Document Type : Original Article


Nature Wise Consulting, 308 Uganda Avenueو Victoria B.C., V9A 5X7, Canada


A growing number of ecological studies suggest ratio-dependence may be common in many predator-prey systems. I assessed Eberhardt’s ratio-dependent wolf-ungulate model using 22 years of wolf (Canis lupus) and elk (Cervus elaphus) survey data, and 28 years of elk hunter harvest data from the northern range of Yellowstone National Park, USA. This model required estimation of eight parameters, of which four were determined from other studies and four were evaluated by statistical model fitting. Modelled trends in wolf and elk abundance were correlated with survey estimates, and modelled trends in wolf predation rates were correlated with empirical estimates. The equilibrium ratio of wolves to elk was 0.0063 and the equilibrium ratio of elk to habitat carrying capacity was 0.680. While this simple model provided a realistic portrait of wolf-elk dynamics in a complex predator-prey system, the performance of the model could be improved by including the sex and age structure of the elk population, a wolf functional response, and temporal variation in wolf and elk demographic rates. This model provides additional support for ratio-dependence in large mammal predator-prey systems.


Arditi R., Ginzburg, L.R. 2012. How species interact: altering the standard view of trophic ecology. Oxford University Press. Oxford.

Barber-Meyer S.M., Mech L.D., White P.J. 2008. Elk calf survival and mortality following wolf restoration to Yellowstone National Park. Wildlife Monographs 169:1-30.

Becker M.S. 2008. Applying predator-prey theory to evaluate large mammal dynamics: Wolf predation in a newly-established multiple-prey system. PhD thesis, Montana State University.  Bozeman, MT.

Burnham K.P., Anderson D.R. 1998. Model selection and inference: a practical information-theoretic approach. Springer-Verlag, New York.

Caughley G. 1977. Analysis of vertebrate populations. John Wiley and Sons, New York. 

Cunningham J. 2016. Elk populations in Montana’s Region 3. Pittman-Robertson Federal Aid in Wildlife Restoration Report. 

Eberhardt L.L. 1997. Is wolf predation ratio-dependent? Canadian Journal of Zoology 75:1940-1944.

Eberhardt L.L. 1998. Applying difference equations to wolf predation. Canadian Journal of Zoology 76:380-386.

Eberhardt L.L. 2000. Reply: Predator-prey ratio dependence and regulation of moose populations. Canadian Journal of Zoology 78:511-513.

Eberhardt L.L., Peterson R.O. 1999. Predicting the wolf-prey equilibrium point. Canadian Journal of Zoology 77:494-398.

Eberhardt L.E., Eberhardt L.L., Tiller B.L., Cadwell L.L. 1996. Growth of an isolated elk population. Journal of Wildlife Management 60:369-373.

Eberhardt L.L., Garrott R. A., Smith D.W., White P.J., Peterson R.O. 2003. Assessing the impact of wolves on ungulate prey. Ecological Applications 13:776-783.

Eberhardt L.L., White P.J., Garrott R.A., Houston D.B. 2007. A seventy-year history of trends in Yellowstone’s northern elk herd. Journal of Wildlife Management 71:594-602.

Haddon M. 2001. Modelling and quantitative methods in fisheries. Chapman and Hall/CRC. New York.

Hebblewhite M. 2013. Consequences of ratio-dependent predation by wolves for elk population dynamics. Population Ecology 55:511-522.

Holling C.S. 1959. The components of predation as revealed by a study of small-mammal predation on the European sawfly. The Canadian Entomologist 91:234-261.

Jost C., Devulder G., Vucetich J.A., Peterson R.O., Arditi R. 2005. The wolves of Isle Royale display scale-invariant satiation and ratio-dependent predation on moose. Journal of Animal Ecology 74:809-816.

Keith L.B. 1983. Population dynamics of wolves, in: L.N. Carbyn, (Ed.). Wolves in Canada and Alaska: their status, biology and management. Canadian Wildlife Service Research Series No. 45. Canadian Circumpolar Institute, University of Alberta, Edmonton, Alberta, Canada. pp. 66-77.

Loveless K. 2017. 2017 Annual winter trend count of Northern Yellowstone elk. Montana Fish, Wildlife and Parks, Bozeman, Montana.

Messier F. 1994. Ungulate population models with predation: a case study with the North American Moose. Ecology 75:478-488.

Metz M.C., Smith D.W., Vucetich J.A., Stahler D.R., Peterson R.O. 2012. Seasonal patterns of predation for gray wolves in the multi-prey system of Yellowstone National Park. Journal of Animal Ecology 81:553-563.

Middleton A.D., Morrison T., Fortin-Noreus J., Robbins C.T., Profitt K.M., White P.J., McWhirter D.E., Koel T.M., Brimeyer D.G., Fairbanks W.S., Kauffman M.J. 2013. Grizzly bear predation links the loss of native trout to the demography of migratory elk in Yellowstone. Proceedings of the Royal Society B 280: 20130870.

Murphy K.M. 1998. The ecology of the cougar (Puma concolor) in the northern Yellowstone ecosystem: Interaction with prey, bears, and human. PhD. Dissertation, University of Idaho, Moscow, ID.

Peterson R.O., Vucetich J.A., Bump J.M., Smith D.W. 2014. Trophic cascades in a multiclausal world: Isle Royale and Yellowstone. Annual Review of Ecology, Evolution, and Systematics 45:325-345.

Reeve J.D. 1997. Predation and bark beetle dynamics. Oecologia 112:48-54.

Skalski G.T., Gillian J.M. 2001. Functional responses of with predator interference: viable alternatives to the Holling type II model. Ecology 82:3082-3092.

Skalski J.R., Ryding K.E., Millspaugh J.J. 2005. Wildlife demography: Analysis of sex, age and count data. Elsevier Academic Press. New York.

Smith D.W., White P.J., Stahler D.R., Wydeven A., Hallac D.E. 2016. Managing wolves in the Yellowstone area: Balancing goals across jurisdictional boundaries. Wildlife Society Bulletin 40: 436-445.

Smith D., Stahler D., Stahler E., Metz M., Cassidy K., Cassidy B., Koitzsch L., Harrison Q., Cato E., McIntyre R. 2017. Yellowstone National Park wolf project annual report 2016. National Park Service, Yellowstone Center for Resources, Yellowstone National Park, WY,
USA, YCR-2017-02.

Taper M.L., Gogan P.J.P. 2002. The northern Yellowstone elk: density dependence and climatic conditions. Journal of Wildlife Management 66:106-122.

Taylor R.J. 1984. Predation. Chapman Hall, New York.

Varley N., Boyce M.S. 2006. Adaptive management for reintroductions: updating a wolf recovery model for Yellowstone National Park. Ecological Modelling 193:315-339.

Vucetich J.A., Smith D.W., Stahler D.R. 2005. Influences of harvest, climate and wolf predation on Yellowstone elk, 1961-2004. Oikos 111:259-270.

Vucetich J.A., Hebblewhite M., Smith D.W., Peterson R.O. 2011. Predicting prey population dynamics from kill rate, predation rate and predator-prey ratios in three wolf-ungulate systems. Journal of Animal Ecology 80:1236-1245.

White P.J., Proffitt K.M., Lemke T.O. 2012. Changes in elk distribution and group sizes after wolf restoration. American Midland Naturalist 267:174-187.

Wright G.J., Peterson R.O., Smith D.W., Lemke T.O. 2006. Selection of northern Yellowstone elk by gray wolves and hunters. Journal of Wildlife Management 70:1070-1078.