On the Harvesting of Age Structured of Fish Populations

H. Behncke, Sadiq Al-Nassir


The yearly cycle of marine fish suggests to describe such populations by a discrete time dynamical Leslie model. The only nonlinear input into this model is the recruitment functions. Its influence on the stability of equilibria is investigated. We show that a monotone recruitment function leads to globally stable equilibria. Harvesting is introduced as fishing mortality so that the structure of the underlying model remains the same. The impact of the recruitment function is also discussed.


Marine fish; Age class population model; Stability; Harvesting

Full Text:



R.J. Beverton and S.J. Holt, On the dynamics of exploited fish population, Fisheries Investigation Series II, Vol. 19, Ministry of Agriculture, Fisheries and Food, London (1957).

H. Caswell, Matrix Population Models: Construction, Analysis and Interpretation, Sinauer Publishers (2001).

C.W. Clark, Bioeconomic Modeling and Fisheries Management, Wiley, New York (1985).

C.W. Clark and J.D. De Pree, A simple linear model for the optimal exploitation of renewable resources, App. Math. Optim. 5 (1979), 181–196.

D.H. Cushing and J.W. Horwood, The growth and death of fish larvae, Jour. of Plank Res. 16 (3) (1994), 291–300.

D.H. Cushing, Fisheries Biology, A Study in Population Dynamics, University of Wisconsin Press (1981).

J.M. Cushing, An Introduction to Structured Population Dynamics, CBMSNSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia (1998).

J.M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models, Natural Resource Modeling 8 (4) (1994), 297–333.

F. Diekert, D. Hjermann, E. Naevdal and N. Stenseth, Non-cooperative exploitation of multi-cohort fisheries-the role of gear selectivity in the North-East Arctic cod fishery, Resource and Energy Economics 32 (1) (2010a), 78–92.

F. Diekert, D. Hjermann, E. Naevdal and N. Stenseth, Spare the young fish: Optimal harvesting policies for North-East Arctic cod, Environmental and Resource Economics 47 (4) (2010b), 455–475.

R. Froese, Fishery reform slips through the net, Nature 7 (2011), 475.

F. Diekert, The growing value of age: exploring economic gains from age specific harvesting in the Northeast Arctic cod fishery, Canadian Journal of Fisheries and Aquatic Sciences 70 (9) (2013), 1346–1358.

S. Eladyi, An Introduction to Difference Equations, Springer (2000).

W.M. Getz, Optimal harvesting of structured population, Math. Biosciences 44 (1979), 269–291.

W.M. Getz and R.G. Haight, Population Harvesting, Princeton University Press (1989).

M.P. Hassell, Density dependence in single species populations, J. Anim. Ecol. 44 (1975), 283–295.

R. Hilborn and C.J. Walters, Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty, Chapman & Hall Inc., London (1992).

R. Hilborn, Overfishing: What Everyone Needs to Know?, Oxford University Press, Oxford (2012).

P.J.B. Hart and J.D. Reynolds, Handbook of Fish Biology and Fisheries, Vols. 1-2, Blackwell Science Ltd. (2002).

P.H.Leslie,.On the use of matrices in certain population Mathematics Biometrike 33 (1945), 183–212.

S.A. Levin and C.P. Goodyear, Analysis of age structured fishery model, J. Math. Bio. 9 (1980), 2455–2274.

R. Law and D.R. Grey, Maximum sustainable yields and the self renewal of exploited populations with age-dependent vital rates, in: Size-Structured Populations, B. Ebenman and L. Persson (Eds.) Springer-Verlag, Berlin—Heidelberg (1988).

C. Li and H. Schneider, Applications of Perron-Frobenius theory to population dynamics, J. Math. Bio. (2002), 450–462.

N. Li and A.A. Yakubu, A juvenile-adult discrete time production model of exploited fishery systems, Nature Resource Modeling 25 (2) (2012), 273–324.

R.A. Myers and G. Mertz, Maximum population growth rates and recovery times for Attlantic cod (Gadus morhva), Fishery Bull. 95 (1997), 762–772.

R.A. Myers and N.J. Barrowman, Still more spawner recruitment curves: The hockey stick and its generalizations, Can. J. Fish. Aquat. Sci. 57 (2000), 665–676.

R.A. Myers, N.J. Barrowman, J.A. Hutchings and A.A. Rosenberg, Population dynamics of exploited fish stocks at low population levels, Science 269 (1995), 1106–1108.

R. Norges, Pressemelding: Minstepriser gjeldende fra 07.05.2007, http://www.rafisklaget.no/ pls/portal/url/ITEM/6D4F5250DAD24D22A026C2F97847477B, June 17, 2007.

M. Quaas, T. Requate, K. Ruckes, A. Skonhoft, N. Vestergaard and R. Voss, Incentives for optimal management of age-structured fish populations, Resource and Energy Economics 35 (2) (2013), 113–134.

W.E. Ricker, Stock and recruitment, J. Fish. Res. Board Can. 11 (1954), 559–623.

W.E. Ricker, Handbook of computations for biological statistics of fish populations, Bull. 191, Fisheries Board Canada (1958).

Sadiq Al-Nassir, Optimal Harvesting of Fish Populations with Age Structure, Dissertation, University of Osnabruek, Germany (2015).

M.B. Shaefer, Some aspects of the dynamics of population important to the management of commercial marine fisheries, Inter. Am. Trop. Tuna Comm. Bull. 1 (1954), 25–26.

M.B. Shaefer, Some considerations of population dynamics and economics in relation to the management of marine fisheries, J. Fish. Res. Board Can. 14 (1957), 669–681.

J.G. Shepherd, A versatile new stock and recruitment relationship for fisheries and construction of sustainable yield curves, J. Cons. Int. Explor. Mer. 10 (1982), 67–75.

O. Tahvonen, Age-structured optimization models in fisheries bioeconomics: a survey, in: R. Boucekkine, N. Hritonenko and Yatsenko, Optimal Control of Age-structured Population in

Economy, Demography, and the Environment, Routledge, Chapter 7, 140–173 (2010).

O. Tahvonen, Optimal harvesting of age structured fish populations, Marine Resource Economics 24 (2009), 147–169.

O. Tahvonen, Economics of harvesting age-structured fish populations, Journal of Environmental Economics and Management 58 (3) (2009), 281–299.

O. Tahvonen, M. Quaas, J. Schmidt and R. Voss, Optimal harvesting of an age-structured schooling fishery, Environmental and Resource Economics 54 (1) (2013), 21–39.

A. Wikan, Dynamical consequences of harvest in discrete age-structured population models, J. Math. Bio. 49 (2004), 35–55.

A. Wikan, Four-periodicity in Leslie matrix models with density dependent survival probabilities, Theoretical Population Bio. 53 (1998), 85–97.

J. Wilen, Renewable resource economists and policy: What differences have we made?, Journal of Environmental Economics and Management 39 (3), 306–327.


  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905