Note that if =K, then (4.1) is a condition for global eradication. It is possible, however, to
provide a more direct condition for global eradication by ensuring that there is no positive steady state.
Proposition 3. If Caa(a,y) + Cay(a,y) ≥ 0 on Ω and if
Ca(f(x)-x,f(x)) < Dχ(x) + δ[Ca(f(x)-x,f(x))+Cy(f(x)-x,f(x))]fχ(x)
for all x e (0,K], then every optimal program converges to zero and eradication is globally optimal.
There are three differences between Propositions 2 and 3. First, Proposition 2 relies on a lower bound on
future marginal damages while this is not necessary in Proposition 3. Second, Proposition 3 is a result
about global eradication so the condition in the Proposition is required to hold for every possible
invasion size. On the other hand, Proposition 2 can be used to evaluate the efficiency of eradication from
small invasions; the conditions need not apply when the invasion is large.
Next, we characterize the circumstances under which immediate eradication is optimal, i.e.,
where the invasion is fully eradicated in the current period.
Proposition 4. Suppose that for some y e (0,K], the invasion is currently controlled from every y0 e
(0,f(y)) and that
+ δ inf
0≤a≤ f (x)-x,0≤x≤ y
(Ca(a,f(x))+Cy(a,f(x)))fx(x)
Ca(y,y)<Dx(0)
Then, immediate eradication is optimal from y.
The criterion for immediate eradication balances the costs of removing the last unit of the invasion
against the current and future damages that would be caused should the invasion be allowed to remain.
The second term on the right hand side of the inequalities is a lower bound on the future damages
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