The name is absent

Assumption A2 implies that once an invasion is eradicated it cannot recur. This paper does not
address situations where re-invasion is a serious concern. Assumption A3 says that the invasion growth
function is increasing in the size of the invasion. To be successful an invasive species must necessarily
be able to sustain an invasion. Assumption A4 implies that an invasion can be sustained from an isolated
occurrence of the species. Assumption A5 simply reflects the fact that the spread of any invasion is
bounded by climatic, geological or ecological factors.

The costs of control and damages caused by the invasion are denoted by C(a,y) and D(x),
respectively. Control costs include both the direct costs of control and any indirect costs that may be
associated with control, such as adverse effects arising from the use of chemicals. Derivatives are
indicated by relevant subscripts, e.g. C_a represents the partial derivative of C with respect to a. Let
Ω ⊂ -¾². be the set defined by {(a,y)∣0≤a≤y≤K}. Costs and damages are assumed to satisfy the
following:

B1. C and D are twice continuously differentiable.

B2. C(0,y) = 0 for all y and D(0) = 0.

B3. Ca(a,y) ≥ 0, Cy(a,y) ≤ 0, and C_β(a,y) + Cy(a,y) ≥ 0 on Ω. D_x(x) ≥ 0.

B4. C is convex in a. D is convex.

Assumption B2 rules out fixed costs and it also implies that C_y(0,y) = 0. Assumption B3 implies that
damages are increasing in the size of the invasion, the costs of control increase as control increases, and
that a given amount of control is cheaper to achieve from larger invasions. The assumption that C_a(a,y) +
C_y(a,y) ≥ 0 means that if y ≤ y' it is less costly to reduce the size of the invasion from y to x than it is to
reduce the size of the invasion from y' to x. Assumption B4 gives standard convexity conditions. We do
not assume that C is jointly convex in a and y. Hence, our model allows for nonconvexities in both the
biological growth function and in the control cost function. It is assumed that A1-A5 and B1-B4 hold
throughout the paper.

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