1. Biodiversity of Fishes Death in the Sea Understanding Natural Mortality
Rainer Froese
GEOMAR

2. What is Natural Mortality?
Proportion of fishes dying from natural causes, such as:
Predation
Disease / parasites
Accidents, natural disasters
Old age

3. The M Equation Instantaneous rate of natural mortality of M:
Dt / Nt = Mt
Where
t is the age in years
Dt is the number of deaths at age t
Nt is the population size at age t

4. The M Equation Probability of survival (lt) to age t: lt = e –M t
Where
M is the instantaneous rate of natural mortality
t is the age in years
lt ranges from 1.0 at birth to near zero at maximum age

5. The M Equation Number of survivors N to age t : Nt = N0 e –M t Where
N0 is the number of individuals at start age t=0
Nt is the number of individuals at age t

6. Cohort numbers if M = 0.2

7. Constant Value of M for Adults (in species with indeterminate growth: fishes, reptiles, invertebrates, ..)
M is typically higher for larvae, juveniles, and very old individuals, but reasonably constant during adult life
This stems from a balance between intrinsic and extrinsic mortality:
Intrinsic mortality increases with age due to wear and tear and accumulation of harmful mutations acting late in life
Extrinsic mortality decreases with size and experience

8. The M Equations
If M is different in years 1, 2, 3 and constant thereafter
lt = e –(M1+M2+M3+Mconstant*(t-3))
Nt = N0 e –(M1+M2+M3+Mconstant*(t-3))

9. M is Death Rate in a Stable Population
In a stable, equilibrium population
The number of spawners dying per year must equal the number of ‘new’ spawners per year
Every spawner, when it dies, is replaced by one new spawner, the life-time reproductive rate is
1/1 = 1
If the average duration of reproductive life dr is several years, the annual reproductive rate α is
α = 1 / dr

10. The P/B ratio is M (Allen 1971)
In a stable, equilibrium population
Biomass gained by production (P) must equal biomass lost (Blost) due to mortality
M is the instantaneous loss in numbers relative to the initial number: Nlost / N = M
If we assume an average weight per individual, then we have biomass: Blost / B = M
If Blost = P then P / B = M
Reference: Allen, K.R Relation between production and biomass. Journal of the Fisheries Research Board of Canada, 1971, 28(10):

11. Pauly’s 1980 Equation
log M = – log L∞ log K log T
Where
L∞ and K are parameters of the von Bertalanffy growth function and
T is the mean annual surface temperature in °C
Reference: Pauly, D On the interrelationships between natural mortality, growth parameters, and mean environmental temperature in 175 fish stocks. J. Cons. Int. Explor. Mer. 39(2):

12. Jensen’s 1996 Equation
M = 1.5 K
Where K is a parameter of the von Bertalanffy growth function
Reference: Jensen, A.L Beverton and Holt life history invariants result from optimal trade-off of reproduction and survival. Canadian Journal of Fisheries and Aquatic Sciences:53:

13. M = 1.5 K
Plot of observed natural mortality M versus estimates from growth coefficient K with M = 1.5 K, for 272 populations of
181 species of fishes. The 1:1 line where observations equal estimates is shown. Robust regression analysis of
log observed M versus log(1.5 K) with intercept removed explained 82% of the variance with a slope not significantly different
from unity (slope = 0.977, 95% CL = – 1.03, n = 272, r2 = ). Data from FishBase 11/2006 [File: M_Data.xls]

14. Hoenig’s 1984 Equation ln M = 1.44 – 0.984 * ln tmax
Where tmax is the longevity or maximum age reported for a population
Reference: Hoenig, J.M., Empirical use of longevity data to estimate mortality rates. Fish. Bull. (US) 81(4).

15. Charnov’s 1993 Equation

16. Life History Summary
Note: Blue line is not to scale. Froese and Pauly Fish Stocks, p In Encyclopedia of Biodiversity, Academic Press

17. Fishing Kills Fish Z = M + F Where Z = total mortality rate
F = mortality caused my fishing

18. Total Mortality of Turbot
Numbers at age in survey catches of North Sea turbot (Scophthalmus maximus).
Points at the left are not fully selected by the gear. The point at the right is a
single, rare survivor of fishing. The absolute slope Z = 0.82 represents total mortality
from natural causes M and from fishing F.

19. Conclusions
Natural mortality M is high in early life and near constant in adults
M determines life expectancy, growth and reproduction (and everything else)
Total mortality is Z = M + F
Death rules

20. Exercises
Select a species from FishBase with several estimates of natural mortality (M is under Growth)
Discuss M relative to other species (M-K Graph)
Determine mean M/K ratio
Determine adult life expectancy E