Interspecies Management and Land Use Strategies to Protect Endangered Species

Abstract

We consider an ecosystem management problem where managers can use habitat creation and predator removal to conserve an endangered species. Predator removal may become particularly important in the face of habitat loss, and ecosystem management strategies that ignore the influence of habitat are likely to be inefficient. Using a bioeconomic model, we show that the marginal impact of prey habitat on predators is a key factor in determining the substitutability or complementarity of habitat and removal controls. Applying the model to the case of the endangered Atlantic-Gaspésie Woodland Caribou (rangifer tarandus caribou), we find that the first-best strategy involves extensive caribou habitat protection and a large predator cull initially, and then substituting habitat investments for predator removal as both populations begin to recover, suggesting that habitat protection and predator removal are effectively substitute controls.

Appendices

Appendix 1: Partial Control Strategies

1.1 Predator Removal-Only Strategy (s = $0, h^{*}=h_{SV}$)

This candidate strategy is found by solving (9), (10) and $\partial H/\partial h = 0$ in (7). This system yields $\uplambda _{P}(P)$ as defined in (11), and $\uplambda _{C}(C,P$) as defined in (12). Take the time derivative of (12), $d \lambda _{C}(C,P)$/dt, and substitute this and $\uplambda _{C}(C,P$) into (9) to solve for $h_{SV}(C,P,0$). This is equivalent to the relation in (13), except that $s$ is now set equal to zero. For this strategy, the optimal trajectory is determined by (1) and (2) substituting $h=h_{SV}(C,P,0$) and $s = 0$.

1.2 Conservation Reserve-Only Strategy (s $> 0, h^{*}=0$)

This candidate strategy is determined by solving (8)–(10). In principle, condition (8) could be used to solve for $s(C,P,\uplambda _{C},\uplambda _{P}$). However, it is not possible to then use (9) and (10) to obtain a state-dependent feedback rule for $s$ of the form $s(C,P)$. The reason is that the problem is nonlinear in $s$, and the standard solution to nonlinear control problems is a differential equation, rather than a feedback rule, for the control. Further complicating matters here is that $s$ simultaneously affects both state variables, effectively implying additional adjustment costs as $s$ cannot control either state very efficiently.

We therefore pursue an alternative approach. First, solve (8) for one of the co-states, say $\uplambda _{P}$, to yield $\uplambda _{P}(C,P,s,\uplambda _{C}$). Next, take the time derivative of this expression to obtain a differential equation of the form $\dot{s}=F(C,P,s,\lambda _C )$, which may be solved jointly with (9) and the ecological system (1)–(2) to yield the optimal trajectory.

Appendix 2: Parameter Values

1.1 Total Land: $L$

The Parc National de la Gaspésie conservation reserve is currently $802 \,\hbox {km}^{2}$ (Mosnier et al. 2003). There are calls to add a $214 \,\hbox {km}^{2}$ park enlargement and a $1234 \,\hbox {km}^{2}$ buffer zone; expansion beyond this $2250 \,\hbox {km}^{2}$ area would encroach on land that has been more intensively logged and could not feasibly support caribou for some time (Cadieux and Guay 2010). This suggests a maximum reserve of $2250 \,\hbox {km}^{2}$. While predators can survive on managed land outside of the $2250 \,\hbox {km}^{2}$ area, such animals are likely to be part of a separate population than those residing inside within the area. Indeed, predators outside the $2250 \,\hbox {km}^{2}$ area are unlikely to interact with caribou inside the reserve, as a minority of coyotes are transitory (with a range of $\sim$2600 $\hbox {km}^{2}$; Mosnier et al. 2008). Hence, we only model predators and caribou that interact within a total area of $L = 2250 \,\hbox {km}^{2}$.

1.2 Caribou Carrying Capacity From Managed Land: ${\upalpha }_{C}$

Land managed for human use is generally unsuitable habitat for caribou. The small fraction of caribou ($\sim$17 %) that travel outside the park remain near the park and in land with reserve-like (“conservation zones”) protection (Mosnier et al. 2008; Plan Recovery 2006). This implies $\alpha _{C} = 0$.

1.3 Caribou Carrying Capacity From Conservation Reserve Land: $\upbeta _{C}$

Courtois and Ouellet (2007) use a carrying capacity of 20 caribou per $100 \,\hbox {km}^{2}$. This implies $\upbeta _{C} = 0.2 L$.

1.4 Caribou Growth Parameter: ${r}_{C}$

Courtois and Ouellet (2007) construct a predator–prey model of wolves and Gaspésie Woodland Caribou. They use logistic growth with an intrinsic growth rate of 0.245 for caribou. We adopt a sigmoid growth function similar to that of Courtois and Ouellet (although their model is nonlinear in caribou carrying capacity, while it is linear here). Intrinsic growth in our model, given the initial land use $s_{0} = 802/ L$, is $r_{C}(\upalpha _{C}+\upbeta _{C} s_{0})=r_{C} \times 160.4$. Equating this to 0.245 yields $r_{C} = 0.0015$.

1.5 Caribou Predation Rate: ${\upgamma }$

We calibrate $\upgamma$ so that the model yields a reasonable steady state when $s=s_{0}$ and $h = 0$. The steady state $C$ and $P$ under these conditions are unknown. We assume a steady state of $C = 96$, which is the smallest recorded caribou population (St. Laurent et al. 2009) and, due to a lack of other estimates, a steady state $P = 270$, which is the extant number of predators (Mosnier et al. 2003). Equation (1) then implies $\upgamma = 0.00036$.

1.6 Predator Growth Parameters: ${r}_{{P}}, {\upalpha }_{{P}}$, and ${\upbeta }_{{P}}$

These parameters are calibrated by simultaneously solving three relations. First, Tanner (1975) suggests an intrinsic growth rate of 0.5 for wolves, which we use because there is a lack of published data for coyote and wolves are physiologically similar to coyotes (black bears likely have a smaller growth rate but coyotes compose most of the predators around the park. In any case, the robustness of the growth rate examined via sensitivity analysis). Intrinsic growth in our model, given the initial land use $s_{0}$, is $r_{P}[\upalpha _{P }\!+\!\upbeta _{P} s_{0}]$. Setting this equal to 0.5 yields $r_{P }\!=\! 0.5/[ \upalpha _{P }\!+\!\upbeta _{P} s_{0},]$. As predators prefer managed lands, we assume $\upbeta _{P} \!=\! -\upalpha _{P}/2$. Finally, we calibrate $\upalpha _{P}$ to ensure the steady state described above ($C \!=\! 96$ and $P \!=\! 270$ when $h \!=\! 0$ and $s_{0} \!=\! 0.356$) exists. In a steady state, Eq. (2) implies $[0.5/(\upalpha _{P }\!-\! \upalpha _{P} (0.356)/2)]\!\times \! [\upalpha _{P }\!-\! \upalpha _{P} (0.356)/2 \!-\! 270] \!-\! 0.00036\!\times \!96 \!=\! 0 \!\Rightarrow \! \upalpha _{P}= 353$. In turn, we can solve for $\upbeta _{P} = -176.5$ and $r_{C} = 0.0017$. These values imply approximate densities of 14 predators/$100\,\hbox {km}^{2}$ on managed land and 7 predators/100 $\hbox {km}^{2}$ in a reserve, which are within observed bounds for coyotes (Patterson and Messier 2001) and black bears (Rogers and Allen 1987).

1.7 Value of Caribou and Predators: ${u}_{{C}}, {u}_{{P}}$

Published estimates of household annual willingness to pay, in 2005 dollars (all dollars are in U.S currency), for a population of woodland caribou ($44.71) and a population of coyotes ($5.49) come from Martín-Lopez et al. (2008); we are not aware of a published nonmarket, nonuse value for black bears, which may differ substantially from coyote and would therefore affect the computation of predator existence values. We take the number of households to be those in the provinces of Quebec (3.16 million, given a population of 7.9 million and an average Canadian household size of 2.5) and New Brunswick (0.3 million, given a population of 0.75 million). Given the total of 3.46 million households and converting monetary values to 2012 dollars using an inflation factor of 1.17, then, $u_{C}$ ln(140) = 3,460,000 $\times$ 44.71 $\times$ 1.17 $\Rightarrow u_{C} = 36{,}766{,}035$, and $u_{P}$ ln(270) = 3,460,000 $\times 5.49 \times 1.17 \Rightarrow u_{P} = 3{,}986{,}302$.

1.8 Cost of Predator Removal: ${c}_{{h}}$

An average coyote hunt costs $17,200 (1983 dollars) and yields a 26.33 % population reduction (Smith et al. 1986). Adjusting to 2012 dollars using an inflation factor of 2.303, this implies $c_{h} \times 0.2633 = 17{,}200 \times 2.303 \Rightarrow c_{h} = 150{,}433$.

1.9 Cost of Conservation Reserve: ${c}_{{R1}}, {c}_{{R2}}$

The opportunity cost of placing land into the reserve is the foregone benefits of activities such as timber and hunting. To our knowledge, there are no reports on the value land in the area. Instead, at the current reserve size, we assume the annual benefit of one more hectare of managed land is $250 (or $25000/$\hbox {km}^{2})$, which from an internet search of real estate on the Gaspé peninsula is within the range of market values for forested land. This implies $c_{R1}+2c_{R2} s_{0} L=c_{R1}+2c_{R2} \times 802 = 25{,}000$. We also assume a cost elasticity of 1.25, yielding $25{,}000/[c_{R1} +c_{R2} s_{0} L] = 1.25$. These conditions imply $c_{R1}= 15{,}000$ and $c_{R2} = 6.23$.