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Generation-Generation Models

(Stock-Recruitment Models)

Fish 458, Lecture 20

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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.

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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.

( ) ( ) exp( )t t L t L t L tR N s N f N w

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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!

1 exp( )t t tR R f s w

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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 amongst 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 maximise 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)

0

200

400

600

800

0 200 400 600 800Spawners

Recruits

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)

0

200

400

600

800

0 200 400 600 800Spawners

Recruits

Survival depends on site; individuals always choose the unoccupied site with the highest

expected survival rate.

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Case 3 : Site-dependent survival (random site selection)

0

200

400

600

800

0 200 400 600 800Spawners

Recruits

Survival depends on site; individuals choose sites randomly until an unoccupied site is found.

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Case 4 : Site-independent survival (random site selection)

0

200

400

600

800

0 200 400 600 800Spawners

Recruits

Survival is independent of site; individuals choose sites randomly but die / fail to breed if a occupied

site is chosen.

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0

100

200

300

400

500

0 500 1000 1500 2000

Spawners

Recruits

Case 5 : Site-independent survival (competition among

occupiers).

Survival is independent of site; individuals choose sites randomly but if two (or more)

individuals choose the same site they all die / fail to breed.

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Numerical Example(Overview of results)

Depending on the hypothesis for density-dependence: Recuitment 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.

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Selecting and Fitting Stock-Recruitment Relationships

Skeena River sockeye

0

1,000

2,000

3,000

4,000

0 500 1,000 1,500

Spawners

Recruites

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The Beverton-Holt Relationship

The survival rate of a cohort depends on the size of the cohort, i.e.:

This can be integrated to give:

( ) ; (0)

( ) is the number of recruits at time ,

is the number (biomass) of spawners.

dRq pR R R a S

dtR t t

S

31

1 2 2 31

a Sa S SR

b S b a S b S

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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:

( ) ; (0)

( ) is the number of recruits at time ,

is the number (biomass) of spawners.

dRq pS R R a S

dtR t t

S

1 1 2 2exp( ) exp( / )R a S b S S a S b

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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):1( ) wR aS b S e

Ricker : limit

Beverton-Holt : 1

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The Many Shapes of the Generalized Curve

0

0.3

0.6

0.9

1.2

0 1 2 3 4 5 6

Spawners

Recruits

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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.;w bS wS

R e R a S e ea b S

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The fits !

0

1,000

2,000

3,000

4,000

0 500 1,000 1,500

Spawners

Recruits

Beverton-Holt

Ricker

Negative log-likelihoodBeverton-Holt: -11.92Ricker: -12.13

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Readings Burgman et al. (1993); Chapter 3. Hilborn and Walters (1992);

Chapter 7. Quinn and Deriso (1999); Chapter

3.