Transcript Death in the Sea Understanding Natural Mortality

Death in the Sea
Understanding Mortality
Rainer Froese
IFM-GEOMAR
(SS 2008)
What is Natural Mortality?
Proportion of fishes dying from natural
causes, such as:
• Predation
• Disease
• Accidents
• Old age
The M Equation
Instantaneous rate of mortality M:
Dt / N t = 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
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
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
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 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
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
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
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]
Froese’s (in prep.) Equation
L∞ = C
-0.45
M
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
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).
Froese’s (in prep.) Equation
M = 4.5 / tmax
tmax = 4.5 / M
Charnov’s 1993 Equation
E=1/M
Where E is the average life expectancy of adults
Reference: Charnov, E.L. 1993. Life history invariants: some
explorations of symmetry in evolutionary ecology. Oxford University
Press, Oxford, 167 p.
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
Life History Summary
Fishing Kills Fish
Z=M+F
Where
Z = total mortality rate
F = mortality caused my fishing
Size at First Capture Matters
140
100
Impact on cohort biomass
120
80
60
80
60
40
Catch of 40 t
40
20
20
0
0
20
40
60
80
Length (cm)
100
120
0
140
Catch (% Bt)
Cohort Biomas (t)
100
Size at First Capture Matters
Impact on Size-Structure (F=M)
120
Cohort Biomas (t)
100
Lc = 80 cm
80
60
40
Lc = 24 cm
20
0
0
20
40
60
Length (cm)
80
100
120
What You Need to Know
Nt = N0 e
–M (t)
Bt = Nt * Wt
Z=M+F
Thank You