The name is absent

The condition in the corollary implies that for any invasion of size below , it is less costly to reduce the
size of the invasion than to incur the current and future damages should the invasion be allowed to grow.
When discounted growth exceeds one from every invasion size below , then eradication is optimal even
if marginal damages are very low. The rationale is that if a fast growing invasion increases future
damages and control costs more rapidly that the rate of discount then it makes sense not to allow the
invasion to grow and the invasion should be eradicated from its current level. The condition for
eradication becomes stronger with higher values of . Thus, it is generally more likely for eradication to
be optimal from small invasions than from large invasions.

If C is submodular and the marginal costs of control vary more with the amount of control than
with the size of the invasion, then the optimal policy for controlling the invasion is monotone (Lemma
2.a). In that case, the efficiency of eradication depends on the economic and biological conditions at
steady states and a tighter condition for eradication is possible.

Proposition 2. Assume C_aa(a,y) + C_ay(a,y) ≥ 0 on Ω. If there exists a ζ >0 such that the invasion is
currently controlled from every y e (0,f(ζ)], and if

Q(f(x)-x,f(x)) < Dx(x) + δCa(0,f(x))fx(x)                                                     (4.1)

for all x e (0,ζ], then (eventual) eradication is optimal from every invasion of size y₀ e (0,ζ].

A steady state policy is one where a = f(x)-x and the invasion remains constant over time. The condition
in Proposition 2 balances the marginal costs of steady state control against the current marginal damages
plus the lower bound on future marginal damages and control costs. If the current and future marginal
damages from steady state control are higher for all invasions smaller than then eradication is optimal
from all such invasions.

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