Rapp. P.-v. Réun. Cons. int. Explor. M er, 191: 324-329. 1989
Using growth histories to estimate larval fish mortality rates
Pierre Pepin
Pepin, Pierre. 1989. Using growth histories to estimate larval fish mortality rates. Rapp. P.-v. Réun. Cons. int. Explor. M er, 191: 324-329.
A simple stochastic model is presented to show the potential influence of variable
growth rates on survival within a cohort of larval fish. The results show that mortality
will decrease as the mean and variance in growth rates increase. Changes in the
distribution of growth histories (i.e. mean and variance), estimated by contrasting
the past growth rates of survivors, taken at time t , , with samples of the population
taken at an earlier time, t 0, can be used to estimate the mortality rates of the
population. Application of the model requires individual growth histories as input.
This provides an alternative to estimating mortality rates by extensive field surveys
of abundance.
Pierre Pepin: Department o f Fisheries and Oceans, Science Branch, P.O. Box 5667,
St J o h n ’s, Newfoundland, Canada, A 1 C 5X1.
Introduction
To understand the causes of variations in larval fish
survival and subsequent recruitment, it is necessary to
determine which individual pre-recruits die at different
stages in early life. It is reasonable to assume that
slow-growing individuals are more likely to die before
reaching metamorphosis, or some other critical stage,
than are faster-growing individuals, in view of the evi­
dence that daily mortality rates decrease with increasing
size (Peterson and Wroblewski 1984; McGurk 1986;
Pepin, in preparation). The implication is that the varia­
bility in growth rates within a year class is important to
consider in order to forecast survivorship, regardless of
the ultimate causes of mortality.
The analysis of daily growth increments in otoliths of
larval fish allows determination of past growth rates
(see Campana and Neilson, 1985, for a review). What
can growth histories, measured at different stages in the
life cycle, tell us about factors influencing differences in
cumulative mortality between cohorts? There is evi­
dence that individual fish larvae that have a size or
growth advantage at one stage tend to maintain that
advantage over at least several weeks and have a higher
probability of survival over such an interval (Ricker,
1969; Rosenberg and Haugen, 1982; Evans et al. , 1984).
By sampling a population at different times, we can
estimate growth rates of survivors through back-calculation. Houde (1987) argues that changes in popu­
lation characteristics due to external selective factors
should be reflected in otoliths of a representative group
of the population. Therefore, by contrasting the distri­
324
bution of growth rates, we can determine which portion
of the population survived and thus estimate mortality
rates. If we know the susceptibility of larval fish to
different environmental factors (e.g., predators, advection), then we may infer factors responsible for observed
changes in distribution at different stages.
It was previously shown that growth histories can be
used to study how changes in predator abundance, and
hence mortality rates, influence growth and survival
rates of population (Pepin, in press). H ere, it is shown
that the distribution (i.e ., mean (n) and variance ( a 2)) of
growth rates of fish within a cohort must be considered if
we are to estimate mortality rates and uncover the
causes of variations in survival during early life. It is
further demonstrated that changes in distribution of
growth rates, over time, can be used to estimate the
survivorship of a cohort.
Length is the most frequently used measure of size in
studies of larval fish (Pepin, in preparation). Therefore
the model presented is based on length although it does
not differ in general format from other size-dependent
models (see Beyer, 1988).
The model
A simple model of growth during the larval phase is
considered in which the size-specific growth of an indi­
vidual larva can be described as:
where L is the length of the larva (mm), t is time, g(L)
is the coefficient of daily instantaneous growth, and
where:
A t(L 0, L
r Li dx
(2 )
l ) = J ,„ göö
is the time required to grow from L 0 to L ,. If mortality
is size-specific, such that:
J_dN
N ~dT
(3)
= -M (L )
where M is the coefficient of instantaneous mortality,
then size-specific survivorship is:
Ni
S(L0, L , ) = — = exp
I'1!)
M(x)
(/:
dx
(4)
g(x)
which states that survivorship decreases with increasing
mortality rates and decreases with decreasing growth
rates.
For many species of larval fish, growth in length is
approximately linear (M ethot and Kramer, 1979;
Hunter and Kimbrell, 1980; Bolz and Lough, 1983;
Munk et al., 1986) such that:
g(L) = a
(5)
where a, the growth rate, is constant in mm/day.
Pepin’s (in preparation) summary of size-dependent
characteristics in larval fish shows that mortality rates
( d ' 1) are inversely proportional to length:
M(L) = cL-
(6 )
where c is the mortality coefficient, reflecting the mag­
nitude of the external source of mortality (e.g., pred­
ators). Length-specific survivorship (eq. 4) yields:
S(L0, L , ) = O ' c/a
(7)
where Q = L t/Lø. Survivorship increases as the growth
rate (a) increases and decreases as the mortality coef­
ficient (c) increases.
The model outlined above forms the basis of many
studies dealing with larval fish ecology: the growth rate
determines At, for a given size interval (L 0, L ,), which
in turn determines survivorship, when multiplied by
a given mortality rate. Many ecologists contend that
knowledge of the causes of variations in growth [g(L)]
and mortality [M(L)] rates will permit the forecasting
of variations in survivorship (see Rothschild, 1986 for a
review). This assumes that the mean growth rate of
a cohort is an adequate measure of an individual’s
probability of growing through a size interval (L u, Li).
The simple model presented above would be
adequate if all larvae were identical. However, there is
ample evidence that all individuals within a cohort do
not have the same growth rate (Beyer and Laurence,
1980; Rosenberg and Haugen. 1982; Evans et al. 1984;
Chambers and Leggett, 1987). Because the mean
growth rate of an individual determines its probability
of survival over a length interval, the distribution of
growth rates of a cohort determines the survival poten­
tial of that cohort for a given set of conditions. To
demonstrate this, consider a cohort which has an initial
probability distribution of growth rates F 0(aj). It follows
that the mean growth rate of that cohort is:
ao — 2 a:F 0(ai)
i= i
(8)
A sample of the cohort, at a sequential time tj, should
have a mean growth rate:
ai = 'Z aiFi(ai)
(9)
i= 1
where:
FM ocQ^Fote)
( 10)
and for which survivorship is:
S (L o ,L 1) = E O “c/a-Fo(ai)
i= 1
(ID
Because survivorship decreases as aj decreases, the
mean growth rate of survivors will increase as the m or­
tality coefficient (c) increases, as stated earlier. It fol­
lows that the difference in mean growth rates will be
proportional to survivorship over a length interval (L (l,
L,) or a time interval (t0, t t), for a given initial distri­
bution of growth histories. Also, the difference in the
mean growth rate of survivors will increase, as Q
increases (i.e. the critical size interval), for a given
mortality coefficient (c),
because survivorship
decreases. However, the difference in growth rate of
survivors is directly proportional to the cohort’s sur­
vivorship, independent of Q (Fig. 1). It is therefore
possible to estimate survivorship, based on sequential
samples of a cohort taken at times t0 and t, via the
relationship of S to Aä, for a given F0(aj).
The variance in the initial distribution of growth rates
[F(l(aj)] has an important influence on the relationship
between the mean growth rates of survivors and sur­
vivorship. As the variance in a cohort’s growth rates
increases, survivorship of the cohort will increase for a
given mean growth rate and mortality coefficient
because the potential cumulative mortality decreases
exponentially as the growth rate increases (Fig. 2). The
325
Q.
.38
LL_
o (/)
Cd
ÜJ O
H -> .36
x
-2
iQ£
—4
V)
QL
ZD
~
3
CO
.34
o -5
o
o
.32
-6
-7
-8
LO G , n
S U R VIV O R S H IP
Figure 1. Simulation results showing the mean growth rate of
the surviving portion of a cohort in relation to the logarithm
of the relative survivorship. The simulations were performed
by generating an approximately continuous normal distribution
of initial growth rates with mean ([i = 0.3 m m /d) and variance
( a 2 = 0.002). The survivorship of each 0.001 m m /d interval
was calculated using equation 7, for a given O and c, and total
survivorship was calculated by summing over all intervals (eq.
11). The mean growth rate of survivors was calculated by
scaling the cumulative frequency distribution of survivors to a
unit area and summing according to equation 9. Variations in
survivorship were obtained by altering the value of Q or c. The
results are independent of whether Q or c was changed.
addition of individuals to the fast-growing end of the
distribution has a greater effect on the cohort's potential
survivorship relative to individuals moved an equal dis­
tance from the mean towards the slow-growing end of
the distribution. Consequently, the change in the mean
growth rate of survivors, in relation to survivorship, will
increase as the variance in the initial distribution of
growth rates increases (Fig. 3).
The variance in growth rates can be caused partly
by inherent environmental factors (e.g. random prey
encounters) and partly by sampling error. Although it
is essential that sample size be adequate to minimize
sampling error, the inherent variability in growth rates
will largely determine the separation in time which will
yield accurate estimates of survivorship. Establishing
useful confidence limits from surveys of abundance
requires a stable population and a representative sample
from that population, neither of which can be assumed
in normal environmental conditions (Zweifel and Smith,
1981). Estimates of mortality coefficients derived from
extensive field surveys of abundance are often based on
regressions of the natural logarithm of the m ean number
in relation to the average size (Smith and Richardson,
1977). The confidence intervals of the slope of such
regressions are seldom reported (Zweifel and Smith,
1981). However, it must be remembered that they are
326
MORTALITY C O EFFICIENT
Figure 2. Simulation results showing survivorship of a cohort
in relation to the mortality coefficient (c) for different values in
the variance of the initial distribution of growth rates (dashed,
dotted and solid lines represent values o f a 2 of 0.001, 0.003,
and 0.01 respectively). See caption to Figure 1 for details of
the simulations.
estimated from samples which typically have a variance
which is greater than the mean (i.e. the coefficient of
variations of estimated abundance is greater than 30%)
(Zweifel and Smith, 1981; Ware and Lam bert, 1985).
Thus confidence intervals of the estimated mortality
cc
-7
-6
-5
L 0 G 10
-4
-3
-2
0
SURVI VO RSH IP
Figure 3. Simulation results showing the m ean growth rate of
survivors in relation to survivorship of the cohort for different
values in the variance of the initial distribution of growth rates
(dashed, dotted and solid lines represent values of o 2 of 0.001,
0.003, and 0.01 respectively). See caption to Figure 1 for details
of the simulations.
coefficient, based on surveys of abundance, are likely
to be large.
The accuracy of using growth histories to estimate
the mortality coefficient and survivorship is dependent
on the accuracy of the difference between the estimated
mean growth rates which is approximated by:
(Sokal and Rohlf, 1981, p. 228) where Y„ s? and n, are
the mean and variance in growth rates and the sample
size, at time ti; respectively. The confidence intervals
of the estimated difference in mean growth rates is
approximated by:
z = (Ÿ, —Ÿ„) ± ta[n]sv, - Ÿo
(13)
where t j n ] is the value of Student’s t for a significance
level of a with n = n , + n 0 - 2 degrees of freedom. Thus
by using the confidence intervals of the difference in the
means, the confidence intervals of S(L(), L ,) can be
obtained.
There are potentially im portant sources of error in
using growth histories to estimate survivorship. Esti­
mating the growth rate of individuals using analysis of
otolith microstructure relies on (1) an empirical relation­
ship between the age of an individual and the number
of increments on the otolith, and (2) an empirical
relationship between length of fish and otolith size.
Each of the dependent variables (age and length) have
confidence intervals dependent on the accuracy of the
relationships, often derived under laboratory con­
ditions. Because growth rates are ratios, the error
associated with the combined measurements will be
greater than the error of the components (Sokal and
Rohlf, 1981). This can be particularly critical when
growth rate estimates are calculated over short time
intervals (Rice, 1985). It is therefore essential that vali­
dation experiments yield clear and accurate estimates
of age and length in order for the proposed model to
be useful.
Example
To illustrate the application use of the model as well as
limitations of the m ethod, a simple example using data
from a previous study is presented. Although there are
many studies of fish growth, it is difficult to find an
example which incorporates the essential elements of
the model. These are: the mean size (L;) of larvae at
times t 0 and t); the distribution of growth histories for
larvae of age x at time t0[F0(ai)]; the distribution of
growth histories for larvae of age x at time t 0 caught at
time ti[F 1(ai)].
Although sample size is limited and the study time is
short, work by Rosenberg and H augen (1982) on larval
turbot (Scophthalmus m axim us) provides the elements
necessary for the model as well as the growth trajectories
of individual larvae. Furtherm ore, the study was con­
ducted in an experimental mesocosm which provided an
estimate of the survivorship of the cohort and provides a
basis for comparison with the method presented above.
I contrasted the growth rates of larvae before and after
yolk absorption (day 7 of their study) which was associ­
ated with a period of high mortality. The experiment
lasted 12 days. The mean growth rate of larvae prior to
yolk absorption (see their Table 3) was 0.23 mm /day
(a2 = 0.004) whereas the mean growth rate of survivors
was 0.28 mm /day, for which the variance was not
reported. Assuming that the distribution of initial
growth rates is normal, we can project survivorship and
mean growth rate of survivors using equation 7 (Fig. 4).
oc
GROWTH RATE ( m m /d a y )
.35 r
cr
i—
-4
-3
LOG
-2
-1
SU R VIV O R S H IP
Figure 4. Relative distribution of growth rates for a cohort of
larval turbot before (solid line) and after (dotted line) the
period of yolk absorption (from Rosenberg and Haugen,
1982). The distribution of growth rates after yolk absorption
was estimated from a simulated projection (see caption to
Figure 1 for details) which resulted in a change of 0.05 m m /d
in the mean growth rate of survivors relative to the initial
distribution of growth rates. The projections assume that
growth is linear throughout the study period. The bottom panel
shows the mean growth rate of survivors in relation to predicted
survivorship (eq. 7).
327
The simulation results show that in order to achieve a
mean growth rate of 0.28 m m /day in survivors, a cohort
survivorship of 3% is necessary. Rosenberg and Haugen
(1982) observed a survivorship of approximately 9%
during the 12-day experiment. Although the difference
may appear substantial, the standard deviation of the
difference between the mean growth rates is 0.016 which
results in my estimated survivorship being well within
the 95% confidence intervals of survivorship (0.0010.30).
The analysis is consistent with model predictions,
although the confidence intervals of this example are
poor due to the small sample size (n0 = 30, n, = 32).
The results indicate that the method could be used to
estimate survivorship based on differences in growth
histories of the survivors relative to a previous sample
of the cohort. However, the application of this method
may be better suited to cases in which back-calculated
growth rates can be estimated accurately (i.e ., estimated
over longer time intervals (Rice, 1985)).
Discussion
The study of growth and mortality of larval fish has
yielded few results which allow accurate prediction of
survivorship during early life. This may be due partly
to the assumption that a cohort’s characteristics are
accurately reflected by the average of a variable such as
growth rate. A principle of evolutionary ecology is that
the effect of selective forces is dependent on variability
within a cohort. We consider that recruitment is depen­
dent on the outcome of selective processes acting on
variance within and between cohorts; thus it is not only
average growth rate of a cohort which defines its survival
potential but also distribution of the population around
the mean.
Results from previous studies are consistent with the
principles presented here. In a re-analysis of M ethot’s
(1983) results, Pepin (in press) found that northern
anchovy, Engraulis m ordax, have a high mean backcalculated growth rate in a year with low mean sur­
vivorship (i.e., high mortality) relative to the mean
growth rates in a year with relatively high survivorship.
D ata from Chambers and Leggett (1987) indicate that
mean growth rates of winter flounder larvae, Pseudopleuronectes americanus, reared under identical
laboratory conditions, increase with decreasing sur­
vivorship. The mean back-calculated growth rate of
American shad, A losa sapidissima, during the first 56
days of life of 6 cohorts is also correlated (rs = 0.76,
p < 0.05) with the relative mortality during that period
(Crecco and Savoy, 1985). Although limited, these
observations are consistent with the m odel’s predictions
for the late larval and early juvenile stages of fish.
The significance of size-selection in estimating growth
and mortality rates of fish has been discussed by Jones
(1958) and Ricker (1969). They determined a simple
328
relationship between changes in back-calculated length
and the mortality rate of the population. Jones and
Ricker concluded that mean back-calculated growth
rates of a cohort would decrease as survivorship
decreased. The simple assumptions of their model (line­
arly increasing mortality with size, normal distribution
of lengths within a cohort) are not applicable directly
to larval fish, however. T here is clear evidence that
mortality of larval fish decreases exponentially with
increasing size (Peterson and Wroblewski, 1984;
M cGurk, 1986; Pepin in preparation). Furtherm ore,
the assumption of a normal distribution of lengths or
growth rates may not always be reasonable. Never­
theless, the general principles presented by Jones (1958)
and Ricker (1969) are the same as those proposed
herein: back-calculated growth histories can be used to
estimate the magnitude of selective forces influencing
the survival of fish. The requirem ent is that we obtain
a representative and unbiased sample of the population
at different stages.
I have presented arguments that a simplistic use of
average larval fish growth rates is likely to limit the rate
at which our understanding of the causes of variations
in survival progresses. The variability of growth rates
within a cohort determines its overall survival potential.
How the variance in growth rates changes over time
can let us infer which individuals are being removed. It
is possible that the m ethod presented in this paper may
be of limited use over short time intervals because of
the accuracy required and the labour involved in analysis
of otolith microstructure. The method may be par­
ticularly useful when abundance estimates are poor
and differences in growth rates can be estimated more
accurately, such as in studies of late larvae and early
juveniles.
A cknowledgem ents
Suggestions by G. T. Evans were valuable in clarifying
the ideas presented in this paper. The criticism by two
anonymous referees was greatly appreciated. J. C. Rice
and J. T. Anderson provided useful comments.
References
Beyer, J. E. 1988. The stock and recruitment problem . ICES
1988 Early Life History Symposium. Abstracts.
Beyer. J. E ., and Laurence, G. C. 1980. A stochastic model
of larval fish growth. Ecological Modelling, 8: 109-132.
Bolz, G. R .. and Lough, R. G. 1983. G row th of larval cod,
Gadus m orhua, and haddock, M elanogrammus aeglefinus,
on Georges Bank, spring 1981. Fish. Bull., 81: 827-836.
Campana, S. E ., and Nielson, J. D. 1985. M icrostructure of
fish otoliths. Can. J. Fish. A quat. Sei., 42: 1014-1032.
Chambers, R. C., and Leggett, W. C. 1987. Size and age at
m etamorphosis in m arine fishes: an analysis of laboratoryreared winter flounder (Pseudopleuronectes americanus)
with a review of variation in other species. Can. J. Fish.
Aquat. Sei., 44: 1936-1947.
Crecco, V. A ., and Savoy, T. F. 1985. Effects of biotic and
abiotic factors on growth and relative survival of young
American shad, A losa sapidissima. Can. J. Fish. Aquat.
Sei., 42: 1640-1648.
Evans, G. T ., Rice, J. C., and Chadwick, M. P. 1984. Patterns
in growth and smolting of Atlantic salmon (Salmo salar)
parr. Can. J. Fish. A quat. Sei., 41: 1513-1518.
Houde, E. D. 1987. Fish early life dynamics and recruitment
variability. Am. Fish. Soc. Symp., 2: 17-29.
H unter, J. R ., and Kimbrell, C. A. 1980. Early life history of
Pacific mackerel, Scomber japonicus. Fish. Bull., 78: 89101 .
Jones, R. 1958. L ee’s phenom enom of “apparent change in
growth rate” , with particular reference to haddock and
plaice. Intern. Comm. Northwest Atlantic Fish. Spec. Publ.,
1: 229-242.
Methot, R. D. 1983. Seasonal variation in survival of northern
anchovy estimated from the age distribution of juveniles.
Fish. Bull., 81: 741-750.
M ethot, R. D., and Kram er, D. 1979. Growth of northern
anchovy, Engraulis m ordax, larvae in the sea. Fish. Bull.,
77: 413-423.
M cGurk. M. D. 1986. Natural mortality of marine pelagic fish
eggs and larvae: role of spatial patchiness. Mar. Ecol. Prog.
Ser., 34: 227-242.
Munk, P. M ., Christensen, V., and Paulsen, H . 1986. Studies
of a larval patch in the Buchan area. II. Grow th, mortality
and drift of larvae. Dana, 6: 11-24.
Pepin, P. In press. Predation and starvation of larval fish: a
numerical experiment of size- and growth-dependent sur­
vival. Biol. Oceanogr.
Peterson, I., and Wroblewski, J. S. 1984. Mortality rate of
fishes in the pelagic ecosystem. Can. J. Fish. Aquat. Sei.,
41: 1117-1120.
Rice, J. A. 1985. Reliability of age and growth-rate estimates
derived from otolith analysis. In Age and growth of fish, pp.
167-176. Ed by R. C. Summerfelt and G. E. Hall. Iowa
State University Press, Ames.
Ricker, W. E. 1969. Effects of size-selective mortality and
sampling bias on estimates of growth, mortality, production
and yield. J. Fish. Res. Bd. C an., 26: 479-534.
Rosenberg, A. A ., and H augen, A. S. 1982. Individual growth
and size-selective mortality of larval turbot, Scophlhalmus
m axim us, reared in enclosures. Mar. Biol., 72: 73-77.
Rothschild, B. J. 1986. Dynamics of marine fish populations.
Harvard University Press, Cambridge. 277 pp.
Smith, P. E . . and R ichardson, S. A. 1977. Standard techniques
for pelagic fish egg and larva surveys. F A O Fisheries Tech.
Paper No. 175.
Sokal, R. R., and Rohlf, F. J. 1981. Biometry. W. H . Freeman
and Company, San Francisco. 859 pp.
Ware, D. M. and Lam bert, T. C. 1985. Early life history of
Atlantic mackerel. Scomber scombrus, in the southern Gulf
of St. Lawrence. Can. J. Fish. Aquat. Sei., 42: 577-592.
Zweifel, J. R., and Smith, P. E. 1981. Estimates of abundance
and mortality of larval anchovies (1951-75): application of
a new method. Rapp. P.-v. Réun. Cons. int. Explor. Mer,
178: 248-259.
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