Harvesting and seeding of stochastic populations: analysis and numerical approximation

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

“…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).…”

Harvesting of a stochastic population under a mixed regular-singular control formulation

“…The function f : R + × U → R is the harvesting-renewing rate corresponding to the control C(•). In [8], the authors consider the case f (x, c) = c for any c ∈ U. We can also consider f (x, c) = cx for (x, c) ∈ [λ, κ] × U, where κ is a positive constant.…”

“…The paper [6] focuses on sustainable harvesting policies under longrun average criteria, which are further studied in [18]; see also [21] for a related work on a predator-prey system and [15] for an ergodic two-sided singular control formulation. In [7,8], the authors study ecosystems that allow for the modelling of both renewing and harvesting. Optimal exploitation problems of renewable natural resources, which are harvesting-type problems, are studied in [14,27].…”

“…Moreover, we also consider the control objective associated with renewing and harvesting. It should be noted that in optimal harvesting formulations in [1,2,25,7,8], the control cost is simply combined with the price functions. As a result, it is frequently seen that when the manager decides to harvest (resp., renew), she should do that with the maximum possible harvesting rates (resp.…”

“…As a result, it is frequently seen that when the manager decides to harvest (resp., renew), she should do that with the maximum possible harvesting rates (resp. renewing rates); see [1,7,8]. In [14,27], interesting phenomena appear when the control objective is taken into account.…”