Bulletin of Mathematical Biology

, Volume 66, Issue 6, pp 1685–1704 | Cite as

An analysis of a nonlinear stage-structured cannibalism model with application to the Northeast Arctic cod stock

  • Arild Wikan
  • Arne Eide


A two-dimensional stage-structured population model with nonlinear cannibalism terms is studied. We show that there is a large parameter interval where the nontrivial equilibrium of the model is the only stable attractor, but that there also exist parameter intervals where we find quasiperiodic, periodic and chaotic dynamics. Moreover, in the interplay between increasing the fecundity and increasing the cannibalism pressure, the former turns out to be a destabilizing effect while the latter tends to act in a stabilizing fashion. Finally, we have applied the model to the North Atlantic cod stock using ICES biomass estimates. Our main conclusion from this study is that the combined effect of recruitment and cannibalism may not serve as an explanation of the observed fluctuations in the cod stock.


Hopf Bifurcation Saddle Node Bifurcation Invariant Curve Instability Threshold Stock Biomass 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Anon (2001). Advisory Committee on Fisheries Management (ACFM): Report of the Arctic fisheries Working Group. ICES CM 2001/ACFM:19.Google Scholar
  2. Bogstad, B., G. R. Lilly, S. Mehl, Ó. K. Pálsson and G. Stefánsson (1994). Cannibalism and year-class strength in Atlantic cod (Gadus morhua) in Arcto-boreal ecosystems (Barents Sea, Iceland, and eastern Newfoundland). ICES Mar. Sci. Symp. 198, 576–599.Google Scholar
  3. Caswell, H. (2001). Matrix Population Models, Sunderland, Massachusetts: Sinauer Ass. Inc. Publishers.Google Scholar
  4. Costantino, R. F., R. A. Deshamais, J. M. Cushing and B. Dennis (1997). Chaotic dynamics in an insect population. Science 275, 389–391.CrossRefGoogle Scholar
  5. Cushing, J. M., R. F. Costantino, B. Dennis, R. A. Deshamais and S. M. Henson (1998). Nonlinear population dynamics: models experiments and data. J. Theor. Biol. 194, 1–9.CrossRefGoogle Scholar
  6. Cushing, J. M., B. Dennis, R. A. Deshamais and R. F. Costantino (1996). An interdisciplinary approach to understanding nonlinear ecological dynamics. Ecol. Model 92, 111–119.CrossRefGoogle Scholar
  7. Dennis, B., R. A. Deshamais, J. M. Cushing and R. F. Costantino (1997). Transition in population dynamics: equilibria to periodic cycles to aperiodic cycles. J. Anim. Ecol. 6b, 704–729.Google Scholar
  8. Fox, L. R. (1975). Cannibalism in natural populations. Ann. Rev. Ecol. Syst. 6, 87–106.CrossRefGoogle Scholar
  9. Guckenheimer, J. and P. Holmes (1990). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Berlin, Heidelberg, New York, Tokyo: Springer Verlag.Google Scholar
  10. Hjort, J. (1914). Fluctuations in the great fisheries of Northern Europe, in Const. Int. Explor.-Mer., Copenhagen.Google Scholar
  11. Jørgensen, T. (1992). Long-term changes in growth of North-east Arctic cod (Gadus morhua) and some environmental influences. ICES J. Mar. Sci. 49, 263–277.Google Scholar
  12. Levin, S. A. (1981). Age-structure and stability in multiple-age spawning populations, in Renewable Resource Managements, T. L. Vincent and J. M. Skowronski (Eds), Heidelberg: Springer Verlag.Google Scholar
  13. Levin, S. A. and P. H. Goodyear (1980). Analysis of an age-structured fishery model. J. Math. Biol. 9, 245–274.MathSciNetCrossRefGoogle Scholar
  14. Linehan, J. E., R. S. Gregpry and D. C. Schneider (2001). Predation risk of age-0 cod (Gadus) relative to depth and substrate in coastal waters. J. Exp. Mar. Biol. Ecol. 263, 25–44.CrossRefGoogle Scholar
  15. Murray, J. D. (1993). Mathematical Biology, 2nd edn corrected, Berlin, Heidelberg, New York: Springer.Google Scholar
  16. Myers R. A., W. Blanchard, K. R. Thompson, 1990. Summary of North Atlantic fish recruitment 1942–1987. Can. Tech. Rep. Fish. & Aquat. Sci. 1743.Google Scholar
  17. Ottersen, G. (1996). Environmental impact on variability in recruitment, larval growth and distribution of Arcto-Norwegian cod, Dr Scient thesis, Geophysical Institute, University of Bergen.Google Scholar
  18. Polis, G. A. (1981). The evolution and dynamics of intraspecific predation. Ann. Rev. Ecol. Syst. 12, 225–251.CrossRefGoogle Scholar
  19. Ricker, W. E. (1954). Stock and recruitment. J. Fish. Res. Board Can. 11, 559–623.Google Scholar
  20. Tjelmeland, S. and B. Bogstad (1998a). Biological modelling, in Models for Multispecies Management, T. Rødseth (Ed.), Heidelberg, New York: Physica, pp. 69–91.Google Scholar
  21. Tjelmeland, S. and B. Bogstad (1998b). MULTSPEC—a review of a multispecies modelling project for the Barents Sea. Fisheries Research. 37, Elsevier, pp. 127–142.CrossRefGoogle Scholar
  22. Wikan, A. and E. Mjølhus (1996). Overcompensatory recruitment and generation delay in discrete age-structured population models. J. Math. Biol. 35, 195–239.MathSciNetCrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2004

Authors and Affiliations

  1. 1.Harstad University CollegeHarstadNorway
  2. 2.Norwegian College of Fishery ScienceUniversity of Tromsø, BreivikaTromsøNorway

Personalised recommendations