The name is absent

When the size of the invasion has a large effect on the marginal cost of control, as in part (b), this
may result in a non-monotonic optimal policy for the size of the invasion.³ An example of this occurs
when the marginal costs of control for a small invasion are sufficiently high that the optimal policy
involves no control while the invasion is small. At the invasion grows larger, marginal costs decrease
and at some point it may become optimal to reduce the invasion back to very small levels, from which no
control is once again optimal.

Some invasions cause minimal damage and control is not cost effective. It is therefore useful to
first identify the circumstances under which control makes sense. There are different ways to view the
control of an invasion. One may be interested in control from an invasion of a particular size, control of
an invasion of any size, or one may be concerned about control of the invasion immediately or at some
future date. This motivates the following definitions.

Definition. (a) An invasion is a controlled invasion from y if there exists some t such that A(y_t) > 0,
where y_t is optimal from y. (b) An invasion is currently controlled from y if A(y) > 0. (c) An invasion
is controlled globally if A(y) > 0 for all y. (d) An invasion is interior if it is controlled globally and
X(y) > 0 for all y.

Each successive definition of control is more restrictive in the sense that (d) → (c) → (b) → (a).

The next result characterizes the economic conditions that are sufficient for different types of
control. Define the t^th iterate of f(∙) and its derivative by f⁰(y) = y, f^t(y) = f^t-1(f(y)), t = 1,... and f^t_x(y) =
df^t(x)/dx.

³This is true even in a convex model. The non-monotonicity of the optimal policy arises solely from the
fact that C_aa + C_ay ≤ 0.

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