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18 Economic Control of Invasive Species diminished by damages D(X(t)), where the invader stock is given by X(t), where X(t) ¼ before the invasion at time t¼ t 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 VB,n,tị ẳ Btị ntị ẵ1 These expenditures also reduce the probability of invasion and so the realization of damages Following Reed (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 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 control 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 &Z t ' maxs Et ert Vn,tịdt ỵ ert JX Xtịị ẵ7 cntị,btịị ẳ limDt-0 fPrInvasion int,t ỵ Dtịjno invasion at tị=Dtg ẵ2 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 subscripts indicate partial derivatives in what follows) The probability of no invasion having occurred up to time t is given by the survivor function Z t cntị,btịịdv ẳ eytị Sp tị ẳ e ½3 where the change of variables allows one to define y ẳ cn,bị, y0ị ẳ so that Sp(0)¼ 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 FðXðtÞÞ À K ðhðtÞÞ, XðtÞ ¼ Xt , t A ft,Ng ½4 More money spent on control can be expected to remove more of the invader, so Kh40 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 DX40 and Dao0 Let the flow of social net benefits from time t onward be given by VX so that VXB,X,a,h,tị ẳ Btị DXtị,atịị htị atị ẵ5 The expected net present value of net benefits earned over an infinite horizon is given by &Z t ' Z N ert VXX,a,h,tịdt ẵ6 ert Vn,tịdt ỵ J ẳ Et t 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 manageà ment from R N t through time to be given by JX Xtịị ẳ max h,a t eÀrt VX ðX,a,h,t Þ dt (where the star indicates the function has been optimized by the optimal choices of hÃ(t) and aÃ(t) through time), it can be seen that JXà depends 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 Ã(X(t)) is the solution to the standard renewable resource model of an optimally controlled invasion as developed in Eiswerth and Following the steps laid out in the Appendix A, eqn [7] can be rewritten in a manner similar to that provided by Reed and Heras (1992) Z N ẵVn,tị ỵ cn,bịJX Xtịịertytị dt ẵ8 J ẳ max n subject to y ẳ cn,bị, y0ị ẳ 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 Pontryagin’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 conditional current value Hamiltonian H ẳ Vnị ỵ cn,bịJX ỵ rcn,bị ½9 where r is the costate variable for y This deterministic formulation 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 ẳ V nị ỵ cn ẵ JX ỵ r ẳ qn ẵ10 given the evolution of y follows y ẳ cn,bị, y0ị ẳ ẵ11 and the evolution of r is r ẳ rr ỵ cn,bịị ỵ Vnị ỵ cn,bịJX which has solution (Horan and Fenichel, 2007) Z N rtị ẳ ẵVnị ỵ cn,bịJX erstịystị ds t ẵ12 ẵ13 The implication (Reed and Heras, 1992) is that À 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 JXÃ(X,aÃ,h Ã) (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 behavior – prevention, control, and adaptation The problem is now one defined by endogenous risk
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