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. Predictin g 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 proje ct 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: t t 

### MN N   1 (1) where N

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

# memorandumAbt Associates Inc., December 16, 2009 Fisheries matrix population models 2 

###                


k k k

k k k



### s s s s s

m s m s m s m M



   
0 0 0

0 0 0 0

0 0 0

0 0 2 1 2

2 1 1 (2) . Within the transition matrix M, m i is the number of young produced by a female of age i , and s i is 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 N t will converge on a stable age distribution (SAD) (i.e., a stable proporti on 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 N t. 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: t t t 

### MN N D N ) ( 1 

###   (4) where D(N

t) is a density dependent function acting on the population vector N t (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 dep endent 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 N K N D t

t L 

###   ) ( (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 byAbt Associates Inc., December 16, 2009 Fisheries matrix population models 3 

### ) ( 1 1 ) ( 2 1 0 k t t t t t H B 

### N N N N c N D      

 

###  (6) , where N

t 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 suff er 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 limiti ng), and is defined by: ) ( 2

1 0 

### ) ( k t t t t N N N N c t R 

### e N D      

###   ( 8 ). 

### 2.2 Model parameterization Fecundity schedules in Leslie matrices (m i) 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 (s i) 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 (M k), fisheries - (F

k) , and I & E -related (IE k) mortality . Because mortality occurs throughout the year, estimates o f 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 IE M F i 

### e s 

  

###  ( 9 ) . For all modeled species, initial estimates of mortality were obtained:

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

- F k – defined as F

MSY (i.e., instantaneous fishing pressure at maximum sustainable yield) as found in NOAA NMFS species - specific management plans (Mayo et al. 2006)

- IE

k – 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 IE

k. 

### 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 c ases, 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 (a ij) (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 combinati ons, 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 speci es, 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 res ults from these two model categories while avoiding over parameterization, 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 popu lation 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 les s 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 dep endence 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 occur s is likely to play a substantial role in determining the effects of I & E losses. Unfortunately, due to theAbt 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 dep endent 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 dep endent 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 i s 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 assumption s 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) cou ld 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 egg s 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 re solution 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 f ish 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 bioma ss 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 betweenAbt 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 population s 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 estimate s of local fish stocks (Danila 2000; Lorda et al. 2000) . 

## 5 Key Findings and Futu re 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. Com bined 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 0.19 9 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 0.98 7 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 0.312Abt Associates Inc., December 16, 2009 Fisheries matrix population models 9

Table 2 : Average projected incre ase s 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 S pecies S tage 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 0.1 70 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 0.605Abt Associates Inc., December 16, 2009 Fisheries matrix population models 10

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