Optimizing strategies for slowing the spread of invasive species
Fig 4
The annual cost of treatment decreases with the target speed v and increases with α (simple model). The main (middle) panel shows the annual cost of treatment associated with the optimal treatment, ACT*, as a function of the target propagation speed of the population, v, for various choices of α (Eq (5)). ACT* decreases as v increases, until v equals the natural speed at which the species propagate without treatment v = v0≈50, where ACT* = 0. The ACT* is also higher when α is larger. The sub-panels demonstrate the shape of the optimal treatment profile, Aopt, and the population density, nopt, as a function of the location, x. The three sub-panels on the top row show results for α = 0.6, and those on the bottom row for α = 0. Each sub-panel shows the optimal solution for a given target speed v, where both the treatment and the population density continuously move leftward at that speed (Eqs (9, 11)). The sub-panels demonstrate that, if α = 0, treatment is applied only where n(x) = 0, whereas if α = 0.6, treatment is distributed over broader areas, including some areas where n(x) > 0. If v < 0 (left sub-plots), a large concentration of treatment is peaked at the border between (a) the region where the population density is large and (b) the region where the rest of the treatment is applied. All the parameters except α and v are the same in all the panels: Eq (1) is considered, where b and d are given by Eq (3) with r = 2 year‒1, k = 2, and γ = 1 year‒1; G is given by Eq (4) with σ = 25 km‒1; and R given by Eq (5) with β = 1/(1 ‒ α) USD‒1 year‒1. The raw data with the results of all the simulations for each α and v can be found on Dryad [60].