Ecological Archives E088-073-A2

Spencer R. Hall, Mathew A. Leibold, David A. Lytle, and V. H. Smith. 2007. Grazers, producer stoichiometry, and the light : nutrient hypothesis revisited. Ecology 88:1142–1152.

Appendix B. Description of model system.

In this Appendix, we describe the model that we simulated to generate Fig. 3 of the main text. It turns out one cannot fully solve for the equilibria of this model system analytically. Therefore, the patterns shown in Fig. 3 were produced using a standard, adaptive-step integrator (Mathworks 1999).

This system of ordinary differential equations closely resembles similar models (Hessen and Bjerkeng 1997, Klausmeier et al. 2004, Hall et al. 2005, 2006). It represents change in algal biomass (A), nutrient quota (Qj) of phosphorus (P) and nitrogen (N), grazer biomass (G), and freely available N or P (Rj), as the balance of gains and losses (see also Table B1):



where j = N, P. The balance equation for algal growth (Eq. B.1a) indicates that per capita productivity is a function of the nitrogen and phosphorus quota, , and degree of light limitation, B. This nutrient quota function follows the standard Droop formulation (Grover 1997, Klausmeier et al. 2004):


where min(…) is the minimum of the arguments, is the physiologically maximal growth rate, and kQ,j is the minimal quota of nutrient j. In the Droop model, the plant’s birth rate increases with nutrient quota of the plant in a saturating, hyperbolic fashion. The minimum function reflects the general belief that plant growth can be limited by only one nutrient at a time (i.e., Leibig’s Law of the Minimum; cf. Rhee 1974, 1978, and Klausmeier et al. 2004). The true limiting nutrient had the smallest value of (kQ,j / Qj).

Meanwhile, the degree of light limitation (B, Hall 2004, Hall et al. 2006) follows Huisman and Weissing’s (1995) model:


which represents light (L) as a uni-directionally supplied resource. If light attenuates with depth due to self-shading by algae (A) and background sources, the integral tracks attenuation of incoming light, Lin, as it passes through a water column at depth s, L(s), following Lambert-Beer’s law:




Here, incoming light is extinguished exponentially with depth at rate k(A), which depends upon the density of algae (A), the per unit absorption of light by algae (kA), and absorption of light by background sources (kbg). The 1/z term averages the light environment over the water column depth, z, and the solution to this integral is contained on the right hand side of the equation. Equation B.5 was used to estimate the parameters kA and kbg from tanks that lacked crustacean grazers (Fig. B1).

Change in nutrient quota, Qj, (Eq. B.1b) follows the classic Droop (1968) model and reflects the balance between nutrient uptake and “dilution by growth” (Grover 1997). The nutrient uptake portion follows the Monod formulation, where vj is the maximal uptake rate and hj is the half-saturation constant for the nutrient, Rj. The “dilution by growth” portion is simply the per capita production rate of the producer, , times nutrient quota.

Grazer growth rate (Eq. B.1c) depends upon the balance between conversion of consumed plants into new grazer biomass and losses at rate d. (Here, we follow Hessen and Bjerkeng [1997] and ignore respiration rate; see Hall 2004 for a discussion of this choice). Realized conversion efficiency (eR) depends upon which resource limits growth of grazing – carbon, phosphorus, or nitrogen. This realized efficiency follows the function:


where the function min(…) is the minimum of its arguments, Qj is the quota of the producer for nutrient j, and qj is the grazer’s quota for nutrient j. A nutrient becomes more limiting as the plant quota for that nutrient decreases relative to grazer’s quota, or as Qj/qj becomes smaller. The first part of function eR selects the resource most limiting to grazers (i.e., the one with lowest Qj/qj or 1, whichever is smaller). The second case adjusts that conversion efficiency according to maximal conversion efficiency of nutrients (eN,P) and carbon (eC). If carbon limits grazing, eR simplifies to eC; otherwise, conversion drops as Qj/qj drops. See Hessen and Bjerkeng (1997) for full derivation of this Eq. and for variations on it.

Finally, change in availability of free nutrients (Eq. B.1d) follows a balance between gains and losses from several sources. First, nutrient j, with concentration Rj, is inputted at concentration Sj and lost due to dilution (both at rate a). Producers take up nutrients (with Monod kinetics), but nutrients contained in dead producers are instantaneously recycled (at rate mQjA). Finally, grazers contribute to the free nutrient pool via recycling of food (at rate ρj), and dead grazers are instantaneously recycled (released at rate dqj nutrients per unit carbon). The recycling term also follows the derivation by Hessen and Bjerkeng (1997):


where recycling of consumed food (fA) depends upon: nutrient content of that food (Qj); realized conversion efficiency of the food (eR), which itself depends upon the resource limiting grazing (see Eq. B.6); and the minimum of the nutrient quotas for plants (Qj) and grazers (qj). Importantly, nutrient recycling approaches but never drops below zero when Qj << qj; thus, this function assures us that grazers never directly uptake nutrients from the freely available pool.

TABLE B1. Variables and parameters in the stoichiometrically explicit, plant-grazer model.





State Variables



mg C·m-3

Algal carbon (biomass)


mg C·m-3

Grazer carbon (biomass)


mg·(mg C)-1

Cell quota (content) of algae, nutrient j


mg ·m-3

Dissolved concentration of nutrient j




Resource Supply Parameters



μmol photons·m-2·s-1

Incident light supply



μmol photons·m-2·s-1

Light intensity at depth s




Total supply of nutrient j

5-300 P,

N at 5:1 & 50:1

Other Parameters



day -1

Dilution rate of chemostat


μmol photons·m-2·s-1

Half-saturation constant for light



Degree of light limitation



Mortality rate

0.15 b


Maximal conversion efficiency, carbon

0.6 e


Maximal conversion efficiency, nutrient

1.0 e


Realized conversion efficiency



Half saturation constant, nutrient j

0.7, 5 *,b



Light extinction coefficient, background

0.38 d


m2·(mg C)-1

Light extinction coefficient, algae

2.5 x 10-4 d


mg·(mg C)-1

Minimum absolute quota of algae, nutrient j

0.004, 0.031 *,b


day -1

Density-independent losses of algae

0.05 b


mg·(mg C)-1

Nutrient content, grazer

0.03, 0.19 *,b


day -1

Growth rate at infinite quota



mg·(mg C)-1·day-1

Maximum uptake rate of algae for nutrient j

0.04, 0.122 *,c


mg·(mg C)-1·day-1

Recycling rate, grazers

* Parameters listed for phosphorus (P), then nitrogen (N).

a Source: Huisman et al. (1999).

b Source: Andersen (1997).

c Calculation assumes steady-state conditions (Grover 1997).

d Estimated from this study (see Fig. B1).

e From Hessen and Bjerkeng (1997).


   FIG. B1. Relationship between algal carbon, A, and empirically-estimated light extinction coefficient, k(A) in experimental mesocosms that lacked crustacean grazers. Following Eq. B5, we fit the model k(A) = kA A + kbg + ε, where εs are normally-distributed errors. Parameter estimates are included in Table B1.



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