Harvesting of interacting stochastic populations

Abstract

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.

A survey of numerical solutions for stochastic control problems: Some recent progress

“…Assumption (15) guarantees that the transition probabilities in (17) below are well defined. The definition of p h ((x, α), •|ξ) for (x, α) / ∈ G × M is not important since in the analysis of the control problem, the chain will be stopped at the first time it exits G. Using the transition probabilities, we can check that the localconsistency conditions of {(X h n , Λ h n )} in ( 16) are satisfied.…”

“…To proceed, we define the transition probability function p h (•) so that the evolution of {(X h n , Λ h n )} well approximates the local behavior of {(X(t), Λ(t))} given by (7). Under assumption (15), at the diffusion steps, p h (•|0, ξ) and ∆t h (•, •, 0, ξ) are defined as in (17). As a singular control step of the ith component, we define…”

“…Theorem 3.6. Let the chain {(X h n , Λ h n )} be constructed with transition probabilities defined in (17) and (27), X h (•), Λ h (•), B h (•), M h (•), and Z h (•) be the continuoustime interpolation defined in (28)-( 29), {u h n } be an admissible control, and m h (•) be its relaxed control representation. Let T h (•) be the rescaled time processes given by Definition 3.5, X h (•), Λ h (•), B h (•), M h (•), m h (•), τ h , Z h (•) be the corresponding rescaled processes and denote…”

“…Singular Control Formulation. We consider d species interacting nonlinearly in a stochastic environment with harvesting; see [53,54] for details and see also [17,18]. Let X i (t) be the population size of species i at time t and The payoff functional is defined by For each (x, α), the set of admissible controls A x,α can be defined as that for (7).…”

Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

“…1 Stochastic control is a branch of the optimal control specialized for problems with noise-driven dynamics. 2 Management problems not limited to but include those of energy and resources, 3,4 population, 5,6 environment, 7,8 finance and economics, 9,10 and planning 11,12 have been analyzed as stochastic control problems.…”