�9 1987 by The Humana Press Inc. All rights of any nature whatsoever reserved. 0163-4992/87/1100-0057502.20

Modeling ProductMty of a Salt Marsh

EDWARD PAUL

Stockton State College, Pomona NJ 08240

Received January 23, 1987; Accepted March 10, 1987

ABSTRACT

A model of primary productivity in a salt marsh is developed and compared to a regression analysis study of data showing depend- ence of growth on growing season, mean tidal height, and average monthly temperatures for several grass species.

Index Entries: Salt marsh; net primary productivity; model; plant growth.

INTRODUCTION

This paper is intended to provide an insight into the process of de- veloping a mathematical model of system behavior, following the style and approach of Terrell Hill (1). The system of interest is the annual growth of plant species in salt marshes.

Salt marshes are located between the mainland and the bays formed by ocean barrier islands along the eastern and southern coasts of the United States and other locations. They appear to be extensive grassy fields that flood with the daily tides. They are characterized ecologically by a wide range of flora and fauna; their importance is in providing habi- tat and nutrient for many diverse species. Historically, they served as grazing land during the seventeenth and eighteenth centuries in the United States and continue as habitat for hunting of wildfowl and fur- bearing game.Economically, the marshes are important because of their role in the commercial and sport fishing, hunting, and boating industries (2-5).

The net primary productivity (NPP) of a salt marsh is the mass of plant material produced annually. Since it occupies a significant position on the food chain, the rate of plant growth is important. Studies of marsh

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58 Paul

production show that the rates are unusually high compared to terrestrial or phytoplankton communities (6).

The predominant salt marsh plants include several grass species, each with its own growth rate and characteristics. Stiven and Plotecia (7) studied the correlation of the growth of seven of these species, and a mixed group, with various environmental factors, including growing season, mean tidal height, average temperatures, and latitude. The re- sult of the study was a best fit of data to Eq. (1):

log(NPP) = a + bG + cH + d A T + eL + f [1]

where G is the length of the growing season, H is the average height of the mean high tide, AT is the difference between mean July and January (or freezing point of water) temperatures, and L is the latitude of the growth site. The fitting parameter, f, depends on the specific species con- sidered and parameters a, b, c, d, and e are fit to pooled data for all spe- cies.

The study was designed to use regression analyses and data fitting techniques to estimate the productivity of a salt march as a function of the parameters considered. Although some of the results made intuitive sense, there was no effort to derive or model the system. Despite the number of fitting parameters, Eq. (1) explained only 69% of the variance of 23 test data points. The complexity and presence of wide individual variations in biological systems are responsible for more data scatter than is common for physical or chemical systems (8).

THE MODEL

Mathematical modeling is a process that reduces sets of observations to a limited number of idealized behaviors and then uses relationships between them to predict general outcomes. It is successful to the extent that the real systems behave in ideal ways. Most modeling assumes that the parameters chosen to describe behavior are independent of each other and that interactions between them are small. In physical and chemical systems this is often true. In biological systems, however, vari- ables often do affect each other, making simple models of limited quanti- tative use. Modeling does, however, provide insight into behaviors and relationships that can guide understanding of the systems under consid- eration.

Eq. (1) offers a starting point for a model of salt marsh growth. It summarizes a substantial amount of empirical data without providing an understanding of the behavior or of why the observed variable depend- ances appear. From a modeling viewpoint, however, the logarithmic de- pendence suggests a place to begin the description.

Systems in which growth is proportional to the presence of starting material are characterized by logarithmic behavior, which serves as the

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hlodeling Productivity o f a Salt Harsh 59

basis for most growth studies. In this case, dM (the change in dry weight of a given species of plant) is proportional to M (the amount material present) and to dt (the time available for growth). Thus:

dM = aMdt [2]

where a is the rate proportionality factor. The growth rate factor, a, is itself dependent on several variables,

including the availability of nutrients, energy, and sunlight. From an ideal modeling viewpoint, these factors should act independently on a plant and can be considered separately. These were taken into account in Eq. (1) through their association with the tidal height, temperature, and latitude parameters. The regression analyses (7) showed that their rela- tive importance in predictions of the NPP is in the order given in Eq. (1), with growing season responsible for about twice as much effect as lati- tude.

Eq. (2) serves as the basis for a model of the system. To continue the process, it is necessary to show how the factors described above can be included in factor a. Since they are taken to be independent, a can be thought of as a product of the different effects.

Nutrients for plants in a salt marsh come from the soil, which is washed and replenished by the tides. Nutrient availability is affected most strongly by the action of tides on the marshes. More water brings in more nutrient. Thus, the nutrient portion of a can be described by:

n = 11o + nlH = no (1 + bH) [31

where no accounts for the effect of the soil itself, and n~H reflects the ef- fects of the mean tidal height, H, on the availability of nutrients. It is con- venient to define b = nJno for manipulating the resulting equations. Ra- tio b gives the relative importance of tidal flux to ground nutrient for these systems. Although it is possible to devise experiments to deter- mine the values of no and nl, their specific values are not needed for this model.

The role of energy in plant growth is complex. One simple modeling approach is to observe that the total available energy must be propor- tional to the difference between the ambient temperature and some base point. This is similar to the use of degree-day data for heating and cool- ing statistics. In the case of salt marsh growth, the relevant baseline is either the temperature in January (when no growth occurs) or the freez- ing point of water (since cellular processes do not proceed below this temperature). Because degree-day data are not available for the sites in question, the cruder parameter of average monthly temperature can be used. Although some accuracy could be gained by calculating monthly energy contributions, it is sufficient for the purpose of this model to fol- low the approach (7) of looking at AT, the difference between mean July and January (or freezing temperatures), to account for the effect of en- ergy on plant growth, as shown in Eq. (4).

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60 Paul

E = EoAT [4]

The effect of sunlight on plant growth is difficult to describe mathe- matically. The amount of sunlight incident on a salt marsh depends on the latitude of its location in a complex way. One modeling technique that avoids these difficulties is the use of a parameter that broadly covers these questions. Ideally, the parameter should be related to available data. In this case, the concept of growing season is helpful. For the pur- pose of the model, the growth period can be modeled through the func- tion k(t), which equals 1 during the growing season and 0 outside of the growing season. Growing season information is useful in part because data on growing seasons at different locations have been tabulated by the US Department of Agriculture and are generally available (9).

Because these various effects are taken to be independent, parame- ter a in Eq. (2) can be replaced by the product of Eq. (3), Eq. (4), and k(t) to give:

dM = kn0(1 + bH) EoATMdt = k(t)(A)(1 + bH)ATMdt [5]

where the time-independent parameters A = noEo and b need to be fit for this model. Both A and b may be species specific. A includes the effects of nutrient and temperature on growth rates, whereas b balances soil and tidal nutrient.

Equation (5) can be divided by M and integrated over a 1-yr time pe- riod to give the annual plant growth:

fdM/M = ln(M) - ln(M0) = fkA(1 + bH)ATdt = A(1 + bH)ATG [61

where G = f k(t)dt is the length of the growing season and M0 is the plant mass in January. Since M0 < < M, the term ln(M0) may be dropped and the expression rearranged to give:

C = ln(M)/(ATG) = A(1 + bH) [71

indicating that C is proportional to H. Equation (7) is useful since it pre- dicts a linear relationship between C and H, allowing parameters A and b to be determined. Plots of C against H showed the expected behavior, with some data scatter. The parameters A and b were least-squares fit for each species. Slopes for b were reasonably close for different species, sug- gesting that the tidal effect on nutrient availability does not depend on particular plant types. The intercepts showed small species differences, implying that the productivity does not depend significantly on which species of plants are growing. The intercepts were averaged to give a single-species independent value for A. The final result, given in Eq. (8), shows how the NPP varies with the length of the growing season (G), mean July and January (or freezing) temperature difference (AT), and mean tidal height (H).

ln(NPP) = AGAT(1 + bH) [8]

where A = 8.00 x 10-4/d ~ (or A = 1.44 x 10-3/d ~ b = 2.48 x

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Modeling Productivity of a Salt Marsh 61

lO-5/ft (or b = 8.13 x 10-5/m). The parameters are given in English units to conform with the available data presentations (7,9,10).

D I S C U S S I O N

The predictions of the current model are included in Table 1. Al- though they do not fit data quite as well as the values given by the previ- ous study (7), the model's simplicity and insights make it useful. For ex- ample, the present work uses only 2 data-fitting parameters rather than 13 parameters, as in the previous study.

A major advantage of the model is that it suggests specific questions to pursue in order to better understand the system. The question of nu- trient ratios is one example: The model helps understand the role of tidal fluxes on plant growth. Effects of energy and light are other quantitative

TABLE 1 Measured and Predicted Values of Net Primary Productivity (NPP)

and Corresponding Environmental Data"

NPP,

NPP, predicted Growing Temperature, Mean observed (~ (~') season, d ~ tide, ft Species' Reference

560 780 1470 249 32.5 4.1 G (11) 492 374 371 185 39.5 0.4 B (12) 518 455 371 185 39.5 0.4 H (12)

1296 1803 735 249 32.5 4.1 A (13) 461 1012 1470 249 32.5 4.1 B (13) 329 488 1470 249 32.5 4.1 C (13) 796 970 1619 257 29.7 6.8 G (14)

2883 1259 1619 257 29.7 6.8 B (14) 1028 1028 1196 257 29.7 5.2 E (14) 2092 2092 1196 257 29.7 5.2 F (14) 3990 2242 1619 257 29.7 6.8 A (14) 1246 1113 735 187 42.3 1.4 H (15) 1207 915 735 187 42.3 1.4 B (15) 445 621 537 183 42.3 0.5 B (16) 985 1276 1573 256 29.7 6.8 B (17) 840 1075 262 176 35.7 3.5 A (18) 432 291 262 176 35.7 3.5 C (18) 430 510 262 176 35.7 3.5 D (18) 680 734 262 176 35.7 3.5 H (18) 754 606 1722 269 33.3 1.3 G (19)

1297 1093 2638 289 32.1 2.0 D (20) 1334 1574 2638 289 32.1 2.0 H (20) 1361 997 2638 289 32.1 2.0 G (20)

"From ref. (7). ~Present Work 'Species key: A. Spartina alterniflora (tall); B. S. alterniflora (medium); C. S. alterniflora (short); D. S.

patens; E. S. cynosuroides (tall); F. S. cynosuroides (medium); G. Juncus roemerianus; H. mixed species.

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62 Paul

questions that can be investigated. The present study could be improved quantitatively by fitting parameters A or b to specific species and by using weather bureau records on the daily temperatures for the particular years for which data in Table 1 were collected. These steps could be important in detailed correlations of growth to give better quantitative predictions.

The previous study (7) was undertaken to predict growth in marsh- lands at a variety of locations for several common species. It deliberately used tabulated data for mean temperatures and growing seasons rather than determining year-specific information because its purpose was to correlate average behavior to available data for quick and easy use. The method of data analysis does not imply an understanding of system mechanisms.

The present model attempts to predict behavior by linking plant growth to specific relationships with nutrient, energy, and light. Each of these dependences is cast into a mathematical form that is susceptible to modification. The advantage of a mathematical model is that it can be changed to reflect qualitative differences in the ways that system behav- ior is understood. The weakness of the model is in its idealization of be- havior, with the implication that the parameters act independently of each other. The real system complexity results in this approach not giv- ing a quantitatively accurate prediction of the productivity. The idealiza- tion, however, is central to the role of modeling in understanding the ba- sis of system behavior.

REFERENCES

1. Gordon, R., and Chen, Y. (1987), Cell Biophys. this issue. 2. Sweet, D. C. (1971), The Economic and Social Importance of Estuaries. Environ-

mental Protection Agency, Water Quality Office, Washington, DC. 3. Day, J. W., Deegan, L. A., Gosselink, J. G., Yanezarancibia, A., Soberon-

chavez, G., and Sanchezgil, P. (1985), Estuaries 8, 3. 4. Lynne, G. D., Conroy, P., and Prochaska, F. J. (1981), J. Environ. Econ.

Manag. 8, 175. 5. Farnworth, E. G., Tidrick, T. H., Jordan, C. F., and Smathers, W. M. (1981),

Environ. Conserv. 8, 275. 6. Keefe, C. W. (1972), Contrib. Mar. Sci. 16, 163. 7. Stiven, A. E., and Plotecia, R. K. (1976), UNC Sea Grant Public. 76-06, NC

State University, Raleigh, NC, p. 21. 8. Dame, R. F., and Kenny, P. D. (1986), Mar. Ecol.: Prog. Ser. 32, 71. 9. Department of Agriculture Yearbook (1941), Climate and Man. US Depart-

ment of Agriculture, Washington, DC. 10. Tide Tables, East Coast, North and South America. (1974), National Ocean

Survey, US Department of Commerce (NOAA), p. 288. 11. Foster, W. A. (1968) Master's Thesis. North Carolina State University. 12. Keefe, C. W., and Boynton, W. (1972) Chesapeake Sci. 32, 117. 13. Stroud, L. M., and Cooper, A. W. (1969) University of North Carolina Water

Resources Research Institute, Report No. 14, 86p. 14. Odum, E. P., and Fanning, M. E. (1973) Bull. Ga. Acad. Sci. 31, 12.

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Modeling Productivity o f a Salt Marsh 63

15. Johnson, M. (1970) University of Maryland Office of Water Resources, (NRI Ref. No. 71-6.) 164.

16. Morgan, M. H. (1961) Master's Thesis. University of Delaware. 17. Smalley, A. E. (1959) Proc. Salt Marsh Conf., Mar. Inst. Univ. Georgia, Sapelo

Isl., Ga. 96. 18. Nixon, S. W., and Oviatt, C. A. (1973) Botanica Marine 32, 103. 19. Williams, R. B., and Murdoch, M. B. (1972) Chesapeake Sci. 13, 69. 20. Waits, E. D. (1967) Ph.D. Dissertation. North Carolina State University.

Cell Biophysics Vol, 11, 1987