Controlling Forest Damage by Dispersive Beaver Populations: Centralized Optimal Management Strategy

The beaver (Castor canadensis) population in the United States has caused severe damage to valuable timberland through dam-building and flooding of bottomland forest. Traditionally, beavers have provided a source of livelihood to a small group of people. However, recent low pelt prices have failed to stimulate adequate trapping pressure, and thus have resulted in increased beaver populations and damage losses. The low trapping pressure has left the burden of nuisance control on property owners. Since beaver populations are mobile, beaver extermination in controlled parcels results in beaver immigration from neighboring less controlled parcels. Beaver migration from less controlled to controlled parcels imposes an external cost (negative diffusion externality) on the owners of controlled parcels because they must incur the future cost of trapping immigrating beavers. Unless all land owners agree to control the beaver population simultaneously, the diffusion externality can decrease the incentive of individual landowners to control nuisance beavers, thereby driving a wedge between social and private needs for such control. This study attempts to develop a bioeconomic model that incorporates dispersive population dynamics of beavers into the design of a cost-minimizing trapping strategy. Attention is focused on the situation where all landowners in a given habitat share a common interest in controlling beaver damages, and thus collectively agree to place the area-wide control decision in the hands of a public agency on a cost-sharing basis. The public manager is assumed to minimize the present value of combined timber damage and trapping costs over a finite period of time, subject to spatiotemporal dynamics of beaver population. These dynamics are summarized by a parabolic diffusive Volterra-Lotka partial differential equation, and the population control problem is cast in the framework of a distributed-parameter-control model. The cost-minimizing area-wide trapping model accounts for net migration at each location and time, and characterizes the beaver-control strategy that leaves sufficient beavers to strike an optimal balance between timber damage and trapping costs. The marginality condition governing this trade-off requires that avoided timber damage (measured as the imputed nuisance value, or "shadow price," of the beaver stock in the area) be balanced by trapping cost. The optimality system for this problem is solved numerically. The validity of the theoretical model is empirically examined using the bioeconomic data collected for the Wildlife Management Regions of the New York State Department of Environmental Conservation. Empirical simulation generates discrete values for optimal beaver densities and trapping rates across all individual operational units over time. The optimal trapping program causes the initially uneven population distribution to eventually smooth out across the habitat. The sensitivity analysis alternates trapping-cost and timber-damage parameters between high and low values. Increased trapping costs decrease the level of trapping in the initial years of the optimal program, thereby leaving more beavers in the habitat. This triggers more intensive trapping during the later years of the program, requires more beavers to be trapped over the entire time horizon, and results in a higher overall program cost. Alternatively, increased timber-damage potential calls for increased trapping in the initial years of the program. Fewer beavers are maintained in the habitat and less trapping is required in the later years. Perhaps surprisingly, this results in a smaller number of beavers trapped over the entire time horizon.