The name is absent

A policy, = ( ₁, ₂,...), is a sequence of decision rules, _t, that specify a plan for controlling the
biological invasion as a function of the previous history, h_t = (y₀,a₀,x₀,...,a_t-1,x_t-1,y_t). That is, a_t = _t(h_t) and
x_t = y_t - _t(h_t). A stationary Markov policy is associated with a pair of decision rules that specify the
control and the size of the invasion that remains at the end of each period as a function of the size of the
invasion at the beginning of the period. Associated with each initial state, y₀, and each policy is a
∞

discounted sum of social costs V_π(y₀) = ^ δ^t[C(a_t,y_t) + D(x_t)], where the sequence {a_t,x_t} is
t=0

generated by the invasion growth function, f, and the policy, , in an obvious manner. The objective of
the dynamic optimization problem is to minimize the discounted sum of costs and damages over time
subject to the transition equation that governs the growth and spread of the invasion. The optimal value
satisfies:
∞

V(y₀) = Min ∑ δ^t [ C ( a_t, y_t ) + D ( x_t )] subject to y_t = a_t+x_t and y_t+1 = f(x_t).                (2.1)

t=0

Under A1-A5 and B1-B4, standard dynamic programming arguments imply that there exists a stationary
optimal value that satisfies the recursion V(y_t) = Min [C(a_t,y_t) + D(x_t) + δV(f(x_t))] subject to 0 ≤ x_t ≤ y_t, y_t
= a_t+x_t and y_t+1 = f(x_t), and that there exists a stationary Markov optimal policy whose decision rules are
X(y) = Arg Min{[C(y-x,y) + D(x) + δV(f(x))^0≤x≤y} and A(y) = y - X(y). A sequence (y_t,x_t,a_tV that
solves (2.1) is an optimal program from y₀. Given an initial invasion of size y₀ = y and a selection x(y)
from the stationary optimal policy X(y), an optimal program is defined recursively by y_t+1 = f(x(y_t)), x_t =
x(y_t), a_t = a(y_t), t =0, 1,2,...

3. Controlled Invasions and their Basic Properties

This section characterizes the basic properties of an optimal policy and the optimal value. The
initial results characterize the sensitivity of the optimal value V(y) and optimal policy X(y) to the size of
the invasion.

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