GenerationGeneration Models StockRecruitment Models

1
Generation-Generation Models (Stock-Recruitment
Models)

• Fish 458

2
Readings

• Burgman et al. (1993) Chapter 3.
• Hilborn and Walters (1992) Chapter 7.
• Quinn and Deriso (1999) Chapter 3.
• Myers, R. A., A. A. Rosenberg, P. M. Mace, N.
  Barrowman, and V. R. Restrepo. 1994. In search of
  thresholds for recruitment overfishing. ICES J.
  Mar. Sci. 51 191-205.
• Gilbert, D. J. 1997. Towards a new recruitment
  paradigm for fish stocks. Can. J. Fish. Aquat.
  Sci. 54 969-977.

3
Recruitment

• Annual recruitment is defined as the number of
  animals added to the population each year.
• However, recruitment is also defined by when
  recruitment occurs
• at birth (mammals and birds)
• at age one (mammals and birds, some fish)
• at settlement (invertebrates / coral reef
  fishes)
• when it is first possible to detect animals using
  sampling gear and
• when the animals enter the fishery.
• All of these definitions are correct but you
  need to be aware which one is being used.

4
Stock and Recruitment - Generically (the single
parental cohort case)

• The generic equation for the relationship between
  recruitment and parental stock size (spawner
  biomass in fishes) is
• Recruitment equals parental numbers multiplied by
  survival, fecundity and environmental variation.
• The functional forms allow for density-dependence.

5
Stock and Recruitment - Generically (the single
parental cohort case)

• Consider a model with no density-dependence
• The population either grows forever (at an
  exponential rate) or declines asymptotically to
  extinction.
• The must be some form of density-dependence!

6
Some Hypotheses for Density-Dependence

• Habitat
• Some habitats lead to higher survival of
  offspring than others (predators / food).
  Selection of habitat may be systematic (nest
  selection) or random (location of settling
  individuals).
• Fecundity
• Animals are territorial the total fecundity
  depends on getting a territory.
• Feeding
• Given a fixed amount of food, sharing of food
  among spawners will occur.

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A Numerical Example-I

• Assume we have an area with 1000 settlement (or
  breeding) sites.
• Only one animal can settle on (breed at) each
  site.
• The factors that impact the relationship between
  the number attempting to settle (breed) and the
  number surviving (breeding) depends on several
  factors.

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A Numerical Example-II

• Hypothesis factors
• Sites are selected randomly / to maximize
  survival (breeding success).
• Survival differs among sites (from 1 to 0.01) or
  is constant.
• Attempts by more than one animal to settle on a
  given site leads to finding another site (if one
  is available), death (failure to breed) for all
  but one animal, death of all the animals
  concerned.
• How many more can you think of??

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Case 1 No density-dependence (below 1000 or
unlimited habitat/food)
Survival is independent of site individuals
always choose unoccupied sites (or they choose
randomly until they find a free site).
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Case 2 Site-dependent survival (optimal site
selection habitat gradation)
Survival depends on site individuals always
choose the unoccupied site with the highest
expected survival rate.
All the habitat is equally suitable
Most suitable habitats get occupied, so the
survival decreases
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Case 3 Site-dependent survival (random site
selection)
Survival depends on site individuals choose
sites randomly until an unoccupied site is found.
The increase in RPS will depend on the
probability of find a good habitat
12
Case 4 Site-independent survival (random site
selection)
Survival is independent of site individuals
choose sites randomly but die / fail to breed if
a occupied site is chosen.
13
Numerical Example(Overview of results)

• Depending on the hypothesis for
  density-dependence
• Recruitment may asymptote.
• Recruitment may have a maximum and then decline
  to zero.
• We shall now formalize these concepts and provide
  methods to fit stock-recruitment models to data
  sets.

14
Selecting and Fitting Stock-Recruitment
Relationships
Skeena River sockeye
4,000
3,000
Recruits
2,000
1,000
0
0
500
1,000
1,500
Spawners
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Selecting and Fitting Stock-Recruitment
Relationships
Skeena River sockeye
4,000
3,000
Recruits
2,000
1,000
0
0
500
1,000
1,500
Spawners
16
The Beverton-Holt Relationship

• The survival rate of a cohort depends on the size
  of the cohort at any time i.e.
• This can be integrated to give

17
The Ricker Relationship

• The survival rate of a cohort depends only on the
  initial abundance of the cohort, i.e
• This can be integrated to give

18
A More General Relationship

• The Ricker and Beverton-Holt relationships can be
  generalized (even though most stock-recruitment
  data sets contain very little information about
  the shape of the stock-recruitment relationship)

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The Many Shapes of the Generalized Curve
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Fitting to the Skeena data

• We first have to select a likelihood function to
  fit the two stock-recruitment relationships. We
  choose log-normal (again) because recruitment
  cannot be negative and arguably whether
  recruitment is low, medium or high (given the
  spawner biomass) is the product of a large number
  of independent factors.

21
The fits !
Negative log-likelihood Beverton-Holt
-11.92 Ricker -12.13
22
SRR and age-structured models

• R is measured in terms of recruits to some
  specific age, and S is often a total egg
  production.
• We may wish to explore the potential yield when
  recruitment is more (or less) sensitive to
  spawning stock.
• Can be done by changing a, but using
• any change in a will result in a change in
  the equilibrium unfished stock size

23
The steepness parameter (I)

• E0 egg production in virgin conditions
• eggs produced per recruit when the stock
  is
• unfished
• R0 recruitment in unfished conditions
• h the fraction of R0 obtained
• at spawning stock 0.2 E0.

0.2 E0
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The steepness parameter (II)

• If h approaches 1 then recruitment is nearly
  constant over a broad range of spawning stocks
• If h is slightly larger than 0.2 then recruitment
  is proportional to spawning stock.
• The parameter is easily derived from the natural
  mortality and fecundity-at-age schedules.
•  
• The following relationships therefore hold
•  
•  
• We can solve for a and b in terms of h and R0

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The steepness parameter (III)