1. Death in the Sea Understanding Mortality
Rainer Froese
IFM-GEOMAR
(SS 2008)

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

3. The M Equation Instantaneous rate of mortality M: Dt / Nt = M 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): 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 0.0x 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 specimens at start age t=0
Nt is the number of specimens at age t

6. 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 specimens, 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. M is Death Rate in Unfished Population
In an unfished, stable 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 is α = 1 / dr

9. The P/B ratio is M (Allen 1971)
In an unfished, stable population
Biomass B gained by production P must equal biomass lost due to mortality
M is the instantaneous loss in numbers relative to the initial number: Nlost / N = M
If we assume a mean weight per individual, then we have biomass: Blost / B = M
If Blost = P then P / B = M

10. 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):

11. 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:

12. 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]

13. Froese’s (in prep.) Equation
L∞ = C M-0.45
This is the L∞ – M trade-off, where L∞ is the asymptotic length of the
von Bertalanffy growth function and C is an indicator of body plan,
environmental tolerance and behavior, i.e., traits that are relatively
constant in a given species.
If C is known e.g. from other populations of a species, M corresponding
to a certain L∞ can be obtained from
M = (L∞ / C)-2.2

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. Froese’s (in prep.) Equation
M = 4.5 / tmax
tmax = 4.5 / M

16. Charnov’s 1993 Equation E = 1 / M
Where E is the average life expectancy of adults
Reference: Charnov, E.L Life history invariants: some explorations of symmetry in evolutionary ecology. Oxford University Press, Oxford, 167 p.

17. Froese’s (in prep.) Equation
dr = 1 / M
Where dr is the mean duration of the reproductive phase
If mortality is doubled then reproduction is shortened by half

18. Life History Summary

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

20. Size at First Capture Matters

21. Size at First Capture Matters Impact on Size-Structure (F=M)

22. What You Need to Know
Nt = N0 e –M (t)
Bt = Nt * Wt
Z = M + F

23. Thank You