Rapp. P.-v. Réun. Cons. int. Explor. Mer, 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. Mer, 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, t0, 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 John’s, Newfoundland, Canada, A 1C 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 evi­dence that daily mortality rates decrease with increasing size (Peterson and Wroblewski 1984; McGurk 1986; Pepin, in preparation). The implication is that the varia­bility 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 evi­dence 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-cal- culation. Houde (1987) argues that changes in popu­lation characteristics due to external selective factors should be reflected in otoliths of a representative group of the population. Therefore, by contrasting the distri­

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, advec- tion), 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). Here, 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 indi­vidual larva can be described as:

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where L is the length of the larva (mm), t is time, g(L) is the coefficient of daily instantaneous growth, and where:

At(L0, LrLi dx

l ) = J ,„ göö(2)

is the time required to grow from L0 to L ,. If mortality is size-specific, such that:

J_dN

N ~dT= -M (L ) (3)

where M is the coefficient of instantaneous mortality, then size-specific survivorship is:

NiS(L0,L , ) = — = exp

I'1!) (/:M(x)

g(x)dx

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 (Methot 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 mag­nitude of the external source of mortality (e.g., pred­ators). Length-specific survivorship (eq. 4) yields:

S(L0,L , ) = O ' c/a

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 poten­tial of that cohort for a given set of conditions. To demonstrate this, consider a cohort which has an initial probability distribution of growth rates F0(aj). It follows that the mean growth rate of that cohort is:

ao — 2 a:F 0(ai)i= i

(8)

(4) A sample of the cohort, at a sequential time tj , should have a mean growth rate:

ai = 'Z aiFi(ai)i= 1

where:

F M o c Q ^ F o t e )

and for which survivorship is:

S (L o ,L 1) = E O “c/a-Fo(ai) i = 1

(9)

(10)

(ID

(7)

where Q = L t/Lø. Survivorship increases as the growth rate (a) increases and decreases as the mortality coef­ficient (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 (L0, 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 (Lu, Li).

Because survivorship decreases as aj decreases, the mean growth rate of survivors will increase as the mor­tality coefficient (c) increases, as stated earlier. It fol­lows 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 distri­bution 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 sur­vivorship, 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 sur­vivorship. 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

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.38LL_

o (/)Cd ÜJ O H -> .36

.34

.32

LOG, n SURVIVORSHIP

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 (a2 = 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 dis­tance 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 mean 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

Q.x - 2V)QL

i ~ 3Q£ —4ZDCO

o - 5oo - 6

- 7

- 8

MORTALITY COEFFICIENT

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 of 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 Lambert, 1985). Thus confidence intervals of the estimated mortality

cc

- 7 - 6 - 5 - 4 - 3 - 2 0L 0 G 10 SURVIVORSHIP

Figure 3. Simulation results showing the mean 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.

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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 , + n0 - 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 important sources of error in

using growth histories to estimate survivorship. Esti­mating the growth rate of individuals using analysis of otolith microstructure relies on (1) an empirical relation­ship 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 con­ditions. 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 vali­dation 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 method, 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 t0 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 t0 caught at time ti[F1(ai)].

Although sample size is limited and the study time is short, work by Rosenberg and Haugen (1982) on larval turbot (Scophthalmus maximus) provides the elements necessary for the model as well as the growth trajectories of individual larvae. Furthermore, the study was con­ducted 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 associ­ated 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 ay )

.35 r

cr

i—

- 2 - 1

SURVIVORSHIP- 3

LOG- 4

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).

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The simulation results show that in order to achieve a mean growth rate of 0.28 mm/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 (0.001- 0.30).

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 depen­dent 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 mordax, have a high mean back- calculated growth rate in a year with low mean sur­vivorship (i.e., high mortality) relative to the mean growth rates in a year with relatively high survivorship. Data from Chambers and Leggett (1987) indicate that mean growth rates of winter flounder larvae, Pseu- dopleuronectes americanus, reared under identical laboratory conditions, increase with decreasing sur­vivorship. The mean back-calculated growth rate of American shad, Alosa 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 model’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

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 (line­arly increasing mortality with size, normal distribution of lengths within a cohort) are not applicable directly to larval fish, however. There is clear evidence that mortality of larval fish decreases exponentially with increasing size (Peterson and Wroblewski, 1984; McGurk, 1986; Pepin in preparation). Furthermore, the assumption of a normal distribution of lengths or growth rates may not always be reasonable. Never­theless, 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 requirement 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 method 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 par­ticularly 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.

Acknowledgements

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.

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