diminished by damages DXt, where the invader stock isgiven by Xt, where Xt ¼0 before the invasion at time t¼t and Xt40.. Expenditures on prevention nt reduce net benefits before the inva
Trang 1diminished by damages D(X(t)), where the invader stock is
given by X(t), where X(t) ¼0 before the invasion at time t¼t
and X(t)40 Greater levels of X lead to greater damages so
that ðq D=q XÞ40 Expenditures on prevention n(t) reduce net
benefits before the invasion, V, so that
V ðB,n,tÞ ¼ BðtÞ nðtÞ ½1
These expenditures also reduce the probability of invasion
and so the realization of damages FollowingReed (1987), the
probability the invader is introduced at any time t, given that it
has not been introduced up to that point in time, is given by
the effective hazard function
cðn ðtÞ,bðtÞÞ ¼ limDt-0fPrðInvasion inðt,t þ DtÞ no invasion at tÞ=Dtgj
½2
where b(t) is the background hazard or probability of invasion
in the absence of prevention The greater the background
probability of invasion, the greater the hazard cb40 and the
more the prevention reduces the hazard cno0 (where
sub-scripts indicate partial derivatives in what follows) The
probability of no invasion having occurred up to time t is
given by the survivor function
S p ðtÞ ¼ e
Z t
0
cðn ðtÞ,bðtÞ Þdv
¼ eyðtÞ ½3
where the change of variables allows one to define
˙y ¼ cðn,bÞ, yð0Þ ¼ 0 so that S p(0)¼1
Following an invasion at time t, the invader stock grows
following a density-dependent growth function F(X(t)).
Control expenditures h(t) reduce or reverse the growth in the
invader following an invasion through a ‘‘kill function’’
K(h(t)) so that
˙
X ¼ F XðtÞð Þ K hðtÞð Þ, XðtÞ ¼ Xt, tAft,Ng ½4
More money spent on control can be expected to remove
more of the invader, so K h40 Adaptation expenditures a(t)
simply lower realized damages while allowing the invader
stock to remain so that D(X(t), a(t)) with partial derivatives
D X40 and Dao0 Let the flow of social net benefits from time
tonward be given by VX so that
VX ðB,X,a,h,tÞ ¼ BðtÞ D XðtÞ,aðtÞð Þ hðtÞ aðtÞ ½5
The expected net present value of net benefits earned over
an infinite horizon is given by
J ¼ Et
Z t
0
ert V ðn,tÞdt þ
Z N t
ert VX ðX,a,h,tÞdt
½6
where the expectations operator reflects the uncertainty of
invasion time t and r is the discount rate If one defines the
present value of an optimal program of ex post
manage-ment from t through time to be given by JXðX tð ÞÞ ¼
maxh,aR
N
t ert VX X,a,h,tð Þdt (where the star indicates the
function has been optimized by the optimal choices of h(t)
and a(t) through time), it can be seen that JXdepends on
ex ante management through prevention before t through
both the point in time that damages begin to accrue and
the initial stock of the invader in t JX(X(t)) is the
solution to the standard renewable resource model of an
optimally controlled invasion as developed inEiswerth and
Johnson (2002),Brown et al (2002),Lichtenberg and Lynch (2006),Burnett et al (2006,2008),Potapov et al (2007), and
Finnoff et al (2010b) First the ex ante management schemes
are discussed, and then the insight provided by optimal
con-trol for ex post management is explored.
Ex ante Management: Prevention Rewriting eqn [6] allows the objective function for ex ante
prevention to be written as
maxs Et
Z t 0
e rt V ðn,tÞdt þ e rt JXðXðtÞÞ
½7
Following the steps laid out in the Appendix A, eqn [7] can
be rewritten in a manner similar to that provided byReed and Heras (1992)
J¼ maxn
Z N 0
V ðn,tÞ þ cðn,bÞJXðX ðtÞÞ
subject to ˙y ¼ cðn,bÞ, yð0Þ ¼ 0, and the equation of motion in
eqn [4] The method results in a problem of deterministic optimization (Reed, 1987) that can be solved applying Pon-tryagin’s maximum principle (see Kamien and Schwartz,
1991) The beauty of the method is that it incorporates the endogenous risk of invasion directly into the associated con-ditional current value Hamiltonian
H ¼ VðnÞ þ cðn,bÞJXþ rcðn,bÞ ½9
where r is the costate variable for y This deterministic
for-mulation allows the application of the maximum principle (Pontryagin et al., 1962) The associated necessary conditions require prevention to be chosen along the optimal path to maximize the conditional current value Hamiltonian
qH
qn ¼ V0ðnÞ þ c n½JXþ r ¼ 0 ½10
given the evolution of y follows
˙y ¼ cðn,bÞ, yð0Þ ¼ 0 ½11
and the evolution of r is
˙r ¼ r r þ cðn,bÞð Þ þ VðnÞ þ cðn,bÞJX ½12
which has solution (Horan and Fenichel, 2007)
rðtÞ ¼
Z N
t ½VðnÞ þ cðn,bÞJXerðstÞyðstÞ ds ½13 The implication (Reed and Heras, 1992) is that r(t) is
the expected present value of net benefits from the current
time onward, or the ex ante value of an optimally managed
system facing the threat of invasion This value depends on the
state of the system before V(n) and after an invasion
JX(X,a,h) (i.e., the severity) and the probability of invasion c(n,b), where each in turn depends on both the ecological baseline risk of invasion and the key aspects of manager be-havior – prevention, control, and adaptation The problem is now one defined by endogenous risk
18 Economic Control of Invasive Species