Transcript Biodiversity of Fishes: Life-History Allometries and Invariants Rainer Froese 29.01.2015 What is Life History? • The stages of life an organism passes through from birth to.

Biodiversity of Fishes:
Life-History Allometries and
Invariants
Rainer Froese
29.01.2015
What is Life History?
• The stages of life an organism passes
through from birth to death
• The study of the timing of life cycle events
such as maturity, max growth and death
• Keywords: life span, longevity, mortality,
survival, reproduction, fecundity, eggs,
larvae, juveniles, adults, natural selection
of adaptive traits
Life History Allometries
Typically a power function describing how
one trait changes in relation to another.
Example: How body weight scales with length
W = a Lb
where a is a proportionality factor and
b ~ 3 is the typical scaling of weight with length
Body Weight Allometries
•
Y = a W 0.75
– where Y is a whole body rate such as oxygen consumption,
ingestion, heat production, blood flow and W is body weight
•
Y=aW1
– where Y is another weight or volume such as weight at maturity,
gonad weight, heart volume
•
Y = a W 0.25
– where Y is age such as age at maturity, life span, longevity
•
Y = a W -0.25
– where Y is a rate per year such as natural mortality, annual
reproductive rate, growth rate (individual and population)
Traits that change with
body weight
The von Bertalanffy Growth Function
•
dW/dt = H W 2/3 – k W 1
–
where H W 2/3 stands for anabolism assumed proportional to resorbing
surfaces scaling as 2/3 = 0.666 with weight
– and k W 1 stands for catabolism scaling proportional to weight
• Integrating, rearranging and simplifying gives
•
Wt = W∞ (1 – e-K(t – to))3
– where K = 3 k.
Life History Invariants:
Maximum growth (weight of add-on tissue)
is obtained at
0.296 Winf
if b~3
0.667 Linf
Average Adult Life Expectancy

l d
y
Ex 
y
x
lx
where Ex is the average life expectancy after reaching age x and l are
the probabilities of reaching x and subsequent ages y. If the mortality rate
is constant then
1
Em 
M
Mortality and Growth
• In species that grow throughout their lives,
maximum size is determined by life span
• Life span is determined by mortality
Therefore
• Maximum size and growth is determined
by mortality
• K ~ 2/3 M
Growth and Mortality
1
Weight
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
Age (years)
14
16
18
20
22
24
Growth and Mortality
Winf
1
Weight
0.8
0.6
0.4
max. growth rate
0.2
0
0
2
4
6
8
10
12
Age (years)
14
16
18
20
22
24
Growth and Mortality
1
Weight
0.8
0.6
0.4
max. growth rate
0.2
Probability
of survival
0
0
2
4
6
8
10
12
Age (years)
14
16
18
20
22
24
Growth and Mortality
1
M/K > 3/2
Peak left
and smaller
Expected weight
M/K < 3/2
Peak right
and smaller
0.8
0.6
Weight
M/K = 3/2
0.4
max. growth rate
0.2
Probability
of survival
0
0
2
4
6
8
10
12
Age (years)
14
16
18
20
22
24
M observed vs M = 1.5 K
1:1
100.00
M observed
10.00
1.00
0.10
0.01
0.01
0.1
1
M Mfrom
1.5KK
from 1.44
10
100
Life History Invariants:
Length at Maximum Reproductive Biomass
Lopt
topt 
3
2
 L
 L
M
3
3
K
1
3K
1.1
ln(1 
)  t0 
 t0
K
M
K
Western Baltic Cod Life History
18
asymptotic weight
16
max age
Gadus morhua , Linf = 120 cm,
Winf = 16.2 kg, K = 0.14, M = 0.2
14
Weight (kg)
12
10
max reproductive
biomass of cohort
8
6
max growth
4
maturity
2
average adult life span
0
0
5
10
15
Age (years)
20
25
Reproductive Strategies
Froese & Pauly 2013, Fish Stocks, Encyclopedia of Biodiversity, Academic Press
Length at Maturity for Different Reproductive
Strategies
Froese & Pauly 2013, Fish Stocks, Encyclopedia of Biodiversity, Academic Press
Variability in Maturity
10000
L∞
0.67 L∞
Lm (cm)
1000
0.35 L∞
100
10
1
1
10
100
L∞ (cm)
1000
10000
Longevity as Size Invariant
• Taylor (1958) suggests maximum age is
reached at 95% Linf -> tmax = 3/K
• A good fit is obtained at 96% Linf
Longevity vs Age at 96% Linf
1000
1:1
Observed longevity (years)
100
10
1
0.1
0.1
1
10
t max = 3.22/K (years)
100
1000
Approximate Relation of Key
Parameters
rmax ≈ 2 M ≈ 3 K ≈ 9 / tmax
where rmax is the maximum intrinsic rate of population increase
M is the rate of natural mortality
K is the somatic growth rate
tmax is maximum age
Summary
• Growth, average adult lifespan, maximum
reproductive biomass, and longevity have
co-evolved so that maximum reproductive
output is reached as fast as possible and
maximum lifespan is reached near
maximum size
• Maturity may start before Lopt if successful
reproduction is uncertain
Exercise
• Find species with growth and maturity data
and high versus low fecundity
• Compare Lm/Linf with 0.67 and discuss
differences