Ky Tran

Stochastic competitive Lotka–Volterra ecosystems under partial observation: Feedback controls for permanence and extinction

Abstract This paper is concerned with Lotka–Volterra models formulated using stochastic differential equations with regime switching represented by a continuous-time Markov chain. Different from the existing literature, the Markov chain is hidden and can only be observed in a Gaussian white noise in our work. For such partially observed problems, we use a Wonham filter to estimate the Markov chain from the observable evolution of the given process, and convert the original system to a completely observable one. We then establish the regularity, positivity, stochastic boundedness, and sample path continuity of the solution. Moreover, stochastic permanence and extinction using feedback controls are investigated. Numerical experiments are conducted to validate the theoretical findings and demonstrate how feedback controls perform in practice.

Optimal harvesting strategies for stochastic competitive Lotka-Volterra ecosystems

This work develops optimal harvest strategies for Lotka-Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. To better reflect reality, a continuous-time Markov chain is used to model the random environment. The underlying systems are thus controlled regime-switching diffusions that belong to the class of singular control problems. Starting with a model having multiple species, we construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. Moreover, in some cases, the lower bounds provide a good approximation of the value functions.

Hybrid competitive Lotka–Volterra ecosystems with a hidden Markov chain

This work concerns Lotka–Volterra models that are formulated using stochastic differential equations with regime-switching. Distinct from the existing formulations, the Markov chain that models random environments is unobservable. For such partially observed systems, we use Wonham’s filter to estimate the Markov chain from the observable evolution of the population, and convert the original system to a completely observable one. We then show that the positive solution of our model does not explode in finite time with probability 1. Several properties including stochastic boundedness, finite moments, sample path continuity and large-time asymptotic behaviour are also obtained. Moreover, stochastic permanence, extinction and feedback controls are also investigated.

Numerical methods for optimal harvesting strategies in random environments under partial observations

This work is concerned with optimal harvesting problems in random environments. In contrast to the existing literature, the Markov chain is hidden and can only be observed in a Gaussian white noise in our work. We first use the Wonham filter to estimate the Markov chain from the observable evolution of the given process so as to convert the original problem to a completely observable one. Then we treat the resulting optimal control problem. Because the problem is virtually impossible to solve in closed form, our main effort is devoted to developing numerical approximation algorithms. To approximate the value function and optimal strategies, Markov chain approximation methods are used to construct a discrete-time controlled Markov chain. Convergence of the algorithm is proved by weak convergence method and suitable scaling. A numerical example is provided to demonstrate the results.

Asymptotic expansions of solutions for parabolic systems associated with transient switching diffusions

Abstract This work develops asymptotic properties for a parabolic system with two-time scales associated with a transient switching diffusion. Although the problem is motivated by stochastic systems, the techniques that we are using are purely analytic. Asymptotic expansions are constructed; their validity is justified.