Advertisement

Theoretical Ecology

, Volume 7, Issue 1, pp 3–20 | Cite as

Functional responses and predator–prey models: a critique of ratio dependence

  • Frédéric Barraquand
REVIEW ARTICLE

Abstract

Arditi and Ginzburg (2012) propose ordinary differential equations (ODEs) with ratio-dependent functional responses as the new null model for predation, based on their earlier work on ratio-dependent food chains and a number of functional response measurements. Here, I discuss some of their claims, arguing for a flexible and problem-driven approach to predator–prey modeling. Models to understand population cycles and models to predict the effect of basal enrichment on food chains need not be the same. While ratio-dependent functional responses in ODE models might sometimes be useful as limit cases for food chains, they are not intrinsically more useful than prey-dependent models to understand the effect of a given predator on prey population dynamics—and sometimes less useful, given the small temporal scales considered in many models. “Instantism” is showed to be an invalid criticism when ODEs are interpreted as describing average trajectories of stochastic birth–death processes. Moreover, other modeling frameworks with strong ties to time series statistics, such as stochastic difference equations, should be promoted to improve the feedback loop between field and theoretical research. The main problems of current trophic ecology do not lie in a wrong null model, as ecologists have already several at their disposal. The loose connection of ODE models with empirical data and spatial/temporal scaling up of empirical measurements constitute more serious challenges to our understanding of trophic interactions and their consequences on ecosystem functioning.

Keywords

Consumer dependence Autoregressive models Kill rate Predation rate Trophic communities Exploitation ecosystems 

Notes

Acknowledgments

For exchanges on the concepts discussed here, I thank D. J. Murrell, J.A. Henden, J. Matthiopoulos, L. New, S. Redpath, G. Gauthier, and N.G. Yoccoz. D.J. Murrell provided detailed comments on a previous version. Two referees and the editor made constructive suggestions that improved the manuscript. Thanks to the participants of the ResearchGate Stochastic Processes list for help with model formulation in Fig. 2.

References

  1. Abrams PA (1994) The fallacies of “ratio-dependent” predation. Ecology 75(6):1842–1850Google Scholar
  2. Abrams P (1997) Anomalous predictions of ratio-dependent models of predation. Oikos 80(1):163–171Google Scholar
  3. Abrams P, Ginzburg L (2000) The nature of predation: prey dependent, ratio dependent or neither? Trends Ecol Evol 15(8):337–341PubMedGoogle Scholar
  4. Abrams P, Roth J (1994) The effects of enrichment of three-species food chains with nonlinear functional responses. Ecology 75(4):1118–1130Google Scholar
  5. Abrams PA, Walters CJ (1996) Invulnerable prey and the paradox of enrichment. Ecology 77(4):1125–1133Google Scholar
  6. Ahrens RN, Walters CJ, Christensen V (2011) Foraging arena theory. Fish Fish 13(1):41–59Google Scholar
  7. Akcakaya H (1992) Population cycles of mammals: evidence for a ratio-dependent predation hypothesis. Ecol Monogr 62(1):119–142Google Scholar
  8. Andersson M, Erlinge S (1977) Influence of predation on rodent populations. Oikos 29(3):591–597Google Scholar
  9. Arditi R, Ginzburg L (1989) Coupling in predator–prey dynamics: ratio-dependence. J Theor Biol 139(3):311–326Google Scholar
  10. Arditi R, Ginzburg LR (2012) How species interact: altering the standard view on trophic ecology. Oxford University Press, New YorkGoogle Scholar
  11. Barraquand F, Murrell DJ (2012) Evolutionarily stable consumer home range size in relation to resource demography and consumer spatial organization. Theor Ecol 5(4):567–589Google Scholar
  12. Barraquand F, Murrell D (2013) Scaling up predator–prey dynamics using spatial moment equations. Methods Ecol Evol 4(3):276–289Google Scholar
  13. Barraquand F, Inchausti P, Bretagnolle V (2009) Cognitive abilities of a central place forager interact with prey spatial aggregation in their effect on intake rate. Anim Behav 78(2):505–514Google Scholar
  14. Bartlett MS (1957) On theoretical models for competitive and predatory biological systems. Biometrika 44(1/2):27–42Google Scholar
  15. Beddington J (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44:331–340Google Scholar
  16. Begon M, Bennett M, Bowers R, French N, Hazel S, Turner J (2002) A clarification of transmission terms in host-microparasite models: numbers, densities and areas. Epidemiol Infect 129(1):147–153PubMedGoogle Scholar
  17. Berryman A, Michalski J, Gutierrez A, Arditi R (1995) Logistic theory of food web dynamics. Ecology 76(2):336–343Google Scholar
  18. Bêty J, Gauthier G, Giroux JF, Korpimäki E (2001) Are goose nesting success and lemming cycles linked? Interplay between nest density and predators. Oikos 93(3):388–400Google Scholar
  19. Bêty J, Gauthier G, Korpimäki E, Giroux J (2002) Shared predators and indirect trophic interactions: lemming cycles and arctic-nesting geese. J Anim Ecol 71(1):88–98Google Scholar
  20. Bjørnstad O, Stenseth N, Saitoh T (1999) Synchrony and scaling in dynamics of voles and mice in northern Japan. Ecology 80(2):622–637Google Scholar
  21. Bolker B (2004) Continuous-space models for population dynamics. In: Hanski I, Gaggiotti O (eds) Ecology, genetics, and evolution in metapopulations. Academic Press, San Diego, pp 45–69Google Scholar
  22. Brose U, Williams RJ, Martinez ND (2006) Allometric scaling enhances stability in complex food webs. Ecol Lett 9(11):1228–1236PubMedGoogle Scholar
  23. Charnov E, Orians G, Hyatt K (1976) Ecological implications of resource depression. Am Nat 110(972):247–259Google Scholar
  24. Cosner C, DeAngelis D, Ault J, Olson D (1999) Effects of spatial grouping on the functional response of predators. Theor Popul Biol 56(1):65–75PubMedGoogle Scholar
  25. Cox D, Isham V (1980) Point processes. Chapman & Hall/CRC, LondonGoogle Scholar
  26. de Roos A, McCauley E, Wilson W (1991) Mobility versus density-limited predator–prey dynamics on different spatial scales. Biol Sci 246(1316):117–122Google Scholar
  27. DeAngelis DL, Goldstein RA, O’Neill RV (1975) A model for trophic interaction. Ecology 56:881–892Google Scholar
  28. Elmhagen B, Ludwig G, Rushton S, Helle P, Linden H (2010) Top predators, mesopredators and their prey: interference ecosystems along bioclimatic productivity gradients. J Anim Ecol 79(4):785–794PubMedGoogle Scholar
  29. Erbach A, Lutscher F, Seo G (2013) Bistability and limit cycles in generalist predator–prey dynamics. Ecol Complex 14:48–55Google Scholar
  30. Fenton A, Spencer M, Montagnes D (2010) Parameterising variable assimilation efficiency in predator–prey models. Oikos 119(6):1000–1010Google Scholar
  31. Fryxell J, Lundberg P (1998) Individual behavior and community dynamics. Chapman & Hall, New YorkGoogle Scholar
  32. Fryxell J, Mosser A, Sinclair A, Packer C (2007) Group formation stabilizes predator–prey dynamics. Nature 449(7165):1041–1043PubMedGoogle Scholar
  33. Fulton E, Smith A, Johnson C (2003) Effect of complexity on marine ecosystem models. Mar Ecol Prog Ser 253:1–16Google Scholar
  34. Fussmann G, Blasius B (2005) Community response to enrichment is highly sensitive to model structure. Biol Lett 1(1):9–12PubMedCentralPubMedGoogle Scholar
  35. Fussmann GF, Weithoff G, Yoshida T (2005) A direct, experimental test of resource vs. consumer dependence. Ecology 86(11):2924–2930Google Scholar
  36. Fussmann G, Weithoff G, Yoshida T (2007) A direct, experimental test of resource vs. consumer dependence: reply. Ecology 88(6):1603–1604Google Scholar
  37. Garrott R, Bruggeman J, Becker M, Kalinowski S, White P (2007) Evaluating prey switching in wolf-ungulate systems. Ecol Appl 17(6):1588–1597PubMedGoogle Scholar
  38. Gatto M (1991) Some remarks on models of plankton densities in lakes. Am Nat 137(2):264–267Google Scholar
  39. Gauthier G, Berteaux D, Bêty J, Tarroux A, Therrien J, McKinnon L, Legagneux P, Cadieux M (2011) The tundra food web of Bylot Island in a changing climate and the role of exchanges between ecosystems. Ecoscience 18(3):223–235Google Scholar
  40. Geritz S, Kisdi E (2004) On the mechanistic underpinning of discrete-time population models with complex dynamics. J Theor Biol 228(2):261–269PubMedGoogle Scholar
  41. Getz WM (1984) Population dynamics: a per capita resource approach. J Theor Biol 108(4):623–643Google Scholar
  42. Gilg O, Hanski I, Sittler B (2003) Cyclic dynamics in a simple vertebrate predator–prey community. Science 301(5646):866–868Google Scholar
  43. Gilg O, Sittler B, Sabard B, Hurstel A, Sané R, Delattre P, Hanski I (2006) Functional and numerical responses of four lemming predators in high arctic Greenland. Oikos 113(2):193–216Google Scholar
  44. Ginzburg L, Jensen C (2004) Rules of thumb for judging ecological theories. Trends Ecol Evol 19(3):121–126PubMedGoogle Scholar
  45. Hansen T, Stenseth N, Henttonen H (1999) Multiannual vole cycles and population regulation during long winters: an analysis of seasonal density dependence. Am Nat 154(2):129–139Google Scholar
  46. Hanski I, Korpimäki E (1995) Microtine rodent dynamics in northern Europe: parameterized models for the predator–prey interaction. Ecology 76(3):840–850Google Scholar
  47. Hanski I, Henttonen H, Korpimaki E, Oksanen L, Turchin P (2001) Small-rodent dynamics and predation. Ecology 82(6):1505–1520Google Scholar
  48. Hassell M (2000) Host-parasitoid population dynamics. J Anim Ecol 69(4):543–566Google Scholar
  49. Higgins K, Hastings A, Sarvela J, Botsford L (1997) Stochastic dynamics and deterministic skeletons: population behavior of dungeness crab. Science 276(5317):1431–1435Google Scholar
  50. Hone J, Krebs C, O’Donoghue M, Boutin S (2007) Evaluation of predator numerical responses. Wildl Res 34(5):335–341Google Scholar
  51. Hu H, Nigmatulina K, Eckhoff P (2013) The scaling of contact rates with population density for the infectious disease models. Math Biosci 244:125–134PubMedGoogle Scholar
  52. Huxley J (1934) A natural experiment on the territorial instinct. Brit Birds 27(10):270–277Google Scholar
  53. Ives A, Dennis B, Cottingham K, Carpenter S (2003) Estimating community stability and ecological interactions from time-series data. Ecol Monogr 73(2):301–330Google Scholar
  54. Ives AR, Einarsson Á, Jansen VA, Gardarsson A (2008) High-amplitude fluctuations and alternative dynamical states of midges in Lake Myvatn. Nature 452(7183):84–87PubMedGoogle Scholar
  55. Jost C, Arditi R (2001) From pattern to process: identifying predatorprey models from time-series data. Popul Ecol 43(3):229–243Google Scholar
  56. Jost C, Ellner SP (2000) Testing for predator dependence in predatorprey dynamics: a non-parametric approach. Proc Roy Soc B: Biol Sci 267(1453):1611–1620Google Scholar
  57. Jost C, Devulder G, Vucetich J, Peterson R, Arditi R (2005) The wolves of Isle Royale display scale-invariant satiation and ratio-dependent predation on moose. J Anim Ecol 74(5):809–816Google Scholar
  58. Kendall B, Prendergast J, Bjørnstad O (1998) The macroecology of population dynamics: taxonomic and biogeographic patterns in population cycles. Ecol Lett 1(3):160–164Google Scholar
  59. Kendall B, Briggs C, Murdoch W, Turchin P, Ellner S, McCauley E, Nisbet R, Wood S (1999) Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80(6):1789–1805Google Scholar
  60. Kharitonov S, Volkov A, Willems F, van Kleef H, Klaassen R, Nowak D, Nowak A, Bublichenko A (2008) Brent goose colonies near snowy owls: internet distances in relation to abundance of lemmings and arctic foxes. Biol Bull 35(3):270–278Google Scholar
  61. King C, Powell R (2007) The natural history of weasels and stoats: ecology, behavior, and management. Oxford University Press, New YorkGoogle Scholar
  62. King A, Schaffer W (2001) The geometry of a population cycle: a mechanistic model of snowshoe hare demography. Ecology 82(3):814–830Google Scholar
  63. Klemola T, Korpimäki E, Norrdahl K, Tanhuanpää M, Koivula M (1999) Mobility and habitat utilization of small mustelids in relation to cyclically fluctuating prey abundances. Ann Zool Fenn 36(2):75–82Google Scholar
  64. Krebs CJ (2011) Of lemmings and snowshoe hares: the ecology of northern Canada. Proc R Soc B Biol Sci 278(1705):481–489Google Scholar
  65. Krebs CJ (2013) Population fluctuations in rodents. University of Chicago Press, ChicagoGoogle Scholar
  66. Lande R, Engen S, Sæther B (2003) Stochastic population dynamics in ecology and conservation. Oxford University Press, USAGoogle Scholar
  67. Leslie P (1948) Some further notes on the use of matrices in population mathematics. Biometrika 35(3/4):213–245Google Scholar
  68. Lundberg P, Fryxell J (1995) Expected population density versus productivity in ratio-dependent and prey-dependent models. Am Nat 146(1):153–161Google Scholar
  69. Matthiopoulos J, Graham K, Smout S, Asseburg C, Redpath S, Thirgood S, Hudson P, Harwood J (2007) Sensitivity to assumptions in models of generalist predation on a cyclic prey. Ecology 88(10):2576–2586PubMedGoogle Scholar
  70. Matthiopoulos J, Smout S, Winship A, Thompson D, Boyd I, Harwood J (2008) Getting beneath the surface of marine mammal–fisheries competition. Mamm Rev 38(2–3):167–188Google Scholar
  71. May R (1973) Stability and complexity in model ecosystems. Princeton University Press, PrincetonGoogle Scholar
  72. Maynard Smith J, Slatkin M (1973) The stability of predator–prey systems. Ecology 54(2):384–391Google Scholar
  73. McCallum H, Barlow N, Hone J (2001) How should pathogen transmission be modelled? Trends Ecol Evol 16(6):295–300PubMedGoogle Scholar
  74. McCann K (2011) Food webs. Princeton University Press, PrincetonGoogle Scholar
  75. McCauley E, Wilson W, de Roos A (1993) Dynamics of age-structured and spatially structured predator–prey interactions: individual-based models and population-level formulations. Am Nat 142(3):412PubMedGoogle Scholar
  76. McCauley E, Wilson W, de Roos A (1996) Dynamics of age-structured predator–prey populations in space: asymmetrical effects of mobility in juvenile and adult predators. Oikos 76(3):485–497Google Scholar
  77. McKane A, Newman T (2005) Predator–prey cycles from resonant amplification of demographic stochasticity. Phys Rev Lett 94(21):218102PubMedGoogle Scholar
  78. Merrill E, Sand H, Zimmermann B, McPhee H, Webb N, Hebblewhite M, Wabakken P, Frair JL (2010) Building a mechanistic understanding of predation with GPS-based movement data. Philos Trans R Soc B Biol Sci 365(1550):2279–2288Google Scholar
  79. Moran PA (1953) The statistical analysis of the canadian lynx cycle. 1. Structure and prediction. Aust Can J Zool 1(2):163–173Google Scholar
  80. Murdoch W, Briggs C, Nisbet R (2003) Consumer-resource dynamics. Princeton University Press, PrincetonGoogle Scholar
  81. Murrell D (2005) Local spatial structure and predator–prey dynamics: counterintuitive effects of prey enrichment. Am Nat 166(3):354–367PubMedGoogle Scholar
  82. New L, Matthiopoulos J, Redpath S, Buckland S (2009) Fitting models of multiple hypotheses to partial population data: investigating the causes of cycles in red grouse. Am Nat 174(3):399–412PubMedGoogle Scholar
  83. New L, Buckland S, Redpath S, Matthiopoulos J (2012) Modelling the impact of hen harrier management measures on a red grouse population in the UK. Oikos 121(7):1061–1072Google Scholar
  84. Nielsen Ó (1999) Gyrfalcon predation on ptarmigan: numerical and functional responses. J Anim Ecol 68(5):1034–1050Google Scholar
  85. Nilsson I, Nilsson S, Sylven M (1982) Diet choice, resource depression, and the regular nest spacing of birds of prey. Biol J Linn Soc 18(1):1–9Google Scholar
  86. Nisbet R, Gurney W (1976) A simple mechanism for population cycles. Nature 263:319–320PubMedGoogle Scholar
  87. Nisbet R, Gurney W (1982) Modelling fluctuating populations. Wiley, New YorkGoogle Scholar
  88. Oksanen L, Oksanen T (2000) The logic and realism of the hypothesis of exploitation ecosystems. Am Nat 155:703–723PubMedGoogle Scholar
  89. Oksanen L, Fretwell S, Arruda J, Memela P (1981) Exploitation ecosystems in gradients of primary productivity. Am Nat 118(2):240–261Google Scholar
  90. Oksanen T, Oksanen L, Schneider M, Aunapuu M (2001) Regulation, cycles and stability in northern carnivore-herbivore systems: back to first principles. Oikos 94(1):101–117Google Scholar
  91. Oksanen T, Oksanen L, Dahlgren J, Olofsson J (2008) Arctic lemmings, Lemmus spp. and Dicrostonyx spp.: integrating ecological and evolutionary perspectives. Evol Ecol Res 10(3):415–434Google Scholar
  92. Pachepsky E, Nisbet R, Murdoch W (2008) Between discrete and continuous: consumer-resource dynamics with synchronized reproduction. Ecology 89(1):280–288PubMedGoogle Scholar
  93. Peckarsky B, Abrams P, Bolnick D, Dill L, Grabowski J, Luttbeg B, Orrock J, Peacor S, Preisser E, Schmitz O (2008) Revisiting the classics: considering nonconsumptive effects in textbook examples of predator–prey interactions. Ecology 89(9):2416–2425PubMedGoogle Scholar
  94. Pineda-Krch M, Blok J, Dieckmann U, Doebeli M (2007) A tale of two cycles-distinguishing quasi-cycles and limit cycles in finite predator–prey populations. Oikos 116(1):53Google Scholar
  95. Poggiale J, Michalski J, Arditi R (1998) Emergence of donor control in patchy predator–prey systems. Bull Math Biol 60(6):1149– 1166Google Scholar
  96. Post E, Stenseth N, Peterson R, Vucetich J, Ellis A (2002) Phase dependence and population cycles in a large-mammal predator–prey system. Ecology 83(11):2997–3002Google Scholar
  97. Rall BC, Guill C, Brose U (2008) Food-web connectance and predator interference dampen the paradox of enrichment. Oikos 117:202–213Google Scholar
  98. Redpath S, Mougeot F, Leckie F, Elston D, Hudson P (2006) Testing the role of parasites in driving the cyclic population dynamics of a gamebird. Ecol Lett 9(4):410–418PubMedGoogle Scholar
  99. Renshaw E (1993) Modelling biological populations in space and time, vol 11. Cambridge University Press, CambridgeGoogle Scholar
  100. Ripa J, Ives AR (2003) Food web dynamics in correlated and autocorrelated environments. Theor Popul Biol 64(3):369–384PubMedGoogle Scholar
  101. Rohani P, King AA (2010) Never mind the length, feel the quality: the impact of long-term epidemiological data sets on theory, application and policy. Trends Ecol Evol 25(10):611–618PubMedCentralPubMedGoogle Scholar
  102. Rosenzweig M (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171(3969):385–387PubMedGoogle Scholar
  103. Rosenzweig M, MacArthur R (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 97:209–223Google Scholar
  104. Roughgarden J (1998) Primer of ecological theory. Prentice Hall Upper Saddle, New JerseyGoogle Scholar
  105. Royama T (1992) Analytical population dynamics. Chapman and Hall, LondonGoogle Scholar
  106. Sabelis M, Janssen A, Diekmann O, Jansen V (2005) Global persistence despite local extinction in acarine predator–prey systems: lessons from experimental and mathematical exercises. Adv Ecol Res 37:183–220Google Scholar
  107. Sheriff M, Krebs C, Boonstra R (2010) The ghosts of predators past: population cycles and the role of maternal programming under fluctuating predation risk. Ecology 91(10):2983–2994PubMedGoogle Scholar
  108. Skalski G, Gilliam J (2001) Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 82(11):3083–3092Google Scholar
  109. Smout S, Lindstrøm U (2007) Multispecies functional response of the minke whale Balaenoptera acutorostrata based on small-scale foraging studies. Mar Ecol Prog Ser 341:277–291Google Scholar
  110. Stenseth NC (1999) Population cycles in voles and lemmings: density dependence and phase dependence in a stochastic world. Oikos 87(3):427–461Google Scholar
  111. Stenseth NC, Bjørnstad ON, Falck W (1996) Is spacing behaviour coupled with predation causing the microtine density cycle? A synthesis of current process-oriented and pattern-oriented studies. Proc R Soc Lond B Biol Sci 263(1376):1423–1435Google Scholar
  112. Stenseth NC, Falck W, Chan KS, Bjørnstad ON, O’ Donoghue M, Tong H, Boonstra R, Boutin S, Krebs CJ, Yoccoz NG (1998) From patterns to processes: phase and density dependencies in the canadian lynx cycle. Proc Natl Acad Sci 95(26):15430–15435PubMedGoogle Scholar
  113. Stenseth NC, Leirs H, Mercelis S, Mwanjabe P (2001) Comparing strategies for controlling an African pest rodent: an empirically based theoretical study. J Appl Ecol 38(5):1020–1031Google Scholar
  114. Stephens D, Brown J, Ydenberg R (2007) Foraging: behavior and ecology. University of Chicago Press, ChicagoGoogle Scholar
  115. Taylor R (1984) Predation. Chapman and Hall, New YorkGoogle Scholar
  116. Taylor RA, Sherratt JA, White A (2012) Seasonal forcing and multi-year cycles in interacting populations: lessons from a predator–prey model. J Math Biol 1–24. doi: 10.1007/s00285-012-0612-z
  117. Tornberg R, Lindèn A, Byholm P, Ranta E, Valkama J, Helle P, Lindèn H (2013) Coupling in goshawk and grouse population dynamics in Finland. Oecologia 171(4):863–872PubMedGoogle Scholar
  118. Turchin P, Batzli G (2001) Availability of food and the population dynamics of arvicoline rodents. Ecology 82(6):1521–1534Google Scholar
  119. Turchin P, Ellner S (2000) Living on the edge of chaos: population dynamics of fennoscandian voles. Ecology 81(11):3099–3116Google Scholar
  120. Turchin P, Hanski I (1997) An empirically based model for latitudinal gradient in vole population dynamics. Am Nat 149(5):842– 874PubMedGoogle Scholar
  121. Turchin P, Ostfeld R (1997) Effects of density and season on the population rate of change in the meadow vole. Oikos 78(2):355–361Google Scholar
  122. Tyson R, Haines S, Hodges K (2010) Modelling the Canada lynx and snowshoe hare population cycle: the role of specialist predators. Theor Ecol 3(2):97–111Google Scholar
  123. Tyutyunov Y, Titova L, Arditi R (2008) Predator interference emerging from trophotaxis in predator–prey systems: an individual-based approach. Ecol Complex 5(1):48–58Google Scholar
  124. Vik JO, Brinch CN, Boutin S, Stenseth NC (2008) Interlinking hare and lynx dynamics using a century’s worth of annual data. Popul Ecol 50(3):267–274Google Scholar
  125. Vonesh JR, Bolker BM (2005) Compensatory larval responses shift trade-offs associated with predator-induced hatching plasticity. Ecology 86(6):1580–1591Google Scholar
  126. Vucetich J, Peterson R (2004a) The influence of top–down, bottom–up and abiotic factors on the moose (Alces alces) population of Isle Royale. Proc R Soc Lond B Biol Sci 271(1535):183– 189Google Scholar
  127. Vucetich J, Peterson R (2004b) The influence of prey consumption and demographic stochasticity on population growth rate of isle royale wolves Canis lupus. Oikos 107(2):309–320Google Scholar
  128. Vucetich J, Peterson R, Schaefer C (2002) The effect of prey and predator densities on wolf predation. Ecology 83(11):3003–3013Google Scholar
  129. Vucetich J, Hebblewhite M, Smith D, Peterson R (2011) Predicting prey population dynamics from kill rate, predation rate and predator–prey ratios in three wolf-ungulate systems. J Anim Ecol 80(6):1236–1245PubMedGoogle Scholar
  130. Walters C, Kitchell JF (2001) Cultivation/depensation effects on juvenile survival and recruitment: implications for the theory of fishing. Can J Fish Aquat Sci 58(1):39–50Google Scholar
  131. Wilson D, Bromley R (2001) Functional and numerical responses of predators to cyclic lemming abundance: effects on loss of goose nests. Can J Zool 79(3):525–532Google Scholar
  132. Wilson W, De Roos A, McCauley E (1993) Spatial instabilities within the diffusive Lotka–Volterra system: individual-based simulation results. Theor Popul Biol 43(1):91–127Google Scholar
  133. Wilson W, McCauley E, De Roos A (1995) Effect of dimensionality on Lotka–Volterra predator–prey dynamics: individual based simulation results. Bull Math Biol 57(4):507–526Google Scholar
  134. Yodzis P (1994) Predator–prey theory and management of multispecies fisheries. Ecol Appl 4(1):51–58Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Arctic and Marine BiologyUniversity of TromsøTromsøNorway

Personalised recommendations