Fig 1.
Flow diagram summarizing the algorithm for finding the optimal treatment.
Fig 2.
The algorithm finds the optimal treatment, which slows the population’s propagation to a given target speed, v, while minimizing the annual cost of treatment (ACT) (simple model). (A) The algorithm begins with a population front, , that has evolved naturally when the species has propagated at a speed v0 (v0 > v; blue line). The algorithm then finds the treatment function
for which the speed of the leftward movement of the front does not exceed v in any location. (B-D) The algorithm finds new front shapes, as well as the treatment functions that hold these fronts propagating leftward at a speed v. In each iteration, the algorithm changes
and
to those for which the ACT is lower. (E-H) The same algorithm as in (A-D) is executed, only here it begins with a piecewise-linear population front (E). In both simulation no. 1 (A-D) and no. 2 (E-H) of the algorithm,
and
converge to a similar shape ((D) is similar to (H)), and we denote
and
of this final outcome of the algorithm (shown in (D) and (H)) as nopt and Aopt, respectively. Parameters are the same in all panels: The target speed is v = 10 km/year, and the dynamics of n follow Eq (1), 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 = 1 km; and R given by Eq (5) with β 1.25 USD‒1 year‒1 and α = 0.2.
Fig 3.
The algorithm finds the optimal treatment, which slows the population’s propagation to a given target speed, v, while minimizing the annual cost of treatment (ACT) (spongy moth population model). The description of the panels is similar to that in Fig 2. (A-D) The algorithm begins with a population front that has evolved naturally (A); it finds front shapes that could be slowed with lower annual costs (B-D), until reaching a population front
for which ACT is minimized. (E-H) The same algorithm as in (A-D) is executed, only here it begins with a piecewise-linear population front (E). As in Fig 2, in both simulations no. 1 (A-D) and no. 2 (E-H) of the algorithm, the population front and the corresponding treatment function converge to a similar shape ((D) is similar to (H)). Units: Distance (x) is shown in units of σ = 10 km: population size (
) is in units of its carrying capacity; treatment cost (
) is in USD per hectare per year; and ACT is given in thousands of USD per one-kilometer strip of land. Parameters are the same in all panels: The target speed is v = 220 m/year, and the dynamics of n follow Eq (2), where b is given by Eqs (6–8) with r = 2, kλ0 = 100, and a = 0.08 USD‒1.
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].
Fig 5.
The annual cost of treatment decreases with the target speed v and increases with kλ0 (spongy moth population model). As in Fig 4, the main (middle) panel shows the annual cost of treatment associated with the optimal treatment, ACT*, as a function of the target propagation speed v for three choices of kλ0. ACT* decreases as v increases, and reaches zero when v equals the species’ natural propagation speed without treatment (v = v0≈12 km year−1). Higher kλ0 values, indicating a lower Allee threshold relative to carrying capacity, resulting in an increased ACT*, where ACT* is measured in units of thousands of USD per one-kilometer strip of land. The dashed black lines show, for each line, where the marginal cost of slowing the spread (slope of ACT*) equals the marginal benefit of slowing the spread by one kilometer over a one-kilometer strip, estimated as 16K USD per year by [62]. In particular, the optimal speed according to this estimation is v = ‒2.4 km/year if kλ0 = 100; v = 4.8 km/year if kλ0 = 300; and v = 9.6 km/year if kλ0 = 1000. In turn, the sub-panels demonstrate the optimal treatment profile (Aopt), and population density (nopt) across locations (x in units of σ = 10 km). Each sub-panel shows the optimal solution for a specific target speed v, with both treatment and population density advancing leftward at this speed (Eqs (9, 11)). All the parameters except α and v are the same in all the panels: Eq (2) is considered, where b is given by Eqs (6–8) with r = 2 and a = 0.08 USD‒1, and G is given by Eq (4). The raw data with the results of all the simulations for each kλ0 and v can be found Dryad [60].