The name is absent

Lemma 1. V(y) is continuous and non-decreasing.

Lemma 1 formalizes the intuitive notions that incremental changes in the size of an invasion are
associated with small changes in social cost and that larger invasions involve higher social costs.

Sensitivity of the optimal policy depends on how the costs of control vary with control and the
invasion size. In a nonconvex model the optimal policy may be multivalued. That is, there may be more
than one optimal control from a given invasion size. Consequently, our characterization is based on the
properties of a correspondence. Let x e X(y) and x' e X(y') where y ≤ y'. A correspondence X(y) is an
ascending correspondence if min[x,x'] e X(y) and max[x,x'] e X(y'). Similarly, X(y) is descending if
max[x,x'] e X(y) and min[x,x'] e X(y').

Lemma 2. (a) If C_aa(a,y) + C_ay(a,y) ≥ 0 on Ω, then X(y) is an ascending correspondence and the
maximal and minimal selections from X are non-decreasing functions. If the inequality is strict then
every selection from X is non-decreasing. (b) Assume C_aa(a,y) + C_ay(a,y) ≤ 0 on int Ω. If there exists
some y < K such that 0 < X(y) < y then there is a neighborhood N(y) of y such that X(∙) is descending on
N(y) and the maximal and minimal selections from X are non-increasing functions on N(y).

The economic requirement of the first part of Lemma 2 is that a change in control has a larger
effect on marginal costs than a change in the size of the invasion. This provides an economic criterion
for the optimal size of the invasion to evolve monotonically over time. Since the optimal invasion size is
bounded, every invasion with a monotonic time path must necessarily converge to a positive steady state
or zero (eradication). If two invasions differ only in their initial size, then the invasion that is larger
today will be (weakly) larger at all points in the future.

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