The name is absent

Proposition 7.a. Suppose there exists an X such that every invasion of size y ≥ f (X) is currently
controlled and that

Ca(y - X, y) > Dx(X) + δ     sup     [(Ca(a, f (X)) + Cy(a, f (X))) fx(x)]

0≤ x < X,0≤ a ≤ f ( x )
.

Then from every initial invasion size y ≥ f (X) , the invasion size in every period is bounded below by
f ( X )

b. Assume C_aa(a,y) + C_ay(a,y) ≥ 0 on Ω. Suppose there exists an X such that for every X ∈ (0, X) ,

C_a(f(x)-x,f(x)) > Dχ(x) + δ[C_a(f(x),f(x))+Cy(f(x),f(x))]f_x(x).

Then from every initial invasion size y ≥ f (X) , the invasion size in every period is bounded below by
f ( X ).

If the marginal costs of reducing the size of the invasion over time exceed the current and future marginal
damages for every invasion larger than f (X) , then it can never be efficient to reduce the invasion size

below f (X) .

6. Application of the Results

This section uses the case of exponential control costs and damages to illustrate the application
of the results. The aim is to demonstrate that the conditions are internally consistent and may be easily
applied when costs and damages belong to specific functional classes. Consider costs and damages given
by C(a,y) = (exp(αa)- 1)exp(-βy) and D(x) = exp(γx). Control costs increase exponentially in the amount
of control, but decrease exponentially with the invasion size. The parameter a represents intrinsic

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