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Entrainment Impact Estimate- A!gG 21 11975 

Using the Equivalent Adult A b *­

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L 

3 

1 
The Biological Services Program was established wi hin th U.S. 

Fish and Wildlife Service to supply scientific informat on and methodologies on key environmental issues which-impact fish and wi dl ife 

resources and their supporting ecosystems. The mission of the Program 

is as follows: 

1. To strengthen the Fish and Wildlife Service in its role as a 

primary source of information on national fish and wildlife 

resources, particularly in respect to environmental impact 

assessment. 

2. To gather, analyze, and present information that will aid . 

decision makers in the identification and resolution of 

problems associated with major land and water use changes. 

3. To provide better ecological information and evaluation for 

Department of the Interior development programs, such as those 

relating to energy development. 

Information developed by the Biological Services Program is intended for use in the planning and decision making process to 'prevent or 

minimize the impact of development on fish and wildlife. Biological

Services research activities and technical assistance services are based 

on an analysis of the issues, the decision makers involved and their 

information needs, and an evaluation of the state of the art to identify

information gaps and determine priorities. This is a strategy to assure 

that the products produced and disseminated will be timely and useful. 

Biological Services projects have been initiated in the following 

areas : 

Coal extraction and conversion 

Power plants 

Geothermal , mineral , and oil shale development 

Water resource analysis , including stream a1 terations and 

western water allocation 

Coastal ecosystems and Outer Continental Shelf development 

Systems and inventory, including National Wetlands Inventory,

habitat classification and analysis, and information transferThe Program consists of the Office of Biological Services in Wash- j 

ington, D.C., which is responsible for overall.planning and management;

National Teams which provide the Prosram's central scientific and technical expertise and who arrange for contracting Biological Services 

studies with States, universities, consulting firms, and others; Regiona I 

staff who provide a link to problems at the operating level; and staff 

at certain Fish and Wildlife Service research facilities who conduct in-
-.Bi ol ogi cal ~___ Services Progan __ 

FbiS/OGS-78/65

July 1978 

US f NG THE EQUIVALENT ADULT APPROACE 

C. Phi 11i p Goodycar

Mational Power Plant Team 

1451 Green Road 

Ann Arbor., Michigan 48105 

Power Plant Project

Office of Biological Services 

Fish and Wildlife Service 

U.S. Department of the Interior 

. . . . . . . ... . .... . : t’ Fish and Wildlife -~ Service

1J.S. Department of the Interior 
--- 

-- 

ENTRAINMENT IMPACT ESTIMATES 

! 

USING THE EQUIVALENT ADULT APPROACH 

! 

Abstract 

One approach to analyzing the importance of entrainment of fish 

I eggs and larvae by power plants has been to estimate the number of 

adults which would have resulted from the entrained larvae. Certain 

limitations of a commonly used model can lead to serious underestimates 

of the actual impact. These limitations are discussed and an alternative formulation is presented which will reduce the bias. The approach

is also expanded to provide estimates of the reduction in numbers of 

fish captured by fishermen. 

INTRODUCT I0N 

One approach to assessing the importance of entrainment mortality 

of eggs and larvae of fish and shellfish by power plants is to estimate 

the number of adult organisms which this loss represents. This concept 

was iptroducpd by Horst I1975) as a "simplistic approach'' that could be 

used when it is not possible to employ more sophisticated methods because 

of the lack of available data. He formulated the model.$? follows: 

Na = S, Ne ... .................. .(1) 

where Na = number of adults that would have resulted from the entrained 

1arvae, 

S,? = survivorship from larvae to adulthood, and * 

Ne = number of larvae killed by entrainment. 

The paramater S, of equation (1) can be defined for a population

that is in equilibrium if the average lifetime fecundity of adult 

females (Fa) and the survival probability from a newly spawned egg to 

larvae are known. At population equilibrium each adult female produces 

a sufficient number of eggs during her lifetime to replace herself;

thus, the probability of an egg surviving to produce an average adult 

female is equal to the reciprocal of the average lifetime fecundity. If 

it is assumed, in addition, that one adult male is produced for each 

adult female, then the probability (S) of an egg surviving to become an

adult is \

---2 ­ e s=- .................... .(2) 

II Fa 

I r 

, ­ .. 

. j'.I .. . : 

. -.. ..... 9 --:..-:. ..i..r.. .......... .. I ...... . ., . ~ ........... _.= .. . .:. _I. 
-- 

2 

and the probability of a larva surviving to become an adult is 

s, = s/s, = 2 ............... . (3) 'eF, 

where Se = the survival probability iron a newly spawned egg to an 

entrained larvae. 

Thus the estimate of the nunber of adiiits lost using the'tiorst 

formulation is dependent on the pararneters S , F. , and N . The value of 

the parameter N must be determined froin fie?d sgmpling %ndmay be 

subject to cons7derable bias or error becai.!se of inappropriate sampling 

or data reduction procedures. However, it is assumed here that the '. 

number of larvae killed by entrainment can be accurately determined. 

1n H0rSA& rmu1-a.t.isa-t.he-..e.s..tSmate.._of_-slJr~-val...~l?__~~-_~~-.en-~-r~

1arva e,r)-,-he- cans.La.nL3ar-a 11 en1;ra i n_ed-laryag, and th i s 

d 

i s.,-a.ss~me.$-.-to.­

is--a--po-fefitial source of substantial errcr, as noted by SaundeTC(-(1477)­

and-D-a7;ri-b~1~7~~~-The\/f-~-~~-actuai ly a function of the aye

of the affected larvae. This point 7s important because of the low 

survival rate typically associated with the younger larval stages. Most

of the larvae that are entrained tend to be the younger ages, but most 

of the impact may be associated with the older, less abundant larvae. 

In this situation an estimate of the mean S which would be representative for most of the entrained larvae, the young& ones, would underestimate 

the impact. 

A LTERN AT I V E E40DEL 

iclNiSi]. .............. (4) 

where n = number of larval age classes that are subject to entrainment 

mortality, 

Ni = number of age class i larvae that are killed by enirainment, 

Si = survival probability from age class i to adulthood, 

2 s. = 

'e,i Fa ................ * (5) 

Se,i = survival-probability front egg to age class i, and -

Fa = average lifetime fecundity of a female recruit. 
--- 

. -. -3 - .......... 

The use of equation (4) requires est.irnates.of the number of larvae 

entrained for each larval age class (Ni) and the rate of survival from 

the egg stage to each age cla.ss (S .) . If the duratioils of the selected 

age class intervals are sufficiently long that the survival probabilities

of older individuals of the age class are significantly higher than the 

younger ones, than the nuinber of adults predicted using equation (4) may

still underestiinate the true impact. This tendency can be an important

consideration because it is generally very difficult to determine the 

age of larvae precisely enough to allow the age class interval to be 

reduced sufficiently to eliminate this bias. 

ES'rII;ll.\TES OF EGG TO LARVA SURVIVAL 

Because of the difficulty in aging larval fish it is often desirable 

to describe mortality as a function of length rather than age (Swedberg

and Walburg 1970). Data on abundaiice of larvae, grouped by body length

categories, can be used to estiinate a probability of survival from one 

length class to the next during the period that larvae are vulnerable to 

entrainment. Using this atter approach, the nuinber of adults that 

would have resulted froin arvae killed by entrainment can be estimated 

by rewriting equation (4) as follows: 

k 

Na=X NS . ................... .(6) 

R = 1 P' K. 

where k = nutnber of larval length classes that are subject to 

entrainriletit nior-tali ty 

N, = tiutrtber of length class R larvae that are killed by

eiitrai illnetit 

S, = survival probability froiri length class R to adulthood 

- 2 - --___ .................... .(7) 

si 'e,, Fa >_ 

S

e,% = survival probability froiri egg to length class a. 

The estimate of survival fraction by length can be obtained from 

cumulative catch data segregated into length classes. If the plot 'of 

the log of the cuniulative catch against length is a straight line (Figure

1) then mortality is a constant function of length and the instantaneous 

rate of mortality with increasitig leng-th is defined as: -_ . . . . . . ,'. --._ 

. . . . 

c 
4 

- . 

Cumulative Abundance of Larvae 

-I 

Y 

I-1 0 d 0 0 

0 0 0 

7 

2. 

tn 

S 

-5 

0 

r 

0 

S 

tn 

rt 

3 

3

3 

U 

3 
... 

5 

where d = instantaneous total mortality during length class R, 

NL+l = cumulative catch for length class R + 1 

i 

N, = cumulative catch for length class R 

and, 

Se,, = He 

-d( L,-h) ... .................(9) 

where -H = fraction of eggs that hatch, 

L, = length cf length class R, and 

h = length at hatching. 

If the plot of the log of the cumulative catch against length is 

not a straight line (Figure Z), then mortality is not a constant function 

of length and equations (8) and (9) are.not applicable. In this case, 

the value of S can be estimated as follows: 

e,z 

s =H-.Nz .................... (10) 

e,, NO 

where H = fraction of eggs that hatch, 

i 

N, = number of larvae of length class R, and 

No = initial number of larvae. 

Equations (9) and (10) both assume that the change in cumulative 

catch as a function of length reflects the true population change in 

abundance. This assumption requires the sampling to be non-selective by

length and the sampling effort to be constant throughout the period that 

larvae grow through the various length classes (i.e., the probability of 

an individual being captured at a given length must be the same for all 

1ength classes). This method assumes that length increases ljnearly

with time during the entrainment period. Violation of this assumption

will introduce some bias into the impact estimates. 

ESTIMATES OF AVERAGE LIFETIME FECUNDITY OF "ADULTS" 

In addition to the estimate of Se the parameter S, of equation 

(6) requires--an-<stimate of the averagg 1ifetime fecundity of adult 

females (F ). For-species which spawn more than' once' during their 

lifetime, the value of Fa is not likely to be the same as the average

fecundity of the extant adults, but is a' function of age of recruitment 

to the spawning population, frequency of spawning, relative fecundity of . ::%: - . . .... .. ..... .... ;I_ . . . .... . I . ._-. . ........ . . . . .. .. - . ... ... . -. ... 
. -.- . -

6. . 

Cumulative Abundance ..­

--I 0

" 4 0 0 0 0 0 0 u 0 a 

0 

P 

cn 

Q, 

--I 

0 

-.) 

N 

N 

N 

N 

P 

N 

o\ 

N 

(*, 

W' 

0' 

. 4. 

= -
-- 

7 

each age class, and total survival between age classes. Ttw average

lifetime Fecundity equals the average fecuridity of adults in the population

orily when the age structure of the adult population conforms to the 

stable age distribution that \crould exist; at equilibriuiii. 

To determine average lifetime fecundity of an adult feiiiale it is 

necessEry to define the age at which the individual becorries an adult. 

Two alternative definitions are ponsidered here: (1) an adult is a 

recruit to the age class in which spawiling is first obset*ved, or (2) an 

"adult" is an itidividual in the first aye class tliat is recruited to a 

fisfrery. If \E zssurne that feriiales spwn only once each year, then for 

either Jef iiiitiori, the average lifetiiiie fecuridity of a fmale recruit to 

the "adul t" poptrlatiori is yiveri by: 

C P.S-E 

J ................... .(I'l) j=z J ~ 

aye of recruitincnt of the "adult" population, 

111 = tiuiiiber of age classes in the populatioti, 

P .

J = .fk-action 0.f females that are iiiatuve iti age class j, 

sj - survival probabi'l ity froiii recrui tilierrt to age class j , and 

E. = average fecundity 0-F inature feiriales of age class j . J 

if rscruitirtent to ttie ''adult" population is cons-ideved to occur at 

the instant that spawning occurs then ttie value of S .

J 

is given by: 

Sj = e -Z( j-a) .................. .(12) 

where Z is the instantaneous rate of total niortality. 

E. in equation (llihis the average riumber of eggs deposited by

niatureJfeina1es OF ttie j age class. Fecundity estiiriates based 011 

counts of eggs it1 the ovaries of females are generally soiiiewtiat higher 

thati the riuinber of eggs that are actually spawned because soiiie of the 

ova reiiiain in the ovary after spawning. The unspawned eggs can be 

accounted for by t-educing the fecundity estimates derived in this manher 

by ttie preportioii of eggs that reniain in spent feriiales. 

PROJECTIONS Of IMPACT -. -._._I ... 

.After the values for S N,, arid Fa have been determined, the 

nuniber of potential adults bt are el - i m W-.b,y-enLr&mt can be 

estiiria-ted using equation (6): .. 

. ­

-_ . . - ..... .: -

...­ .. . . ~.... . .. . . ..... .

....... ... 
--. 

8 

k 

Na = C M, S, 

R'l 

n 

The reduction in numbers of fish captured by the fishery (Nf) can 

be estimated as: 

Nf -

- -NaF ..

. . . . . . . . . . . . . . . . . . . . .(13) 

where F is the instantaneous rate of fishing mortality. 

L I M ITAT IONS 

E_stima tes of the ,J.~ss~e~Qf adu 1ts uslugitheaquiya-lent-adu-1 t-appxoac h 

pr$-s_ume ~ 

that"-__. there --.- "___.-__ is no **..--..r----i--. densia--.-dependent -.-. mortality C.__^_ ~ occurring between

the tJme tKa.t..the 1ar;va -is killed by. ent.rainment and"Wt-iijiCX.-would 

have C.____^__.__,.,.__. .been -recruited, _. -. .. to' the "adul t'!...p. opul,atio,n_. However, if these estimates

are assumed to represent the long term impact of entrainment losses then 

density dependent effects are assumed to operate between generations.

This conclusion results from the fact that a reduction in adults in one 

generation is imp1ici tly- asSumed .to not reduce the number of adults 

recruited to the following generations.. Otherwise there would be feedback 

between generations that would cause the impact of the power plant to 

become rriore severe with time. 

There is for some populations, of course, a possibility-of feedback 

between generations. Thus, the equivalent adult methodology provides -_­... . 

1i tt1e in s i 9ht i&to th~~~.~.~~~~~~.i~~--~~-~.h~--a.�f~.s 

and should be._~us~~_,~nly~..t.o~~.~~~~~~.~s~~.a~~,~~of the severi ty

omntial ~ _____ 1Oss.e~. In applications, special consideraiio%-'sXGI?!-be

given to the accuracy of the estimates of the number of larvae killed by

entrainment, and to the fraction of eggs that survive to become larvae. 

EXAMP L E 

A hypothetical power plant withdraws its cooling water from a lake 

which supports a popular sport fishery. An intensive sampling program

in the lake and at the power station has provided the data needed to use 

the equivalent adult approach outlined herein. Fish are recruited to 

the fisherywhen they become 3 years of age and are exposed to an annual 

instantaneous 'fishing mortality of 0.29 thereafter. Some are mature at 

age 4, all are mature by age 8, and they spawn once each year. The 

reduction in numbers of fish captured by the fishery is estimated as - follows: - -
9 

Step 1. The average lifetime fecunaitvf a 3 year-old-female

recruit is estimated by use of equation (11) to be 534,507 eggs (Table 

119 

Step 2. The instantaneous rate of larval mortality with increasing

length was estimated to be 0.43 (Table Z), 

Step 3. Estimates of survival from egg to the midpoint of each 

larval length class (Se,%) were made using equation (9) (Table 3), 

Step 4. Estimates of survival from the midpoint of each larval 

length class to 3 year-old recruit were made using equation (7) (Table 

3) 3 

Step 5. The estimate of the reduction in 3 year-old recruits as a 

result of entrainment mortality was made using equation (6) (Table 4),

and 

Step 6. The number of catchable fish lost to the fishery as a 

result of entrainment mortality is determined as follows: 

- (836500) (0.29)

Nf - 0.5108 

Nf = 470,000 

.. 

.. . - . . ., .. .. 

.... , ,: , ..-. .. . 

. . . .

. . .. .. . .... . 

. .. . '4 ... .. 

. . - ..-

---- 

-- 

...... 

.: ;. . ,..,. - 7 . .- . 11 .. , 

. - . - .. I 

.%?' . 

.. 

Table 2.: Seasonal length frequency distribution of larvae necdcd to . -.1 estimate the instantaneous rate of mortality on length (d). 

Length"c1 ass Length (mm) Number m1-iccted 

( 4 

6.0 - 6.9 952 I 

2.- .I .7.0 - 7.9 61S 

3 8.0 - 8.9 38'3 

4 9.0 - 9.9 163 

5 10.0 - 10.9 235 I _ 

6 ll.@ - 11.9 216 

7 12.0 - 12.9 66 

8 13.0 - 13.9 53 

9 14.0 - 14.9 20 

9 

c N, * 

d.= - .In a=2 

9 

618+389+168+235+216+66+59+20 

= .-In ( 952+618+389+168+235+216+66+59+20 

.. i 5 

d. = 0..43 . .,

* 

This equation was adapted from Section 2.7.4 of Ricker (1975). 

a 

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0 0 0 0 0 

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