Abstract
The nutrient uptake length, the average displacement of a nutrient in a stream before being taken up by the biota, is an important quantity to characterize and compare streams and rivers, or to quantify certain aspects of their related ecosystems. This concept has been widely used for almost 30 years now, and uptake lengths have been estimated for several nutrients in many systems, but it also suffers from a number of limitations, one of them being the requirement of a spatially homogeneous stream or river. We combine recently advocated, transport-based models of stream processes with current concepts of dispersal theory into a novel framework for nutrient uptake length. The framework is based on the theory for dispersal kernels in terrestrial systems, where the entire distribution of dispersal distances is calculated and not only the average. Within this framework, we can re-derive all previous results and formulae for uptake length, and we can include spatially heterogeneous stream environments. In addition, we propose a number of new characteristic quantities that can complement nutrient uptake length when evaluating the health of a stream system or the impact of a source of nutrients. We illustrate our method with two examples of spatially non-homogeneous streams: point-source input of nutrients (e.g., wastewater treatment plant) and diffuse lateral input (e.g., agricultural run-off), and we show how to measure the relative contribution of the two sources to the uptake length and other characteristics.
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References
Anderson KE, Nisbet RM, McCauley E (2008) The spatial-scale dependence of transient dynamics in streams and rivers. B Math Biol 70:1480–1502
Bencala KE, Walters RA (1983) Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model. Water Resour Res 19(3):718–724
Bouraoui F, Grizzetti B (2008) An integrated modelling framework to estimate the fate of nutrients: application to the Loire (France). Ecol Model 212:450–459
Carr GM, Morin A, Chambers, PA (2005) Bacteria and algae in stream periphyton along a nutrient gradient. Freshwater Biol 50:1337–1350
Chambers PA, Meissner R, Wrona FJ, Rupp H, Guhr H, Seeger J, Culp JM, Brua RB (2006) Changes in nutrient loading in an agricultural watershed and its effects on water quality and stream biota. Hydrobiologia 556:399–415
Earl SR, Valett HM, Webster JR (2006) Nitrogen saturation in stream ecosystems. Ecology 87(12):3140–3151
Earl SR, Valett HM, Webster JR (2007) Nitrogen spiraling in streams: comparisons between stable isotope tracer and nutrient addition experiments. Limnol Oceanogr 52(4):1718–1723
Ensign SH, Doyle MW (2006) Nutrient spiraling in streams and river networks. J Geophys Res 111:G04009
Gibson CA, Meyer JL (2007) Nutrient uptake in a large urban river. J Am Water Resour As 43(3):576–587
Green MB, Fritsen CH (2006) Spatial variation of nutrient balance in the Truckee River, California–Nevada. J Am Water Resour As 42(3):659–674
Grimm NB, Sheibley RW, Crenshaw CL, Dahm CN, Roach WJ, Zeglin LH (2005) N retention and transformation in urban streams. J N Am Benthol Soc 24(3):626–642
Grizzetti B, Bouraoui F, de Marsily G, Bidoglio G (2005) A statistical method for source apportionment of riverine nitrogen loads. J Hydrol 304:302–315
Gücker B, Brauns M, Pusch MT (2006) Effects of wastewater treatment plant discharge on ecosystem structure and function of lowland streams. J N Am Benthol Soc 25(2):313–329
Haggard BE, Stanley EH, Storm DE (2005) Nutrient retention in a point-source-enriched stream. J N Am Benthol Soc 24(1):29–47
Lutscher F, Pachepsky E, Lewis MA (2005) The effect of dispersal patterns on stream populations. SIAM Rev 47(4):749–772
Lutscher F, Lewis MA, McCauley E (2006) The effects of heterogeneity on population persistence and invasion in rivers. B Math Biol 68(8):2129–2160
Lutscher F, McCauley E, Lewis MA (2007) Spatial patterns and coexistence mechanisms in rivers. Theor Popul Biol 71(3):267–277
McNair JN, Newbold JD, Hart DD (1997) Turbulent transport of suspended particles and dispersing benthic organisms: how long to hit the bottom? J Theor Biol 188:29–52
Merseburger GC, Martí E, Sabater F (2005) Net changes in nutrient concentrations below a point source input in two streams draining catchments with contrasting land uses. Sci Total Environ 347:217–229
Meyer JL, Paul MJ, Taulbee WK (2005) Stream ecosystem function in urbanizing landscapes. J N Am Benthol Soc 24(3):602–612
Mulholland PJ, DeAngelis DL (2000) Surface-subsurface exchange and nutrient spiraling. In: Jones Jr JB, Mulholland PJ (eds) Streams and ground waters. Academic, New York, pp 149–166
Mulholland PJ, Helton AM, Poole GC, Hall RO, Hamilton SK, Peterson BJ, Tank JL, Ashkenas LR, Cooper LW, Dahm CN, Dodds WK, Findlay SEG, Gregory SV, Grimm NB, Johnson SL, McDowell WH, Meyer JL, Valett HM, Webster JR, Arango CP, Beaulieu JJ, Bernot MJ, Burgin AJ, Crenshaw CL, Johnson LT, Niederlehner BR, O’Brien JM, Potter JD, Sheibley RW, Sobota DJ, Thomas SM (2008) Stream denitrification across biomes and its response to anthropogenic nitrate loading. Nature 452:202–205
Muneepeerakul R, Bertuzzo E, Lynch HJ, Fagan WF, Rinaldo A, Rodriguez-Iturbe I (2008) Neutral metacommunity models predict fish diversity patterns in Mississippi–Missouri basin. Nature 453:220–222
Neal C, Jarvie HP, Love A, Neal M, Wickham H, Harman S (2008) Water quality along a river continuum subject to point and diffuse sources. J Hydrol 350:154–165
Neubert MG, Kot M, Lewis MA (1995) Dispersal and pattern formation in a discrete-time predator–prey model. Theor Popul Biol 48(1):7–43
Newbold JD, Elwood JW, O’Neill RV, Van Winkle W (1981) Measuring nutrient spiraling in streams. Can J Fish Aquat Sci 38:860–863
Newbold JD, O’Neill RV, Elwood JW, Van Winkle W (1982) Nutrient spiralling in streams: implications for nutrient limitation and invertebrate activity. Am Nat 120:628–652
Newbold JD, Elwood JW, O’Neill RV, Sheldon AL (1983) Phosphorus dynamics in a woodland stream ecosystem: a study of nutrient spiraling. Ecology 64(5):1249–1265
Newbold JD, Bott TL, Kaplan LA, Dow CL, Jackson JK, Aufdenkampe AK, Martin LA, Van Horn DJ, de Long AA (2006) Uptake of nutrients and organic C in streams in New York City drinking-water-supply watersheds. J N Am Benthol Soc 25(4):998–1017
Pachepsky E, Lutscher F, Nisbet R, Lewis MA (2005) Persistence, spread and the drift paradox. Theor Popul Biol 67:61–73
Payn RA, Webster JR, Mulholland PJ, Valett HM, Dodds WK (2005) Estimation of stream nutrient uptake from nutrient addition experiments. Limnol Oceanogr Methods 3:174–182
Runkel RL (2007) Toward a transport-based analysis of nutrient spiraling and uptake in streams. Limnol Oceanogr Methods 5:50–62
Speirs DC, Gurney WSC (2001) Population persistence in rivers and estuaries. Ecology 82(5):1219–1237
Tank JL, Rosi-Marshall EJ, Baker MA, Hall RO (2008) Are rivers just big streams? A pulse method to quantify nitrogen demand in a large river. Ecology 89(10):2935–2945
Vasilyeva O, Lutscher F (2012) Competition of three species in an advective environment. Nonlinear Anal R World Appl 13:1730–1748
Vis C, Hudon C, Carignan R, Gagnon P (2007) Spatial analysis of production by macrophytes, phytoplankton and epiphyton in a large river system, under different water level conditions. Ecosystems 10:293–310
Wakelin SA, Colloff MJ, Kookana RS (2008) Effect of wastewater treatment plant effluent on microbial function and community structure in the sediment of a freshwater stream with variable seasonal flow. Appl Environ Microb 74(9):2659–2668
Wollheim WM, Vörösmarty CJ, Peterson BJ, Seitzinger SP, Hopkinson CS (2006) Relationship between river size and nutrient removal. Geophys Res Lett 33:L06410
Acknowledgements
We thank Mark Lewis and Roger Nisbet for insightful discussions and two anonymous reviewers for their detailed comments. FL gratefully acknowledges support from an NSERC Discovery Grant and an Early Researcher Award from the Ontario Ministry of Research and Innovation. EMreceived support from the Canada Research Chairs program, theAlberta Water Research Institute, and Le Studium, Institute for Advanced Studies (France).
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Appendices
Appendix 1: Derivation of the uptake distribution
To make this paper self-contained, we briefly outline how to calculate the uptake distribution K(x) in the case of constant coefficients (11). We follow the exposition in Lutscher et al. (2005). We multiply Eq. 9 by λ and integrate with respect to time from 0 to ∞ . Then we obtain a second-order differential equation for K, namely
The characteristic equation is Dα 2 − uα − λ = 0, which has the two real roots
with α − < 0 < α + . Since K has to be bounded as x→± ∞ , we obtain
The value of the constant A is given by the condition that K integrates to unity, i.e.,
Appendix 2: Parameters and dimensions
We choose biological parameters as in Newbold et al. (1982) and hydrological parameters similar to Runkel (2007). The dimensions in our model differ from the one by Newbold et al. (1982) in that we consider linear densities whereas they considered nutrient density as grams per volume and microbe density as grams per area. To convert between the two approaches, we choose a stream width of 2 m and a depth of 0.25 m for a cross-sectional area of 0.5 m2. Then the biological parameters in Newbold et al. (1982) convert to the ones in Table 1. The total downstream nutrient flux, N, is also taken straight from Newbold et al. (1982).
For the hydrological parameters, we chose values similar to Runkel (2007), who, in turn, based his parameter values on previous work by Newbold et al. (1981). The range of dispersion explored by Runkel (2007) (between 0 and 2 m2 s − 1) is slightly larger than here. Choosing a very large value of D requires simulations over a large spatial extent in order to adequately capture the uptake distribution.
Appendix 3: Equations with storage exchange
The flow velocity in the water column typically decreases from surface to benthos. The advection–diffusion model replaces this velocity profile with some average flow speed. A slightly more detailed model considers a moving water zone coupled to a transient storage zone, in which the water velocity is zero. Nutrients move between these two zones by simple first-order decay kinetics. Such a model gives a much better fit to experimental data from conservative tracer experiments (Bencala and Walters 1983) with relatively minor increase in model complexity and number of parameters, see also Runkel (2007).
The storage exchange model derived by Mulholland and DeAngelis (2000) is based on discrete spatial patches, connected by water flow. The dynamics in each patch is given by a set of ordinary differential equations, describing nutrient uptake and recycling, and the dynamics between patches are simple linear exchange terms. When we take the continuum limit as the patch length decreases, we obtain the following system of partial differential equations for the nutrient concentration in the water and storage zone (C w , C s ) and the microbes in the water and storage zone (B w , B s ).
As was the case with the original series of papers by Newbold and coworkers, Mulholland and DeAngelis (2000) do not consider longitudinal dispersion. Also similar to the above, uptake length is independent of dispersion as long as the environment is homogeneous. Our calculations are not anymore complicated than in the absence of dispersion.
In this system of equations, the functions F s,w and G s,w are the nutrient uptake and release functions as before, with the subscript indicating that parameters may differ between the water and the storage zone. The parameters k s,w are the transfer rates of nutrients between the water and the storage zone. The rate k d stands for microbes settling from the water into the storage zone (the reverse process would be easy to incorporate into the model, but was not considered by Mulholland and DeAngelis (2000)). The parameter e stands for export from the system into long-term storage. It can be shown, however, that for e > 0, the only longitudinally homogeneous steady state of the system is zero; hence, we will only consider e = 0. (We speculate that since the results in Mulholland and DeAngelis (2000) are based on numerical simulations and since their value of e > 0 was very small, the effect could not be observed there.)
To find the steady-state values, we set the time derivatives on the left-hand side and the spatial derivatives on the right-hand side equal to zero. We obtain a system of algebraic equations, the solutions of which depend on the precise form of the uptake and release kinetics and shall not be explored further here.
The most interesting step is to calculate the uptake distribution. We obtain this distribution by considering the uptake equations with uptake rates λ s,w fixed by the steady-state values above. The uptake rates will, in general, be different for the water and storage compartment. The equations read
with initial conditions
describing the situation that a particle starts in the p w compartment at x = 0. The distribution of uptake locations is given by
K w and K s are the distributions of uptake locations in the water and storage compartment, respectively.
We multiply the first equation by λ w and the second by λ s and integrate with respect to time between 0 and ∞ to obtain the following equations for K w,s .
We can now multiply by x and integrate with respect to x to obtain the average uptake distance in water and storage, \(\hat x_w,\hat x_s\), respectively, as the solution of
As before, the term containing K′′ is zero after integration by parts. However, the integral of K w is not unity as was the case earlier for the integral over K. In fact, the quantity \(P_w=\int K_w(x) {\rm d}x,\) is the probability that the particle is taken up in the water compartment (rather than in the storage compartment). Analogously, we introduce \(P_s=\int K_s(x){\rm d}x.\) Additional equations for these two quantities can be obtained from integrating the equations for K s,w (Eqs. 40, 41) directly, which gives
From the last Eq. 45, we get
which we plug into the P w equation (Eq. 44) to obtain
We check that, indeed, P w + P s = 1. Putting this result into Eq. 42, we get
This is exactly the same system for the P s,w as we had above for the \(\hat x_{s,w}\) except for the left hand side. The solution is
From this we get the uptake length as
This is precisely the formula found by Mulholland and DeAngelis (2000), albeit by completely different means. Also, if we set k w = k s = 0, then we obtain the uptake length S upt = u/λ w , which corresponds to the initial formula (13) already given by Newbold et al. (1982).
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Lutscher, F., McCauley, E. A probabilistic framework for nutrient uptake length. Theor Ecol 6, 71–86 (2013). https://doi.org/10.1007/s12080-012-0161-5
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DOI: https://doi.org/10.1007/s12080-012-0161-5