∞
Lemma 3. a. If there is an n ≥ 0 such that Ca(0,fn(y)) < ^δiDx(fi(y))fxi (y)then the invasion is a
i=n
controlled invasion from y.
b. If Ca(0,y) < Dx(y) then the invasion is currently controlled from y.
c. If Ca(0,y) < Dx(y) + δ[infα {Ca(a,f(y)) + Cy(a,f(y))}]fx(y) for all y > 0 then the invasion is controlled
globally.
The first part of Lemma 3 provides a weak criteria for control to be optimal at some point. It says that if
the invasion is allowed to grow unrestricted and, if at some future date the marginal cost of starting to
control the invasion is less than the discounted stream of future marginal damages from that time onward
into the indefinite future, then it is optimal to control the invasion at some point. The second part of
Lemma 3 says that if the current marginal damages from an invasion of size y exceed the current
marginal costs of starting to control the invasion at size y then control is optimal when an invasion is size
y. This is because even a myopic social planner would undertake positive control in such a situation. If
the inequality in part (b) is true for all y > 0, then the invasion is globally controlled. The latter holds for
example, if the costs of the initial increment in control are negligible or Ca(0,y) = 0, and if the marginal
damages from an invasion are always strictly positive, i.e. Dx(y) > 0 for all y > 0. Part (c) of Lemma 3
provides a somewhat weaker condition for an invasion to be globally controlled by comparing the
marginal cost of an arbitrarily small control to the savings in current marginal damage and future
marginal damages and control costs. When a change in control has a larger effect on marginal costs than
a change in the invasion size, then the requirement of Lemma 3c simplifies to Ca(0,y) < Dx(y) +
δCa(0,f(y))fx(y).4 A final observation is that in many instances the marginal costs of control will be
4When Caa+Cay ≥ 0 then infa Ca(a,y)+Cy(a,y) = Ca(0,y) + Cy(0,y) = Ca(0,y), where the last equality follows
from B2.
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