1. Biodiversity of Fishes: Life-History Allometries and Invariants
Rainer Froese

2. 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, …

3. 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

4. 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]
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, somatic growth rate
Note: these are empirical scalings typically observed in plots across many species

5. Traits that change with body weight

6. 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 = 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 is close to the expected 0.75 scaling

7. 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

8. Average Adult Life Expectancy
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

9. 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

10. Growth and Mortality

11. Growth and Mortality
Winf

12. Growth and Mortality

13. Growth and Mortality M/K > 3/2 M/K < 3/2 Peak left Peak right
and smaller
M/K < 3/2
Peak right
and larger
M/K = 3/2

14. 2016
“Thus, from an evolutionary perspective, maximum growth performance including
the production of gonad tissue is combined with the peak in expected offspring
production if M/K = 1.5 (Froese and Pauly 2013). This provides a fitness advantage
because with this ratio, natural selection has ‘economize[d ...] the organization’ of
reproduction (Darwin 1859).”

15. M observed vs M = 1.5 K
1:1
M from 1.5 K

16. Optimum Length and Age at Reproduction (for semelparous species)
Roff 1984
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.

17. Western Baltic Cod Life History
max age
max reproductive
biomass of cohort
max growth
maturity
average adult life span

18. Reproductive Strategies
Froese & Pauly 2013, Fish Stocks, Encyclopedia of Biodiversity, Academic Press

19. Length at Maturity for Different Reproductive Strategies
Froese & Pauly 2013, Fish Stocks, Encyclopedia of Biodiversity, Academic Press

20. 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

21. Longevity vs Age at 96% Linf
1:1

22. 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
Note: ongoing research shows that these relations hold for typical fish (cod & herring),
but not for small (e.g. gobies) and large (e.g. tunas) fish or for low fecundity (e.g. sharks)

23. Summary
Growth, average adult lifespan, maximum reproductive biomass, and longevity have co-evolved as a trade-off between maximum reproductive output and not too long generation time

24. Exercise
Find species with growth and maturity data and high versus low fecundity
Compare Lm/Linf with 0.67 and discuss differences