Modeling Species Invasions in Ecopath with Ecosim: an Evaluation Using

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Ecological Modelling 247 (2012) 251–261

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Ecological Modelling

jo urnal homepage: www.elsevier.com/locate/ecolmodel

Modeling species invasions in Ecopath with Ecosim: An evaluation using

Laurentian Great Lakes models

a,∗ b c

Brian J. Langseth , Mark Rogers , Hongyan Zhang

Quantitative Fisheries Center, 153 Giltner Hall, Michigan State University, East Lansing, MI 48824, USA

U.S. Geological Survey, Great Lakes Science Center, Lake Erie Biological Station, 6100 Columbus Avenue, Sandusky, OH 44870, USA

Cooperative Institute for Limnology and Ecosystems Research, University of Michigan, Ann Arbor, MI 48108, USA

a r t i c l e i n f o a b s t r a c t

Article history: Invasive species affect the structure and processes of ecosystems they invade. Invasive species have been

Received 7 May 2012

particularly relevant to the Laurentian Great Lakes, where they have played a part in both historical and

Received in revised form 20 August 2012

recent changes to Great Lakes food webs and the ﬁsheries supported therein. There is increased interest

Accepted 21 August 2012

in understanding the effects of ecosystem changes on ﬁsheries within the Great Lakes, and ecosystem

models provide an essential tool from which this understanding can take place. A commonly used model

Keywords:

for exploring ﬁsheries management questions within an ecosystem context is the Ecopath with Ecosim

Invasive species

(EwE) modeling software. Incorporating invasive species into EwE models is a challenging process, and

Food web models

Ecopath descriptions and comparisons of methods for modeling species invasions are lacking. We compared four

Ecosim methods for incorporating invasive species into EwE models for both Lake Huron and Lake Michigan based

Great Lakes on the ability of each to reproduce patterns in observed data time series. The methods differed in whether

invasive species biomass was forced in the model, the initial level of invasive species biomass at the

beginning of time dynamic simulations, and the approach to cause invasive species biomass to increase at

the time of invasion. The overall process of species invasion could be reproduced by all methods, but ﬁts to

observed time series varied among the methods and models considered. We recommend forcing invasive

species biomass when model objectives are to understand ecosystem impacts in the past and when time

series of invasive species biomass are available. Among methods where invasive species time series were

not forced, mediating the strength of predator–prey interactions performed best for the Lake Huron

model, but worse for the Lake Michigan model. Starting invasive species biomass at high values and then

artiﬁcially removing biomass until the time of invasion performed well for both models, but was more

complex than starting invasive species biomass at low values. In general, for understanding the effect

of invasive species on future ﬁsheries management actions, we recommend initiating invasive species

biomass at low levels based on the greater simplicity and realism of the method compared to others.

1. Introduction quagga (D. bugensis) mussel (hereafter referred to as dreissenids),

and round goby (Neogobius melanostomus) have all received exten-

Non-native species are a continual threat to the maintenance sive attention for their perceived ability to alter Great Lakes food

of Great Lakes ecosystems. Approximately 185 non-native species webs. Bythotrephes invaded the Great Lakes during the 1980s

have invaded the Laurentian Great Lakes (Ricciardi, 2006) with (Vanderploeg et al., 2002), and preys heavily on mesozooplankton

effects ranging from mild to severe. Non-native species that become (Bunnell et al., 2011). Dreissenids were ﬁrst observed in Lake St.

established, increase in abundance, and alter system processes are Clair in 1988 (Vanderploeg et al., 2002), and have been suggested

termed “invasive” (Williamson and Fitter, 1996) and are the focus to contribute to shifts in the zooplankton community (Barbiero

of policy, research, and restoration efforts within the Great Lakes. et al., 2009; Bunnell et al., 2011), declines in prey ﬁsh abundances

Several non-native species from the Ponto-Caspian region of (Riley et al., 2008), reductions in the spring phytoplankton bloom

Eurasia have recently invaded the Great Lakes. The spiny-water (Barbiero and Tuchman, 2004), and declines in abundance of the

ﬂea (Bythotrephes longimanus), zebra (Dreissena polymorpha) and native benthic amphipod (Diporeia spp.; Nalepa et al., 2007). Round

goby were ﬁrst observed in the St. Clair River in 1990 (Vanderploeg

et al., 2002) and compete with other benthic ﬁsh, predominantly

∗ sculpins.

Corresponding author. Tel.: +1 517 355 0126; fax: +1 517 355 0138.

Developing ecosystem models facilitate with assessment of

E-mail addresses: [email protected] (B.J. Langseth), [email protected]

recent changes to Great Lakes food webs, including the effect

(M. Rogers), [email protected] (H. Zhang).

252 B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261

of invasive species. Ecosystem model development also supports 2. Methods

ecosystem based approaches to ﬁsheries management, which are

becoming more prevalent (Pikitch et al., 2004). A popular tool for 2.1. Ecopath with Ecosim models

exploring food-web dynamics and assessing multi-species man-

agement objectives is the Ecopath with Ecosim (EwE) computer We developed two Ecopath with Ecosim models for lakes Huron

software (Halfon and Schito, 1993; Halfon et al., 1996; Kitchell et al., and Michigan, to evaluate alternative methods for modeling inva-

2000; Cox and Kitchell, 2004; Stewart and Sprules, 2011). Ecopath sive species. Both models focused on the offshore ﬁsh communities,

with Ecosim requires a mass-balance description of a food web at an so groups primarily inhabiting nearshore areas were excluded.

initial point in time (Ecopath; Christensen and Pauly, 1992), which Within the offshore ﬁsh community, only groups hypothesized as

is then used as the basis for forward-projecting time-dynamic sim- most important to food-web function were included in the mod-

ulations (Ecosim; Christensen and Walters, 2004). els. The Lake Huron Ecopath model was parameterized using data

Simulating species invasions in EwE models is challenging. collected around 1981 for 21 unique species or groups of species,

Species included in Ecosim time-dynamic simulations must have while the Lake Michigan Ecopath model was parameterized using

positive biomass when the model is initialized and balanced in Eco- data collected around 1987 for 29 unique species or groups of

path. This presents a problem for species that invade the system species. Invasive species invading after the initial Ecopath year

after the initial Ecopath year. One solution to account for inva- in the Lake Huron model included Bythotrephes, round goby, and

sive species effects has been to construct separate pre-invasion dreissenids, whereas invasive species in the Lake Michigan model

and post-invasion Ecopath food-web models (e.g. Jaeger, 2006; included round goby and dreissenids. Bythotrephes invaded Lake

Stewart and Sprules, 2011). Such an approach only utilizes Eco- Michigan prior to the year we initialized the Lake Michigan model

path, and precludes opportunities to explore dynamic simulations (i.e. 1987), and therefore were modeled as if they were non-invasive

in Ecosim. A solution that utilizes Ecosim is to run simulations in (i.e. actually already present in the system in the initial Ecopath

Ecosim based on an Ecopath model initialized in a year after all year). Age-stanzas were included for some species that underwent

invasive species were present, and therefore when biomass val- ontogenetic diet shifts or were targeted by ﬁsheries, increasing the

ues were truly positive. Time series of data are used by Ecosim total number of modeled groups to 36 for the Lake Huron model

to tune model parameters and evaluate model performance, and and 42 for the Lake Michigan model (Table 1).

therefore initializing Ecopath in a later year excludes available time

series data from the model-ﬁtting process that could inform species

Table 1

interactions prior to the invasion.

Species or species groupings in the Lake Huron (H) and Lake Michigan (M) Ecopath

Another approach to simulate species invasions in Ecosim mod-

with Ecosim models. Age stanzas for multi-stanza groups are provided in years.

els has been used, and does not reduce the length of data time

Group/species name (age stanzas) Scientiﬁc name Model

series. With this approach, invasive species are initialized at some

positive biomass value in Ecopath, prior to actual invasion, arti- Sea lamprey Petromyzon marinus H, M

Lake whiteﬁsh Coregonus clupeaformis M

ﬁcially maintained at negligibly low biomasses until the year of

(0, 1–3, 4+) H

invasion, and then afterwards allowed to proliferate. For exam-

Lake trout Salvelinus namaycush

ple, Pine et al. (2007) simulated the invasion of ﬂathead catﬁsh

(0, 1, 2–4, 5+) H

(Pylodictis olivaris) into an inland reservoir by artiﬁcially increas- (0, 1–5, 6+) M

Stocked Chinook salmon Oncorhynchus tshawytscha

ing ﬁshing mortality on catﬁsh, to explore what the system might

(0, 0.5, 1–5, 6+) H

have looked like prior to invasion, and then reducing ﬁshing mor-

(0, 1–5, 6+) M

tality to allow catﬁsh to invade. Similarly, Espinosa-Romero et al.

Wild Chinook salmon (0–5, 6+) Oncorhynchus tshawytscha M

(2011) initially suppressed biomass of sea otters (Enhydra lutris) Steelhead Oncorhynchus mykiss

by applying high artificial fishing mortality, and then released fish- (0, 1, 2–5, 6+) H

(0, 1–5, 6+) M

ing mortality in order to simulate reintroductions across a spatial

Brown trout (0, 1–5) Salmo trutta M

gradient. Forcing biomass of an invasive species (i.e. specifying a

Coho salmon (0, 1–2, 3+) Oncorhynchus kisutch M

time series rather than dynamically modeling it) has also been

Burbot Lota lota M

successfully employed to model lionﬁsh (Pterois volitans) invasion (0–3, 3+) H

Alewife (0, 1+) Alosa pseudoharengus H, M

in the Caribbean (V. Christensen, Fisheries Centre, University of

Rainbow smelt Osmerus mordax M

British Columbia, pers. comm.). Cox and Kitchell (2004) concluded

(0, 1+) H

that changes to data inputs for prey and predators of invasive

Bloater (0, 1+) Coregonus hoyi H, M

species are required when attempting to model invasion dynamics. Round goby Neogobius melanostomus H, M

Although the methods described above are ad-hoc, they repre- Slimy sculpin Cottus cognatus H, M

Deepwater sculpin Myoxocephalus thompsoni H, M

sent practical approaches to account for the effect of invasive

Ninespine stickleback Pungitius pungitius H, M

species.

Diporeia Diporeia spp. H, M

To date, methods for modeling invasive species in EwE models

Mysis Mysis diluviana H, M

have not been compared to determine their effectiveness for cap- Benthic invertebrates H

Fingernail clams M

turing food-web dynamics through time. Furthermore, trade-offs

Chironomids/Oligochaetes M

among methods or required assumptions for potential methods are

Dreissenid mussels Dreissena polymorpha, D. bugensis H, M

unspeciﬁed. In this study, we explored the relative performance

Predatory zooplankton Bythotrephes longimanus H, M

of four methods for incorporating invasive species into EwE mod- Zooplankton H

els. All methods used both Ecopath and Ecosim components of the Calanoid copepod M

Cyclopoid copepod M

modeling framework. Models for two Laurentian Great Lakes were

Non-predatory cladoceran M

developed and four methods were compared within each model.

Rotifer M

For each method, we evaluated the model’s ability to capture the

Phytoplankton H, M

biomass time-dynamics for both invasive and non-invasive species. Diatoms M

Detritus H, M

We present a quantitative comparison as to which method best

Diatom detritus M

reproduced observed data time series, and provide advantages and

disadvantages to using each method. Modeled as a ﬁshery in the Lake Michigan model.

B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261 253

Estimates of biomass (B), production to biomass ratios (P/B), where aij is the effective search rate of predator j for prey i, and

consumption to biomass ratios (Q/B), diet components (DC), har- vij is the vulnerability exchange rate of prey i to predator j. More

vest, and biomass accumulation (BA) were taken from published complicated forms of Eq. (3) can be found in Christensen and

and non-published sources and used to deﬁne interactions among Walters (2004).

modeled groups and ﬁsheries within Ecopath. Details of how Perhaps the most important parameters governing Ecosim

Ecopath uses these data inputs to deﬁne species interactions interactions are vulnerabilities (Plagányi and Butterworth, 2004;

have been described extensively (e.g. Christensen and Pauly, Ahrens et al., 2012), which control the strength of predator–prey

1992; Christensen and Walters, 2004) however we provide basic interactions. Vulnerabilities are founded in foraging arena the-

equations below. The mass balance equation in Ecopath, for each ory (see Walters and Martell, 2004; Ahrens et al., 2012), which

group i is, acknowledges that prey behaviors (e.g. risk sensitive foraging)

can limit their vulnerability to predation. Although unique vul-

= + + +

Pi Yi (BA)i Bi(M2)i Bi(M0)i, (1)

nerabilities for each predator–prey interaction are possible, using

a single value for all prey to a single predator is recommended

where P is production; Y is total harvest; BA is biomass accumula-

(V. Christensen, Fisheries Centre, University of British Columbia,

tion; B is biomass; M2 is predation mortality rate; and M0 is other

pers. comm.). Vulnerabilities (V) adjusted by the Ecosim user are

mortality rate. The Ecopath user does not provide estimates for

related to vulnerability exchange rates (vij in Eq. (3)) as vij = M2ijVij

all parameters in Eq. (1) directly, but rather provides inputs from

(Walters and Christensen, 2007). Ecosim has a ﬁtting procedure

which parameters in Eq. (1) are derived. Predation mortality (M2)

to estimate vulnerability parameters based on minimizing dif-

on prey i is calculated as the sum of consumption (Bj × Q/Bj × DCij)

ferences between model predicted outputs and observed data

by all j predators, divided by the biomass of prey i. Other mortality

time series (Christensen and Walters, 2004). Evaluating ﬁts of

(M0), the mortality not explained by modeled sources, is the

model predictions to observed data time series provides a metric

difference between total mortality (i.e. P/B ratio, Allen, 1971)

of biological realism; if ﬁts are better, then model predictions of

entered into the model and the sum of predation mortality, ﬁshing

past dynamics are more credible. Data time series can be values

mortality, and BA for each group. During balancing, a negative

of biomass, harvest, average weight, or total mortality. The ﬁtting

M0 for any group indicated imbalance owing to modeled sources

procedure also provides a way to estimate inter-annual variability

of mortality exceeding the production input for the group. If

in primary producer P/B (i.e. production anomalies). Estimating

the model was unbalanced, data inputs were adjusted following

production anomalies often greatly reduces deviations between

recommended practices (Christensen et al., 2005). Data inputs

model predicted biomasses and observed biomasses compared

for the balanced models (excluding diets) are provided in online

to estimating vulnerabilities alone (C. Walters, Fisheries Centre,

Supplementary materials.

University of British Columbia, pers. comm.), thereby enhancing

After food-web linkages and ﬁsheries were deﬁned in Ecopath,

model performance.

Ecosim was used to predict biomasses for each modeled group

through time. Ecosim predictions result from two equations, one

to deﬁne the change in biomass through time, and the other to 2.2. Alternative methods for incorporating invasive species

deﬁne the amount of consumption by any one predator on its prey

We compared four methods of incorporating invasive species

at a speciﬁc point in time. For groups with a single age stanza, the

into EwE models. The methods included: (1) forcing biomass

change in biomass of each group is modeled as

dynamics of invasive species through time (forcing biomass); (2)

i = starting biomass of invasive species at low levels, and allowing

g Q − Q − ((M0) + F )B , (2)

dt i ji ij i i i

biomass to increase at the time of invasion (low initial Ecopath

j j

biomass); (3) starting biomass of invasive species at high levels,

where t is time in months; g is the gross food-conversion efﬁciency reducing biomass until the time of invasion, and then allowing

ratio (GCE), which is the ratio of P/B to Q/B; Qji is consumption of biomass to increase afterwards (high initial Ecopath biomass);

prey j by predator i; Qij is consumption of prey i by predator j; and and (4) starting biomass of invasive species at high levels, and

F is the ﬁshing mortality rate. The solution to Eq. (2) is estimated reducing and then increasing biomass by adjusting the strength

using an Adams–Bashford method (Christensen et al., 2005). More of predator–prey interactions through time (mediating vulner-

complex versions of Eq. (2) are used for groups with multiple age abilities). The methods differed in terms of the assumed initial

stanzas (Walters et al., 2008). The amount of consumption, in its values of biomass for invasive species, either low or high; the

simplest form is modeled as approach used to maintain and “release” invasive species biomass,

either forcing biomass, using an artiﬁcial ﬁshery, or changing

aijvijBiBj

vulnerabilities; and whether observed invasive-species biomass =

Qij , (3) +

2vij aijBj

time-series were forced or ﬁt in the ﬁtting procedure. A summary

Table 2

Summary of methods evaluated for incorporating invasive species into Ecopath with Ecosim models for lakes Huron and Michigan. Details of each method are described in

the text.

Method Time series of Initial biomass of Approach used to Reasoning for use

invasive species invasive species maintain and release

invasive species

1. Forcing biomass Forced High Forced time series Forcing time series allows ﬁtting routine to match dynamics

to other species while invasive species dynamics are ﬁt

without error

2. Low initial Ecopath biomass Fit Low Artiﬁcial ﬁshery Invasive species begin their invasion at low biomass levels,

and thus, should be initialized as such

3. High initial Ecopath biomass Fit High Artiﬁcial ﬁshery Starting biomass at levels more similar to recent years allows

invasive species to reach high biomass levels more easily

4. Mediating vulnerabilities Fit High Changes in Biological processes maintain invasive species biomass until

vulnerabilities the time of invasion and biomass increases thereafter

254 B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261

of each method is provided (Table 2), with details described in the available. The level of ﬁshing mortality was chosen to reﬂect the

sections below. instantaneous rate of population increase (r), as calculated from

observed increases in the biomass time series of invasive species

2.2.1. Method 1 – forcing biomass during the early years of invasion. We ﬁt a model, Bt = B0e , to the

A simple way in which invasive species were included in the years of initial biomass increase, where B0 was the ﬁrst biomass

two EwE models was via biomass forcing. Time series of inva- estimate in the time series, and Bt was the biomass estimate t years

sive species biomass were used to overwrite predicted values from after the year for B0. The number of data points used to estimate

Ecosim equations for years when data were available. Time series r was 3 and 5 for Lake Huron and 11 and 4 for Lake Michigan

for dreissenids and Bythotrephes in Lake Huron did not contain data dreissenids and round gobies, respectively. The biomass time series

from the earliest years of invasion, nor was data available for every for Bythotrephes in Lake Huron did not cover a period of biomass

year once the time series began. As there was no way to know the increase, and so we used the slope of the line connecting the initial

biomass of dreissenids and Bythotrephes prior to when data were biomass input used in Ecopath to the value of the ﬁrst data point

available for the Lake Huron model, we assumed that biomass was in the time series, calculated on a log scale, as our estimate of r.

zero in years prior to the ﬁrst data point. Any gaps in the forcing To offset the added ﬁshing mortality rate, initial P/B ratios of inva-

time series were ﬁlled in with Ecosim predicted estimates. Biomass sive species were increased by an amount equivalent to the level

time series for invasive species in the Lake Michigan model were of ﬁshing mortality. Increasing P/B ratios ensured that estimates

complete and covered all years after invasion. for M0 were similar between Ecopath models with and without

The way in which Ecosim filled in gaps in the time series was the artificial fishing mortality, and that when fishing mortality was

affected by the choice of initial biomass estimates. Low initial removed, Ecosim estimates of P/B ratios were appropriate.

biomass estimates in Ecopath resulted in lower Ecosim biomass

predictions than were observed in the time series. On the other 2.2.3. Method 3 – high initial Ecopath biomass

hand, high initial biomass estimates in Ecopath resulted in Ecosim An alternative to initializing invasive species biomass at low lev-

predictions that more closely matched the observed biomasses for els was to initialize invasive species biomass at higher, recently

years adjacent to the period where the gap occurred. We therefore observed levels and then artiﬁcially reduce biomass until the year

chose to use high initial biomass estimates when forcing invasive of invasion. As for method 2, biomass estimates of invasive species

species biomass. We describe the details for initializing invasive and diet contributions of invasive species to their predators were

species biomass at high values in our description of method three. taken from 2002 in Lake Huron and 2005–2010 for Lake Michigan

Although only time series in the Lake Huron model contained and used as initial data inputs in Ecopath. Contributions of other

gaps, both models used high initial biomass estimates for invasive groups to the diet of predators of invasive species during the ini-

species. tial Ecopath year were proportionally adjusted so that total diet

summed to one. In contrast to method 2, diet and biomass values

2.2.2. Method 2 – low initial Ecopath biomass of invasive species were not reduced by a scaling factor.

In contrast to method 1, methods 2–4 used deviations between We encountered a problem when initializing the model with

Ecosim predictions and observed data time series of invasive high invasive species biomass values. Recent observed biomass

species in the ﬁtting procedure to estimate vulnerability param- values for dreissenids were substantial and caused imbalance for

eters and production anomalies. Initial biomass inputs for invasive groups that dreissenids consumed. To correct the additional pre-

species were set at low values in method 2, which were repre- dation mortality caused by starting dreissenids at high biomass, as

sentative of a pre-invasion state. However, entering biomasses or well as by other invasive species, we added negative BA to prey

diets of invasive species at arbitrary low levels could lead to grossly items that was equivalent to the initial level of consumption of the

incorrect descriptions of trophic interactions between groups. To prey item by the invasive species. Adding negative BA allowed Eco-

describe interactions between invasive species and their prey and path to calculate appropriate levels of M0 that were based on only

predators when invasive species biomass was initialized at low non-invasive species (species actually present in the system in the

levels, we used the following approach. We picked a recent year initial Ecopath year).

(2002 for the Lake Huron model and 2005–2010 for the Lake Michi- Once invasive species were appropriately entered into the initial

gan model) for each invasive species in which their biomass was Ecopath model, artiﬁcial ﬁshing mortality was applied to each inva-

high and diet information for their predators was available. The sive group in Ecosim so that biomass would become zero within the

biomass value for each invasive species from the chosen year was ﬁrst year. Reducing invasive species biomass with ﬁshing allowed

then divided by 1000 and used as the initial biomass value in Eco- invasive species to be present in the time dynamic simulations prior

path. The 1/1000th scaling factor was small enough to make initial to their actual invasions, but without having any effects on the rest

biomass estimates for the invasive species small, but also large of the food web. Negative BA equivalent to the level of ﬁshing was

enough so that when diet contributions of invasive species to their added to each invasive group so that estimates of M0 remained

predators were scaled downward, Ecopath would not round the unchanged by the addition of ﬁshing mortality. As in method 2, the

contributions to zero. Diet contributions of invasive species to their artiﬁcial ﬁshing mortality was released in the year of invasion.

predators were reduced by the same scaling factor so that Ecopath

calculated search rates were maintained at realistic (recent) values. 2.2.4. Method 4 – mediating vulnerabilities

By down-scaling the invasive species component of predator’s diet Method 4 used the same approach as method 3 for setting initial

contributions, we assumed that predator search rates for invasive biomasses and diets, but used a different approach for simulating

species did not change over time. To allow diet contributions of the invasion process. In contrast to methods 2 and 3, where the pro-

predators to sum to one, non-invasive components of their diets, cess of invasion was simulated by the addition and then removal

taken in the initial Ecopath year, were adjusted proportionally. of artiﬁcial ﬁshing mortality, assumptions about changes in the

Once invasive species were entered into Ecopath at low levels, strength of trophic interactions (i.e. vulnerabilities) were used to

they tended to remain at low levels. To simulate an increase in simulate species invasion for method four. Two forcing functions

biomass during the time at which invasion was assumed to occur, were used in Ecosim to mediate vulnerabilities; one to mediate

ﬁshing mortality was applied to each invasive species initially, and the vulnerabilities of prey to invasive species predators (Fig. 1a),

then removed in the year of invasion. The year of invasion cor- and the other to mediate vulnerabilities of invasive species to their

responded to the ﬁrst year in which positive biomass data were predators (Fig. 1b). In both instances, vulnerabilities were forced to

B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261 255

incorporating invasive species into EwE models. Ecosim calculates

residual sum of squared deviations (RSS) between the Ecosim

predicted and the observed data values, calculated on a log scale.

Observed data values can be either relative or absolute values. We

used absolute biomass time series for invasive species and a mix-

ture of absolute and relative time series for non-invasive groups.

Time series of biomasses, growth, and total mortality for some

modeled groups were available from published and unpublished

sources in 1981–2008 for Lake Huron and 1987–2008 for Lake

Michigan, and were used in the Ecosim ﬁtting procedure (Table 3).

Seventeen time series were used for the Lake Huron model and

42 were used for the Lake Michigan model (Table 3). Unique vul-

nerability parameters for every modeled predator and production

anomalies for every modeled year were estimated in both models.

3. Results

Our analysis revealed that the overall process of species invasion

could be reproduced in Ecosim models, but the exact timing and

magnitude of changes in invasive species biomass differed among

the methods and models considered. Timing depended both on

whether invasive species biomass was quickly reduced to, or main-

tained at low levels prior to the time of invasion, and whether

invasive species biomass increased from low levels at the appro-

priate year. As expected, method 1 (i.e. forcing) ﬁtted the timing of

biomass changes for invasive species perfectly (Figs. 2 and 3). Tim-

ing of biomass increases of invasive species between methods 2

and 3 were comparable, although biomass of round goby increased

sooner in method 3 than in method 2 for the Lake Michigan model

and biomass of Bythotrephes increased sooner in method 2 than in

method 3 for the Lake Huron model (Figs. 2 and 3). The primary

difference in timing between method 2 and 3 was reﬂected prior

Fig. 1. Shapes of vulnerability forcing function for (a) prey of invasive species and

to the time of invasion, when biomass levels declined more slowly

(b) invasive species to their predators. For vulnerabilities of prey to invasive species,

for method 3 (Figs. 2 and 3). Similarly, timing for method 4 differed

vulnerabilities began very low for the early simulation years, and increased to a peak

(Y) after the species invaded (time period X1), then stabilized to 1 once the species from the other three methods prior to invasion, when biomass of

began to become established (time period X2). For vulnerabilities of invasive species invasive species took longer to decline to low levels (Figs. 2 and 3).

to their predators, vulnerabilities began very low for the early simulation years, and

The timing for method 4 was similar to the other methods once

increased to 1 once the species invaded.

invasion occurred.

The magnitude of changes in invasive species biomass depended

zero in years prior to the actual invasion, which implies no effect on whether invasive species biomass increased to observed levels

of the invasive species on either their predators or their prey. Prey once invasion occurred. Invasive species biomass was more similar

species were assumed to become more susceptible to the invasive among methods in the Lake Huron model than the Lake Michigan

species predator once invasion occurred, and was simulated by hav- model, especially for round goby and dreissenids (Figs. 2 and 3). As

ing vulnerabilities increase to a peak (Y in Fig. 1a). Over time, we expected, biomass values of invasive species were ﬁtted perfectly

assumed that prey defense mechanisms would be developed and with method 1 (Figs. 2 and 3). Apart from method 1, few patterns

vulnerabilities to invasive species predators would reach stable lev- were present when comparing invasive species biomass across

els (i.e. would return to a relative value of 1 by year X2 in Fig. 1a). methods. Although method 3 appeared to have greater changes in

Similarly, in the initial stages of an invasion (around year X1 in the magnitude of invasive species biomass for the Lake Michigan

Fig. 1b), predators on invasive species may not have developed a model, high biomass values for round goby in the Lake Michi-

search image, and therefore vulnerabilities of invasive species to gan model and high biomass values for Bythotrephes in the Lake

their predators were assumed to be very low. Over time during Huron model were achieved for method 2 (Figs. 2 and 3). Simi-

the invasion, the vulnerabilities of invasive species to their preda- larly, although biomass of dreissenids in method 2 underestimated

tors were assumed to increase to stable levels (i.e. would return to observed values in both the Lake Huron and Lake Michigan models,

a relative value of 1 by year X2 in Fig. 1b). The shapes (Fig. 1) of biomass of round gobies were not underestimated any more than

the vulnerability forcing functions were based on the shape of the in other methods (Figs. 2 and 3).

round goby biomass time series in the Lake Huron model. Parame- In addition to ﬁtting the dynamics of invasive species, we also

ter values for the shape of vulnerabilities of prey to invasive species wanted our methods to adequately ﬁt non-invasive groups. Fits

predators were Y = 5; X1 = 1997 for dreissenids and round goby, and of non-invasive groups ranged from adequate to poor, depending

1998 for Bythotrephes; and X2 = 7 years after X1 (Fig. 1a). The same on the group, model, and method employed. Patterns in biomass

parameter values for X1 were used for the shape of vulnerabilities of dynamics among non-invasive groups, including groups without

invasive species to predators, however X2 = 6 years after X1 (Fig. 1b). biomass time series, from which to draw inferences about method

performance were not observed. Consequently, we present results

2.3. Assessment of invasive species methods for groups of ﬁshery management importance and concern (e.g.

lake whiteﬁsh Coregonus clupeaformis, lake trout Salvelinus namay-

Fits of predicted Ecosim dynamics to observed time series cush) or that are important prey items for other species (e.g. alewife

provided objective criteria to compare the four methods of Alosa psuedoharengus, Diporeia) (Figs. 2 and 3). Fits for other groups

256 B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261

Table 3

Time series, and number of data points with each time series, used for comparing performance of methods to incorporate invasive species into Ecopath with Ecosim models

of lakes Huron and Michigan.

Lake Huron Lake Michigan

Type of time series # of data points Type of time series # of data points

Biomass Total mortality

Age 1–3 lake whiteﬁsh 26 Lake whiteﬁsh 20

Age 4+ lake whiteﬁsh 26 Age 1–5 lake trout 21

Age 2–4 lake trout 25 Age 6+ lake trout 21

Age 5+ lake trout 25 Age 0–5 wild Chinook salmon 21

Age 1–5 Chinook salmon 28 Age 1–5 stocked Chinook salmon 21

Age 2–5 steelhead 25 Age 1–5 steelhead 21

Age 1+ alewife 23 Age 1–5 brown trout 21

Age 1+ rainbow smelt 23 Age 1–2 Coho salmon 21

Age 1+ bloater 23 Biomass

Round goby 9 Lake whiteﬁsh 20

Slimy sculpin 23 Age 1–5 lake trout 21

Deepwater Sculpin 23 Age 6+ lake trout 21

Ninespine stickleback 12 Age 0–5 wild Chinook salmon 21

Diporeia 10 Age 1–5 stocked Chinook salmon 21

Dreissenids 8 Age 1–5 steelhead 21

Predatory zooplankton 8 Age 1–5 brown trout 21

Zooplankton 9 Age 1–2 Coho salmon 21

Burbot 20

Age 1+ alewife 20

Rainbow smelt 13

Age 1+ bloater 20

Round goby 6

Slimy sculpin 20

Deepwater sculpin 20

Ninespine stickleback 20 Diporeia 20

Mysis 6

Dreissenids 11

Predatory zooplankton 13

Calanoid copepod 13

Cyclopoid copepod 13

Non-predatory cladoceran 13

Rotifer 8

Phytoplankton 18

Average weight

Age 1–5 lake trout 21

Age 6+ lake trout 21

Age 0–5 wild Chinook salmon 19

Age 1–5 stocked Chinook salmon 19

Age 1–5 steelhead 21

Age 1–5 brown trout 21

Age 1–2 Coho salmon 21

Age 1+ alewife 17

Age 1+ bloater 21

are provided in online Supplementary materials. Fits for lake white- not contribute toward the RSS, whereas in methods 2–4 invasive

ﬁsh, lake trout, alewife, and Diporeia in the Lake Huron model species time series contributed to the RSS. Consequently, only RSS

(Fig. 2) tended to vary little among the four methods when com- between methods 2–4 could be compared among each other.

pared to the ﬁts in the Lake Michigan model (Fig. 3). Fits in the Performance of each method varied between the models con-

Lake Huron model for lake whiteﬁsh and alewife were most differ- sidered. Among methods 2–4, methods 4 had the lowest RSS for the

ent, and poorest, for method 2 whereas ﬁts for lake trout, although Lake Huron model, followed closely by method 3 and less closely by

similar among methods, varied the most from observed trends method 2 (Table 4). Method 2 performed poorer for the Lake Huron

(Fig. 2). Fits in the Lake Michigan model were variable among model in part because of poorer ﬁts to dreissenids, lake whiteﬁsh,

the four methods considered, making comparisons across meth- and alewife than the other methods. For the Lake Michigan model,

ods more difﬁcult. Fits for lake whiteﬁsh were most similar to method 3 had the lowest RSS, followed by method 2 and then by

observed biomass values among the groups considered, and among method 4 (Table 4). Method 4 performed substantially poorer for

lake whitefish fits, method 2 performed the poorest (Fig. 3). Fits for the Lake Michigan model because of poor fits to invasive species,

lake trout overestimated observed data, and did not capture the but especially round goby, early in the time series. Time series for

declining and then increasing trend in the observed data (Fig. 3). invasive species in the Lake Huron model did not include early

Similarly, with perhaps the exception of method 4, declines during years, and therefore the slow declines for invasive species biomass

the ﬁnal years of observed data for alewife were not captured in in method 4 did not contribute to overall ﬁt.

the Lake Michigan model (Fig. 3).

Residual sum of squared deviations provided a way to quanti- 4. Discussion

tatively assess the performance of each method. Not all RSS values

could be directly compared among methods because the number The ability to incorporate species invasions into time-dynamic

of time series used to ﬁt the models differed. Time series of inva- ecosystem models is helpful in understanding system dynamics.

sive species biomass were forced in method 1, and therefore did Many creative methods can be employed for modeling species

B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261 257

force low high vuln

●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ●●● ●● ● ●●● ●● ● ●●● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ●●●● ●●●● ●●●● ●●●●

0.5 1.5 ● ● ● ● ●●●● ●●●● ●●●● ●●●●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ● ●● ● ●● ● ●● ● ●● ●● ●● ●● ●●● ● ● ●●● ● ● ●●● ● ● ●●● ● ● ●● ●● ●● ●● ●●●●● ●●●●● ●●●●● ●●●●● 0.02 0.06

● ● ● ● ● ● ● ● ● ● ● ● 2.0 ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ●● ● ● ● ● ●●●●● ●●●●● ●●●●● ●●●●● 0.0 1.0

● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ●●● ●●● ●●● ●●● 0204060

● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●● ● ● ● 0.00 0.02 0.04

● ● ● ●

● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● 050100

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 0.8 ●● ● ●● ● ●● ● ●● ● ● ● ● ● bytho●●●●●●●●●●●●●●●●● dreiss goby diporeia alewife trout lake whitefish 0.0 1980 1990 2000 1980 1990 2000 1980 1990 2000 1980 1990 2000 Year

Fig. 2. Fits to a subset of modeled groups for each method from the Lake Huron model from 1981 to 2008. The solid black line represents model predicted biomass (g/m ),

and the open circles represent observed biomass (g/m ). Groups include age 4+ lake whiteﬁsh (whiteﬁsh), age 5+ lake trout (lake trout), age 1+ alewife (alewife), Diporeia

(diporeia), round goby (goby), dreissenids (dreiss), and predatory zooplankton (bytho). Predatory zooplankton, round goby, and dreissenids were modeled as invasive species.

invasions, however the task of comparing methods and consid- provide guidance about the strengths and weaknesses of our four

ering their trade-offs can be challenging given model complexity. proposed methods and help future EwE modelers decide how to

Our analysis provided a ﬁrst step in comparing methods of incor- include invasive species in models of their systems.

porating invasive species into time-dynamic EwE models without The performance of each method as based on RSS depended

limiting the length of available time series. Rather than providing on the model used (Table 4). With the exception of the slow

a rule that all other EwE modelers should use, we instead hoped to decline in invasive species biomass for method 4, the methods

Table 4

Residual sum of squared deviations between observed biomass time series and Ecosim predicted biomass values for each method of incorporating invasive species into

Ecopath with Ecosim models of lakes Huron and Michigan.

Model Method

1 – forcing biomass 2 – low initial Ecopath biomass 3 – high initial Ecopath biomass 4 – mediating vulnerabilities

Lake Huron 103.3 146.5 130.7 129.2

Lake Michigan 197.4 653.2 624.0 2263

258 B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261 force low high vuln

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 01025

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 1.0 2.0

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0102030

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 200 500

● ● ● ● 0.4 0.8 ● ● ● ● ●●●●●●●●●●●●●●●● ● ● ● ●●●●●●●●●●●●●●●● ● ● ● ●●●●●●●●●●●●●●●● ● ● ● ●●●●●●●●●●●●●●●● ● ● ● 0.0

● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●● 0 2000

● ● ● ● 2.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

bytho dreiss● goby diporeia alewife● trout lake whitefish ● ●

0.0

1990 1995 2000 2005 1990 1995 2000 2005 1990 1995 2000 2005 1990 1995 2000 2005

Year

Fig. 3. Fits to a subset of modeled group for each method from the Lake Michigan model from 1987 to 2008. The solid black line represents model predicted biomass (kg/ha),

and the open circles represent observed biomass (kg/ha). Groups include lake whiteﬁsh (whiteﬁsh), age 6+ lake trout (lake trout), age 1+ alewife (alewife), Diporeia (diporeia),

round goby (goby), dreissenids (dreiss), and predatory zooplankton (bytho). Round goby and dreissenids were modeled as invasive species.

generally captured the basic biomass dynamics of invasive species RSS are on a log scale, differences among groups in the magnitude

(Figs. 2 and 3). Although RSS for method 1 was much lower of biomass change throughout the time series can bias the ﬁtting

in both models considered, we could not compare method 1 procedure toward ﬁtting one group over another. Thus, a method

with the other methods because different time series were used. that ﬁts well for important groups but poorly for a single unim-

Comparisons might be made among non-invasive species only, portant group that had highly variable biomass through time may

but we felt that because the invasive species time series con- have a poorer RSS than a method with marginal ﬁts for all groups.

tributed to RSS in methods 2–4, thereby affecting the ﬁts for other In general, careful attention should be given to judging which data

groups, comparing RSS for non-invasive groups only would still sources are most informative about food-web dynamics and how

be incomplete. How best to compare the performance of method such data sources may inﬂuence RSS, and which groups warrant

1 to the performance of methods 2–4 based on RSS remains priority in model ﬁtting given the modeling objective.

uncertain. Differences in the number of time series used for each model

Caution with RSS is warranted even when the number of time provide an opportunity to discuss the challenges with RSS. Val-

series used during ﬁtting is the same. When multiple time series ues of RSS for the Lake Michigan model were higher than those

are used, the weighting of time series to the overall sum of squares for the Lake Huron model. Perhaps the large amount of ﬁtted time

can be important. Although scales in Ecosim are relative, and the series caused difﬁculties for the Lake Michigan model to ﬁt any one

B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261 259

group well. There are tradeoffs between using a few time series that 4.2. Method 2 – low initial Ecopath biomass

can be ﬁt really well, and using many time series that ﬁt less well.

Information theoretic approaches would be effective at inform- The primary advantage of method 2 was its realism with respect

ing how many time series to use, and are under development (V. to biomass. Starting biomass of invasive species at low levels best

Christensen, Fisheries Centre, University of British Columbia, pers. mirrors the process of actual invasion in which invasive species

comm.). However, employing information theoretic approaches enter a system at low biomass levels and then proliferate to higher

requires caution. Data points within time series are correlated, levels. A secondary advantage of method 2 was its simplicity. Con-

which affects the effective number of data points to include in trary to the other three methods, method 2 did not require the

information theoretic calculations. For example, a time series of removal of high initial biomass before the time of invasion. Simi-

20 points may only have 2 informative points if it is linear. We larly, because invasive species biomass was initialized in Ecopath

have approached the problem with time series practically, and have at low levels, fewer adjustments to the diet of predators of invasive

used available time series unless the inclusion of a particular time species, and M0 of prey of invasive species were required than for

series greatly and negatively inﬂuenced model ﬁt. How the num- methods 1, 3, and 4.

ber of time series used for each individual model affected our RSS The primary disadvantage of method 2 was that invasive species

performance metric remains unclear. dynamics had to be maintained by an artiﬁcial ﬁshery. The increase

To help other EwE modelers make their own assessments about in biomass of invasive species was inﬂuenced by the time of release

which method to use, we summarize advantages and disadvan- from ﬁshing and by the level of ﬁshing mortality, and both affected

tages of each method in the sections below. Our methods were the estimated vulnerabilities of prey to invasive species. When

ad-hoc and contained artiﬁcial mechanisms for simulating the pro- released from high ﬁshing mortality rates, invasive species biomass

cess of invasion. Given the constraints of the modeling software, increased quickly and estimated vulnerabilities of prey to inva-

that species cannot be added mid-way through simulations, arti- sive species were lower in order to match observed dynamics.

ﬁcial mechanisms were required. We based our approaches on Our approach of applying ﬁshing mortality equivalent to the rate

biologically realistic values when possible by taking actual diets of biomass increase (as described in Section 2.2.2) at the time

and biomass values from recent periods (when invasive species when data time series began provided a standardized approach for

were established), and using a rate of biomass increase estimated including artiﬁcial ﬁshing mortality.

from observed data to initiate increases in biomass. Furthermore, Concerns about the realism of applying artiﬁcial ﬁshing mor-

we felt that including invasive species, despite artiﬁcial approaches tality can be mitigated if changes in biomass are viewed in terms

in doing so, resulted in a more realistic model of the ecosystem of biologically realistic processes. Low levels of invasive species

compared to excluding invasive species all together. biomass could reﬂect a condition where the species was being

sustained by a constant rain of propagules (Gottelli, 2008). The

increase in biomass (i.e. release from ﬁshing mortality) would then

4.1. Method 1 – forcing biomass reﬂect a scenario where good conditions have caused the constant

rain of propagules to become established, causing invasive species

The primary advantage of method 1 was that forcing biomass biomass to increase.

of invasive species was the best way to match observed biomass

dynamics of invasive species. Ecosim can predict future biomass

4.3. Method 3 – high initial Ecopath biomass

levels of invasive species even when forcing occurs, so method 1

does not eliminate the ability of the model to simulate future sce-

Method 3 was complicated to implement, however performed

narios. As described in Section 2.2.1, predictions of future biomass

well for both models (Table 4). The primary advantage was that

levels of invasive species depend heavily on initial biomass inputs

invasive species were able to reach high levels of biomass very

in Ecopath. We recommend that if simulations in Ecosim are used,

quickly due to the high initial biomass and removal of substan-

and invasive species biomass is forced, that recent biomass esti-

tial ﬁshing mortality at the time of invasion (Figs. 2 and 3). High

mates be used to initialize invasive species biomass in Ecopath. A

ﬁshing mortalities, contrary to the lower calculated rates used in

secondary advantage of method 1 was its simplicity, in that forcing

method 2, were used in method 3 to drive down the high initial

was a direct way to keep calculated biomass levels low until the

biomass of invasive species so that the effect of invasive species on

time of invasion and did not require artiﬁcial ﬁshing mortality or

other modeled groups would be minimal prior to invasion.

adjustments to trophic interactions.

The primary disadvantage for method 3 was its complexity. The

The major disadvantage of method 1 was that because invasive

level of ﬁshing mortality and BA adjustment inﬂuenced how quickly

species were forced, estimates of invasive species vulnerabilities

and to what level invasive species biomass declined, and also inﬂu-

(vulnerabilities of prey to invasive species) were not informed by

enced the estimation of vulnerabilities of prey to invasive species.

the biomass time series of invasive species themselves. Rather, esti-

If ﬁshing mortality rates were not high enough, biomass values

mates of invasive species vulnerabilities were only informed by

for invasive species did not reach low levels prior to the time of

the effect of invasive species on the biomass time series of other

invasion after vulnerabilities were estimated. Consequently, the

modeled groups. Although the exact effect is unknown, having vul-

approach we used of iteratively trying values of ﬁshing mortality

nerabilities be estimated based on biomass ﬁts to other groups may

and BA and then estimating vulnerabilities to achieve an overall

inﬂuence future projections of invasive species. A secondary disad-

best ﬁt was more subjective than methods 1 and 2.

vantage was that method 1 required a fairly complete time series

of observed biomass data for invasive species. Missing data in time

series could be addressed by interpolating missing points before 4.4. Method 4 – mediating vulnerabilities

entering the time series into the model, or by allowing the model

itself to estimate biomass dynamics in years when data are miss- The primary advantage of method 4 was that it used biological

ing during Ecosim simulations. Alternatively, if a large continuous interactions (mediating vulnerabilities) to adjust biomass of inva-

portion of the time series is missing (e.g. the first five years of invasive species rather than using artificial fishing. Although biomass of

sion) the invasion can be assumed to start later than it actually invasive species was slower to decline than when artiﬁcial ﬁshing

occurred, as was the case in the Lake Huron model for Bythotrephes was used, dynamics after the time of invasion were captured by the

and dreissenids. method (Figs. 1 and 2). Method 4 reﬂects an attempt to account for

260 B.J. Langseth et al. / Ecological Modelling 247 (2012) 251–261

theoretical trophic responses rather than using artiﬁcial modeling be preferable. Choosing among the three ﬁtting methods should

methods to control species invasion. depend on model objectives and data availability. If information on

The greatest disadvantage for method 4 was in the sensitivity how species invasion may inﬂuence the strength of predator-prey

of invasive species biomass to adjustments in vulnerability and the interactions were available, then method 4 may be preferred, but

lack of empirical knowledge about the shape of our adjustments. if not, then the simplicity of method 2 may be preferred.

The overall shape of the mediation functions we used made theo- The methods we examined here emerged from a working group

retical sense, but there was no way to know whether the shape was discussion of options for including invasive species in EwE models.

correct. The time over which vulnerabilities stabilized (the differ- Our list is not exhaustive, but we suspect other methods will be

ence between X1 and X2 in Fig. 1a and b) and the overall magnitude related to one or more of the methods examined here, or reﬂect a

of the effect of invasive species on their prey (Y in Fig. 1a) inﬂu- combination of the methods examined here. As the EwE software

enced the estimated biomass values of invasive species. Although moves toward more user-developed plug-ins (Christensen and Lai,

changing the shape could improve biomass ﬁts, doing so would be 2007), additional ways to include invasive species will likely be

similar to forcing biomass ﬁts for invasive species. Consequently, developed and are encouraged. Our comparison was not meant

we did our best to standardize the shapes for our analyses when to be a single recommendation for all modelers, but instead will

using method 4 (as described in Section 2.2.4 and Fig. 1), but with- hopefully foster new approaches to including invasive species in

out empirical evidence other than the biomass time series, these food-web models.

reﬂect one hypothesis about the shapes of the process.

Acknowledgements

4.5. Comparing non-biomass outputs

We would like to thank members of the EwE modelers working

We focused our analysis on reproducing biomass dynamics of

groups for their valuable comments on improving the understand-

our systems, primarily because we had more biomass data with

ing of the methods presented herein and for comments on ways to

which to compare our model predictions. Ecosim provides output of

best present them in a publication. Speciﬁcally, we would like to

other model-estimated values, and while comparing output of Q/B

thank the contributions of Yu-Chun Kao, Michael Jones, Ed Ruther-

ratios, we noticed different patterns among the four methods. For

ford, and David Bunnell, who provided sounding boards for our

method 1 in particular, consumption to biomass ratios for invasive

ideas in settings other than the modelers meeting workshops. We

species increased dramatically (from 8.6 to over one million for the

would like to thank the Great Lakes Fishery Commission (GLFC),

Lake Huron model) when biomass of invasive species was forced to

for providing a grant to MR, to fund travel to the modelers meeting

low values, but then returned near initial values once biomass was

workshops, as well as a separate GLFC grant that funded the work

increased. Consequently, predation mortality rates on prey of inva-

for BJL. This is QFC publication number 2012-09, contribution num-

sive species were similar through time, even when invasive species

ber 1708 of the U.S. Geological Survey Great Lakes Science Center,

biomass was near zero. Changes in Q/B with biomass suggest strong

and contribution number 1636 of NOAA Great Lakes Environmental

density dependence in growth (Walters et al., 2000). The change

Research Laboratory.

in Q/B ratios was less for method 3 (from 8.6 to over 100 for the

Lake Huron model), where biomass was also initialized at high val-

Appendix A. Supplementary data

ues, and even less for method 2 (from 8.6 to 15 for the Lake Huron

model), where biomass was initialized at low values. Therefore,

Supplementary data associated with this article can be found,

biomass may not be a suitable indicator of the effects of invasive

in the online version, at http://dx.doi.org/10.1016/j.ecolmodel.

species on their prey, at least using methods 1–3. Consumption to

2012.08.015.

biomass ratios for method 4, where invasive species biomass was

also initialized at high values, were zero prior to invasion. There-

fore, method 4 appeared to more properly “remove” the effects of References

invasive species on their prey compared to the other methods. As

stated earlier, our focus was on reproducing biomass dynamics, and Ahrens, R.N.M., Walters, C.J., Christensen, V., 2012. Foraging arena theory. Fish and

Fisheries 13, 41–59.

even though Q/B ratios were high, biomass dynamics appeared to

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