Biodiversity of Fishes:Life-History Allometries and

Invariants

Rainer Froese10.12.2015

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, …

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 = a W 1 – where Y is another weight or volume such as weight at maturity,

gonad weight, heart volume [exception: brain weight scales 1/3]

• 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)

Note: these are empirical rates typically observed in plots across many species

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.

Note: the within-population scaling of 2/3 = 0.67, which is close to the expected 0.75

Life History Invariants:Maximum growth (weight of add-on tissue)

is obtained at

0.296 Winf

if b~3 this corresponds to

0.667 Linf the growth curve in length has no inflexion,

growth rate in length is max at origin

Average Adult Life Expectancy

x

x

y

xl

dl

E

y

MEm

1

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 rateis constant then

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

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

Growth and Mortality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

max. growth rate

Winf

Growth and Mortality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

Probabilityof survival

max. growth rate

Growth and Mortality

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20 22 24

Age (years)

Weight

Probabilityof survival

max. growth rate

Expected weight

M/K = 3/2

M/K > 3/2Peak left and smaller

M/K < 3/2Peak right and smaller

M observed vs M = 1.5 K

0.01

0.10

1.00

10.00

100.00

0.01 0.1 1 10 100

M from 1.44 K

M o

bse

rved

M from 1.5 K

1:1

Life History Invariants:Length at Maximum Reproductive Biomass

L

KM

LLopt 3

2

3

3

00

1.1)

31ln(

1t

Kt

M

K

Ktopt

Note: Since cohort biomass and fecundity peak at topt, this is also the most common age of parents, which is the definition of generation time.

Western Baltic Cod Life History

0

2

4

6

8

10

12

14

16

18

0 5 10 15 20 25

Age (years)

Wei

gh

t (k

g)

asymptotic weight

Gadus morhua , Linf = 120 cm,Winf = 16.2 kg, K = 0.14, M = 0.2

maturityaverage adult life span

max growth

max reproductivebiomass of cohort

max age

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

1

10

100

1000

10000

1 10 100 1000 10000

L∞ (cm)

Lm

(cm

)

L∞

0.67 L∞

0.35 L∞

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

0.1

1

10

100

1000

0.1 1 10 100 1000

t max = 3.22/K (years)

Ob

serv

ed lo

ng

evit

y (y

ears

)

1:1

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