded Med-VQA research. The dataset and evaluation tools are publicly available at huggingface.co/datasets/Dangindev/VinDR-CXR-VQA.

cs.CV↗

arXiv2024

Long-term behavior of stochastic SIQRS epidemic models

In this paper we analyze and classify the dynamics of SIQRS epidemiological models with susceptible, infected, quarantined, and recovered classes, where the recovered individuals can become reinfected. We are able to treat general incidence functional responses. Our models are more realistic than what has been studied in the literature since they include two important types of random fluctuations. The first type is due to small fluctuations of the various model parameters and leads to white noise terms. The second type of noise is due to significant environment regime shifts in the that can happen at random. The environment switches randomly between a finite number of environmental states, each with a possibly different disease dynamic. We prove that the long-term fate of the disease is fully determined by a real-valued threshold $λ$. When $λ< 0$ the disease goes extinct asymptotically at an exponential rate. On the other hand, if $λ> 0$ the disease will persist indefinitely. We end our analysis by looking at some important examples where $λ$ can be computed explicitly, and by showcasing some simulation results that shed light on real-world situations.

q-bio.PE↗

arXiv2021

Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

We consider a Lotka-Volterra food chain model with possibly intra-specific competition in a stochastic environment represented by stochastic differential equations. In the non-degenerate setting, this model has already been studied by A. Hening and D. Nguyen. They provided conditions for stochastic persistence and extinction. In this paper, we extend their results to the degenerate situation in which the top or the bottom species is subject to random perturbations. Under the persistence condition, there exists a unique invariant probability measure supported by the interior of $\mathbb{R}_+^n$ having a smooth density. Moreover, we study a more general model, in which we give new conditions which make it possible to characterise the convergence of the semi-group towards the unique invariant probability measure either at an exponential rate or at a polynomial one. This will be used in the stochastic Lotka-Volterra food chain to see that if intra-specific competition occurs for all species, the rate of convergence is exponential while in the other cases it is polynomial.

math.PR↗

arXiv2021

Variational Bayes on Manifolds

Variational Bayes (VB) has become a widely-used tool for Bayesian inference in statistics and machine learning. Nonetheless, the development of the existing VB algorithms is so far generally restricted to the case where the variational parameter space is Euclidean, which hinders the potential broad application of VB methods. This paper extends the scope of VB to the case where the variational parameter space is a Riemannian manifold. We develop an efficient manifold-based VB algorithm that exploits both the geometric structure of the constraint parameter space and the information geometry of the manifold of VB approximating probability distributions. Our algorithm is provably convergent and achieves a convergence rate of order $\mathcal O(1/\sqrt{T})$ and $\mathcal O(1/T^{2-2ε})$ for a non-convex evidence lower bound function and a strongly retraction-convex evidence lower bound function, respectively. We develop in particular two manifold VB algorithms, Manifold Gaussian VB and Manifold Neural Net VB, and demonstrate through numerical experiments that the proposed algorithms are stable, less sensitive to initialization and compares favourably to existing VB methods.

cs.LG↗

arXiv2021

A classification of the dynamics of three-dimensional stochastic ecological systems

The classification of the long-term behavior of dynamical systems is a fundamental problem in mathematics. For both deterministic and stochastic dynamics specific classes of models verify Palis' conjecture: the long-term behavior is determined by a finite number of stationary distributions. In this paper we consider the classification problem for stochastic models of interacting species. For a large class of three-species, stochastic differential equation models, we prove a variant of Palis' conjecture: the long-term statistical behavior is determined by a finite number of stationary distributions and, generically, three general types of behavior are possible: 1) convergence to a unique stationary distribution that supports all species, 2) convergence to one of a finite number of stationary distributions supporting two or fewer species, 3) convergence to convex combinations of single species, stationary distributions due to a rock-paper-scissors type of dynamic. Moreover, we prove that the classification reduces to computing Lyapunov exponents (external Lyapunov exponents) that correspond to the average per-capita growth rate of species when rare. Our results stand in contrast to the deterministic setting where the classification is incomplete even for three-dimensional, competitive Lotka--Volterra systems. For these SDE models, our results also provide a rigorous foundation for ecology's modern coexistence theory (MCT) which assumes the external Lyapunov exponents determine long-term ecological outcomes.

math.PR↗

arXiv2021

Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles

We study the long-term qualitative behavior of randomly perturbed dynamical systems. More specifically, we look at limit cycles of stochastic differential equations (SDE) with Markovian switching, in which the process switches at random times among different systems of SDEs, when the switching is fast and the diffusion (white noise) term is small. The system is modeled by $$ dX^{ε,δ}(t)=f(X^{ε,δ}(t), α^ε(t))dt+\sqrtδσ(X^{ε,δ}(t), α^ε(t))dW(t) , \ X^ε(0)=x, $$ where $α^ε(t)$ is a finite state space Markov chain with irreducible generator $Q=(q_{ij})$. The relative changing rates of the switching and the diffusion are highlighted by the two small parameters $ε$ and $δ$. We associate to the system the averaged ODE \[ d\bar X(t)=\bar f(\bar X(t))dt, \ X(0)=x, \] where $\bar f(\cdot)=\sum_{i=1}^{m_0}f(\cdot, i)ν_i$ and $(ν_1,\dots,ν_{m_0})$ is the unique invariant probability measure of the Markov chain with generator $Q$. Suppose that for each pair $(ε,δ)$ of parameters, the process has an invariant probability measure $μ^{ε,δ}$, and that the averaged ODE has a limit cycle in which there is an averaged occupation measure $μ^0$ for the averaged equation. We are able to prove that if $\bar f$ has finitely many unstable or hyperbolic fixed points, then $μ^{ε,δ}$ converges weakly to $μ^0$ as $ε\to 0$ and $δ\to 0$. Our results generalize to the setting of state-dependent switching \[ \mathbb{P}\{α^ε(t+Δ)=j~|~α^ε=i, X^{ε,δ}(s),α^ε(s), s\leq t\}=q_{ij}(X^{ε,δ}(t))Δ+o(Δ),~~ i\neq j \] as long as the generator $Q(\cdot)=(q_{ij}(\cdot))$ is bounded, Lipschitz, and irreducible for all $x\in\mathbb{R}^d$. We conclude our analysis by studying a predator-prey model.

math.PR↗

arXiv2021

Stochastic Functional Kolmogorov Equations II: Extinction

This work, Part II, together with its companion Part I develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the problems, it is natural to divide our contributions into two parts to answer a long-standing question in biology and ecology. What are the minimal conditions for long-term persistence and extinction of a population? Part I of our work provides characterization of persistence, whereas in this part, extinction is the main focus. The main techniques used in this paper are combination of the newly developed functional It^o formula and a dynamical system approach. Compared to the study of stochastic Kolmogorov systems without delays, the main difficulty is that infinite dimensional systems have to be treated. The extinction is characterized after investigating random occupation measures and examining behavior of functional systems around boundaries. Our characterizations of long-term behavior of the systems reduces to that of Kolmogorov systems without delay when there is no past dependence. A number of applications are also examined.

math.PR↗

arXiv2021

Stochastic Functional Kolmogorov Equations (I): Persistence

This work (Part (I)) together with its companion (Part (II) [45]) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the results, it seems to be instructive to divide our contributions to two parts. In contrast to the existing literature, our effort is to advance the knowledge by allowing delay and past dependence, yielding essential utility to a wide range of applications. A long-standing question of fundamental importance pertaining to biology and ecology is: What are the minimal necessary and sufficient conditions for long-term persistence and extinction (or for long-term coexistence of interacting species) of a population? Regardless of the particular applications encountered, persistence and extinction are properties shared by Kolmogorov systems. While there are many excellent treaties of stochastic-differential-equation-based Kolmogorov equations, the work on stochastic Kolmogorov equations with past dependence is still scarce. Our aim here is to answer the aforementioned basic question. This work, Part (I), is devoted to characterization of persistence, whereas its companion, Part (II) [45], is devoted to extinction. The main techniques used in this paper include the newly developed functional It^o formula and asymptotic coupling and Harris-like theory for infinite dimensional systems specialized to functional equations. General theorems for stochastic functional Kolmogorov equations are developed first. Then a number of applications are examined to obtain new results substantially covering, improving, and extending the existing literature. Furthermore, these conditions reduce to that of Kolmogorov systems when there is no past dependence.

math.PR↗

arXiv2021

A general theory of coexistence and extinction for stochastic ecological communities

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (Journal of Mathematical Biology '19) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Annals of Applied Probability '18) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

q-bio.PE↗

arXiv2020

General Nonlinear Stochastic Systems Motivated by Chemostat Models: Complete Characterization of Long-Time Behavior, Optimal Controls, and Applications to Wastewater Treatment

The paper considers a chemostat model describing an activated sludge process in wastewater treatment. The model is assumed to be subject to environment noise in terms of both white noise and color noise. The paper fully characterizes the asymptotic behavior of the model that is a hybrid switching diffusion. We show that the long-term properties of the system can be classified using a value $λ$. More precisely, if $λ\leq 0$, the bacteria in the sewage will die out, which means that the process does not operate. If $λ>0$, the system has an invariant probability measure to which the transition probability of the solution process converges exponentially fast. One of the distinctive contributions of this paper is that the critical case $λ=0$ is considered. Numerical examples are given to illustrate our results.

math.PR↗

arXiv2020

A method to deal with the critical case in stochastic population dynamics

In numerous papers, the behaviour of stochastic population models is investigated through the sign of a real quantity which is the growth rate of the population near the extinction set. In many cases, it is proven that when this growth rate is positive, the process is persistent in the long run, while if it is negative, the process converges to extinction. However, the critical case when the growth rate is null is rarely treated. The aim of this paper is to provide a method that can be applied in many situations to prove that in the critical case, the process congerves in temporal average to extinction. A number of applications are given, for Stochastic Differential Equations and Piecewise Deterministic Markov Processes modelling prey-predator, epidemilogical or structured population dynamics.

math.PR↗

arXiv2019

The competitive exclusion principle in stochastic environments

In its simplest form, the competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. However, it has been observed empirically that in some settings it is possible to have coexistence. One example is Hutchinson's `paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times. In this paper we show in various settings how a variable (stochastic) environment enables a set of competing species limited by a smaller number of resources or other density dependent factors to coexist. If the environmental fluctuations are modeled by white noise, and the per-capita growth rates of the competitors depend linearly on the resources, we prove that there is competitive exclusion. However, if either the dependence between the growth rates and the resources is not linear or the white noise term is nonlinear we show that coexistence on fewer resources than species is possible. Even more surprisingly, if the temporal environmental variation comes from switching the environment at random times between a finite number of possible states, it is possible for all species to coexist even if the growth rates depend linearly on the resources. We show in an example (a variant of which first appeared in Benaim and Lobry '16) that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.

q-bio.PE↗

arXiv2019

Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate

In this paper, we consider a stochastic SIRS model with general incidence rate and perturbed by both white noise and color noise. We determine the threshold $λ$ that is used to classify the extinction and permanence of the disease. In particular, $λ<0$ implies that the disease-free $(K, 0, 0)$ is globally asymptotic stable, i.e., the disease will eventually disappear. If $λ>0$ the epidemic is strongly stochastically permanent. Our result is considered as a significant generalization and improvement over the results in \cite{HZ1, GLM1, LOK1, SLJJ1, ZJ1}.

math.PR↗

arXiv2018

Persistence in Stochastic Lotka--Volterra food chains with intraspecific competition

This paper is devoted to the analysis of a simple Lotka-Volterra food chain evolving in a stochastic environment. It can be seen as the companion paper of Hening and Nguyen (J. of Math. Biol. `18) where we have characterized the persistence and extinction of such a food chain under the assumption that there is no intraspecific competition among predators. In the current paper we focus on the case when all the species experience intracompetition. The food chain we analyze consists of one prey and $n-1$ predators. The $j$th predator eats the $j-1$st species and is eaten by the $j+1$st predator; this way each species only interacts with at most two other species - the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on the invasion rates of the predators (which we can determine from the interaction coefficients of the system via an algorithm), which species go extinct and which converge to their unique invariant probability measure. We obtain stronger results than in the case with no intraspecific competition because in this setting we can make use of the general results of Hening and Nguyen (Ann. of Appl. Probab.). Unlike most of the results available in the literature, we provide an in depth analysis for both non-degenerate and degenerate noise. We exhibit our general results by analysing trophic cascades in a plant--herbivore--predator system and providing persistence/extinction criteria for food chains of length $n\leq 3$.

math.PR↗

arXiv2018

Asymptotic harvesting of populations in random environments

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never be such that the population is harvested to extinction -- instead the harvested population converges to a unique invariant probability measure. When the yield function is the identity, we show that the optimal strategy has a bang-bang property: there exists a threshold $x^*>0$ such that whenever the population is under the threshold the harvesting rate must be zero, whereas when the population is above the threshold the harvesting rate must be at the upper limit. We provide upper and lower bounds on the maximal asymptotic yield, and explore via numerical simulations how the harvesting threshold and the maximal asymptotic yield change with the growth rate, maximal harvesting rate, or the competition rate. We also show that, if the yield function is $C^2$ and strictly concave, then the optimal harvesting strategy is continuous, whereas when the yield function is convex the optimal strategy is of bang-bang type. This shows that one cannot always expect bang-bang type optimal controls.

math.PR↗

arXiv2017

Stability of Regime-Switching Diffusion Systems with Discrete States Belonging to a Countable Set

This work focuses on stability of regime-switching diffusions consisting of continuous and discrete components, in which the discrete component switches in a countably infinite set and its switching rates at current time depend on the continuous component. In contrast to the existing approach, this work provides more practically viable approach with more feasible conditions for stability. A classical approach for asymptotic stabilityusing Lyapunov function techniques shows the Lyapunov function evaluated at the solution process goes to 0 as time $t\to \infty$. A distinctive feature of this paper is to obtain estimates of path-wise rates of convergence, which pinpoints how fast the aforementioned convergence to 0 taking place. Finally, some examples are given to illustrate our findings.

math.PR↗