Carrying Capacity
Spawning
habitat
Summer
rearing
Winter
rearing
Photo by John
McMillan
Smolt
Definition
Carrying Capacity = Number of
individuals (K) the resources of a given
area can support usually through the most
unfavorable period of the year.
– Tied to Limiting Factors
2
Population Regulation
• Density Independent Regulation =
Population growth is not affected by
population density; population persistence is
explained by unpredictable environmental
variability (Andrewartha and Birch).
• Density Dependent Regulation =
Population growth is affected by mechanisms
whose effectiveness increases as population
size increases (Nicholson, Lack, and Elton).
3
Evidence of Density
Dependence
Chiwawa Spring Chinook
0.16000
0.14000
Yearling Smolts/Eggs
• Plot of population size
and population growth
rate (or surrogates such
as mortality, natality,
productivity, recruits,
individual growth rates,
movement).
• There is a negative
relationship between
population size and
growth rate, which
means that carrying
capacity may be
estimated.
0.12000
0.10000
0.08000
0.06000
0.04000
0.02000
0.00000
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
Number of Eggs
4
Methods for Estimating
Carrying Capacity
• Time series of population growth over
time (logistic function)
• Habitat modeling (maximum number of
fish within each habitat type; PHS)
• Stock-recruitment modeling (Ricker,
Beverton-Holt, or Smooth Hockey Stick)
5
Assumptions
• Assume we can define a population
unambiguously.
• Assume that we can measure
population size accurately.
• Assume that we have a biologically
relevant time-step over which to
measure the population growth rate.
• Assume a uniformity of nature.
6
Habitat Models
• Habitat capacity can be
estimated as the product of
habitat area and fish/habitat
relationships.
• Percent Habitat Saturation
Model (PHS)
𝑃𝐻𝑆 = 100 𝑥 Σ 𝐷𝑖 𝑥 𝑇𝑖
• Estimate the number of fish
at maximum spawning
escapement.
7
Stock-Recruitment Modeling
•
Fit Ricker, Beverton-Holt, and
Smooth Hockey Stick models to
stock (eggs) and recruitment
(fry, parr, smolts) data.
Ricker:
Ricker Model
300000
200000
150000
100000
50000
𝑬(𝑹) = 𝜶𝑺𝒆−𝜷𝑺
𝜶 −𝟏
𝑲=
𝒆
𝜷
•
Recruits 1
Recruits 2
Recruits 3
Recruits 4
250000
Recruits
•
0
0
200
400
600
800
1000
800
1000
Stock
Beverton-Holt:
𝑬 𝑹 =
𝜶𝑺
𝜷+𝑺
Smooth Hockey Stick Model
2500
𝜶=𝑲
Recruits
• Smooth Hockey Stick:
𝑬(𝑹) = 𝑹∞ 𝟏 − 𝒆
𝑹∞ = 𝑲
Recruits 1
Recruits 2
Recruits 3
Recruits 4
2000
−
𝜶
𝑹∞
𝑺
1500
1000
500
0
0
200
400
600
Stock
8
Chiwawa Spring Chinook
Habitat Model
• No habitat modeling has
been completed in the
Chiwawa River basin.
• No estimates of PHS have
been calculated.
• Maximum parr and
yearling smolt production
likely occurred in 2002
when 1,725 spawners
escaped in 2001.
• One can also select the
maximum number of parr
and smolts.
BY
Eggs
Resident Parr
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
478,400
1,570,098
556,394
485,686
66,248
106,835
374,740
218,325
166,090
642,944
4,984,672
1,605,630
648,684
1,156,559
1,436,564
1,284,228
1,256,803
3,163,888
1,925,233
2,165,628
2,157,420
3,716,240
45,483
79,113
55,056
55,240
5,815
16,066
68,415
41,629
114,617
134,874
91,278
45,177
49,631
79,902
60,752
82,351
106,705
128,220
141,510
103,940
158,407
Number of
Yearlings
(Smolts)
42,525
39,723
8,662
16,472
3,830
15,475
28,334
23,068
10,661
40,831
86,482
90,948
16,755
72,080
69,064
45,050
25,809
35,023
30,959
47,511
37,185
9
Chiwawa Spring Chinook
Stock-Recruitment Model
Number of Resident Parr
180,000
160,000
140,000
120,000
100,000
80,000
60,000
40,000
20,000
0
0
1,000,000
2,000,000
3,000,000
4,000,000
5,000,000
6,000,000
Number of Eggs
B-H Model
Ricker Model
Hockey Stick
Chiwawa Spring Chinook
100,000
90,000
Number of Smolts
• Stock-recruitment
functions were fit
successfully to parr
and yearling smolt
data.
B-H Model
Ricker Model
Hockey Stick
Chiwawa Spring Chinook
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
1,000,000
2,000,000
3,000,000
4,000,000
Number of Eggs
5,000,000
6,000,000
10
Chiwawa Spring Chinook
Stock-Recruitment Model
Parr:
Model
Ricker
Hockey Stick
Beverton-Holt
Parameter
Carrying
capacity
Productivity
Maximum
stock
0.1190490 0.0000004
11.6628
0.1418
155,585.92 989,480.74
109,489
116,167
155,586
0.1190
0.1418
0.1572
2,500,000
2,455,000
∞
Parameter
Carrying
capacity
Productivity
Maximum
stock
57,541
49,016
63,292
0.0626
0.0804
0.0916
2,500,000
1,825,000
∞
A
B
Smolt:
Model
Ricker
Hockey Stick
Beverton-Holt
A
B
0.0625650 0.0000004
10.7999
0.0804
63,292.40 691,129.47
11
12