458
Generation-Generation Models
(Stock-Recruitment Models)
Fish 458, Lecture 20
Recruitment
458
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Annual recruitment is defined as the number of
animals “added to the population” each year.
However, recruitment is also defined by when
recruitment occurs:

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



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.
Stock and Recruitment - Generically
(the single parental cohort case)
458

The generic equation for the relationship
between recruitment and parental stock size
(spawner biomass in fishes) is:
Rt  Nt  L s( Nt  L ) f ( Nt  L ) exp( wt )

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Recruitment equals parental numbers
multiplied by survival, fecundity and
environmental variation.
The functional forms allow for densitydependence.
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Stock and Recruitment - Generically
(the single parental cohort case)

Consider a model with no densitydependence:
Rt  Rt 1 f s exp( wt )
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The population either grows forever (at
an exponential rate) or declines
asymptotically to extinction.
The must be some form of densitydependence!
Some Hypotheses for
Density-Dependence
458

Habitat:

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Fecundity
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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).
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.
A Numerical Example-I
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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.
A Numerical Example-II
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Hypothesis factors:
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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??
Survival is independent of site; individuals always
choose unoccupied sites (or they choose randomly until
they find a free site).
800
600
Recruits
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Case 1: No density-dependence
(below 1000)
400
200
0
0
200
400
Spawners
600
800
Survival depends on site; individuals always choose the
unoccupied site with the highest expected survival rate.
800
600
Recruits
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Case 2 : Site-dependent survival
(optimal site selection)
400
200
0
0
200
400
Spawners
600
800
Survival depends on site; individuals choose sites
randomly until an unoccupied site is found.
800
600
Recruits
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Case 3 : Site-dependent survival
(random site selection)
400
200
0
0
200
400
Spawners
600
800
Survival is independent of site; individuals choose sites
randomly but die / fail to breed if a occupied site is chosen.
800
600
Recruits
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Case 4 : Site-independent survival
(random site selection)
400
200
0
0
200
400
Spawners
600
800
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.
500
400
Recruits
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Case 5 : Site-independent survival
(competition among occupiers).
300
200
100
0
0
500
1000
Spawners
1500
2000
Numerical Example
(Overview of results)
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Depending on the hypothesis for
density-dependence:
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Recuitment may asymptote.
Recruitment may have a maximum and
then decline to zero.
We shall now formalize these concepts
and provide methods to fit stockrecruitment models to data sets.
Selecting and Fitting StockRecruitment Relationships
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Skeena River sockeye
Recruites
4,000
3,000
2,000
1,000
0
0
500
1,000
Spawners
1,500
The Beverton-Holt Relationship
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The survival rate of a cohort depends
on the size of the cohort, i.e.:
dR
 (q  pR ) R; R(0)  a S
dt
R (t ) is the number of recruits at time t ,
S is the number (biomass) of spawners.

This can be integrated to give:
a3 S
a1 S
S
R


b1  S b2  a2 S 1  b3 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:
dR
 (q  pS ) R; R(0)  a S
dt
R (t ) is the number of recruits at time t ,
S is the number (biomass) of spawners.

This can be integrated to give:
R  a1 S exp(b1S )  S exp(a 2 S / b2 )
A More General Relationship
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The Ricker and Beverton-Holt relationships
can be generalized (even though most stockrecruitment data sets contain very little
information about the shape of the stockrecruitment relationship):
R  aS (b  1 S ) ew
Ricker : limit   
Beverton-Holt :   1
The Many Shapes of the
Generalized Curve
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Recruits
1.2
0.9
0.6
0.3
0
0
1
2
3
Spawners
4
5
6
Fitting to the Skeena data
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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.
S
R
ew ;
abS
R  a S e bS e w
The fits !
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Negative log-likelihood
Beverton-Holt: -11.92
4,000
Ricker: -12.13
Beverton-Holt
Recruits
Ricker
3,000
2,000
1,000
0
0
500
1,000
Spawners
1,500
Readings
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Burgman et al. (1993); Chapter 3.
Hilborn and Walters (1992); Chapter 7.
Quinn and Deriso (1999); Chapter 3.