Theoretical Ecology

, Volume 8, Issue 1, pp 1–13 | Cite as

A generalized perturbation approach for exploring stock recruitment relationships

  • Justin D. Yeakel
  • Marc Mangel


Models of stock-recruitment relationships (SRRs) are often used to predict fish population dynamics. Commonly used SRRs include the Ricker, Beverton-Holt, and Cushing functional forms, which differ primarily by the degree of density-dependent effects (compensation). The degree of compensation determines whether recruitment respectively decreases, saturates, or increases at high levels of spawning stock biomass. In 1982, J.G. Shepherd united these dynamics into a single model, where the degree of compensation is determined by a single parameter. However, the difficulty in relating this parameter to biological data has limited its usefulness. Here, we use a generalized modeling framework to show that the degree of compensation can be related directly to the functional elasticity of growth, which is a general quantity that measures the change in recruitment relative to a change in biomass. We show that the elasticity of growth can be calculated from perturbations in fish biomass, is robust to observation error, and can be used to determine general attributes of the SRR in both continuous time production models, as well as discrete time age-structured models.


Compensatory dynamics Generalized modeling Stock-recruitment relationships Shepherd function Neimark-Sacker bifurcation 



We thank S. Allesina, M.P. Beakes, D. Braun, T. Gross, C. Kuehn, T. Levi, A. MacCall, J.W. Moore, S. Munch, M. Novak, C.C. Phillis, and A.O. Shelton for many helpful discussions and comments. We also thank the Dynamics of Biological Networks Lab at the Max-Planck Institute for the Physics of Complex Systems and the University of Bristol for sharing the ideas and knowledge that inspired this work. This project was partially funded by the Center for Stock Assessment and Research, a partnership between the Fisheries Ecology Division, NOAA Fisheries, Santa Cruz, CA and the University of California, Santa Cruz and by NSF grant EF-0924195 to M.M.


  1. Auger P, Poggiale JC (1996) Emergence of population growth models: Fast migration and slow growth. J Theor Biol 182(2):99–108CrossRefPubMedGoogle Scholar
  2. Beverton RJH, Holt SJ (1957) On the dynamics of exploited fish populations. Springer, New YorkGoogle Scholar
  3. Brooks EN, Powers JE (2007) Generalized compensation in stock-recruit functions: Properties and implications for management. ICES J Mar Sci 64(3):413–424CrossRefGoogle Scholar
  4. Calder III WA (1996) Size, function, and life history. Courier Dover Publications, CambridgeGoogle Scholar
  5. Cushing DH (1973) The dependence of recruitment on parent stock. J Fish Res Bd Can 30:1965–1976CrossRefGoogle Scholar
  6. Cushing DH (1988) The problems of stock and recruitment. In: Fish population dynamics: the implications for management. Wiley, New YorkGoogle Scholar
  7. Dirac PAM (1958) The principles of quantum mechanics. Clarendon Press, OxfordGoogle Scholar
  8. Dorner B, Peterman RM, Haeseker SL (2008) Historical trends in productivity of 120 Pacific pink, chum, and sockeye salmon stocks reconstructed by using a Kalman filter. Can J Fish Aquat Sci 65(9):1842–1866CrossRefGoogle Scholar
  9. Fell DA (1992) Metabolic control analysis: a survey of its theoretical and experimental development. Biochem J 286(Pt 2):313–330CrossRefPubMedPubMedCentralGoogle Scholar
  10. Fell DA, Sauro HM (1985) Metabolic control and its analysis. Eur J Biochem 148(3):555–561CrossRefPubMedGoogle Scholar
  11. Feynman R (1948) Space-time approach to non-relativistic quantum mechanics. Rev Mod Phys 20(2):367–387CrossRefGoogle Scholar
  12. Gross T, Feudel U (2006) Generalized models as a universal approach to the analysis of nonlinear dynamical systems. Phys Rev E 73(1 Pt 2)016:205Google Scholar
  13. Gross T, Rudolf L, Levin SA, Dieckmann U (2009) Generalized models reveal stabilizing factors in food webs. Science 325(5941):747–750CrossRefPubMedGoogle Scholar
  14. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New YorkCrossRefGoogle Scholar
  15. Guill C, Drossel B, Just W, Carmack E (2011a) A three-species model explaining cyclic dominance of Pacific salmon. J Theor Biol 276(1):16–21CrossRefPubMedGoogle Scholar
  16. Guill C, Reichardt B, Drossel B, Just W (2011b) Coexisting patterns of population oscillations: the degenerate Neimark-Sacker bifurcation as a generic mechanism. Phys Rev E 83(2):021,910CrossRefGoogle Scholar
  17. Gulland JA (1988) The analysis of data and development of models. In: Fish population dynamics: the implications for management. Wiley, New YorkGoogle Scholar
  18. Hilborn R, Mangel M (1997) The ecological detective: confronting models with data. Princeton University Press, PrincetonGoogle Scholar
  19. Horvitz C, Schemske DW, Caswell H (1997) The relative “importance” of life-history stages to population growth: prospective and retrospective analyses. In: Structured-population models in marine, terrestrial, and freshwater systems. Chapman and Hall, New York, pp 247–271CrossRefGoogle Scholar
  20. Johnson DW, Grorud-Colvert K, Sponaugle S, Semmens BX (2014) Phenotypic variation and selective mortality as major drivers of recruitment variability in fishes. Ecol Lett Early editionGoogle Scholar
  21. Kleiber P, Argue AW, Kearney RE (1987) Assessment of Pacific Skipjack Tuna (Katsuwonus pelamis) resources by estimating standing stock and components of population turnover from tagging data. Can J Fish Aquat Sci 44(6):1122–1134CrossRefGoogle Scholar
  22. Krkosek M, Hilborn R, Peterman RM, Quinn TP (2011) Cycles, stochasticity and density dependence in pink salmon population dynamics. Proc Roy Soc B 278(1714):2060–2068CrossRefGoogle Scholar
  23. Kuehn C, Siegmund S, Gross T (2013) Dynamical analysis of evolution equations in generalized models. IMA J Appl Math 78(5):1051–1077CrossRefGoogle Scholar
  24. Kuznetsov Y (1998) Elements of applied bifurcation theory. Springer, New YorkGoogle Scholar
  25. MacCall AD (2002) Use of known-biomass production models to determine productivity of west coast groundfish stocks. N Am J Fish Manage 22(1):272–279CrossRefGoogle Scholar
  26. Mangel M (2006a) An introduction to some of the problems of sustainable fisheries. In: The theoretical biologist’s toolbox: Quantitative methods for ecology and evolutionary biology. Cambridge University Press, Cambridge, pp 1–38CrossRefGoogle Scholar
  27. Mangel M (2006b) The theoretical biologist’s toolbox: Quantitative methods for ecology and evolutionary biology. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  28. Mangel M, Marinovic B, Pomeroy C, Croll D (2002) Requiem for Ricker: Unpacking MSY. B Mar Sci 70(2):763–781Google Scholar
  29. Mangel M, Levin P, Patil A (2006) Using life history and persistence criteria to prioritize habitats for management and conservation. Ecol Appl 16(2):797–806CrossRefPubMedGoogle Scholar
  30. Mangel M, Brodziak J, DiNardo G (2010) Reproductive ecology and scientific inference of steepness: a fundamental metric of population dynamics and strategic fisheries management. Fish Fish 11(1):89–104CrossRefGoogle Scholar
  31. Mangel M, MacCall AD, Brodziak J, Dick EJ, Forrest RE, Pourzand R, Ralston S (2013) A perspective on steepness, reference points, and stock assessment. Can J Fish Aquat Sci 70(6):930–940CrossRefGoogle Scholar
  32. Marshall CT (2009) Fish reproductive biology: implications for assessment and management. In: Jakobsen T, Fogarty M J, Megrey B A, Moksness E (eds) Fish reproductive biology: Implications for assessment and management. Wiley-Blackwell, West Sussex, pp 395–420CrossRefGoogle Scholar
  33. May RM (1974) Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos. Science 186(4164):645–647CrossRefPubMedGoogle Scholar
  34. Mchich R, Auger PM, Bravo de la Parra R, Raissi N (2002) Dynamics of a fishery on two fishing zones with fish stock dependent migrations: Aggregation and control. Ecol Model 158(1–2):51–62CrossRefGoogle Scholar
  35. Moore JW, Yeakel JD, Peard D, Lough J, Beere M (2014) Life-history diversity and its importance to population stability and persistence of a migratory fish: Steelhead in two large North American watersheds. J Anim EcolGoogle Scholar
  36. Morgan MJ, Perez-Rodriguez A, Saborido-Rey F, Marshall CT (2011) Does increased information about reproductive potential result in better prediction of recruitment? Can J Fish Aquat Sci 68(8):1361–1368CrossRefGoogle Scholar
  37. Munch SB, Kottas A, Mangel M (2005) Bayesian nonparametric analysis of stock-recruitment relationships. Can J Fish Aquat Sci 62(8):1808–1821CrossRefGoogle Scholar
  38. Murdoch WW (1994) Population regulation in theory and practice. Ecology 75(2):271–287CrossRefGoogle Scholar
  39. Myers RA, Barrowman NJ, Hutchings JA, Rosenberg AA (1995) Population dynamics of exploited fish stocks at low population levels. Science 269(5227):1106–1108CrossRefPubMedGoogle Scholar
  40. Perretti CT, Munch SB, Sugihara G (2013a) Model-free forecasting outperforms the correct mechanistic model for simulated and experimental data. Proc Natl Acad Sci USA 110(13):5253– 5257CrossRefPubMedPubMedCentralGoogle Scholar
  41. Perretti CT, Sugihara G, Munch SB (2013b) Nonparametric forecasting outperforms parametric methods for a simulated multispecies system. Ecology 94(4):794–800CrossRefGoogle Scholar
  42. Radchenko VI, Temnykh OS, Lapko VV (2007) Trends in abundance and biological characteristics of pink salmon (Oncorhynchus gorbuscha) in the North Pacific Ocean. North Pac Anadromous Fish Comm Bull 4:7–21Google Scholar
  43. Ricker H (1954) Stock and recruitment. Can J Fish Aquat Sci 11(5):559–623Google Scholar
  44. Shelton AO, Mangel M (2011) Fluctuations of fish populations and the magnifying effects of fishing. Proc Natl Acad Sci USA 108(17):7075–7080CrossRefPubMedPubMedCentralGoogle Scholar
  45. Shepherd J (1982) A versatile new stock-recruitment relationship for fisheries, and the construction of sustainable yield curves. J Conseil 40(1):67CrossRefGoogle Scholar
  46. Sissenwine MP, Shepherd JG (1987) An alternative perspective on recruitment overfishing and biological reference points. Can J Fish Aquat Sci 44(4):913–918CrossRefGoogle Scholar
  47. Stiefs D, van Voorn GAK, Kooi BW, Feudel U, Gross T (2010) Food quality in producer-grazer models: a generalized analysis. Am Nat 176(3):367–380CrossRefPubMedGoogle Scholar
  48. Sydsaeter K, Hammond PJ (1995) Essential mathematics for economic analysis. Prentice-Hall Inc., New JerseyGoogle Scholar
  49. Yeakel JD, Stiefs D, Novak M, Gross T (2011) Generalized modeling of ecological population dynamics. Theor Ecol 4(2):179–194CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Center for Stock Assessment Research and Department of Ecology and Evolutionary BiologyUniversity of California Santa CruzSanta CruzUSA
  2. 2.Santa Fe InstituteSanta FeUSA
  3. 3.Center for Stock Assessment Research and Department of Applied Mathematics and StatisticsUniversity of California Santa CruzSanta CruzUSA
  4. 4.Department of BiologyUniversity of BergenBergenNorway

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