and for all x e (f'1(y),y), Ca(y - x, y) < Dx(χ) + δ
inf
0≤a≤ f (x)-x
Ca(a,f(x))+Cy(a,f(x))
fx(x).
Then every optimal program (yt,xt,atV satisfies y1 = f(x0) < y0.
To interpret this result, first examine the static case where = 0. Consider the set of controls that
lead to an increase in the size of the invasion next period. If marginal damages exceed marginal costs for
every control in this set, then can never be optimal to allow the invasion to increase in size.5 In the
dynamic case, marginal control costs are compared to the current and future marginal damages, adjusted
for the effect of the invasion size on future control costs. The second term on the right hand side of the
inequality in Proposition 1 is a lower bound on the effect that a reduction in the invasion today has on
future damages and control costs.
We now characterize the economic and biological conditions under which eradication is optimal
in the general sense where the size of invasion is reduced to zero in the long run. Whether eradication is
optimal or not depends on the initial size of the invasion. If Proposition 1 holds for every invasion whose
size is between zero and some positive level, , then if the initial invasion size is below it is optimal to
reduce the size of invasion in every period and over time the invasion is eradicated.
Corollary to Proposition 1. If there exists a >0 such that the invasion is currently controlled from
every y e (Q,f(ζ)), and if
for every y e (Q,ζ] and for all x e (f -1(y),y). Then, (eventual) eradication is optimal from every invasion
of size y e (Q,ζ]. If this condition holds for ζ = K, then global eradication is optimal.
Ca(y-x,y)<Dx(x)+δ
inf
Q≤a≤ f (x)-x
Ca(a,f(x))+Cy(a,f(x))
fx(x)
5The non-convex structure of the dynamic optimization problem is the reason the inequality has
to hold over the entire range of control that would lead to an increase in the size of the invasion.
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