Modeling Environmental Crime in Protected Areas Using the Level Set Method

Control-Theoretic Models of Environmental Crime

“…In [4] this optimization is performed implicitly by using a level set method originally developed for tracking the propagation of interfaces [29]. The authors show that, in an Albers-style radially symmetric setting with unit walking speed of perpetrators, their approach produces the same P (x) as predicted by formula (2.2).…”

“…Arnold et al [4], these restrictions were largely removed, with numerical simulations on arbitrary two-dimensional domains and the terrain directly affecting the speed of perpetrators. The key mathematical tool employed in [4] is the level-set method, developed to track the propagation of interfaces by solving time-dependent Hamilton-Jacobi PDEs [29]. We begin by briefly reviewing both of these models in Section 2.…”

“…A recent paper by D.J. Arnold et al in [4] extended the above general approach to more realistic protected areas. Their model allows for general domain geometries, resource distributions, terrain effects, and PAMs' strategies.…”

“…Since there is no extraction along the path, the motivation for this exclusion might perhaps be based on extraction-unrelated damage to the protected area. While [4] does not present a systematic method for optimizing PAMs' choice of ψ, the authors use the above interpretation to compare the effectiveness of several (ad hoc) patrol strategies staying within the specified patrol budget. Their simulations (using γ = 1, α = 1, and ε = 0.085) are based on the terrain data from Yosemite National Park in California and Kangaroo Island in South Australia, which they have also kindly shared with us to conduct the numerical experiments presented in section 6.…”

“…But Arnold et al describe path-planning that strikes a balance between the time of travel and risk of detection all the way to the boundary -with no provision for switching to quicker/easier paths if authorities manage to intercept the extractors en route and confiscate their haul. Thus, we believe that the model in [4] is more suitable for the setting where the extractors remain oblivious of their detection (perhaps in the case of aerial surveillance) and are only intercepted upon reaching ∂Ω.…”