Biodiversity of Fishes
Death in the Sea
Understanding Natural Mortality
Rainer Froese
GEOMAR
08.01.2015
What is Natural Mortality?
Proportion of fishes dying from natural
causes, such as:
• Predation
• Disease / parasites
• Accidents, natural disasters
• Old age
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
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
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
Cohort numbers if M = 0.2
1200
Nt = Nts * exp(-M*(t - ts))
Cohort numbers
1000
800
600
400
200
0
0
5
10
15
Cohort age (years)
20
25
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
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))
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
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. 1971. Relation between production and biomass. Journal of the Fisheries
Research Board of Canada, 1971, 28(10): 1573-1581
Pauly’s 1980 Equation
log M = -0.0066 – 0.279 log L∞ + 0.6543 log K + 0.4634 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. 1980. On the interrelationships between natural mortality, growth
parameters, and mean environmental temperature in 175 fish stocks. J. Cons. Int. Explor.
Mer. 39(2):175-192.
Jensen’s 1996 Equation
M = 1.5 K
Where K is a parameter of the von Bertalanffy
growth function
Reference: Jensen, A.L. 1996. Beverton and Holt life history invariants
result from optimal trade-off of reproduction and survival. Canadian
Journal of Fisheries and Aquatic Sciences:53:820-822
M = 1.5 K
100
1:1
M observed
10
1
0.1
0.01
0.01
0.1
1
10
100
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 = 0.923 – 1.03, n = 272, r2 = 0.8230). Data from FishBase 11/2006 [File: M_Data.xls]
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., 1984. Empirical use of longevity data to estimate mortality
rates. Fish. Bull. (US) 81(4).
Charnov’s 1993 Equation
Life History Summary
Note: Blue line is not to scale. Froese and Pauly 2013. Fish Stocks, p. 477-487 In Encyclopedia of Biodiversity, Academic Press
Fishing Kills Fish
Z=M+F
Where
Z = total mortality rate
F = mortality caused my fishing
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.
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
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