b. If Ca(0,y) >Dx(y)+δ sup
0≤a≤ f (x),0≤x≤y
(Ca(a,f(x))+Cy(a,f(x)))fx(x)
then X(y) = y.
If the damages from an arbitrarily small invasion are less than the marginal costs of removing the entire
invasion, then it is always optimal to allow some of the invasion to remain. There should be no control
in the current period if, given the current invasion size, the marginal costs of initiating control exceed the
maximum current and future marginal damages that can occur. This proposition focuses on the optimal
policy from a given invasion size. For example Proposition 5b can be used to provide a condition for no
control for small invasions. In general, however, the proposition does not rule out the possibility that
eradication is optimal in the long run. That question is the focus of the next result.
Proposition 6. If Caa(a,y) + Cay(a,y) ≥ 0 on Ω and Dx(0) + δCa(0,0)fx(0) < Ca(0,0) then X(y) > 0 for all y
and, in addition, for all y sufficiently close to zero A(y) = 0 and X(y) = y.
For an arbitrarily small invasion, if the damages compounded indefinitely at the discounted expected
intrinsic growth rate are less than the marginal costs of eradicating the invasion then the optimal policy is
not to control the invasion at all when it is sufficiently small. This implies that eventual eradication is not
an optimal strategy from an invasion of any size.
Proposition 6 comes very close to providing necessary and sufficient conditions for sufficiently
small invasions to be uncontrolled. This can be seen by a comparison of Proposition 6 with Proposition 1
evaluated as the invasion size approaches zero.
A final possibility is that eradication is optimal from small invasions but that it is not optimal if
the invasion grows to be large. The last result can be used to help identify such outcomes.
18
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