Abt Associates Inc., December 16, 2009 Fisheries matrix population models 1

Environment and Resources

55 Wheeler Street, Cambridge, MA  Tel: 617 492-7100  www.abtassociates.com

Date December 16, 2009

To Tom Born, Josh Hall, and Erik Helm, U.S. Environmental Protection Agency

From James Palardy, David Mitchell and Elena Besedin, Abt Associates, Inc.

Subject WA 2-09 Task 4 - Matrix models of I & E effects on fish populations

1 Introduction

Cooling water intake structures (CWIS) withdraw trillions of gallons of water from streams, rivers,

estuaries, and coastal waters on an annual basis. This process results in the impingement and entrainment

(I & E) of large numbers of aquatic organisms. The effect of these losses may be a substantial decrease in

the stock size of various species of fish. Predicting the magnitude of these losses is a necessary step in

conducting meaningful cost-benefit analyses.

Here, we summarize projections from a general population model developed to estimate effects of I & E

losses on fish populations. We use the model to project the spawning stock biomass and population size

of 8 commercially-harvested fish species (American Plaice, Atlantic Cod, Atlantic Herring, Atlantic

Mackerel, Bluefish, Striped Bass, Summer Flounder, and Winter Flounder) into the future, given various

policy options that reduce regional I& E losses. We identify the underlying assumptions of the model, its

limitations, and major sources of uncertainty. As such, the document contains the following sections:

· Model specification and parameterization

· Model results

· Model limitations

· Key findings and future directions

2 Model specification and parameterization

2.1 Model framework

Age structured population models have been an effective tool in population biology since Leslie (1945)

first developed a matrix model to describe change in population age structure over time. The basic form

of Leslie’s (1945) model, incorporating reproduction and mortality as a discrete time process, is:

Nt1  MNt(1)

where Ntis the vector of age structure at time t and M is the population projection matrix:

memorandum
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 2



































k k

k k k k

s s

s

s

m s m s m s m s

M











0 0

0 0 0 0

0 0 0

0 0 0

2

1

1 1 2 2

(2).

Within the transition matrix M, miis the number of young produced by a female of age i, and siis the

probability of survival from age i-1 to i (Caswell 2001; Gotelli 2001).

For organisms with rapid development through early life history (e.g., birth to age 1 for many fish

species), the basic form of the transition matrix is not suitable. The model framework is able to deal with

this limitation, however, by allowing for the construction of a composite survival and/or fertility function.

In our model, the first year of life history is broken down into three stages: egg, larval, and juvenile

stages. For these life history stages, survival probabilities were computed as:





3

1

1 1

x

x

s s (3),

where x represents the life history stages of egg, larvae, and juvenile.

Several features of this transition matrix are notable. First, if a population is growing with constant birth

and death rates, the population vector Nt will converge on a stable age distribution (SAD) (i.e., a stable

proportion of individuals within each age class)(Caswell 2001; Gotelli 2001). Once SAD is attained, the

relative numbers of individuals in each age class remains constant, although absolute numbers will vary.

Thus, the transition matrix M can be expressed as a scalar multiple of Nt. This scalar multiple is known as

the population growth rate, , and is the dominant right-hand eigenvalue of the transition matrix.

Likewise, the SAD can be computed from M: it is the first eigenvector (Caswell 2001).

Although commonly used in population viability analyses (e.g., Crouse et al. 1987; Boyce 1992), the

Leslie matrix model and its many variations have been of limited use in population biology because they

do not incorporate density dependence, the idea that birth rates decrease and/or death rates increase with

increased population size because of competition for limited resources. Assuming that density dependent

effects act the same way on every age class, the Leslie model becomes:

Nt D Nt MNt( )

1  (4)

where D(Nt) is a density dependent function acting on the population vector Nt (Jensen 1995). Density

dependence need not act on all transition matrix probabilities, however, and may instead be applied to

individual life history stages within the M matrix.

For the purposes of fishery population modeling, three density dependent functions were modeled: the

logistic (Gotelli 2001), Beverton-Holt (Beverton and Holt 1993), and Ricker (Ricker 1954) density

dependent functions. In the logistic function,

K

K N

D N

t

L t



( )  (5),

survival and reproductive values decrease in a linear fashion as the population size approaches its

carrying capacity (K) (Gotelli 2001). The Beverton-Holt function (Beverton and Holt 1993; Caswell

2001) in age-structured matrix models is defined by
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 3

1 ( )

1

( )

0 1 2 k

t t t t

B H t

c N N N N

D N

    

 



(6),

where Nt

x

is the number of individuals in age class x at time t, and c is related to the carrying capacity as:

K

c

1





(7).

The Beverton-Holt function is non-linear, and represents an extreme of scramble competition (where all

competing individuals suffer equally as resources become limiting). In contrast, the Ricker density

dependent function (Ricker 1954; Caswell 2001) represents an extreme example of non-linear contest

competition (where there are individual winners and losers are resources become limiting), and is defined

by:

( )

0 1 2

( )

k Nt Nt Nt Nt

c

R t D N e

    





(8).

2.2 Model parameterization

Fecundity schedules in Leslie matrices (mi) represent the number of eggs produced by each female in the

population. Whole-organism fecundity values were obtained from the published literature (Froese and

Pauly 2009), and converted to fecundity rates per kg, assuming that published values of “whole

organism” fecundity occurred for individuals at the mean mass of the spawning stock (defined as all fish

that have attained reproductive age). For several species, multiple independent estimates of fecundity

were obtained: the average of these values was used in initial model runs.

Survival probabilities in Leslie matrices (si) represent the probability of an individual surviving to age i

from age i-1. As such, s is a composite parameter in context of fisheries-related population modeling, and

must represent all sources of mortality, including natural (Mk), fisheries- (Fk), and I & E-related (IEk)

mortality. Because mortality occurs throughout the year, estimates of these values are reported in the

literature as rates of instantaneous mortality. As such, composite values were converted to rates of

discrete mortality by the identity:

( ) i i i F M IE

i

s e

  

 (9).

For all modeled species, initial estimates of mortality were obtained:

- Mk – from Appendix D1 of USEPA (2003) and Fishbase (Froese and Pauly 2009)

- Fk – defined as FMSY (i.e., instantaneous fishing pressure at maximum sustainable yield) as found

in NOAA NMFS species-specific management plans (Mayo et al. 2006)

- IEk – was calculated from estimates of regional I & E from model output

Values of IEk were calculated from regional I & E estimates (for baseline and all regulatory options) as

follows:

1. A standard Leslie population transition matrix was constructed that incorporated natural mortality

and fisheries-related mortality

2. The transition matrix M was deconstructed to obtain the population’s SAD. These values were

then multiplied by the average mass for each life history stage, then standardized to obtain

proportional biomasses in each reproductive age class at SAD.

3. Spawning-stock proportional biomass at SAD (from 2.) was then multiplied by region-wide

estimates of spawning stock biomass (SSBM) (Mayo et al. 2006) to obtain spawning-stock

biomass for all life history stages at SAD.
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 4

4. Spawning-stock biomass values were back-transformed to obtain population numbers of all life

history stages at SAD.

5. Annual estimates of I & E losses were then subtracted from the appropriate life history stage, and

the proportionate annual I&E mortality calculated. This natural-logarithm of this value was taken

to obtain IEk.

2.3 Model calibration

Once all estimates of mortality were calculated, a transition matrix that did not incorporate density

dependence was assembled for each species, and values of  were calculated for each transition matrix.

These values were then compared to published values of the population growth rate for small population

sizes,  (Myers et al. 1999; Myers 2001; Mangel et al. 2009). In all cases, values of  derived solely from

life history data were substantially smaller than  (i.e., independent estimates of transition probabilities

underestimated the ability of each fish stock to grow), and in many cases < 1.0 (meaning that populations

without fishing or I & E pressure would shrink). To compensate for this discrepancy, we first calculated

the elasticity for all transition probabilities. Elasticity is the proportional sensitivity of  to change in each

transition probability (aij) (Caswell 2001), and is defined by:

ij

ij

ij a

a

a 







 

 

ln

(9).

We examined all elasticity matrices to determine if the survival rates of early life history traits (i.e., those

most likely to be influenced by I & E) had a dominant influence on . They did not. As such, a calibration

procedure was run that increased all ‘natural’ transition probabilities (i.e., natural mortality and fecundity

rates, and excluding fisheries and I & E mortality) proportionately, at 0.1% increments, until  from the

density independent Leslie matrix was equal to 

2.4 Model runs

Calibrated Leslie transition matrices were used to project population estimates into the future. For each

species, models were specified and populations projected 10 years into the future for all combinations of

policy option (10 options, including baseline and complete cessation of I & E losses), functional form of

density dependence (3 factors, as in section 2.1), and stages of life history at which density dependence

might occur [12 combinations, including only egg, only larvae, only juvenile, egg & larvae, larvae &

juvenile, egg & larvae & juvenile, each with and without adult density dependence (Lorenzen and Enberg

2002; Lorenzen 2008)]. For all species, N0 was set according to current NOAA NMFS estimates of

SSBM (Mayo et al. 2006).

Estimates of SSBM and population sizes output from all models (360 models per species) were examined

using exploratory data analysis techniques to determine the effect of density dependence at each life

history stage. For all species, models fell into two categories: models that incorporated density

dependence at the egg stage (e.g., at egg, egg & larvae, egg & larvae & adult, etc.), and those that did not

(e.g., at larvae, larvae & juvenile, larvae & adult, etc). In general, models that included a density

dependent age stage had lower population values and SSBM. This observation matches theoretical work

which has shown that density dependence of later life history stages is unlikely to control populations

when reproductive output is high (as in most fish species) (Neubert and Caswell 2000; Jang and Diamond

2003). To obtain representative results from these two model categories while avoiding overparameterization, all further analyses were conducted on two models that incorporated density

dependence in either (1) the egg, or (2) the larval life history stage.
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 5

3 Model Results

On a stock-wide basis, estimated changes to SSBM and population size as a result of I & E losses over 10

years vary substantially, from a decrease of 0.01% to an increase in excess of 25% relative to baseline

conditions (Tables 1-2). These observed changes are strongly influenced by species, density dependent

form (Table 1), stage, and time (Table 2). Generally, however, the projected effects of I & E on regional

stock biomass and population sizes were minimal. Differences between estimates of population size and

SSBM were also minimal. This is unsurprising given the underlying model dynamics that do not suggest

a fundamental change in SAD that would be required for large differences to occur between SSBM and

population size. Because differences between SSBM and absolute population sizes were minimal, only

changes to SSBM will be discussed further.

Under all density dependent forms and policy options for American Plaice, Atlantic Cod, Atlantic

Herring, and Atlantic Mackerel, changes with respect to baseline conditions were less than 0.01% (Table

2). Consequently, these four species will not be considered in further discussion. For the remaining

species, however, the population-scale impacts of I & E losses are much larger, and indicate patterns that

are relevant to regional extrapolation and model generalization.

3.1 Effect of Density Dependent Form

The three density dependent forms used in model runs vary substantially in the extent to which they

modify survival probabilities as a function of population size (Table 1). In over 95% of cases, the use of

logistic density dependence in the model led to the largest increase in fish SSBM (Table 1) and

population numbers (not shown) over baseline conditions. In many cases, the projected increases in

biomass and population numbers relative to baseline data were an order of magnitude greater when using

the logistic form of density dependence than when using the Beverton-Holt or Ricker forms of density

dependence (Table 1). Because this level of model variability exists depending on the form of density

dependence used, and because it is not clear that any of the density dependent forms are ecologically

superior, model averaging procedures (Burnham and Anderson 2002) should be used when using results

in further economic analyses. Because necessary data for such procedures are unavailable, and no

convincing argument can be made to support any of the density dependent forms above the others, model

output from the three density dependent models were averaged in all further discussion of model results

(Tables 2).

3.2 Effect of Life History Stage of Density Dependence

The life history stage at which density dependence operates has a large effect on the proportionate impact

of I & E losses on SSBM and fish populations (Table 2). The magnitude and directionality of this impact,

however, varies with both species and time. For example, Bluefish biomass showed a greater response to

I & E reductions at 5 years if density dependence occurred at the egg stage, but showed a greater response

at 10 years if density dependence occurred at the larval stage (Table 2). The directionality of this change

is opposite of that observed for Winter Flounder, where greater changes occurred at 5 years when density

dependence was modeled during the egg stage, but an increased response is projected at 10 years when

density dependence occurs during the larval stage (Table 2).

The magnitude of change in population projections that occurred by altering the life history stage of

density dependence (i.e., shifting from egg to larval stage) was variable by species and time, and varied

from less than 5% (Winter Flounder, 5 years, all options except Option 11), to more than 300% (Bluefish,

10 years), with an average change across species of 85%±107% (average ± standard deviation).

Together, these observations suggest that the actual life history stage at which density dependence occurs

is likely to play a substantial role in determining the effects of I & E losses. Unfortunately, due to the
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 6

poorly-characterized nature of early life history traits, we do not know at what stage density dependent

mortality occurs for the majority of species (Myers 2001). Moreover, because most studies do not include

individuals younger than age 1, there is little direction from in the literature.

3.3 Effect of Time

In all cases but one (Winter Flounder, larval stage density dependent), the magnitude of changes in fish

populations and SSBM relative to baseline conditions increased through time from 5 to 10 years, varying

from 14% to >500% the magnitude of the effect size at 5 years (Table 2). For 5 of 8 combinations of

species and density dependent life history stage combinations, this difference in magnitude exceeded

150% of the difference observed at 5 years (Table 2). This pattern is not surprising: incremental increases

in fish populations will grow exponentially through time, until carrying capacity is reached. For many

stocks, this occurs after 10 years.

4 Model Limitations

Our model projects future fish populations assuming that the (1) underlying transition matrix does not

change through time, and (2) the stock begins at SAD. These are not a realistic assumptions given

variable fishing pressure, and annual variability within the environment. As such, the model should be

used solely as a tool for estimating the population-level implications of policy with the expressed

limitation of ‘assuming all else is equal’, and realizing that fundamental changes may occur within the

foundational transition probabilities through time. Although not currently implemented, some of these

uncertainties (i.e., changes to fishing pressure, fecundity and survival rates) could be incorporated during

future model development.

• The accuracy of our model projections are limited by the accuracy of life history values. The survival

probabilities of early life history stages of fishes are notoriously difficult to estimate because of the

distance traveled by fish eggs and larvae during development and the high number of eggs initially laid.

Field-based estimates of natural mortality rates in early life history stages often result in such a low yield

of marked fish (Jones et al. 1999) that variance surrounding these estimates are large. Consequently, the

few data that do exist in the literature have low resolution and statistical power, and are infrequently used

in population models. The poor predictive power of life-history table based approaches can be seen in

initial estimates of the population growth rate, , early in model development. For 6 of the 8 fish species

modeled, estimates of  were less than 1 (suggesting population decreases) when assuming no density

dependence or fishing mortality. This disconnect between life history table projections and observed

growth rates at low populations, partially explains why the majority of fisheries population models avoid

counting fish younger than 1 year of age, but instead focus on relative rates of age 1 recruitment (Myers

2001; Lorenzen 2008; Mangel et al. 2009). Because estimates of  were substantially lower than observed

population growth rates at low population sizes (Myers et al. 1999; Myers 2001), model calibration was

necessary. Although suggesting that model projections of absolute biomass should be treated with

caution, this calibration procedure does not change the model’s ability to make relative predictions about

the magnitude of I & E effects, so long as comparisons are made using the same life history data.

• Beyond the limitations imposed by early life history data, the accuracy of population projections is

limited by estimates of I & E losses as a proportion of population size. In the case of the model presented,

two inputs have the greatest influence on projections of population-level effects of I & E: estimates of

population size and estimates of age-specific I & E losses. Although these values are linked in model

output (i.e., absolute I & E losses increase with increased population sizes), they are not linked as inputs.

Instead, while fish population estimates can be obtained (for many fish species) on an annual basis from

NOAA, regional estimates of I & E losses do not vary in time. Consequently, because many of the data

used to drive the regional I &E model are >25 years old, there may be a fundamental disconnect between
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 7

I & E loss data and fish population data. As such, the size of initial population estimates plays a strong

role in determining the effects of I &E on populations through time. Moreover, because raw I &E data is

poorly replicated in both space (i.e., number of facilities) and time (i.e., number of sampling dates at each

facility), and because fish populations do not cycle synchronously, attempts to obtain a ‘baseline year’ for

the I & E losses of all species are unlikely to be fruitful. Obtaining baseline I & E mortality for single

species, however, may be possible.

• Perhaps the biggest limitation to the model runs described is the attempt to estimate regional-scale

effects of I & E losses, which are by their nature inherently localized. Because of the scale-independent

nature of the model framework, however, this is a limitation of data input, rather than a limitation of the

model itself. Consequently, the model may be useful in exploring specific stocks that are heavily

impacted by I & E at local scales (i.e., within a single embayment or estuary) to create a number of case

studies where annual estimates of I & E losses from a single facility (or several facilities near to each

other) coincide with annual estimates of local fish stocks (Danila 2000; Lorda et al. 2000).

5 Key Findings and Future Directions

On a regional scale, the effects of I & E losses on fish populations is small, with substantial model

variability. However, several patterns are clear: effects of reducing I & E losses grow with time, and are

species-specific, ranging from <0.01% change among policy options to a model average of > 4% change

(Striped Bass). Consequently, conducting a more detailed study of a small number of species at a local

level may provide a better representation of ‘true’ I & E impacts to local fish stocks.

The accuracy of projecting I & E effects on fish populations is strongly limited by data availability, most

notably a meaningful understanding of the effects of density dependence on early life history, and

accurate estimates of fish stocks. Consequently, overcoming these shortcomings may not be possible

using any regional-scale modeling approach for many species. Interestingly, despite the limitations of the

model, it is notable that the magnitude of the effects of I & E losses on fish populations reported here are

similar to those reported at regional scales using different modeling approaches (Newbold and Iovanna

2005; Palardy et al. 2009).

Although, the model does not currently incorporate demographic or environmental stochasticity, the

Leslie matrix provides a framework to which such an extension is easily added. Such an extension could

be developed as a function of measured variance in stock/recruitment relationships, or simply as an added

random variable to mean transition probabilities. Such an analysis may be more effective at measuring

effects at smaller scales. Combined with a case-study approach in a locale with annual estimates of I & E

and local stock sizes, a more comprehensive model might be assembled that addressed many limitations

inherent to regional modeling studies.
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 8

Table 1: Projected increase (from models incorporating Beverton-Holt, Ricker, and Logistic forms of density dependence) on fish spawning stock biomass

10 years following the implementation of I & E policy options for 4 species of fish from Mid- and North Atlantic regions.

DD DD Percent increase in SSBM relative to baseline

Species Stage Form Opt. 11 Opt.12 Opt.13 Opt.14 Opt.15 Opt.16 Opt.17 Opt.18 No I&E

Bluefish Egg B-H 0.084 0.162 0.159 0.161 0.161 0.162 0.162 0.147 0.172

Logistic 0.471 0.910 0.889 0.900 0.900 0.909 0.910 0.822 0.966

Ricker -0.044 -0.085 -0.083 -0.084 -0.084 -0.085 -0.085 -0.077 -0.090

Larvae B-H 0.084 0.162 0.159 0.161 0.161 0.162 0.162 0.147 0.172

Logistic 1.957 3.811 3.721 3.769 3.769 3.807 3.811 3.437 4.049

Ricker 0.103 0.199 0.194 0.197 0.197 0.199 0.199 0.180 0.211

Striped Bass Egg B-H 0.189 1.311 1.243 1.286 1.286 1.310 1.311 1.086 1.384

Logistic 1.467 11.505 10.870 11.273 11.273 11.492 11.505 9.429 12.175

Ricker 0.212 0.411 0.401 0.406 0.406 0.410 0.411 0.371 0.436

Larvae B-H 0.195 1.038 0.987 1.019 1.019 1.037 1.038 0.869 1.096

Logistic 3.643 24.049 22.720 23.558 23.558 24.021 24.049 19.685 25.543

Ricker 0.212 0.409 0.400 0.405 0.405 0.409 0.409 0.370 0.435

Summer Flounder Egg B-H 0.058 0.113 0.110 0.112 0.112 0.113 0.113 0.102 0.120

Logistic 0.504 0.975 0.952 0.964 0.964 0.974 0.975 0.880 1.035

Ricker 0.079 0.152 0.148 0.150 0.150 0.152 0.152 0.137 0.161

Larvae B-H 0.059 0.115 0.112 0.114 0.114 0.115 0.115 0.104 0.122

Logistic 1.337 2.597 2.536 2.569 2.569 2.595 2.597 2.343 2.758

Ricker 0.079 0.152 0.148 0.150 0.150 0.152 0.152 0.137 0.161

Winter Flounder Egg B-H 0.140 0.449 0.424 0.427 0.435 0.449 0.449 0.387 0.499

Logistic 0.397 1.329 1.253 1.261 1.285 1.329 1.329 1.142 1.477

Ricker 0.043 0.169 0.158 0.159 0.163 0.169 0.169 0.143 0.188

Larvae B-H 0.139 0.446 0.421 0.423 0.431 0.445 0.446 0.384 0.494

Logistic 0.063 0.888 0.819 0.823 0.846 0.887 0.888 0.722 1.009

Ricker 0.164 0.289 0.279 0.282 0.284 0.289 0.289 0.264 0.312
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 9

Table 2: Average projected increases of fish spawning stock biomass (SSBM) for 8 species of fish, 5 and 10 years after the implementation of I & E policy

options. Data represent the average of models that incorporate Beverton-Holt, Ricker, and Logistic forms of density dependence.

DD Time Percent increase in SSBM relative to baseline

Species Stage Yrs Opt. 11 Opt.12 Opt.13 Opt.14 Opt.15 Opt.16 Opt.17 Opt.18 No I&E

American Plaice egg 5 <0.001 0.004 0.003 0.003 0.004 0.004 0.004 0.003 0.004

10 <0.001 0.009 0.009 0.009 0.009 0.009 0.009 0.007 0.011

larvae 5 <0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.004

10 <0.001 0.013 0.012 0.012 0.013 0.013 0.013 0.011 0.015

Atlantic Cod egg 5 <0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

10 <0.001 0.003 0.002 0.002 0.002 0.003 0.003 0.002 0.003

larvae 5 <0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

10 <0.001 0.003 0.003 0.003 0.003 0.003 0.003 0.002 0.003

Atlantic Herring egg 5 <0.001 0.004 0.004 0.004 0.004 0.004 0.004 0.003 0.004

10 <0.001 0.002 0.001 0.001 0.001 0.002 0.002 0.001 0.002

larvae 5 <0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

10 <0.001 0.003 0.002 0.003 0.003 0.003 0.003 0.002 0.003

Atlantic Mackerel egg 5 <0.001 0.001 0.001 0.001 0.001 0.001 0.001 <0.001 0.001

10 <0.001 0.002 0.002 0.002 0.002 0.002 0.002 0.001 0.002

larvae 5 <0.001 0.001 <0.001 <0.001 <0.001 0.001 0.001 <0.001 0.001

10 <0.001 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002

Bluefish egg 5 0.148 0.286 0.279 0.283 0.283 0.286 0.286 0.258 0.304

10 0.170 0.329 0.321 0.326 0.326 0.329 0.329 0.297 0.349

larvae 5 0.112 0.216 0.211 0.214 0.214 0.216 0.216 0.195 0.229

10 0.715 1.391 1.358 1.375 1.376 1.389 1.391 1.255 1.478

Striped Bass egg 5 0.234 1.094 1.043 1.075 1.075 1.093 1.094 0.921 1.156

10 0.623 4.409 4.171 4.322 4.322 4.404 4.409 3.629 4.665

larvae 5 0.244 2.078 1.964 2.037 2.037 2.076 2.078 1.706 2.195

10 1.350 8.499 8.036 8.327 8.327 8.489 8.499 6.975 9.025

Summer Flounder egg 5 0.078 0.151 0.148 0.150 0.150 0.151 0.151 0.137 0.161

10 0.214 0.413 0.403 0.409 0.409 0.413 0.413 0.373 0.438

larvae 5 0.086 0.165 0.161 0.163 0.163 0.165 0.165 0.149 0.175

10 0.492 0.954 0.932 0.944 0.944 0.954 0.954 0.861 1.014

Winter Flounder egg 5 0.115 0.369 0.348 0.350 0.357 0.368 0.369 0.318 0.409

10 0.193 0.649 0.612 0.616 0.627 0.649 0.649 0.558 0.721

larvae 5 0.145 0.375 0.357 0.359 0.364 0.375 0.375 0.329 0.413

10 0.122 0.541 0.506 0.509 0.520 0.540 0.541 0.457 0.605
Abt Associates Inc., December 16, 2009 Fisheries matrix population models 10

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