Maximum Sustainable Yields and the Self-Renewal of Exploited Populations with Age-Dependent Vital Rates

  • R. Law
  • D. R. Grey


This paper examines the relationship between two problems in the theory of harvesting self-renewing populations with age-dependent vital rates. The first problem is to find a method of harvesting such that the yield sustainable from a population is maximized. Because of the obvious practical importance of this problem, it has been the subject of many investigations (e.g., Beverton and Holt 1957; Beddington and Taylor 1973; Doubleday 1975; Reed 1980; Botsford 1981; Horwood and Whittle 1986). The second problem is to find a method of harvesting which will maximize the capacity of a population to renew itself, given that a certain minimum crop must be taken. At first sight this problem looks relatively unimportant, and it is perhaps understandable that it has received relatively little attention.


Maximum Sustainable Yield Primal Constraint Harvest Pattern Newborn Cohort Minimum Crop 
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  1. Beddington JR, Taylor DB (1973) Optimum age-specific harvesting of a population. Biometrics 29:801–809.CrossRefGoogle Scholar
  2. Beverton RJH, Holt SJ (1957) On the dynamics of exploited fish populations. Fish Invest London Ser II 19:1–533.Google Scholar
  3. Borisov VM (1978) The selective effect of fishing on the population structure of species with a long life cycle. J Ichthyol 18:896–904.Google Scholar
  4. Botsford LW (1981) Optimal fishery policy for size-specific, density-dependent population models. J Math Biol 12:265–293.CrossRefGoogle Scholar
  5. Charlesworth B (1980) Evolution in age-structured populations. Univ Press, Cambridge.Google Scholar
  6. Cushing DH (1966) The arctic cod. Pergamon, Oxford New York.Google Scholar
  7. Cushing DH, Horwood JW (1977) Development of a model of stock and recruitment. In: Steele JH (ed) Fisheries Mathematics. Academic Press, New York London, pp 21–35.Google Scholar
  8. Doubleday WG (1975) Harvesting in matrix population models. Biometrics 31:189–200.CrossRefGoogle Scholar
  9. Garrod DJ (1967) Population dynamics of the Arcto-Norwegian cod. J Fish Res Board Can 24:145–190.CrossRefGoogle Scholar
  10. Garrod DJ (1977) The North Atlantic cod. In: Gulland JA (ed) Fish population dynamics. John Wiley & Sons, New York London, pp 216–242.Google Scholar
  11. Garrod DJ, Jones BW (1974) Stock and recruitment relationship in the Northwest Arctic cod stock. J Con Perm Int Explor Mer 36:35–41.Google Scholar
  12. Gass SI (1969) Linear programming methods and applications. McGraw-Hill, New York.Google Scholar
  13. Goodman D (1980) Demographic intervention for closely managed populations. In: Soulé ME, Wilcox BA (eds) Conservation biology. Sinauer, Sunderland, Mass, pp 171–195.Google Scholar
  14. Grey DR (1984) Non-negative matrices, dynamic programming and a harvesting problem. J Appl. Probab 21:685–694.CrossRefGoogle Scholar
  15. Grey DR, Law R (1987) Reproductive values and maximum yields. Funct Ecol 1:327–330.CrossRefGoogle Scholar
  16. Horwood JW, Whittle P (1986) The optimal harvest from a multicohort stock. IMA J Math Appl Med Biol 3:143–155.CrossRefGoogle Scholar
  17. Law R (1979) Harvest optimization in populations with age distributions. Am Nat 114:250–259.CrossRefGoogle Scholar
  18. MacArthur RH (1960) On the relation between reproductive value and optimal predation. Proc Natl Acad Sci USA 46:143–145.CrossRefGoogle Scholar
  19. Reed WJ (1980) Optimum age-specific harvesting in a nonlinear population model. Biometrics 36:579–593.CrossRefGoogle Scholar
  20. Ricker WE (1954) Stock and recruitment. J Fish Res Board Can 11:559–623.CrossRefGoogle Scholar
  21. Rollefsen G (1953) Observations on the cod and cod fisheries of Lofoten. Rapp P V Reun Cons Perm Int Explor Mer 136:40–47.Google Scholar
  22. Rorres C, Fair W (1975) Optimal harvesting policy for an age-specific population. Math Biosci 24:31–47.CrossRefGoogle Scholar
  23. Slobodkin LB (1968) How to be a predator. Am Zool 8:43–51.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • R. Law
    • 1
  • D. R. Grey
    • 2
  1. 1.Department of BiologyUniversity of YorkYorkEngland
  2. 2.Department of Probability and StatisticsThe UniversitySheffieldEngland

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