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A probabilistic framework for nutrient uptake length

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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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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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Correspondence to Frithjof Lutscher.

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

$$ -\delta(x)=DK^{\prime\prime}-uK^{\prime}-\lambda K. $$
(28)

The characteristic equation is 2 −  − λ = 0, which has the two real roots

$$ \alpha^\pm=\frac{u}{2D}\pm\sqrt{\frac{u^2}{4D^2}+\frac{\lambda}{D}}, $$
(29)

with α − < 0 < α  + . Since K has to be bounded as x→± ∞ , we obtain

$$ K(x) = A\exp(\alpha^+ x), x\leq 0,\quad K(x)=A\exp(\alpha^- x), x\geq 0. $$
(30)

The value of the constant A is given by the condition that K integrates to unity, i.e.,

$$ \int_{-\infty}^\infty K(x){\rm d}x=1,\quad {\rm or}\quad A=\frac{\lambda}{\sqrt{u^2+4D\lambda}}. $$
(31)

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 ).

$$ \begin{array}{rll} \frac{\partial C_w}{\partial t}&=&D_{\rm{c}} \frac{\partial^2 C_w}{\partial x^2} -u \frac{\partial C_w}{\partial x} - k_w C_w + k_s C_s \\ &&-F_w(C_w,B_w)+G_w(B_w), \end{array} $$
(32)
$$ \begin{array}{rll} \frac{\partial B_w}{\partial t}&=&D_{\rm{b}}\frac{\partial^2 B_w}{\partial x^2} -u \frac{\partial B_w}{\partial x} +F_w(C_w,B_w) \\ && -G_w(B_w)-k_{d} B_w, \end{array} $$
(33)
$$ \frac{\partial C_s}{\partial t} = k_w C_w - k_s C_s - F_s(C_s,B_s)+G_s(B_s), $$
(34)
$$ \frac{\partial B_s}{\partial t} = F_s(C_s,B_s)-G_s(B_s) -e B_s + k_{\rm{d}} B_w. $$
(35)

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

$$ \frac{\partial p_w}{\partial t} = D_{\rm c} \frac{\partial^2 p_w}{\partial x^2}-u \frac{\partial p_w}{\partial x} - k_w p_w + k_s p_s -\lambda_w p_w, $$
(36)
$$ \frac{\partial p_s}{\partial t} = k_w p_w - k_s p_s -\lambda_s p_s, $$
(37)

with initial conditions

$$ p_w(0,x)=\delta(x),\qquad p_s(0,x)=0, $$
(38)

describing the situation that a particle starts in the p w compartment at x = 0. The distribution of uptake locations is given by

$$ K(x)=\int_0^\infty \lambda_w p_w(t,x)+\lambda_s p_s(t,x) {\rm d}t =K_w(x)+K_s(x). $$
(39)

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 .

$$ \begin{array}{rll} -\lambda_w \delta(x)&=& D_{\rm c} K_w^{\prime\prime}(x)-u K_w^{\prime}(x) - k_w K_w(x) \\ && + \, k_s \frac{\lambda_w}{\lambda_s} K_s(x) -\lambda_w K_w(x), \end{array} $$
(40)
$$ 0 = k_w \frac{\lambda_s}{\lambda_w} K_w(x) - k_s K_s(x) -\lambda_s K_s(x), $$
(41)

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

$$ 0 = -u \int K_w(x) {\rm d}x - k_w \hat x_w+ k_s \frac{\lambda_w}{\lambda_s} \hat x_s -\lambda_w \hat x_w, $$
(42)
$$ 0 = k_w \frac{\lambda_s}{\lambda_w} \hat x_w - k_s \hat x_s-\lambda_s \hat x_s. $$
(43)

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

$$ \lambda_w = - k_w P_w + k_s \frac{\lambda_w}{\lambda_s} P_s -\lambda_w P_w, $$
(44)
$$ 0 = k_w \frac{\lambda_s}{\lambda_w} P_w - k_s P_s -\lambda_s P_s, $$
(45)

From the last Eq. 45, we get

$$ P_s=\frac{\frac{\lambda_s}{\lambda_w}k_w}{\lambda_s+k_s} P_w, $$
(46)

which we plug into the P w equation (Eq. 44) to obtain

$$ P_w=\frac{\lambda_w}{\lambda_w+k_w\frac{\lambda_s}{\lambda_s+k_s}}. $$
(47)

We check that, indeed, P w  + P s  = 1. Putting this result into Eq. 42, we get

$$ u P_w = - k_w \hat x_w+ k_s \frac{\lambda_w}{\lambda_s} \hat x_s -\lambda_w \hat x_w, $$
(48)
$$ 0 = k_w \frac{\lambda_s}{\lambda_w} \hat x_w - k_s \hat x_s-\lambda_s \hat x_s. $$
(49)

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

$$ \hat x_w=\frac{u}{\lambda_w+k_w\frac{\lambda_s}{\lambda_s+k_s}}P_w, \quad \hat x_s=\frac{u}{\lambda_w+k_w\frac{\lambda_s}{\lambda_s+k_s}}P_s. $$
(50)

From this we get the uptake length as

$$ S_{\rm upt}=\hat x_w+\hat x_s=\frac{u}{\lambda_w+k_w\frac{\lambda_s}{\lambda_s+k_s}}. $$
(51)

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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