Working Together: Optimal Control of Wolf Management Across Multiple States

Abstract

The reintroduction of the gray wolf (Canis lupus) has been largely successful in the upper Rocky Mountain region (URM). This led the federal government to hand over the responsibility of managing the species to the individual states of Idaho, Montana, and Wyoming. As each state currently works mainly independently, this study examines if there are any spillover effects to jointly managing wolves in the region. We develop theoretical optimal control and system dynamics bioeconomic models to determine the steady states for the number of wolves, their management, and corresponding net benefits for Idaho, Montana, Wyoming, and the region as a whole from 2000 to 2030. Results from the models show potential benefits when states work together in the form of greater economic efficiencies in management and potentially larger wolf populations. Using a system dynamics model, we find the optimal management path under three different management scenarios with the possibility of improving net benefits by almost $1 million per year when states work together. Our results provide meaningful insights for policymakers which could potentially impact how states approach management of a species that can be both expensive and controversial.

Appendices

Appendix A: Example of Hypothetical Growth Function

In this Appendix we show a graph of a hypothetical logistic growth function with respect to the wolf population dynamics. The y-axis represents the wolf population while the x-axis represents time.

In this hypothetical logistic function example, the initial wolf population starts at W₀ with low growth rates. After a period of time the population experiences increasing growth rates, reaches maximum growth and the growth rates decrease until the population reaches the carrying capacity (Fig. 6).

Fig. 6

Hypothetical growth function

Appendix B: Full Solution to the Joint Management Model

From Sect. 3.2 we obtain the Hamiltonian of the joint management model:

$$ H = \mathop \sum \limits_{n = 1}^{N} \left[ {\left( {B\left( {w\left( t \right)} \right)_{n} - D\left( {w\left( t \right)} \right)_{n} - C\left( {m\left( t \right)} \right)_{n} } \right) + \mu_{n} \left[ {f_{n} \left( {w_{n} , \ldots ,w_{N} } \right) - m_{n} } \right]} \right] $$

Taking the first order conditions we get:

$$ \begin{array}{*{20}c} { H_{{m_{n} }} = C_{mn} - \mu_{n} = 0\quad \forall n = 1, \ldots , N} \\ \end{array} $$

(12)

$$ \begin{array}{*{20}c} { - H_{{w_{n} }} = \dot{\mu }_{n} - \delta \mu_{n} = - B_{{w_{n} }} + D_{wn} - \mathop \sum \limits_{n = 1}^{N} \mu_{n} f_{{w_{n} }} \quad \forall n = 1, \ldots , N} \\ \end{array} $$

(13)

Taking the time derivative of Equations (A.1) we obtain the following:

$$ \begin{array}{*{20}c} {C_{{m_{n} m_{n} }} \dot{m}_{n} = \dot{\mu }_{n} \to \dot{m}_{n} = \frac{{\dot{\mu }_{n} }}{{C_{{m_{n} m_{n} }} }}} \\ \end{array} $$

(14)

substituting in $\mu_{n}$ and $\dot{\mu }_{n}$ into Equation (14) and rearranging gives the optimal management paths given by Eq. (6) in the text:

$$ \dot{m}_{n} = \frac{{C_{{m_{n} }} \delta - B_{wn} + D_{{w_{n} }} - \mathop \sum \nolimits_{n = 1}^{N} C_{mn} f_{{n,w_{n} }} }}{{C_{{m_{n} m_{n} }} }} \quad \forall n = 1, \ldots , N $$

Appendix C: Phase Diagram of the Single and Joint State Management Model

In this Appendix we assume a shape of the $\dot{m} = 0$ isocline to illustrate some of the important dynamics of the single and joint state model. Rondeau (2001) shows through very similar dynamics that the shape of the $\dot{m} = 0$ isocline is two discontinuous curves, and we use for illustrations purposes.⁴ Figure 7 shows the phase diagram in the population (w)-management (m) space with three potential equilibria (when $\dot{m} = 0$ and $\dot{w} = 0$ cross each other). The discontinuous isocline m represents that of the single state and $m_{n}$ is that of the joint-state problem. In the joint problem the wolf population is now based on all wolves in the region, the $\dot{w} = 0$ isocline gets shifted upward and outward compared to the single state problem. As mentioned in the text, the equilibrium condition for $\dot{m} = 0{ }$ given by Eq. (11) suggests that investment in wolves can come from surrounding states as well. Thus, it seems plausible that this spillover effect would shift the individual $\dot{m}$ isocline curves in the joint problem downward compared to the single state problem. Under these assumptions the optimal solution of the individual state management (harvesting) level would drop, from $m^{*}$ to $m_{n}^{*}$, and theoretically costs less. This all relies on the assumption that the number of wolves is high enough that the path is heading towards the furthest equilibrium. However, near the middle equilibrium it can be seen that individual management may actually increase with cooperation.

Fig. 7

Comparison of two management options. Individual management is denoted by a solid line and the joint management is in the dashed line

Appendix D: Empirical Model Variables and Additional Results

In Table

Table 2 Calibrated and Estimated variables used in the empirical models

2 we provide all the variables that were used in the empirical model. Inputted data came from a variety of sources listed in the last column of the table and ranged from individual state monitoring reports to previous estimations in the literature. Estimated variables were generated using regression models and historical data. Finally, we make assumptions on the management levels for the three different scenarios based on the cumulative management costs of the three states over time.