arXiv subjects

Alexandru Hening

Publications and source records attributed to Alexandru Hening.

At least 19 recordsLinked to original sources

arXiv2025

Population size in stochastic multi-patch ecological models

We look at the interaction of dispersal and environmental stochasticity in $n$-patch models. We are able to prove persistence and extinction results even in the setting when the dispersal rates are stochastic. As applications we look at Beverton-Holt and Hassell functional responses. We find explicit approximations for the total population size at stationarity when we look at slow and fast dispersal. In particular, we show that if dispersal is small then in the Beverton-Holt setting, if the carrying capacity is random, then environmental fluctuations are always detrimental and decrease the total population size. Instead, in the Hassell setting, if the inverse of the carrying capacity is made random, then environmental fluctuations always increase the population size. Fast dispersal can save populations from extinction and therefore increase the total population size. We also analyze a different type of environmental fluctuation which comes from switching environmental states according to a Markov chain and find explicit approximations when the switching is either fast or slow - in examples we are able to show that slow switching leads to a higher population size than fast switching. Using and modifying some approximation results due to Cuello, we find expressions for the total population size in the $n=2$ patch setting when the growth rates, carrying capacities, and dispersal rates are influenced by random fluctuations. We find that there is a complicated interaction between the various terms and that the covariances between the various random parameters (growth rate, carrying capacity, dispersal rate) play a key role in whether we get an increase or a decrease in the total population size. Environmental fluctuations turn to sometimes be beneficial -- this show that not only dispersal, but also environmental stochasticity can lead to an increase in population size.

q-bio.PE↗

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

Population dynamics under random switching

Populations interact non-linearly and are influenced by environmental fluctuations. In order to have realistic mathematical models, one needs to take into account that the environmental fluctuations are inherently stochastic. Often, environmental stochasticity is modeled by systems of stochastic differential equations. However, this type of stochasticity is not always the best suited for ecological modeling. Instead, biological systems can be modeled using piecewise deterministic Markov processes (PDMP). For a PDMP the process follows the flow of a system of ordinary differential equations for a random time, after which the environment switches to a different state, where the dynamics is given by a different system of differential equations. Then this is repeated. The current paper is devoted to the study of the dynamics of $n$ populations described by $n$-dimensional Kolmogorov PDMP. We provide sharp conditions for persistence and extinction, based on the invasion rates (Lyapunov exponents) of the ergodic probability measures supported on the boundary of the positive orthant. In order to showcase the applicability of our results, we apply the theory in some interesting ecological examples.

math.PR↗

arXiv2025

Population size in stochastic discrete-time ecological dynamics

We study how environmental stochasticity influences the long-term population size in certain one- and two-species models. The difficulty is that even when one can prove that there is persistence, it is usually impossible to say anything about the invariant probability measure which describes the persistent species. We are able to circumvent this problem for some important ecological models by noticing that the per-capita growth rates at stationarity are zero, something which can sometimes yield information about the invariant probability measure. For more complicated models we use a recent result by Cuello to explore how small noise influences the population size. We are able to show that environmental fluctuations can decrease, increase, or leave unchanged the expected population size. The results change according to the dynamical model and, within a fixed model, also according to which parameters (growth rate, carrying capacity, etc) are affected by environmental fluctuations. Moreover, we show that not only do things change if we introduce noise differently in a model, but it also matters what one takes as the deterministic `no-noise' baseline for comparison.

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arXiv2024

Population dynamics under demographic and environmental stochasticity

The present paper is devoted to the study of the long term dynamics of diffusion processes modelling a single species that experiences both demographic and environmental stochasticity. In our setting, the long term dynamics of the diffusion process in the absence of demographic stochasticity is determined by the sign of $Λ_0$, the external Lyapunov exponent, as follows: $Λ_0<0$ implies (asymptotic) extinction and $Λ_0>0$ implies convergence to a unique positive stationary distribution $μ_0$. If the system is of size $\frac{1}{ε^{2}}$ for small $ε>0$ (the intensity of demographic stochasticity), demographic effects will make the extinction time finite almost surely. This suggests that to understand the dynamics one should analyze the quasi-stationary distribution (QSD) $μ_ε$ of the system. The existence and uniqueness of the QSD is well-known under mild assumptions. We look at what happens when the population size is sent to infinity, i.e., when $ε\to 0$. We show that the external Lyapunov exponent still plays a key role: 1) If $Λ_0<0$, then $μ_ε\to δ_0$, the mean extinction time is of order $|\ln ε|$ and the extinction rate associated with the QSD $μ_ε$ has a lower bound of order $\frac{1}{|\lnε|}$; 2) If $Λ_0>0$, then $μ_ε\to μ_0$, the mean extinction time is polynomial in $\frac{1}{ε^{2}}$ and the extinction rate is polynomial in $ε^{2}$. Furthermore, when $Λ_0>0$ we are able to show that the system exhibits multiscale dynamics: at first the process quickly approaches the QSD $μ_ε$ and then, after spending a polynomially long time there, it relaxes to the extinction state. We give sharp asymptotics in $ε$ for the time spent close to $μ_ε$.

math.PR↗

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.

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arXiv2023

Stability and Fluctuations in Complex Ecological Systems

From 08-12 August, 2022, 32 individuals participated in a workshop, Stability and Fluctuations in Complex Ecological Systems, at the Lorentz Center, located in Leiden, The Netherlands. An interdisciplinary dialogue between ecologists, mathematicians, and physicists provided a foundation of important problems to consider over the next 5-10 years. This paper outlines eight areas including (1) improving our understanding of the effect of scale, both temporal and spatial, for both deterministic and stochastic problems; (2) clarifying the different terminologies and definitions used in different scientific fields; (3) developing a comprehensive set of data analysis techniques arising from different fields but which can be used together to improve our understanding of existing data sets; (4) having theoreticians/computational scientists collaborate closely with empirical ecologists to determine what new data should be collected; (5) improving our knowledge of how to protect and/or restore ecosystems; (6) incorporating socio-economic effects into models of ecosystems; (7) improving our understanding of the role of deterministic and stochastic fluctuations; (8) studying the current state of biodiversity at the functional level, taxa level and genome level.

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arXiv2023

Stochastic nutrient-plankton models

We analyze plankton-nutrient food chain models composed of phytoplankton, herbivorous zooplankton and a limiting nutrient. These models have played a key role in understanding the dynamics of plankton in the oceanic layer. Given the strong environmental and seasonal fluctuations that are present in the oceanic layer, we propose a stochastic model for which we are able to fully classify the longterm behavior of the dynamics. In order to achieve this we had to develop new analytical techniques, as the system does not satisfy the regular dissipativity conditions and the analysis is more subtle than in other population dynamics models.

math.PR↗

arXiv2022

Stochastic epidemic SIR models with hidden states

This paper focuses on and analyzes realistic SIR models that take stochasticity into account. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington-DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, we look at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. We assume that the direct observation of the percentage of hidden state individuals that are infected, $α(t)$, is not given and only a noise-corrupted observation process is available. Using the nonlinear filtering techniques in conjunction with an invasion type analysis (or analysis using Lyapunov exponents from the dynamical system point of view), this paper proves that the long-term behavior of the disease is governed by a threshold $λ\in \mathbb{R}$ that depends on the model parameters. It turns out that if $λ<0$ the number $I(t)$ of infected individuals converges to zero exponentially fast, or the extinction happens. In contrast, if $λ>0$, the infection is endemic and the system is permanent. We showcase our results by applying them in specific illuminating examples. Numerical simulations are also given to illustrate our results.

math.PR↗

arXiv2021

Random switching in an ecosystem with two prey and one predator

In this paper we study the long term dynamics of two prey species and one predator species. In the deterministic setting, if we assume the interactions are of Lotka-Volterra type (competition or predation), the long term behavior of this system is well known. However, nature is usually not deterministic. All ecosystems experience some type of random environmental fluctuations. We incorporate these into a natural framework as follows. Suppose the environment has two possible states. In each of the two environmental states the dynamics is governed by a system of Lotka-Volterra ODE. The randomness comes from spending an exponential amount of time in each environmental state and then switching to the other one. We show how this random switching can create very interesting phenomena. In some cases the randomness can facilitate the coexistence of the three species even though coexistence is impossible in each of the two environmental states. In other cases, even though there is coexistence in each of the two environmental states, switching can lead to the loss of one or more species. We look into how predators and environmental fluctuations can mediate coexistence among competing species.

math.PR↗

arXiv2021

The effects of random and seasonal environmental fluctuations on optimal harvesting and stocking

We analyze the harvesting and stocking of a population that is affected by random and seasonal environmental fluctuations. The main novelty comes from having three layers of environmental fluctuations. The first layer is due to the environment switching at random times between different environmental states. This is similar to having sudden environmental changes or catastrophes. The second layer is due to seasonal variation, where there is a significant change in the dynamics between seasons. Finally, the third layer is due to the constant presence of environmental stochasticity -- between the seasonal or random regime switches, the species is affected by fluctuations which can be modelled by white noise. This framework is more realistic because it can capture both significant random and deterministic environmental shifts as well as small and frequent fluctuations in abiotic factors. Our framework also allows for the price or cost of harvesting to change deterministically and stochastically, something that is more realistic from an economic point of view. The combined effects of seasonal and random fluctuations make it impossible to find the optimal harvesting-stocking strategy analytically. We get around this roadblock by developing rigorous numerical approximations and proving that they converge to the optimal harvesting-stocking strategy. We apply our methods to multiple population models and explore how prices, or costs, and environmental fluctuations influence the optimal harvesting-stocking strategy. We show that in many situations the optimal way of harvesting and stocking is not of threshold type.

math.PR↗

arXiv2021

Stationary distributions of persistent ecological systems

We analyze ecological systems that are influenced by random environmental fluctuations. We first provide general conditions which ensure that the species coexist and the system converges to a unique invariant probability measure (stationary distribution). Since it is usually impossible to characterize this invariant probability measure analytically, we develop a powerful method for numerically approximating invariant probability measures. This allows us to shed light upon how the various parameters of the ecosystem impact the stationary distribution. We analyze different types of environmental fluctuations. At first we study ecosystems modeled by stochastic differential equations. In the second setting we look at piecewise deterministic Markov processes. These are processes where one follows a system of differential equations for a random time, after which the environmental state changes, and one follows a different set of differential equations -- this procedure then gets repeated indefinitely. Finally, we look at stochastic differential equations with switching, which take into account both the white noise fluctuations and the random environmental switches. As applications of our theoretical and numerical analysis, we look at competitive Lotka--Volterra, Beddington-DeAngelis predator-prey, and rock-paper-scissors dynamics. We highlight new biological insights by analyzing the stationary distributions of the ecosystems and by seeing how various types of environmental fluctuations influence the long term fate of populations.

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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

Optimal ergodic harvesting under ambiguity

We consider an ergodic harvesting problem with model ambiguity that arises from biology. To account for the ambiguity, the problem is constructed as a stochastic game with two players: the decision-maker (DM) chooses the `best' harvesting policy and an adverse player chooses the `worst' probability measure. The main result is establishing an optimal strategy (also referred to as a control) of the DM and showing that it is a threshold policy. The optimal threshold and the optimal payoff are obtained by solving a free-boundary problem emerging from the Hamilton--Jacobi--Bellman (HJB) equation. As part of the proof, we fix a gap that appeared in the HJB analysis of [Alvarez and Hening, {\em Stochastic Process. Appl.}, 2019], a paper that analyzed the risk-neutral version of the ergodic harvesting problem. Finally, we study the dependence of the optimal threshold and the optimal payoff on the ambiguity parameter and show that if the ambiguity goes to 0, the problem converges, to the risk-neutral problem.

math.OC↗

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↗

arXiv2021

Quasi-stationary distributions of multi-dimensional diffusion processes

The present paper is devoted to the investigation of the long term behavior of a class of singular multi-dimensional diffusion processes that get absorbed in finite time with probability one. Our focus is on the analysis of quasi-stationary distributions (QSDs), which describe the long term behavior of the system conditioned on not being absorbed. Under natural Lyapunov conditions, we construct a QSD and prove the sharp exponential convergence to this QSD for compactly supported initial distributions. Under stronger Lyapunov conditions ensuring that the diffusion process comes down from infinity, we show the uniqueness of a QSD and the exponential convergence to the QSD for all initial distributions. Our results can be seen as the multi-dimensional generalization of Cattiaux et al (Ann. Prob. 2009) as well as the complement to Hening and Nguyen (Ann. Appl. Prob. 2018) which looks at the long term behavior of multi-dimensional diffusions that can only become extinct asymptotically. The centerpiece of our approach concerns a uniformly elliptic operator that we relate to the generator, or the Fokker-Planck operator, associated to the diffusion process. This operator only has singular coefficients in its zeroth-order terms and can be handled more easily than the generator. For this operator, we establish the discreteness of its spectrum, its principal spectral theory, the stochastic representation of the semigroup generated by it, and the global regularity for the associated parabolic equation. We show how our results can be applied to most ecological models, among which cooperative, competitive, and predator-prey Lotka-Volterra systems.

math.PR↗

arXiv2019

Coexistence, extinction, and optimal harvesting in discrete-time stochastic population models

We analyze the long term behavior of interacting populations which can be controlled through harvesting. The dynamics is assumed to be discrete in time and stochastic due to the effect of environmental fluctuations. We present extinction and coexistence criteria when there are one or two interacting species. We then use these tools in order to see when harvesting leads to extinction or persistence of species, as well as what the optimal harvesting strategies, which maximize the expected long term yield, look like. For single species systems, we show under certain conditions that the optimal harvesting strategy is of bang-bang type: there is a threshold under which there is no harvesting, while everything above this threshold gets harvested. The second part of the paper is concerned with the analysis of ecosystems that have two interacting species which can be harvested. In particular, we carefully study predator-prey and competitive Ricker models when there are two species. In this setting we show how to find the optimal proportional harvesting strategy. If the system is of predator-prey type the optimal proportional harvesting strategy is, depending on the interaction parameters and the price of predators relative to prey, either to harvest the predator to extinction and maximize the asymptotic yield of the prey or to not harvest the prey and to maximize the asymptotic harvesting yield of the predators. If the system is competitive, in certain instances it is optimal to drive one species extinct and to harvest the other one. In other cases it is best to let the two species coexist and harvest both species while maintaining coexistence. In the setting of the competitive Ricker model we show that if one competitor is dominant and pushes the other species to extinction, the harvesting of the dominant species can lead to coexistence.

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