Skip to main content
Log in

Dynamics and Management of Stage-Structured Fish Stocks

Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

With increasing fishing pressures having brought several stocks to the brink of collapse, there is a need for developing efficient harvesting methods that account for factors beyond merely yield or profit. We consider the dynamics and management of a stage-structured fish stock. Our work is based on a consumer-resource model which De Roos et al. (in Theor. Popul. Biol. 73, 47–62, 2008) have derived as an approximation of a physiologically-structured counterpart. First, we rigorously prove the existence of steady states in both models, that the models share the same steady states, and that there exists at most one positive steady state. Furthermore, we carry out numerical investigations which suggest that a steady state is globally stable if it is locally stable. Second, we consider multiobjective harvesting strategies which account for yield, profit, and the recovery potential of the fish stock. The recovery potential is a measure of how quickly a fish stock can recover from a major disturbance and serves as an indication of the extinction risk associated with a harvesting strategy. Our analysis reveals that a small reduction in yield or profit allows for a disproportional increase in recovery potential. We also show that there exists a harvesting strategy with yield close to the maximum sustainable yield (MSY) and profit close to that associated with the maximum economic yield (MEY). In offering a good compromise between MSY and MEY, we believe that this harvesting strategy is preferable in most instances. Third, we consider the impact of harvesting on population size structure and analytically determine the most and least harmful harvesting strategies. We conclude that the most harmful harvesting strategy consists of harvesting both adults and juveniles, while harvesting only adults is the least harmful strategy. Finally, we find that a high percentage of juvenile biomass indicates elevated extinction risk and might therefore serve as an early-warning signal of impending stock collapse.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Claessen, D., & De Roos, A. M. (2003). Bistability in a size-structured population model of cannibalistic fish—a continuation study. Theor. Popul. Biol., 64, 49–65.

    Article  MATH  Google Scholar 

  • Clark, C. W. (2010). Mathematical bioeconomics: the mathematics of conservation (3th ed.). New York: Wiley.

    Google Scholar 

  • Cushing, J. M. (1989). A competition model for size-structured species. SIAM J. Appl. Math., 49, 838–858.

    Article  MathSciNet  MATH  Google Scholar 

  • De Roos, A. M. (1997). A gentle introduction to physiologically structured population models. In S. Tuljapurkar & H. Caswell (Eds.), Structured population models in marine, terrestrial and freshwater systems (pp. 119–204). New York: Chapman-Hall.

    Chapter  Google Scholar 

  • De Roos, A. M., Schellekens, T., & Van Kooten, T. (2007). Food-dependent growth leads to overcompensation in stage-specific biomass when mortality increases: the influence of maturation versus reproduction regulation. Am. Nat., 170(3), E59–E76.

    Article  Google Scholar 

  • De Roos, A. M., Schellekens, T., Van Kooten, T., De Wolfshaar, K. V., Claessen, D., & Persson, L. (2008). Simplifying a physiologically structured population model to a stage-structured biomass model. Theor. Popul. Biol., 73, 47–62.

    Article  MATH  Google Scholar 

  • Diekmann, O., & Metz, H. (1989). Exploring linear chain trickery for physiologically structured populations. In Bij het afscheid van prof. dr. H.A. lauwerier, TW in beeld (pp. 73–84). Amsterdam: CWI.

    Google Scholar 

  • Farkas, J. Z., & Hagen, T. (2007). Stability and regularity results for a size-structured population model. J. Math. Anal. Appl., 328, 119–136.

    Article  MathSciNet  MATH  Google Scholar 

  • Garcia, S. M., & Newton, C. (1995). Current situation, trends and prospects in world capture fisheries (Fisheries technical paper). Food and Agriculture Organization, Rome.

  • Hamilton, S. L., et al. (2007). Size-selective harvesting alters life histories of a temperate sex-changing fish. Ecol. Appl., 17(8), 2268–2280.

    Article  Google Scholar 

  • Hutchings, J. A., & Myers, R. A. (1994). What can be learned from the collapse of a renewable resource? Atlantic Cod, Gadus morhua, of Newfoundland and Labrador. Can. J. Fish. Aquat. Sci., 51(9), 2126–2146.

    Article  Google Scholar 

  • Metz, J. A. J., & Diekmann, O. (1986). Lecture notes in biomathematics: Vol. 68. The dynamics of physiologically structured populations. Heidelberg: Springer.

    MATH  Google Scholar 

  • Olsen, E. M., Heino, M., Lilly, G. R., Morgan, M. J., Brattey, J., Ernande, B., & Dieckmann, U. (2004). Maturation trends indicative of rapid evolution preceded the collapse of northern cod. Nature, 428, 932–935.

    Article  Google Scholar 

  • Persson, L., Leonardsson, K., De Roos, A. M., Gyllenberg, M., & Christensen, B. (1998). Ontogenetic scaling of foraging rates and the dynamics of a size-structured consumer-resource model. Theor. Popul. Biol., 54, 270–293.

    Article  MATH  Google Scholar 

  • Rijnsdorp, A. D., van Leeuwen, P. I., Daan, N., & Heesen, H. J. L. (1996). Changes in abundance of demersal fish species in the North Sea between 1906–1909 and 1990–1995. ICES J. Mar. Sci., 53, 1054–1062.

    Article  Google Scholar 

  • Walsh, M. R., Munch, S. B., Chiba, S., & Conover, D. O. (2006). Maladaptive changes in multiple traits caused by fishing: impediments to population recovery. Ecol. Lett., 9, 142–148.

    Article  Google Scholar 

Download references

Acknowledgements

Å.B. and X.M. gratefully acknowledge financial support from the Wenner–Gren Foundations. We thank Lennart Persson for suggesting that we work with the simplified stage-structured model and two anonymous reviewers for their valuable comments that helped to improve the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xinzhu Meng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Meng, X., Lundström, N.L.P., Bodin, M. et al. Dynamics and Management of Stage-Structured Fish Stocks. Bull Math Biol 75, 1–23 (2013). https://doi.org/10.1007/s11538-012-9789-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-012-9789-y

Keywords

Navigation