In order to study the impact of organic matter and nutrient loading on food web dynamics, a 1-D, tidally averaged, advection-dispersion model was developed. The governing equation of this model is given by

where

*x*is distance (m)*t*is time (d)-
**C**(*x, t*) is a vector of nutrient or organism concentrations (in mg m^{-3}) *A*(*x*) is river cross sectional area (m^{2})*D*(*x, t*) is dispersion coefficient (m^{2}d^{-1})*u*(*x, t*) is velocity (m d^{-1}) and*A*(*x*)*u*(*x, t*) equals volumetric flow rate**r**(*t*) is a vector of source terms that is determined by the structure of the food web (mg m^{-3}d^{-1})

The dispersion coefficient as a function of location and river discharge has been estimated in the Parker River-Plum Island Sound Estuary, MA, from both dye studies and salinity transects. In the dye study, an inert dye (rodamine WT) was released in the upper Parker River, and its concentration profile was measured with a fluorometer at several subsequent high tides. Parameters in a function used to model the dispersion coefficient were then adjusted so that the dye distribution was accurately reproduced by the advection-dispersion model. This is illustrated by this figure, in which measured dye profiles (shown in open symbols) are compared to model output in sold lines for tidal cycles 1, 3, and 5.

Under steady state conditions, the dispersion coefficient can be estimated from the following equation

where *C*(*t*,*x*)
is
the concentration of an inert tracer
(i.e. salt) and *q*(*t*) is
volumetric flow rate.
Consequently, the
dispersion coefficient can be estimated from salinity profiles. From
salinity
profiles during different river discharge regimes, we have obtained the
following equation to describe dispersion (m^{2} s^{-1})
in
the Plum Island Estuary (see
figure)

where *x*
is distance (m), and *q*
is volumetric flow rate (m^{2}s^{-1})

Cross sectional river area for the Parker River was measured and fit to the following simple function

The food web model is based on the following diagram

which consistes of autotrophs, heterotrophs, dissolved inorganic nitrogen (DIN), and dissolved organic matter (OM) that is represented in both labile and refractory pools. The heterotrophs are able to graze the autotrophs as well as utilize the DIN and labile OM pools and mineralize organic nitrogen. The food web is an example of a "lumped" model, in that the heterotroph pool consists of organisms ranging in size and taxon from bacteria to macrozooplankton. The autotroph pool is similarly represented.

The transient for the autotrophs last about 1 wk (1 tidal cycle equals 12.25 hr), while that for the heterotrophs last about 2 wks. However, predator-prey oscillations are evident in the down stream portion of the estuary. The labile organic matter comes to approximate steady state in only 2 d. Static, 3D images of the state variables versus time (tidal cycles) and distance (km) can also be examined by clicking the thumbnail figures below. Figures were rendered in Mathematica.

Autotrophs | Heterotrophs | DIN |

Salt | OM labile | OM Refractory |

A variation of this
model was used to
examine the autotrophic-heterotrophic nature of estuaries. See
Hopkinson and Vallino (1995) *Estuaries,* **18**:
598-621. Also see Vallino, J.J. and Hopkinson, C.S. (1998). Estimation
of Dispersion and Characteristic Mixing Times in Plum Island Sound
Estuary. *Estuarine, Coastal and Shelf Science* **46**,
333-350.

Code used to solved
the above problem
is written in Fortran, and the
algorithm employed to solve the 1D PDE is a moving-grid method
developed and
written by Blom and Zegeling (1994), *ACM TOMS*
**20**:
194-214. (TOMS731).

Steady state
solutions have also been
obtained using a collocation algorithm
to solve the resulting boundary value problem (COLNEW:
Bader and Ascher (1987), *SIAM J. Sci. Stat. Comp.* **8**:
483-500).

Although not used in this particular project, we have found the BACOLR routine to be very efficient at solving 1D ADV equations (R. Wang, P. Keast, and P. H. Muir. Algorithm 874: BACOLR-spatial and temporal error control software for PDEs based on high-order adaptive collocation. ACM Trans.Math.Softw. 34 (3):1-28, 2008.)

All code is available here; however, there is no documentation.