Optimal management of spatial-dynamical natural resource systems: The addition of dispersal controls to traditional metapopulation models
Bioeconomic systems include systems with an invasive species, endangered species that require protection measures, and/or species harvested for their economic value such as commercial fisheries. Such systems are often characterized by spatial heterogeneity, which can be driven by factors including differences in area-specific benefits and costs as well as linkages across the network of locations (e.g. fish ladders and terrestrial connectivity corridors). In this paper we develop a model to explore the impact of managing linkages between locations of a spatially and temporally heterogeneous resource.
We develop a discrete-space metapopulation model allowing for connectivity over space and patch heterogeneity in the cost parameters, patch values, growth rates, and carrying capacities. There are two types of controls available to the manager: extraction levels in each patch and dispersal control in each direction (from patch 1 to 2 and from 2 to 1).
We solve the problem of optimizing the present value of the resource rent subject to stock dynamics, extraction limits, dispersal control limits, non-negativity constraints, and initial conditions by formulating it as non-linear optimal control problem and using numerical methods. Specifically, we use pseudospectral collocation to solve for the optimal dynamics of extraction and dispersal control over time.
Dispersal controls can increase rent relative to no dispersal controls, both in the steady state and on the path to the steady state. We calculate the optimal steady-state profitability when dispersal controls are available relative to no dispersal controls, varying the extraction costs in patch 1 (high cost patch) and the dispersal cost and find over a 40% increase in profitability in some cases. In traditional models that only utilize extraction controls to address the stock and spatial externalities, the regulator is “forced” to control dispersal with extraction rates and in so doing, there are circumstances where closures are optimal. Under those same conditions, we highlight in the presence of both controls, how optimal extraction rates are no longer zero.
We also find that the gain relative to the no dispersal case is greatest when the difference in extraction costs between the two patches is greatest, making it more attractive to transfer biomass to the low cost patch. This suggests that low or no dispersal control will be optimal (and therefore also the cost associated with ignoring potential dispersal control) when there is little difference in extraction costs between patches and when the cost of dispersal is relatively high.