Exponential growth
(Density Independent):
Population Models in Fisheries
Models for fish populations are similar as those used for birds and mammals.
Some different applications have been developed specific to fisheries.
Nt = No ert
Nt
r = maximum rate of change
Stock – Recruit Relationships
t
Influence of Environmental Variability
Fisheries Harvest Management
Linearize: ln (Nt) = ln (No) + rt
Ln (Nt)
Slope = r
t
This model relates to invasive species, newly created populations.
Currently many fish populations fished to low levels, may exhibit exp. growth.
But it is more realistic that births decline, or deaths increase,
or both occur as population size increases and resources become
limited.
What would figure look like?
Logistic growth
Nt
K
dN/dt = rN(1-N/K)
r starts out high but declines w/ time
Nt+1
Plot Nt+1 vs. Nt
(Density Dependent Models):
instantaneous change
K
Population change over time:
Nt
Max. rate of increase in N
@ K/2
Shape of logistic growth produces a non-linear
(asymptotic) curve that reaches K.
K
Diagonal represents replacement.
Nt+1 = Nt + rNt(1-(Nt/K))
Descrete model (ie annual change)
t
Nt
Max. rate @ K/2
Max. abundance above replacement
at K/2.
K = carrying capacity
r = rate of change
t
t could be months, years, etc.
Population change over time:
Stock-Recruit Relationships
Recruits
Plot Nt+1 vs. Nt
If t is one generation;
Four Desirable Properties of S-R Relationships
Stock
Nt becomes parents or Spawner stock
Nt+1 = offspring or Recruits
1. Should pass through zero – discounts immigration.
2. Should not drop to zero at high stock levels.
3. Rate of recruitment should decline as stock increases.
Compensatory density-dependence.
4. Recruitment must exceed parental stock over some part of the range
of possible parental stocks.
Stock-Recruit Relationships
-Have been used in fisheries management since 1950’s
- Seem more applicable to highly fecund species
Recruits
- Can be some-what controversial
Some camps do not agree that there is a relationship between S & R.
Others want to strictly adhere to a S-R curve to manage a fishery.
Stock
Stock-Recruit Relationships
Stock-Recruit Relationships
Spawner stock – reproductive productivity of mature population
- Number of eggs spawned (estimated from average fecundity per female)
- Biomass of mature adults (females)
- Index of spawners (redds counted)
1. Beverton-Holt
Three examples of S-R Models
1. Beverton-Holt
b+S
R = recruitment
S = spawner stock
a = max. recruits possible
b = stock needed to produce a/2
12000
Subsequent Recruitmen
Recruitment – number of offspring at any point after egg stage.
-Typically use all mature offspring of spawner stock (exclude harvests).
- (should be after stage where density-dependence occurs)
aS : eε
R = ________
10000
8000
6000
4000
error = 0.45
2000
b = 5000
2. Ricker
b = 2000
0
3. Deriso’s Generalized Model
0
2000
4000
6000
8000b = 400 10000
Spawning Stock Size
Stock-Recruit Relationships
R = aSe-bS : eε
3. Deriso’s Generalized Model
18000
b = 0.0004
10000
16000
b = 0.0003
9000
Subsequent Recruitmen
Subsequent Recruitmen
2. Ricker
Stock-Recruit Relationships
R = recruitment
S = spawner stock
a = recruits per spawner at low S
b = rate of decrease as S increases
14000
12000
10000
8000
6000
4000
6000
Stock-Recruit Relationships - Examples
8000
10000
Ricker
5000
Bev-Holt
4000
3000
2000
0
4000
Deriso-Schnute
6000
1000
Spawning Stock Size
α = 10, β = 0.001, γ = -1
7000
0
2000
α = 10, β = 0.0004, γ = 0.25
R = recruitment
S = spawner stock
α,β,γ = parameters
8000
2000
0
R = αS(1-βγS)1/γ
α = 10, β = 0.0004, γ = 0.000001
0
2000
4000
6000
Spawning Stock Size
8000
10000
Returning adults
Ocean entry
Eggs deposited
Fry/parr
Smolts
Ocean entry
Spawners
Spawners
Smolts
Fry/parr
Hypothetical salmon population with Beverton-Holt relationship
Stock – Recruit process is the culmination of the series of survivorship curves
for each life stage of fish.
Eggs deposited
Stock – Recruit process is the culmination of the series of survivorship curves
for each life stage of fish.
Recruits
Recruits
Returning adults
Spawners
Spawners
Plotting the data...
Getting the data...
Columbia River Spring/Summer Chinook Salmon
Counts
Harvest Zone 1-5
Columbia River Spring/Summer Chinook Salmon
Harvest Zone 6
500
01
500
400
Recruits
Chinook salmon x 1,000
400
300
200
100
0
1939
02
52
300 59
56
57
58
55 7262
69 6051
04
64
66 73
67 71636570
00
68
200
46 4943 50
78 77
48
86
75
87 939790
88
45
85 89 74 76
92
44
100
80
81 82
83
8496 91 79
9998
94
95
03
53
54
61
47
06
05
07
0
0
1949
1959
1969
1979
1989
1999
100
200
Spawning stock
300
400
Stock-Recruit Relationships - Examples
Tiger prawns Western Australia Stock-Recruit Relationship :1970-84
Tiger prawns Western Australia Stock-Recruit Relationship :1970-84
With indexes for cyclone intensity during January and February
40
Recruitment Index (Year t+
Recruitment in Year t+
40
B-H SS residual = 1.76
35
30
25
20
15
10
5
Ricker SS residual = 3.63
0
35
30
25
20
15
10
5
0
0
4
8
12
16
20
24
0
3
6
Spawning Stock in Year t
9
12
15
18
21
24
Spawning Stock Size (Year t)
So, what use is a S-R Curve (assuming it holds some reasonable validity)?
- Harvest management
Portion above replacement ≈ harvestable fractions
Recruits
Stock
Harvest near K/2 produce highest yields.
Harvest at populations above K/2 are more stable (resilient to environmental change).
Add harvest to logistic model by adding term Z = M + F
Z = total instantaneous mortality
M = natural mortality
Nt+1 = Nte-Z
F = fishing mortality
How do you maximize fishing yield?
Yield versus Population Size;
K = 100, r = 1.0
What is maximum yield per recruit?
Yield per year
K = 100
3.51
Nt
Need;
Populations growth characteristics (r).
Effects of key environmental variables that influence productivity.
Need to know S-R relationship.
Cohort age structure
Size/age of fish susceptible to fishing pressure.
To produce;
Age or recruitment that produces optimal yield
Level of fishing pressure that produces optimal yield.
6.25
3.8955
3.95
t
Yield
Use model exercises to develop hypothetical curves.
Fishing Mortality F
Yield versus Age;
K = 100, r = 1.0
Cohort
size
Age t