Assessing uncertainty and risk in exploited marine populations

Reliability Engineeringand SystemSafety54 ( 1996 ) ELSEVIER

PII:

50951-8320196)00074-9

183-195 Published by Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0951-8320/96/$15.00

Assessing uncertainty and risk in exploited marine populations M. J. Fogarty', R. K. Mayo', L. O'Brien', F. M. Serchuk" & A. A. Rosenberg b "National Oceanic and Atmospheric Administration, National Marine Fisheries Service, Northeast Fisheries Science Center, Woods Hole, MA 02543, USA hNational Oceanic and Atmospheric Administration, National Marine Fisheries Service, Northeast Region, Gloucester, MA 01930, USA

The assessment and management of exploited fish and invertebrate populations is subject to several types of uncertainty. This uncertainty translates into risk to the population in the development and implementation of fishery management advice. Here, we define risk as the probability that exploitation rates will exceed a threshold level where long term sustainability of the stock is threatened. We distinguish among several sources of error or uncertainty due to (a) stochasticity in demographic rates and processes, particularly in survival rates during the early life stages; (b) measurement error resulting from sampling variation in the determination of population parameters or in model estimation; and (c) the lack of complete information on population and ecosystem dynamics. The first represents a form of aleatory uncertainty while the latter two factors represent forms of epistemic uncertainty. To illustrate these points, we evaluate the recent status of the Georges Bank cod stock in a risk assessment framework. Short term stochastic projections are made accounting for uncertainty in population size and for random variability in the number of young surviving to enter the fishery. We show that recent declines in this cod stock can be attributed to exploitation rates that have substantially exceeded sustainable levels. Published by Elsevier Science Limited.

1 INTRODUCTION

of a risk-theoretic framework for fishery management has recently received considerable attention. I-9 These developments, coupled with concerns about the status of global fisheries, have culminated in calls for the consistent application of a precautionary approach to fishery management. I° In this context, the precautionary principle is based on a recognition of the various sources of uncertainty in fishery systems and the requirement to define sustainable strategies for resource use in consideration of the needs of future generations. Here, we describe fundamental issues in resource assessment and the development of fishery management advice. Three principal sources of error can be identified in the evaluation of resource status: (1) random variation in demographic rates and processes; (2) measurement error arising from sampling variation in the determination of abundance and demographic parameters and in model estimation; and (3) lack of complete information on the population and community dynamics of the system. 6 The first represents a form of aleatory (irreducible) uncertainty which is intrinsic to the system under study; the remaining factors represent forms of epistemic uncertainty.

Marine fish and invertebrate populations characteristically exhibit high level.,, of variability in space and time. This variability is a direct consequence of the effects of environmental stochasticity on survival of the early life stages. The resulting fluctuations in abundance can substantially complicate attempts to determine the status of marine populations. Additional sources of uncer'~ainty include measurement error and incomplete intormation on critical demographic and ecological processes. These issues present important challenges in ~.he development of effective management strategies for exploited marine populations. Uncertainty translates directly into risk in the development and implementation of fishery management advice. We define risk as the probability that exploitation rates will exceed a threshold level where long term sustainability of the stock is threatened. An alternative definition of risk in a fishery management context is the expected loss of benefits from the resource under alternative management actions given a specified loss or utility function. The establishment 183

M.J. Fogarty ct ai.

184

Epistemic uncertainty is reducible (in principle) through improvements in information available about the system, enhanced sampling, etc. Our principal objective is to provide an overview of information requirements, analytical issues, and the implications of uncertainty within a fishery management context. We first describe general principles governing the dynamics of marinc populations with emphasis on the determinants of the stability of exploited populations. Fundamental objectives used to guide management decisions (biological reference points) are then reviewed. We next focus on the dominant source of uncertainty in the dynamics of marine populations: intcrannuai variation in survival rates during the early life stages. These issues are then illustrated in a case study of Atlantic cod (Gadus morhua) on Georges Bank, an historically important fishing ground located off the New England coast. We rely extensively on recent evaluations of the status of the Georges Bank cod stock t ~-~3in this demonstration.

2 CONCEPTUAL

the capacity to affect the viability of the population if exploitation rates exceed limiting levels. Consider thc life cycle diagram depicted in Fig. 1. For the population to persist, a sufficient number of progeny must, on average, survive to replace their parental stock. For the purposes of illustration, we show several life stages including egg, larval, juvenile and adult stages with two substages in both the juvenile and adult forms (e.g. representing different size classes). Here, and throughout this paper, we will assume a closed population in which immigration and emigration are negligible. The transitions between stages represent the probability of surviving and growing into the next stage during a specified time interval. Note that the population becomes vulnerable to exploitation following the first juvenile stage in this example. In the following, it will be convenient to use the size or age at first harvest as a critical demarcation point. We define the number of individuals which survive from the egg stage to this point as recruitment. Individuals hatched in a particular year will be referred to as a cohort or year-class. The life cycle is completed with the production of eggs by the adult component of the population. The fishery reduces the probability of survival in the late juvenile and the adult stages with important consequences for the reproductive output of the population. Below, wc demonstrate that the stability and resilience of the population depends critically on survival probabilities at each life stage, the egg production by the adult stock, and whether the population has any capacity to compensate for increased mortality levels induced by exploitation. In the following, we will focus on populations

FRAMEWORK

An evaluation of risk to an exploited population under alternative management strategies involves a determination of the current condition of the resource relative to critical reference points or management objectives (e.g. biomass thresholds below which the population is considered threatened; ~4 or limiting exploitation rates). With respect to the long-term sustainability of the resource, interest centers on the ability of the population to replenish itself through reproduction. In this context, harvesting clearly has

i I i I

',,:----..,. I i

i!

Adult 11-~1Adult 2

Pre-Recruit Stages

Post-Recruit Stages Vulnerable to Fishery

.,L. Fig. L Hypothetical life cycle diagram for a stage-structured population comprising egg, larva, juvenile and adult phases. The juvenile and adult phases are subdivided into two stanzas each (e.g., size categories). The arrows between cells indicate transitions between stages (a function of the probability of surviving and growing into the next stage). The arrows from the adult stages to the egg stage represent reproductive output of the population. The stage classification can be partitioned into pre-recruit and post-recruit stages: by definition, the latter are vulnerable to the fishery.

185

Uncertainty and risk in marine populations

classified by age groups (a special case of the more general stage-structured approach). We begin by considering the relationship between the reproductive output of a population (total number of eggs produced) in a specified year and the resulting recruitment (i.e., proceases during the pre-recruit stages; see Fig. 1). The total weight (biomass) of the adult stock will be used zs an index of egg production (fecundity is typically a linear function of body weight in many marine organisms). We take the null model relating the adult population and recruitment to be density-independent: R = aS

(1)

where R is the number surviving to the age of recruitment, S is adult biomass, and a is a function of the adult sex ratio, the r,umber of eggs produced per unit of female biomass, and the survival rate (s) during the pre-recruit period [s = exp(/zt,) where /.t is the instantaneous rate of mortality and t, is the duration of the pre-recruit stage]. This model describes a linear relationship (through the origin) between spawning stock and recruitment. This simple model can be contrasted with models exhibiting some form of density-dependence or compensation. We will use the well known Ricker t5 model to illustrate one form of compensatory recruitment dynamics as a point of contrast with the null model. Many other recruitment models have been described; TM the appropriate model depends on the specific compensatory pzocesses operative during the pre-recruit stage. The Ricker model can be written: R = aSe -~s

population during a specified time interval. The control variable is the age-specific fishing mortality rate. Inspection of eqn (3) shows that spawning stock biomass per recruit declines exponentially with increasing fishing mortality. We have now defined the dynamics of a cohort through its lifespan. The equilibrium points of this model can be illustrated graphically. The relationship between adult biomass and recruitment is depicted in Fig. 2a for the null and Ricker models. Next, the relationship between recruitment and the lifetime production of adult biomass for several levels of fishing mortality is provided in Fig. 2b for the null recruitment-spawning stock model. Recall that we can specify the expected level of spawning stock biomass for a given age-specific pattern of fishing mortality. The estimates of spawning biomass per recruit, by definition, specify the slope of the relationship between recruitment and lifetime production of mature biomass for a given level of fishing mortality in this linear model. Increasing fishing mortality results in a monotonic decrease in the slope of the

8

j° lad

nr"

SPAWNING STOCK

(2) v

~o

where /3S is the instantaneous rate of compensatory mortality during the pre-recruit period and all other terms are defined as before. This model describes a convex or domed shaped relationship in the spawning stock-recruitment plane. We next consider post-recruitment processes, with particular emphasis or the lifetime reproductive output of a cohort. Here, we are interested in survival and maturation of a cohort following recruitment to the fishery (see Fig. 1). Again, the null model is taken to be density independent. The spawning biomass produced per recruit over the lifespan of the cohort is given by: Imj~

i -

z

z (/3

~

HIGH F

RECRUITMENT c

I--z i---

5 n-. ,9,

I

S/R = ~.m,w, I l e .;+M,~ i~lr

03 L9

b

(3)

I=1,

where m, is the proport!on mature at age i, w~ is the mean weight at age i, Mj and Fj are the age-specific instantaneous rate of fishing and natural mortality, and t...... is the maximum age (see Gabriel et al. 17 for an overview). The instantaneous rate of fishing mortality is formally defined as the ratio of the number in the catch to the mean number in the

SPAWNING STOCK Fig.

2. Stability

diagram

for

a

life-cycle

model

of

an

exploited population. (a) Relationship between spawning stock biomass and resulting recruitment for the null model (dashed line) and for a Ricker model (solid curve). (b) Relationship between recruitment and lifetime production of spawning stock biomass for several levels of fishing mortality. (c) Superimposition of the above figures to illustrate equilibrium points where the curves intersect.

186

M.J. Fogarty ct al.

relationship between recruitment and subsequent spawning stock. The equilibrium points for the full life cycle model are obtained by superimposing the spawning stockrecruitment and recruitment-spawning stock graphs (Fig. 2c). Under the null recruitment model, a single neutrally stable equilibrium exists when the linear functions between adult stock and recruitment and between recruitment and lifetime production of adult biomass are exactly superimposed. This is a highly precarious situation in which small changes in either the numbers of eggs produced or survival rates during different life history stages disrupts the equilibrium. In contrast, the points of intersection between the compensatory (nonlinear) model and the linear functions relating recruitment with the lifetime production of adult biomass represent globally stable equilibria. As fishing mortality increases, the equilibria shift until there is no intersection between the curves and a recruitment collapse is expected. These points illustrate the central issues in determining the resilience of a population to exploitation. The slopc of the recruitment curve at the origin defines the capacity of the population to persist under exploitation. Further, stable equilibria and, by inference, sustainable fisheries, are only possible if some fi~rm of compensatory process operates at one or more points in the life cycle.

3 BIOLOGICAL REFERENCE POINTS

We next describe several biological referencc points used in fishery management. These can be broadly categorized as (a) target exploitation rates designed to obtain either maximum or optimum yield (maximum yield modified by relevant economic considerations) and (b) limits to exploitation designed to safeguard recruitment or population size from declining below a specified threshold. The most commonly employed target exploitation rates are based on the expected yield of a cohort of fish over its lifespan following recruitment to the fishery. The expected yield from a cohort can be expressed: tm~

Ft,

Y = ~ ~,, {1 - e-Z,}N~w,

rate at which yield-per-rccruit is maximized (Fm~,x) and the fishing mortality rate corresponding to the point on the yield per recruit curve where the rate of change is 10% of the slopc at the origin (EI~). TM Note that F,~ is always less than F,,,,x. A generalized illustration of the relationship between yield per recruit and fishing mortality with associated reference points is provided in Fig. 3. Limits to exploitation have been primarily defined on the basis of spawning biomass per recruit and stock-recruitment theory) '~ The fishing mortality rate resulting in the spawning biomass produced by a cohort just needed to replace the parental generation under the null model is designated F,,.p.2° For the density-independent recruitment model, the number of recruits per unit spawning biomass is a constant ( R / S = a). In practice, a can be measured as the slope of the straight line through the origin in an empirical stock-recruitment relationship. The spawning stock biomass-per-recruit ( S / R ) analysis provides a natural way to define the critical reference point. In particular, persistence of the population under the null model requires that S / R >- 1/a.

4 RECRUITMENT VARIABILITY

Variability in recruitment is the dominant source of uncertainty in determining production from exploited fish populations. 21 Many marine populations are characterized by very high fecundity (number of eggs produced per spawning event) and correspondingly high levels of mortality during the early life stages (egg, larva, juvenile). It is demonstrated below that relatively small changes in mortality rates during these stages can result in orders of magnitude changes in the number of individuals surviving to the age of vulnerability to the fishery. The assumption of time invariant demographic rates in stock-recruitment models can be relaxed to provide stochastic analogues of the deterministic models described earlier. Under the central limit theorem,

Fi"lF~ax YI R

(4)

! - Ir

where Z, = F, + M, and the number at age (N,) is given by: i-I

Ni = R]-le -tp/'+ M)

(5)

'""""" -.

$/R (kg}

i =t,

and all other terms are defined as before. The yield can be normalized to the number of recruits to provide estimates of the yield-per-recruit. Examples of target exploitation rates include the fishing mortality

Fishing Mortmlity (F) Fig. 3. Representative yield (solid line) and spawning stock biomass (dashed line) per recruit curves showing biological reference points.

187

Uncertainty and risk in marine populations

the instantaneous rate of density-independent mortality (ix) can be taken to be a Gaussian random variable. 22 It can be shown that the resulting survival rate is a lognormally distributed random variable. The conditional distribution of recruitment is therefore also lognormally distributed with probability density function (PDF): _

P(RIS)

R

- i

(2~r3i,2 e

[ ,,,~.~.,,,.~2

,.,~. I

(6)

where o-2 is the variance of the process and all other terms are defined as before. The mean recruitment is then: E ( R ) = aSe 'p/2 (7) with variance: V ( R ) = a2S2e"~[e°~ - 1] (8)

variation of 14%), the coeffÉcient of variation of recruitment exceeds 250%. It is therefore not surprising that recruitment is highly variable in many marine populations. Hennemuth et al. 27 provide empirical evidence for the importance of Iognormal distributions in recruitment series. Similar considerations hold for the Ricker model. If the instantaneous rate of density-dependent mortality is a Gaussian random variable, the resulting recruitment is a lognormally distributed random variable with probability density function: - e--.,,; P ( R I S ) - -(21rrr2),2

and the mean recruitment is: E ( R ) = a S e -t3s+'~12

(11)

V ( R ) = a2S2e -2t~s~''~[e " ~ - 1].

(12)

and coefficient of variation: C V ( R ) = [e " ~ - 1] '/2

(9)

(see Fogarty, 22-24 Mertz & Myers 2s for related topics). The mean recruitment under the stochastic model is higher than the deterministic level (equivalent to the median of the stochastic model) by the factor exp(a2/2); the mode is lower than the median by the same factor. The immediate implication of the conditional Iognormai distribution is that most recruitment events will be low to moderate in size with occasional very large year classes (representing the tails of the distribulion; see Fig. 4). Notice also that the coefficient of variation is independent of population size. Bradford 2~' provides empirical support for a nearly constant C V in recruitment with changing population size. Small effects translate into large changes in recruitment given the expected levels of mortality and its variance during the early life history stages. For example, the instantaneous rate of mortality during the first year of life can easily be on the order of /x = 10.0 (equivalent to a survival rate of 4.5 × l0 -5) for highly fecund specie,~,. If we have even a moderate variance in the mortality of 0.2 = 2.0 (a coefficient of

Fig. 4. Illustration of conditional probability density functions for recruitment for given levels of spawning stock size for the null (linear) model. The mean of the distribution is shown in the spawning stock-recruitment plane (dashed line).

(10)

with variance:

The coefficient of variation is given by: C V ( R ) = [e °: - 1] ''2.

(13)

Recall that the coefficient /3 is a measure of stock-dependent mortality and, as in the deterministic case, this form is a simple extension of the null model to incorporate compensatory dynamics. The C V of recruitment is identical to the null model case and the same conclusions hold concerning the effects of random variation in the density independent mortality coefficient. For the Ricker model, the mean and variance of recruitment peak at intermediate levels of adult biomass and decrease with higher or lower levels of spawning stock size.

5 CASE STUDY: A T L A N T I C COD ON GEORGES

BANK

The Atlantic cod has supported important commercial fisherics in New England for over three centuries. Recent decreases in cod stocks under heavy exploitation throughout the Northwest Atlantic 2s have elicited considerable concern for their future viability. Stringent management measures have been imposed in many areas to arrest the decline in stock abundance. In New England, sharp reductions in fishing effort have recently been imposed to permit recovery of the stocks, and strategies for the sustainable use of this valuable resource are being redefined. Our objective is to demonstrate how the implications of recruitment variability and errors in estimates of key variables for thc Georges Bank cod

188

M.J.

Fogarty

stock can be evaluated in a risk assessment framework. Serchuk et al. '~ provided a detailed assessment of this stock for the period 1978-1993 including a spawning biomass per recruit analysis and age-specific estimates of population number, biomass, and fishing mortality. [An overview of the analytical method employed in estimating population size and

I

Hesearch I Vessel

Surveys

mortality rates is given in Appendix A.} Here, we couple the assessment results provided by Serchuk et al. '~ with additional analyses specifying the limiting lishing mortality rate (Fr,,p) to provide an evaluation of risk to the population. An overview of the information requirements, analytical approaches, and end products employed in this evaluation is provided in Fig. 5.

Yield/Recruit

I

Spawning Biomass/Recruit

T

I BiologicalData = Growth/Reproduction Natural Mortality

Fishery I Sampling

!

-" I Relative Abundance Indices

ct al.

of Catch

!

.I Sequential Populationl. "1 Analysis J"

l

1 Catch Per Unit I Effort

Estimates of Population I Size & Fishing Mortality

l

Stock-Recruitment Relationship

1

Biological Reference I,

I

Points

r

1 ,t ishery ManagementAdvic~ IRisk AssessmentI

f

Risk Management

|

Fig. 5. Flow diagram of data requirements and analytical procedures used in risk assessment of Georges Bank cod. Information derived from the commercial fishery and from fishery independent sources of information (e.g., research vessel surveys) is used to evaluate factors such as expected lifetime yield and reproductive output, and age-specific estimates of population size and fishing mortality (see Appendix A). These outputs represent the key ingredients in assessing resource status relative to management criteria (biological reference points) used in risk assessment of this population.

Uncertainty a n d risk in m a r i n e p o p u l a t i o n s

6 MEASURES OF VARIABILITY In the following, we rely extensively on model-based estimation procedures (e.g., sequential population analysis, SPA) with inpu':s from complex, multifactor sampling designs (Appendix A; Fig. 5). In general, analytical estimators of the variance of the output variables cannot be derived for these procedures. We have therefore employed resampling techniques (see Efron ~ for an overview) to evaluate the precision of the estimates. In particular, the analyses provided by Serchuk et al., ~3 and those described here, employ a bootstrap 29 technique in which observations are randomly drawn (with replacement from empirical data or from specified lzrobability density functions) and the process is repeated a large number of times to construct a probability di~;tribution of outcomes. Smith et al. 7 distinguish between several forms of bootstrap analysis including: (a) unconditional nonparametric bootstrap in which observed data are resampled with replacement, (b) unconditional parametric bootstrap in which a statistical distribution is fit to observed data and random observations are drawn from the fitted distribution, (c) conditional nonparametric bootstrap in which observed residuals from a fitted model are resampled and added to predicted values (d) conditional parametric bootstrap in which a specified statistical distribution is fit to observed residuals and random observations fro~'n this distribution are drawn and added to predicted values. Here, we employ examples of (a) and (c). Serchuk et al.t3 examined the variability in estimated population size and fishing mortality rate derived from SPA based on a bootstrap analysis in which the residuals from the fitted model and relative abundance indices from research vessel surveys (used in calibration of the SPA) were resampled using method (c) above. For a detailed description of the methodology for calibrating the SPA see Gavaris 3~ and Conser & Power:~.-~ The variability in the estimated catch for each age class and in the natural mortality rate was considered small relative to the error in the relative abundance index. Mayo (unpublished) estimated that the coefficient of variation in the catch-at-age for the dominant age classes in the fishery (ages 2-6) was of the order of 5-20"/0 based on the method of Gavaris & Gavaris. 32 Direct estimates of the variability in the natural mortality rate are not available for Georges Bank cod but it is reasonable to assume that the variance stabilizes considerably after the first year of life. The estimates of variability in population size and fishing mortality obtained from Serchuk et al. ~ arc assumed to reflect the major sources of uncertainty in the analysis but are understood to represent minimum estimates. Similarly, we have assumed for the purposes of this

189

demonstration, that the spawning biomass per recruit analysis is characterized by negligible error. The inputs to the analysis include the mean weight-at-age, the proportion mature-at-age and the natural mortality rate. Erzini ~3 estimated that the coefficient of variation of length-at-age for cod in the Georges Bank region was on the order of 5-15% (the translation to weight-at-age is accomplished through a length-weight relationship which is typically characterized by low variance). O'Brien et al. ~4 provided estimates of the proportion-mature-at-age for Georges Bank cod. The coefficient of variation in the estimated median age at maturity for females can be estimated to bc less than 15%. Jakobsen 35 demonstrated that biological reference points derived from spawning biomass-perrecruit analyses are relatively stable under variation in the input parameters.

7 RISK ANALYSIS

Serchuk et al. ~ documented declines in estimated adult biomass and recruitment (number of age 1 individuals) for Georges Bank cod for the period 1978-1993 as fishing mortality rates increased steadily (Fig. 6). We are primarily interested in ascertaining whether the declines in recruitment and adult biomass can be attributed to over-exploitation and, in the longer term, prospects for recovery. We begin by examining the relationship between stock and recruitment for Georges Bank cod for the 1978-1993 cohorts (Fig. 7). Note that the age 1 recruitment estimates are highly variable with dominant year classes in 1980 and 1985. Point estimates of recruitment for a given level of spawning stock size varied by a factor of four or greater. This result is consistent with the expected high levels of recruitment variability described earlier. Within the range of spawning stock sizes available for analysis, the relationship between stock and recruitment appeared to be linear. We fit the null recruitment model (eqn (1) to the empirical spawning stock and recruitment data by the method of maximum likelihood under the assumption of lognormally distributed errors. Confidence intervals for the estimated slope of the recruitment curve were estimated using an unconditional nonparametric bootstrap analysis by resampling the empirical stock and recruitment and refitting the model 200 times. Here, the median slope from the bootstrap analysis is used as the estimator of a in the null recruitment model. The resulting linear recruitment model is depicted in Fig. 7. Recall that we wish to determine the limiting value of fishing mortality with reference to the inverse of the slope of the null stock-recruitment relationship. The slope of this is estimated with potentially high error

M.J. Fogartv ct al.

190

100

1.1

90 80 A

E co

O O

v

(o (o

a

E

1

Avg F /

70

J X Adult Blomass

60

\

\,",

, t

50

,

'N~ t \, ~

.'\ ""

, i I

/

/:', ;

r

,

,

,.

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/ ' - 0 . 90.8 _ ,

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40

0 nO 30

•

, Im

mm

Im

't10.3°'-"•

20

0.2

10

0.1

0

-7-- 1

• 1 Biomass

0

77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93

Year Fig. 6. Point estimates of spawning stock (solid line) and recruitment (bars) biomass and mean fishing mortality rates (dashed line) for Georges Bank cod derived from sequential population analysis. '~

due to the stochasticity in recruitment. We first inverted the estimates of a obtained in the bootstrap analysis described above and constructed an empirical probability distribution (Fig. 8). The median value of l/a was 4.6 (see cumulative probability distribution in Fig. 8). With reference to the S/R analysis provided in Serchuk et a/., t3 we can readily derive a fishing mortality rate corresponding to each value of 1/a. The median fishing mortality rate at replacement is estimated to be Fr,,p = 0.42. We are next interested in comparing the most current estimate of fishing mortality with the F~,,plevel. The probability distribution of fishing mortality rates in 1993, as estimated by Serchuk et al., 13 is provided in Fig. 9 in concert with the bootstrapped estimates of Erep derived above. Note that the median estimate of fishing mortality in 1993 (/-93=0.9) is substantially higher than the median Fr,p level. Gabriel 36 provides a similar analysis and conclusion for Georges Bank cod. Although inclusion of all sources of variation in both components of this analysis would undoubtedly increase the dispersion in these distributions, there is very strong evidence that recent levels of fishing mortality have substantially exceeded sustainable levels and we conclude that the risk to the population is high under current rates of exploitation. The estimated trends in population size are consistent

with exploitation rates which have exceeded the replacement level limit over the last decade (see Fig. 6). We next consider the short term prognosis for the stock under two management scenarios: (a) maintenance of fishing mortality rates at the 1993 level and (b) immediate reduction in fishing mortality to the E,.t, level. To capture the uncertainty in the forward projections of catch, we must account for the uncertainty in the number-at-age at the start of the projections (based on thc bootstrapped sequential population analysis) and the uncertainty in the recruitment (number of age 1 fish) in each year of the projection. The structural equation used in the projection is identical to the yield model described earlier in the context of cohort dynamics. We employed a Monte Carlo simulation analysis in which random selections were drawn from the observed ratios of recruitment to spawning stock biomass. These samples were used to generate a random recruitment level for the corresponding adult biomass. This procedure is consistent with the null recruitment model employed in previous analyses. For each year of the projection, 200 replicate simulations were conducted. Stochastic simulations indicate that under the status quo fishing mortality (F9.~=0.91), both yield and spawning stock biomass are predicted to decline over

Uncertainty and risk in marine populations

191

50 85 •

80

~40

":30

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Spawning Stock Biomass (O00's mt) Fig. 7. Relationship between spawning stock biomass and recruitment for Georges Bank cod and fitted null recruitment model based on bootstrap analysis. Three models are depicted corresponding to the median estimate of the slope and the upper and lower 90th percentil,:s. An independent estimate of recruitment for the 1993 year class is used in this analysis. L~

the 5 year time horizon of the projections. The median and upper and lower 90% confidence intervals for these projections are presented in Fig. 10. In contrast, an immediate and sustained reduction in fishing mortality to the Frep lew.q results in a stabilization in the stock, and further declines in both yield and spawning stock biomass are arrested (Fig. 10). We therefore conclude that the risk to the stock under the status quo fishing mortality rate is extremely high and that remedial action to achieve the replacement level is required. Reducing fishing mortality to the F,.p level, however, does not result in stock rebuilding and reductions in fishing mortality below the F,,p would be required to achieve rehabilitation of the stock.

8 DISCUSSION The challenge of defin!,ng sustainable strategies for responsible use of living marine resources has assumed increasing urgency as exploitation pressures have increased on a global basis and many fishery resources have declined. Explicit consideration of the risks to a population under alternative courses of action is essential to guide management strategies.

The decisions made with reference to a formal risk assessment and evaluation represent approaches to risk management. Adherence to the tenets of the precautionary principle in this context will ensure that 'priority will be given to conserving the productive capacity of the resource '1° given uncertainty in the impact of harvesting on the resource. For highly uncertain resource systems, conservative management is essential under the precautionary approach. Finally, a reversal of the traditional burden of proof must be effected. 37 It is not appropriate to assume that the impacts of harvesting on resource populations, habitat, and ecosystems are negligible until proven otherwise under this approach, t° We stress the importance of defining appropriate null models in characterizing the dynamics of exploited marine populations. We have deliberately framed our null population models to define risk averse management reference points. In general, use of a non-compensatory formulation as the null model will lead to more conservative management advice for depleted populations. 6 Further, given the low statistical power likely in testing for compensatory response in natural populations, the burden of proof is shifted toward demonstration that density-dependent

M.J. Fogarty ct al.

192

0.2-

0.75

0.15

it...

g-

II. @ > la m -n

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2.5

3

~ 3.5

4

4.5

o 5

5.5

6

6.5

7

7.5

8

S/R Fig. 8. Probability distribution of the inverse of the slope of the null stock-recruitment model based on bootstrap analysis of empirical spawning stock and recruitment data with cumulative probability distribution (solid line).

0,3-

0.25

Distribution

0.2 >, O lG) O" 4) IILm

Distribution of F 93

0.15

0.1

0.05

O

0

0.25

~ 0.5 0.75 1 Fishing Mortality (F)

1.25

1.5

Fig. 9. Probability distribution of the estimated fishing mortality rate in 1993 and probability distribution of the replacement level of fishing mortality (F,,.~,) based on bootstrap analysis.

Uncertainty and risk in m a r i n e p o p u l a t i o n s 30

30

F= 0.91

25-

~

\,,\

E 200 0 ~1503 O~ C 10

-J

F=

0.42

25-

20-

0

C

193

0

~

15-

.~ 1 0 -

5

~5-

0 1993

199t4

1995

1996

1997

1998

0 1993

1999

1994

1995

Year

1996

1997

1998

1999

1997

1998

1999

Year

50-

50" F = 0.91

F = 0.42

E ~40-

40-

\ 30-

.~ 3 0 -

20-

20-

10-

"~ 1 0 -

cD o "E

0 1993

1994

1995

1996

1997

1998

1999

0 1993

Year

1994

1995

1996 Year

Fig. 10. Projections of short term landings and spawning stock size through 1998 under two management scenarios: maintenance of the status quo fishing mortality rate in 1993 (F=0.91) and reduction in fishing mortality to the F~,.I, level (F = 0.42).

responses are operative and that stock productivity is higher at low population sizes. Conversely, the null model will give overly optimistic advice at higher Icvels of stock abundance if density-dependent factors are important. Accordingly, use of the densityindependent model is not recommended for projections beyond the range of available data. It is clear that extrapolation can lead to overly-optimistic forecasts in this situation. Our analysis suggests that the decline in Georgcs Bank cod can be attributed to exploitation rates that exceed sustainable levels. Wc further note that the status quo fishing mortality rate (E,3) is projected to result in continued sharp declines in yield and spawning stock biomass. An immediate reduction to the F,,~, level of exploitztion is projected to result in stock stabilization and to prevent furthcr declines. However, more aggressive reductions in fishing mortality would be required to allow stock rebuilding and rehabilitation. Despitc the manifold sources of uncertainty in the assessment of this resource and an incomplete accounting of all potential sources of error in the analysis, the dynamics and rccent population trajectories of Georges Bank cod are consistent with the diagnosis of systemic: over-exploitation. Serchuk et al. j3 document an increase in fishing mortality rates from F = 0.3 in 1978 (a sustainable level) to F > 0 . 9 in 1993. Under this escalation in fishing pressure, the estimated adult biomass declined from over 90 thousand mt in 1980 to Icss than 40 thousand mt in 1993.

We havc described risk factors for Atlantic cod in a single species context. This approach implicitly treats other aspects of the ecosystem (including abundance of their prey and predators) as random variables. This random variation is primarily represented in the recruitment dynamics of the stock and incorporated in the forward projections described in the risk analysis. We note, however, that directional changes in the ecosystem must be carefully considered and that changes in productivity in the system as a whole can strongly affect sustainable harvest levels. Indeed, thc recent trcnd in recruitment of cod on Georges Bank suggests a possible shift toward lower productivity which must bc carefully monitored. Note that the 1991-1993 cohorts appear to exhibit low survival rates (negative residuals from the median stock-recruitment line; see Fig. 7). Formal risk assessment provides a mechanism to explicitly account for uncertainty in framing fishery management advice. To complete the process, the issue of risk management must be addressed. Risk management falls within the purview of fishery managers and entails an evaluation of the probability that the stock will decline under alternative management options and the selection of the preferred management strategy. A clear specilication of the various sources of uncertainty and thc implications of various courses of action on yield and future states of the resource is essential to the dcvelopmcnt of effective resource management

194

M. J. Fogarty c t a l .

decisions (Hilborn et al.~: Rosenberg & Restrepo'~). Risk assessment within a fishery management context provides an objective and adaptable framework to meet these needs.

ACKNOWLEDGEMENT Jon Brodziak and Paul Rago graciously made available software for short-term stochastic projections used in this analysis. We are grateful to Larry Barnthouse and Scott Ferson for their many helpful criticisms of an earlier version of this paper.

REFERENCES 1. Brown, B.E. & Patil, G.P., Risk analysis in the Georges Bank haddock fishery - - a pragmatic example of dealing with uncertainty. N. Am. J. Fish. Mgmt., 6 (1986) 183-191. 2. Linder, E., Patil, G.P. & Vaughan, D.S., Application of event tree risk analysis to fishery management. EcoL Model., 36 (1987) 15-28. 3. Shepherd, J.G. (Ed.), Special session on management under uncertainties. Sci. Counc. Stud. 16. Northw. Atl. Fish. Org., Dartmouth. NS, USA, 1991. 4. Francis, R.I.C.C., Use of risk analysis to assess fishery management strategies: a case study using Orange Roughy (Hoplostethus atlanticus) on the Chatham Rise. New Zealand Can. J. Fish. Aquat. Sci., 49 (1992) 922 -930. 5. Restrepo, V.R., Hoenig, J.M., Powers, J.E., Baird, J.W. & Turner, S.C., A simple simulation approach to risk and cost analysis, with applications to swordfish and cod fisheries. Fish. Bull US, 90 (1992) 736-748. 6. Fogarty, M.J., Rosenberg, A.A. & Sissenwine, M.P., Fisheries risk assessment: sources of uncertainty. Environ. Sci. Tech., 26 (1992) 440-447. 7. Smith. S.J., Hunt, J.J., & Rivard, D. (Eds), Risk evaluation and biological reference points for fisheries managemcnt. Can. Spec. Pub. Fish. Aquat. Sci., 120, 1993. 8. Hilborn, R., Pikitch, E.K. & Francis, R.C., Current trends in including risk and uncertainty in stock assessment and harvest decisions. Can. J. Fish. Aquat. Sci., 50 (1993) 874-880. 9. Rosenberg, A.A. & Restrepo, V.R., Uncertainty and risk evaluation in stock assessment advice for US marine fisheries. Can. J. Fish. Aquat. Sci., 51 (1994) 2715-2720. 10. FAO, Precautionary approach to fisheries. FAO Fisheries Technical Paper 350 Part 1, Rome, 1995. 11. Serchuk, F.M. & Wigley, S.E., Assessment and management of the Georges Bank cod fishery: an historical review and evaluation. J. Northw. Atl. Fish. Sci., 13 (1992) 25-52. 12. Serchuk, F.M., Grosslein, M.D., Lough, R.G., Lough. D.G. & O'Brien, L., Fishery and environmental factors affecting trends and fluctuations in the Georges Bank and Gulf of Maine cod stocks: an overview. ICES Mar. Sci. Symp., 198 (1994) 77-109. 13. Serchuk, F.M., Mayo, R.K. & O'Brien, L., Assessment of the Georges Bank cod stock for 1994. Res. Doc. 94-25, Northeast Fish. Sci. Centre, Woods Hole, MA, 1994.

14. Myers, R.A.. Rosenberg, A.A., Mace, P.M., Barrowman, N. & Restrepo, V.R.. In search of thresholds for recruitment overfishing. ICES J. Mar. Sci., 51 (1994) 191-205. 15. Rickcr, W.E., Stock and recruitment.. J. Fish. Res. Board (,'an., 11 (1954) 559-623. 16. Rothschild, B.J., Dynamics of marine fish populations, Harvard Univ. Press, 1986. 17. Gabriel, W.L., Sissenwine, M.P. & Overholtz, W.J., Analysis of spawning stock biomass per recruit: an example for Georges Bank haddock. N. Am. J. Fish. Mgmt, 9 (1989) 383-391. 18. Gulland. J.A. & Boerema, L.K., Scientific advice on catch levels. Fish. Bull. US, 71 (1973) 325-335. 19. Mace, P.M., Relationships between common biological reference points used as thresholds and targets of fishery management strategies. Can. J. Fish. Aquat. Sci., 51 (1993) 110-122. 20. Sissenwine, M.P. & Shepherd, J.G., An alternative perspective on recruitment overfishing and biological reference points. Can. J. Fish. Aquat. Sci., 44 (1987) 913-918. 21. Sissenwine, M.P., Why do fish populations vary? In The Exploitation o f Marine Communities, (ed. R.M. May) Springer-Verlag, Berlin. 1984. 22. Fogarty, M.J., Recruitment distributions revisited. Can. J. Fish. Aquat. Sci., 50 (1993) 2723-2728. 23. Fogarty, M.J., Sissenwine, M.P. & Cohen, E.B., Recruitment variability and the dynamics of exploited marine populations. Trends. Ecol. Evol., 6 (1991) 241-246. 24. Fogarty, M.J., Recruitment in randomly varying environments. ICES J. Mar. Sci., 50 (1993) 247-260. 25. Mertz, G. & Myers, R.A., Estimating the predictability of recruitment. Fish. Bull. US, 3 (1995) 657-665. 26. Bradford, M.J., Precision of recruitment estimates from early life stages of marine fishes. Fish. Bull., 90 (1992) 439-453. 27. Hennemuth, R.C., Palmer, J.C. & Brown, B.E., Statistical description of recruitment in eighteen selected fish stocks. J. Northw. Atl. Fish. Sci., 1 (1980) 101-111. 28. Jakobsson, J. (ed.), Cod and climate change. In Proc. ICES Mar. Sci. Symp., Reykjavik, 23-27 August 1993, vol. 198, 1994. 29. Efron, B., The jackknife, bootstrap, and other resampling plans. Soc. Ind. Appl. Math, PA, 1982. 30. Gavaris, S., An adaptive framework for the estimation of population size. Res. Doc. 88/20, Can. Atl. Fish. Sci. Advis. Comm. (CAFSAC), Miami, FL, 1988. 31. Conser, R.J. & Powers, J.E., Extensions of the adapt VPA tuning method designed to facilitate assessment work on tuna and swordfish stocks. Work. Doe. 89/43, Int. Comm. Cons. Atl. Tuna (ICCAT), Miami, FL, 1989. 32. Gavaris, S. & Gavaris, C.A., Estimation of catch at age and its variance for groundfish stocks in the Newfoundland region. In Sampling commercial catches of marine fish and invertebrates, (eds W.G. Doubleday and D. Rivard) Can. Spec. Pub. Fish. Aquat. Sci., 66 (1983) 178-182. 33. Erzini, K., Variability in length-at-age in marine fishes. Ph.D. dissertation, Graduate School of Oceanography, University of Rhode Island, Narragansett, RI, 1990. 34. O'Brien, L., Burnett, J. & Mayo, R.K., Maturation of nineteen species of finfish of the Northeast coast of the United States. N O A A Tech. Rept. NMFS 113, Woods Hole, MA, 1993.

Uncertainty and risk in marine populations

35. Jakobsen, T., The behaviour of F~,.... F,,,,,,i, and Fh,,.,, in response to variation used in parameters used in their estimation. Can. Spec. Pub. Fish. Aquat. Sci., 120 (1993) 119-126. 36. Gabriel, W.L., A simple method for estimating uncertainty associated with F,,,,,, ICES C.M. 1994/D:5. 1994. 37. Rosenberg, A.A., Fogarty, M.J., Sissenwine, M.P., Beddington, J.R. & Shepherd, J.G. Achieving sustainable use of renewal'le resources. Science. 262 (1993) 828-829. APPENDIX

The population and mortality estimates used in our analyses are based on a simple recursive algorithm employing two fundamental models. The number of individuals alive at age i + 1 (3/,_ ~) at the beginning of the year is related to the number at age i (Ni) at the start of the previous year by: Ni+l =N,e -z'

(A.i)

where Z, is the instanta~aeous rate of total mortality; e-Z~ is the fraction surviving through the year. The total mortality can be decomposed into two components: the instanl:aneous rates of fishing and natural mortality ( Z , : : F + M , ) . We can further specify the expected number in the catch as: C~ = ~ [1 - e Z,]N,

(A.2)

where C~ is the number in the catch of age class i and all other terms are as defined earlier. The term in brackets represents the fraction of the population at age i that will die during the year and the ratio F,/Z, is the fraction of the mortality attributable to fishing. The product of these two terms gives the proportion of the population at age i removed by harvesting. These simple relationships can be used to obtain estimates of the population size-at-age and the fishing mortality rate if the number in the catch for each age is known and an estimate of the natural mortality rate can be provided. Although it would be desirable to estimate the natural mortality rate in this procedure,

195

the system of equations is underdetermined and some parameters must be fixed. Taking the ratio of eqn (A.1) and eqn (A.2) we have: N,~l Z, C, - File z, - 1] (A.3) where all terms are defined as before. If an independent estimate of the fishing mortality rate for the ith age class can be provided, then the number in the population at age i at the start of the year can be determined using eqn (A.2). We can now estimate the fishing mortality rate for the previous age class by substituting the population estimate just derived into the numerator of the right hand size of eqn (A.3) and iteratively solving for the total mortality rate (recall that an estimate of the catch is available for the previous age class). The fishing mortality rate at age is obtained by subtracting the independently provided estimate of the natural mortality rate. We can then estimate the population size for the previous age class using eqn (A.1). This process is followed for successively younger ages starting with the oldest age class for each cohort. This procedure is referred to as Sequential Population Analysis. The age-specific numbers can then be converted to biomass by multiplying by the mean individual weight for each age. Similarly, the biomass of adults can be determined by multiplying the age-specific biomass estimates by the estimated proportion of mature individuals at each age. Uncertainty in these estimates therefore devolves from sampling error in the number in the catch, its age composition, mean individual weight and proportion mature for each age class, natural mortality, and the fishing mortality on the oldest age class. Statistical procedures for estimating the terminal fishing mortality rates and population sizes for the oldest ages in each cohort using auxiliary data derived from research vessel surveys or other sources (calibrating the SPA) are described by Gavaris 3" and Conser & Powers 3~.