A Differential Game Model for Optimal Management of Wolf-Livestock Conflict

Since the early 1990s, the reintroduction of gray wolves (Canis lupus) in the Northern Rocky Mountain region has created a discourse between livestock ranchers and wolf conservationists. Wolf predation of livestock is known to increase costs for ranchers, but conservationists maintain the ecological benefits of wolf reintroduction outweigh these adverse costs to ranchers. Given these conflicting objectives, we study the interaction between ranchers and conservationists in a differential game setting to determine an optimal wolf–livestock conflict management policy in the Northern Rocky Mountain region. The use of differential games allows the optimal strategy to take into account (i) the interaction of conservationists and ranchers and (ii) the influence of this interaction on the population of livestock and wolves. To our knowledge, this approach is novel in the study of wolf (and other endangered species) management as other studies do not consider how the give–and–take relationship between ranchers and conservationists influences wolf–livestock conflict management policies.

Utilizing up–to–date data from federal and state agencies, including wolf hunting and predation, and the model is numerically solved. We obtain optimal solutions of the livestock population, wolf population, number of livestock attacked, and number of wolves hunted over time, which closely match the magnitude and trajectories of the data. In particular, we find optimal hunting effort by ranchers consistently increases throughout the differential game while optimal wolf protection measures by conservationists follows a parabolic path by decreasing in the first portion of the game and increasing in the second portion. These results help to provide policy implications to determine an optimal strategy that balances the damages caused by wolf predation of livestock with the environmental benefits of wolves.

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