Optimal harvesting strategies for stochastic competitive Lotka–Volterra ecosystems

Tran, Ky; Yin, George

doi:10.1016/j.automatica.2015.03.017

Cited by 25 publications

(31 citation statements)

References 16 publications

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“…-There is a growing interest to study the associate harvesting problems [28]. To study the harvesting strategies with systems proposed in this paper has not been done to date and is a worthwhile direction.…”

Section: Discussionmentioning

confidence: 99%

Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

Bui

Yin

2019

Stochastic Analysis and Applications

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This work is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the paper is to reduce the computational complexity by using the two-time-scale systems. Because existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this paper that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods. Lotka [17] and Volterra [29], the well-known Lotka-Volterra models have been investigated extensively in the literature and used widely in ecological and population dynamics, among others. When two or more species live in close proximity and share the same basic requirements, they usually compete for resources, food, habitat, or territory. Initially being posed as a deterministic model, subsequent study has taken randomness into consideration; see [6,10,24] and references therein. Recent effort on the so-called hybrid systems has much enlarged the applicability of Lotka-Volterra systems. One class of such hybrid systems uses a continuous-time Markov chain to model environmental changes and other random factors not represented in the usual stochastic differential equations; see [33] for a comprehensive study of switching diffusions. Introduced byDeterministic Lotka-Volterra systems have been studied by many people. A number of important results were obtained. A set of sufficient conditions for the existence of a globally stable equilibrium point in various models of the n-dimensional Lotka-Volterra system was obtained in [16]; limit cycles for some deterministic competitive three-dimensional Lotka-Volterra systems were treated in [30]. Random perturbations to the Lotka-Volterra model were considered in the literature; see for example [3,12] and many references therein, and also [11,13] for up-to-dated progress on stochastic replicator dynamics. Recently, much effort has been devoted to studying the stochastic Lotka-Volterra with regime-switching; see [7,18,27,34,35]. While most recent works focus on Markov chain with a finite state space, to take into consideration of various factors, it is also natural to consider the Markov chain with a countable state space, which is the effort of the current paper.One of our main aims here is to reduce the computational complexity. Assuming that the Markov chain has a two-time-scale structure so that it has a fast changing part and a slowly varying part, we show that the systems under consi...

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Section: Discussionmentioning

confidence: 99%

Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

Bui

Yin

2019

Stochastic Analysis and Applications

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“…Singular stochastic control problems have been studied extensively in various settings. To mention just a few, we refer to [AS98, Alv00, LØ97, SSZ11, HNUW18, AEH18] for single species ecosystems in random environments and [LØ01, TY15,TY17] for interacting populations. The reader can also find analogous results in the setting of corporate strategy [RS96], and optimal dividend strategies [AT97, JYY13,SS11].…”

Section: Introductionmentioning

confidence: 99%

Harvesting of interacting stochastic populations

et al. 2019

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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 Øksendal (Bernoulli '01), that it is almost surely never optimal to harvest or seed from more than one population at a time.It is usually impossible to find closed-form solutions for the optimal harvesting-seeding strategy. In order to by-pass this obstacle we approximate the continuous-time systems by Markov chains. We 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.

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“…Various harvesting models have been used to investigate the optimal harvesting policies of renewable resources (e.g., Beddington and May [30], Neubert [31]). Recently, many authors explored the optimal harvesting models [32][33][34][35][36][37]. According to the catch-per-unit-effort (CPUE) hypothesis, we consider that predators 1 and 2 are subject to exploitation with harvesting effort rates ℎ 1 > 0 and ℎ 2 > 0, respectively.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Analysis and Optimal Harvesting Strategy for a Stochastic Delayed Predator-Prey Competitive System with Lévy Jumps

Liu

Meng

2019

Mathematical Problems in Engineering

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This paper develops a theoretical framework to investigate optimal harvesting control for stochastic delay differential systems. We first propose a novel stochastic two-predator and one-prey competitive system subject to time delays and Lévy jumps. Then we obtain sufficient conditions for persistence in mean and extinction of three species by using the stochastic qualitative analysis method. Finally, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are derived from Hessian matrix method and optimal harvesting theory of delay differential equations. Moreover, some numerical simulations are given to illustrate the theoretical results.

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Optimal harvesting strategies for stochastic competitive Lotka–Volterra ecosystems

Cited by 25 publications

References 16 publications

Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

Harvesting of interacting stochastic populations

Dynamical Analysis and Optimal Harvesting Strategy for a Stochastic Delayed Predator-Prey Competitive System with Lévy Jumps

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