arXiv subjects

Alexandru Hening

Publications and source records attributed to Alexandru Hening.

At least 37 records · Page 2Linked to original sources

arXiv2019

Harvesting and seeding of stochastic populations: analysis and numerical approximation

It is well known that excessive harvesting or hunting has driven species to extinction both on local and global scales. This leads to one of the fundamental problems of conservation ecology: how should we harvest a population so that economic gain is maximized, while also ensuring that the species is safe from extinction? We study an ecosystem of interacting species that are influenced by random environmental fluctuations. At any point in time, we can either harvest or seed (repopulate) species. Harvesting brings an economic gain while seeding incurs a cost. The problem is to find the optimal harvesting-seeding strategy that maximizes the expected total income from harvesting minus the cost one has to pay for the seeding of various species. We consider what happens when one, or both, of the seeding and harvesting rates are bounded. The focus of this paper is the analysis of these three novel settings: bounded seeding and infinite harvesting, bounded seeding and bounded harvesting, and infinite seeding and bounded harvesting. We prove analytical results and develop numerical approximation methods. By implementing these approximations, we are able to gain qualitative information about how to best harvest and seed species. We are able to show that in the single species setting there are thresholds $0<L_1<L_2<\infty$ such that: 1) if the population size is `low', so that it lies in $(0, L_1]$, there is seeding using the maximal seeding rate; 2) if the population size `moderate', so that it lies in $(L_1,L_2)$, there is no harvesting or seeding; 3) if the population size is `high', so that it lies in the interval $[L_2, \infty)$, there is harvesting using the maximal harvesting rate. Once we have a system with at least two species, numerical experiments show that constant threshold strategies are not optimal anymore.

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

On a predator-prey system with random switching that never converges to its equilibrium

We study the dynamics of a predator-prey system in a random environment. The dynamics evolves according to a deterministic Lotka-Volterra system for an exponential random time after which it switches to a different deterministic Lotka-Volterra system. This switching procedure is then repeated. The resulting process is a Piecewise Deterministic Markov Process (PDMP). In the case when the equilibrium points of the two deterministic Lotka--Volterra systems coincide we show that almost surely the trajectory does not converge to the common deterministic equilibrium. Instead, with probability one, the densities of the prey and the predator oscillate between $0$ and $\infty$. This proves a conjecture of Takeuchi et al (J. Math. Anal. Appl 2006). The proof of the conjecture is a corollary of a result we prove about linear switched systems. Assume $(Y_t, I_t)$ is a PDMP that evolves according to $\frac{dY_t}{dt}=A_{I_t} Y_t$ where $A_0,A_1$ are $2\times2$ matrices and $I_t$ is a Markov chain on $\{0,1\}$ with transition rates $k_0,k_1>0$. If the matrices $A_0$ and $A_1$ are not proportional and have purely imaginary eigenvalues, then there exists $λ>0$ such that $\lim_{t \to \infty} \frac{\log \| Y_t \|}{t} = λ.$

math.PR↗

arXiv2019

The Inverse First Passage Time Problem for killed Brownian motion

The classical inverse first passage time problem asks whether, for a Brownian motion $(B_t)_{t\geq 0}$ and a positive random variable $ξ$, there exists a barrier $b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{P}\{B_s>b(s), 0\leq s \leq t\}=\mathbb{P}\{ξ>t\}$, for all $t\geq 0$. We study a variant of the inverse first passage time problem for killed Brownian motion. We show that if $λ>0$ is a killing rate parameter and $\mathbb{1}_{(-\infty,0]}$ is the indicator of the set $(-\infty,0]$ then, under certain compatibility assumptions, there exists a unique continuous function $b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{E}\left[-λ\int_0^t \mathbb{1}_{(-\infty,0]}(B_s-b(s))\,ds\right] = \mathbb{P}\{ζ>t\}$ holds for all $t\geq 0$. This is a significant improvement of a result of the first two authors (Annals of Applied Probability 24(1):1--33, 2014). The main difficulty arises because $\mathbb{1}_{(-\infty,0]}$ is discontinuous. We associate a semi-linear parabolic partial differential equation (PDE) coupled with an integral constraint to this version of the inverse first passage time problem. We prove the existence and uniqueness of weak solutions to this constrained PDE system. In addition, we use the recent Feynman-Kac representation results of Glau (Finance and Stochastics 20(4):1021--1059, 2016) to prove that the weak solutions give the correct probabilistic interpretation.

math.PR↗

arXiv2018

Harvesting of interacting stochastic populations

We analyze the optimal harvesting problem for an ecosystem of species that experience environmental stochasticity. Our work generalizes the current literature significantly by taking into account non-linear interactions between species, state-dependent prices, and species injections. The key generalization is making it possible to not only harvest, but also `seed' individuals into the ecosystem. This is motivated by how fisheries and certain endangered species are controlled. The harvesting problem becomes finding the optimal harvesting-seeding strategy that maximizes the expected total income from the harvest minus the lost income from the species injections. Our analysis shows that new phenomena emerge due to the possibility of species injections. It is well-known that multidimensional harvesting problems are very hard to tackle. We are able to make progress, by characterizing the value function as a viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equations. Moreover, we provide a verification theorem, which tells us that if a function has certain properties, then it will be the value function. This allows us to show heuristically, as was shown in Lungu and $Ø$ksendal (Bernoulli '01), that it is almost surely never optimal to harvest or seed from more than one population at a time. We approximate the continuous-time systems by Markov chains and show that the optimal harvesting-seeding strategies of the Markov chain approximations converge to the correct optimal harvesting strategy. This is used to provide numerical approximations to the optimal harvesting-seeding strategies and is a first step towards a full understanding of the intricacies of how one should harvest and seed interacting species. In particular, we look at three examples: one species modeled by a Verhulst-Pearl diffusion, two competing species and a two-species predator-prey system.

math.PR↗

arXiv2018

Optimal sustainable harvesting of populations in random environments

We study the optimal sustainable harvesting of a population that lives in a random environment. The novelty of our setting is that we maximize the asymptotic harvesting yield, both in an expected value and almost sure sense, for a large class of harvesting strategies and unstructured population models. We prove under relatively weak assumptions that there exists a unique optimal harvesting strategy characterized by an optimal threshold below which the population is maintained at all times by utilizing a local time push-type policy. We also discuss, through Abelian limits, how our results are related to the optimal harvesting strategies when one maximizes the expected cumulative present value of the harvesting yield and establish a simple connection and ordering between the values and optimal boundaries. Finally, we explicitly characterize the optimal harvesting strategies in two different cases, one of which is the celebrated stochastic Verhulst Pearl logistic model of population growth.

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

Quasistationary distributions for one-dimensional diffusions with singular boundary points

In the present work we characterize the existence of quasistationary distributions for diffusions on $(0,\infty)$ allowing singular behavior at $0$ and $\infty$. If absorption at 0 is certain, we show that there exists a quasistationary distribution as soon as the spectrum of the generator is strictly positive. This complements results of Collet et al. (Ann. Probab. 2009) and Kolb and Steinsaltz (Ann. Probab. 2012) for $0$ being a regular boundary point and extends results by Collet et al. (Ann. Probab. 2009) on singular diffusions.

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

Stochastic population growth in spatially heterogeneous environments: The density-dependent case

This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout $n$ patches whose population abundances are modelled as the solutions of a system of nonlinear stochastic differential equations living on $[0,\infty)^n$. We prove that $r$, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter $r$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $r>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\infty)^n$, while when $r<0$, the population abundances of the patches converge almost surely to $0$ exponentially fast. This generalizes and extends the results of Evans et al (2014 J. Math. Biol.) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. As an example we fully analyze the two-patch case, $n=2$, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al.

math.PR↗

arXiv2017

Coexistence and extinction for stochastic Kolmogorov systems

In recent years there has been a growing interest in the study of the dynamics of stochastic populations. A key question in population biology is to understand the conditions under which populations coexist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of $n$ populations that live in a stochastic environment and which can interact nonlinearly (through competition for resources, predator-prey behavior, etc.). Our models are described by $n$-dimensional Kolmogorov systems with white noise (stochastic differential equations - SDE). We give sharp conditions under which the populations converge exponentially fast to their unique stationary distribution as well as conditions under which some populations go extinct exponentially fast. The analysis is done by a careful study of the properties of the invariant measures of the process that are supported on the boundary of the domain. To our knowledge this is one of the first general results describing the asymptotic behavior of stochastic Kolmogorov systems in non-compact domains. We are able to fully describe the properties of many of the SDE that appear in the literature. In particular, we extend results on two dimensional Lotka-Volterra models, two dimensional predator-prey models, $n$ dimensional simple food chains, and two predator and one prey models. We also show how one can use our methods to classify the dynamics of any two-dimensional stochastic Kolmogorov system satisfying some mild assumptions.

math.PR↗

arXiv2017

Stochastic Lotka-Volterra food chains

We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and $n-1$ predators. The $j$th predator eats the $j-1$th species and is eaten by the $j+1$th 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 an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting. As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction. Another novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.

math.PR↗

arXiv2016

Markov processes conditioned on their location at large exponential times

Suppose that $(X_t)_{t \ge 0}$ is a one-dimensional Brownian motion with negative drift $-μ$. It is possible to make sense of conditioning this process to be in the state $0$ at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to $0$, then the limit of the killed Markov process evolves like $X$ conditioned to hit $0$, after which time it behaves as $X$ killed at the last time $X$ visits $0$. Equivalently, the limit process has the dynamics of the killed "bang--bang" Brownian motion that evolves like Brownian motion with positive drift $+μ$ when it is negative, like Brownian motion with negative drift $-μ$ when it is positive, and is killed according to the local time spent at $0$. An extension of this result holds in great generality for Borel right processes conditioned to be in some state $a$ at an exponential random time, at which time they are killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the "bang--bang" construction for general Markov processes. As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.

math.PR↗

arXiv2015

Transient one-dimensional diffusions conditioned to converge to a different limit point

Let $(X_t)_{t\geq 0}$ be a regular one-dimensional diffusion that models a biological population. If one assumes that the population goes extinct in finite time it is natural to study the $Q$-process associated to $(X_t)_{t\geq 0}$. This is the process one gets by conditioning $(X_t)_{t\geq 0}$ to survive into the indefinite future. The motivation for this paper comes from looking at populations that are modeled by diffusions which do not go extinct in finite time but which go `extinct asymptotically' as $t\rightarrow \infty$. We look at transient one-dimensional diffusions $(X_t)_{t \geq 0}$ with state space $I=(\ell, \infty)$ such that $X_t\rightarrow \ell$ as $t\rightarrow \infty$, $\mathbb{P}^x$-almost surely for all $x\in I$. We `condition' $(X_t)_{t \geq 0}$ to go to $\infty$ as $t\rightarrow \infty$ and show that the resulting diffusion is the Doob $h$-transform of $(X_t)_{t\geq 0}$ with $h=s$ where $s$ is the scale function of $(X_t)_{t\geq 0}$. Finally, we explore what this conditioning does in two examples.

math.PR↗

arXiv2015

When do skew-products exist?

The classical skew-product decomposition of planar Brownian motion represents the process in polar coordinates as an autonomously Markovian radial part and an angular part that is an independent Brownian motion on the unit circle time-changed according to the radial part. Theorem 4 of Liao (2009) gives a broad generalization of this fact to a setting where there is a diffusion on a manifold $X$ with a distribution that is equivariant under the smooth action of a Lie group $K$. Under appropriate conditions, there is a decomposition into an autonomously Markovian "radial" part that lives on the space of orbits of $K$ and an "angular" part that is an independent Brownian motion on the homogeneous space $K/M$, where $M$ is the isotropy subgroup of a point of $x$, that is time-changed with a time-change that is adapted to the filtration of the radial part. We present two apparent counterexamples to Theorem 4 of Liao (2009). In the first counterexample the angular part is not a time-change of any Brownian motion on $K/M$, whereas in the second counterexample the angular part is the time-change of a Brownian motion on $K/M$ but this Brownian motion is not independent of the radial part. In both of these examples $K/M$ has dimension $1$. The statement and proof of Theorem 4 from Liao (2009) remain valid when $K/M$ has dimension greater than $1$. Our examples raise the question of what conditions lead to the usual sort of skew-product decomposition when $K/M$ has dimension $1$ and what conditions lead to there being no decomposition at all or one in which the angular part is a time-changed Brownian motion but this Brownian motion is not independent of the radial part.

math.PR↗

arXiv2015

The free path in a high velocity random flight process associated to a Lorentz gas in an external field

We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.

math.PR↗

arXiv2014

Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments

We consider a population living in a patchy environment that varies stochastically in space and time. The population is composed of two morphs (that is, individuals of the same species with different genotypes). In terms of survival and reproductive success, the associated phenotypes differ only in their habitat selection strategies. We compute invasion rates corresponding to the rates at which the abundance of an initially rare morph increases in the presence of the other morph established at equilibrium. If both morphs have positive invasion rates when rare, then there is an equilibrium distribution such that the two morphs coexist; that is, there is a protected polymorphism for habitat selection. Alternatively, if one morph has a negative invasion rate when rare, then it is asymptotically displaced by the other morph under all initial conditions where both morphs are present. We refine the characterization of an evolutionary stable strategy for habitat selection from [Schreiber, 2012] in a mathematically rigorous manner. We provide a necessary and sufficient condition for the existence of an ESS that uses all patches and determine when using a single patch is an ESS. We also provide an explicit formula for the ESS when there are two habitat types. We show that adding environmental stochasticity results in an ESS that, when compared to the ESS for the corresponding model without stochasticity, spends less time in patches with larger carrying capacities and possibly makes use of sink patches, thereby practicing a spatial form of bet hedging.

math.PR↗

arXiv2014

Killed Brownian motion with a prescribed lifetime distribution and models of default

The inverse first passage time problem asks whether, for a Brownian motion $B$ and a nonnegative random variable $ζ$, there exists a time-varying barrier $b$ such that $\mathbb{P}\{B_s>b(s),0\leq s\leq t\}=\mathbb{P}\{ζ>t\}$. We study a "smoothed" version of this problem and ask whether there is a "barrier" $b$ such that $ \mathbb{E}[\exp(-λ\int_0^tψ(B_s-b(s))\,ds)]=\mathbb{P}\{ζ>t\}$, where $λ$ is a killing rate parameter, and $ψ:\mathbb{R}\to[0,1]$ is a nonincreasing function. We prove that if $ψ$ is suitably smooth, the function $t\mapsto \mathbb{P}\{ζ>t\}$ is twice continuously differentiable, and the condition $0<-\frac{d\log\mathbb{P}\{ζ>t\}}{dt}<λ$ holds for the hazard rate of $ζ$, then there exists a unique continuously differentiable function $b$ solving the smoothed problem. We show how this result leads to flexible models of default for which it is possible to compute expected values of contingent claims.

q-fin.RM↗