International Journal of Oceanography

Volume 2015, Article ID 728753, 6 pages

http://dx.doi.org/10.1155/2015/728753

## Relating Ctenophore Population to Water Mass Indices in the Northeast U.S. Continental Shelf Ecosystem

^{1}Department of Mathematics and Computer Science, Rhode Island College, 600 Mount Pleasant Avenue, Providence, RI 02908, USA^{2}NOAA Fisheries, 1315 East-West Highway, Silver Spring, MD 20910, USA

Received 7 October 2014; Revised 3 February 2015; Accepted 11 February 2015

Academic Editor: Renzo Perissinotto

Copyright © 2015 Rebecca Sparks et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Ctenophores exist throughout the Northeast U.S. Continental Shelf Ecosystem, but the underlying mechanisms that control ctenophore populations at this scale are not clear. Ctenophore population data over the last 30 years coincides with changes in several water masses on the shelf, but discovering which water mass was most influential was problematic without mechanistic clarity. This paper strives to identify the relationship between oceanography and ctenophore populations over the last 30 years. Using a numerical modeling approach, we found a strong relationship between the North Atlantic Oscillation index, percent Labrador Subarctic Slope Water, and ctenophore population. We suggest these results might inform future efforts to develop a predictive capability for major changes in ctenophore population.

#### 1. Introduction

Large increases in gelatinous zooplankton (e.g., medusae, ctenophores, salps, and siphonophores) have been reported in systems such as the Black Sea and the Mediterranean [1, 2], the Caspian Sea [3], and the Bering Sea [4]. Increases like these can drive shifts in ecosystem conditions [5].

A recent analysis of observations by NOAA’s National Marine Fisheries Service (United States Government) on the Northeast U.S. Continental Shelf between 1980 and 2012 describes spikes in ctenophore abundances [6–8]. The percent frequency of occurrence of ctenophores, an index developed by Link and Ford [6], spiked from 6 to 42 in the period from 1990 to 1996.

Similar increases were reported for other gelatinous groups including salps and siphonophores [8]. Frank [9] described the implications of a spike in the abundance of the ctenophore* Pleurobrachia pileus* off Southwestern Nova Scotia in 1983. At times when the levels of ctenophores are high, there can be significant ecosystem effects in terms of the consumption of crustacean zooplankton also required by pelagic fish [10]. In 1983, Frank reported when concentrations of* P. pileus* were high, they consumed 26% of the standing stock of copepods in April and 10% in May. This reduction in copepod stock was associated with extremely low levels of haddock (*Melanogrammus aeglefinus*) larvae produced during the spring of 1983. We feel it is reasonable to assume a similar level of competition between ctenophores and pelagic fish during peaks in our observations. The potential threat to the growth and success of fish populations is an impetus for this study.

We expect the sudden increases and the growth of the ctenophore population on the Northeast Continental Shelf to be associated with changes in oceanographic conditions. The linkage between oceanographic conditions and zooplankton such as copepods is well established and we assume this linkage holds for ctenophores. Inter alia, Greene and Pershing [11], Jossi and Goulet Jr. [12], and Hare and Kane [13] describe these relationships. The oceanographic conditions of the Northeast Continental Shelf can be traced to fluxes in Labrador Subarctic Slope Water (LSSW), Atlantic Temperate Slope Water (ATSW), the position of the north wall of the Gulf Stream, and the phase of the North Atlantic Oscillation.

Potential connections between oceanographic conditions and plankton populations like ctenophores are described in the Ecosystem Status Report [14]. Here, changes in LSSW, the position of the north wall of the Gulf Stream, and the NAO were correlated with a change in the composition of the copepod community on the Northeast Continental Shelf. There was a shift from large copepod species to small copepod species. Since copepods are important prey for ctenophores and ctenophores are not trophically distant from copepods, this correlation supported our expectations that a linkage existed between oceanographic conditions and the ctenophore population.

We attempt to use a new numerical modeling approach to make a first-pass review of the connections that exist between this ctenophore population and the oceanographic conditions of the Northeast Continental Shelf. We will introduce a quantitative technique that will test the fit of thousands of possible models for ctenophore population growth as a function of physical conditions on the shelf. We expect to uncover which oceanographic parameters are influential. By this determination, we expect to make progress toward the development of an indicator of conditions conducive to sudden ctenophore population increase. A leading oceanographic indicator of this type might inform fisheries managers of a shift in the ecosystem.

#### 2. Methods

We constructed a model of the ctenophore population over time in terms of the physical oceanographic influences of the Northeast Continental Shelf from 1986 to 2008. This unique approach involved three parts. First, the form of the logistic growth model served as the basis for finding functions that relate ctenophore population () and physical conditions. A laboratory study of* Pleurobrachia bachei* reported that the relationship of body mass over time was sigmoidal [15]. This suggested ctenophore growth might be curtailed by finite prey resources. While other investigations have uncovered details of the bioenergetics and growth of ctenophores [16, 17], the simpler logistic model appeared to best represent a system where the prey resources of ctenophores are most likely finite. Our model was to serve as a means to evaluate physical oceanographic influences on this population. With the logistic differential equation population model in place, the second step employed a scientific data mining software package, Eureqa [18]. Eureqa searches for mathematical patterns within data and has the ability to identify a relationship among parameters within a specified model. Once a formulation for the differential equation was found (and thus the relevant parameters were identified), the third step utilized the capabilities of Mathematica [19] to create numerical and graphical versions of solutions to the differential equation of the model. Completion of this last step allowed us to illustrate how changes in the various indices would impact the ctenophore population as it was modeled.

An estimate of the abundance of ctenophores (percent frequency of occurrence) was obtained through the quantification of stomach contents of the* Squalus acanthias*, the spiny dogfish (full methodology in [6]). Sampling was conducted across the Northeast Continental Shelf Ecosystem as part of the Food Web Dynamics Program at the NMFS Northeast Fisheries Science Center (NEFSC) in Woods Hole, Massachusetts, USA (http://www.nefsc.noaa.gov/femad/pbb/fwdp/). With this technique, the exact species of ctenophore is not identified and could include species such as* Mnemiopsis leidyi*,* Pleurobrachia pileus*,* Bolinopsis* sp., and* Beroe* sp.

The percent of Labrador Subarctic Slope Water () and the percent Atlantic Temperate Slope Water () are water mass indices based on oceanographic profile observations near the Northeast Channel at the entrance of the Gulf of Maine from NEFSC surveys. Also, we used another formulation of the percent Labrador Subarctic Slope Water () from Mountain [20] based on the same survey data, but including profiles from the Bedford Institute of Oceanography and the World Ocean Database. Since this new formulation of the Labrador Subarctic Slope Water () utilized additional inputs, we included it in our analysis alongside the Labrador Subarctic Slope Water from the NEFSC (). The annual anomaly in the position of the north wall of the Gulf Stream () was taken from time series data by Arnold Taylor’s Gulf Stream North Wall index (http://www.pml-gulfstream.org.uk/). The annual anomaly in the North Atlantic Oscillation () was the Hurrell Station-Based index taken from differences in atmospheric pressure between Lisbon, Portugal, and Reykjavik, Iceland, for the months of December, January, February, and March (https://climatedataguide.ucar.edu/climate-data/hurrell-north-atlantic-oscillation-nao-index-station-based).

##### 2.1. The Logistic Model

The changing size of a population over time is often modeled by a differential equation. A variety of standard approaches are available, including the logistic equation. Letting denote the size of a population at time , this equation takes the formand its solutions approach , the so-called carrying capacity for the population [21]. While the values , are sometimes thought of as constants, this is not a necessary restriction. If and depend on other parameters of interest in a particular application, the underlying dynamics of the equation, in which the rate of change of is either positive or negative depending on whether the value of is, respectively, below or above a certain level, are unchanged. Herein, we seek a logistic model in which , could possibly depend on several parameters of potential relevance in this setting.

##### 2.2. Finding a Model

Models for ctenophore population in terms of these water masses were found using Eureqa [22]. This is a scientific data mining software package that searches for mathematical patterns within data [18].

We entered into Eureqa the time series data for the years 1981 through 2010 for the ctenophore population, North Atlantic Oscillation index (), and for each of the water mass indices Labrador Subarctic Slope Water ( and ), Atlantic Temperate Slope Water (), and Gulf Stream North Wall (). To fill in missing data points within the collected data, we used Eureqa’s interpolation option in which a missing data point is filled in by the mean of two adjacent data points [22]. Eureqa will search for any model format but also allows the user to look for models of a specific type. Since we are interested in how the ctenophore population may be affected by NAO index and the water mass indices specified above, we chose to force the software to search for a logistical model of ctenophore population (). That is, we required the software to search for functions and such thatwhere and are functions of , , , , and . That is, , , and each of , , , , , is dependent on time, . The variable represents the ctenophore population (in year ), represents North Atlantic Oscillation index, and , , , represent the water mass indices for both formulations of Labrador Subarctic Slope Water, Atlantic Temperate Slope Water, and Gulf Stream North Wall, respectively.

Initial software runs used only the raw data, but output models from Eureqa were highly sensitive to small perturbations of data. As a result, we acknowledged the existence of noise in collected data, and before running Eureqa again, we smoothed the data using Eureqa’s smoothing process, which is determined by generalized cross validation among cubic b-splines [18].

##### 2.3. Sensitivity Analysis

Once we select an appropriate model output from Eureqa, the remaining question is, which of the parameters , , , , is the more influential on the ctenophore population ()? We used a standard local sensitivity analysis [23] in which we use small (local) perturbations of each parameter in (2) and examine how each perturbation affects the model output.

For ease of notation let denote the right hand side of (2). That is, let , where , , and let , , , , and represent the standard deviations of the time series data sets of , , , , and , respectively. For example, to illustrate the sensitivity of the ctenophore population to a change in , we solve the differential equation (2) with the known values of modified by adding or subtracting a fraction of . Since this is a local sensitivity analysis, we chose small varying degrees of change in by increasing (and decreasing) the values of by a small fraction of . In our case, we chose fractional changes of 10% of and 40% of . That is, we replaced with and found a numerical solution to to see how this solution deviates from the solution to . We also obtained numerical solutions to , , and . This process would be repeated for each parameter that appears in Eureqa’s model output. Plots of the percent frequency of occurrence of ctenophores () generated from the solution to the differential equation of the model with the perturbed data as a function of year can be compared to the plots of the original, unadjusted model. By changing each parameter one-by-one, we are allowed to see which (if any) parameter has a stronger effect on the model output.

#### 3. Results

Eureqa generated 7.7029 × 10^{7} possible functions, resulting in 18 proposed formulations to the differential equation. These formulations are summarized in Table 1.