Ecological Archives M076-004-A1

Jutta Passarge, Suzanne Hol, Marieke Escher, and Jef Huisman. 2006. Competition for nutrients and light: stable coexistence, alternative stable states, or competitive exclusion? Ecological Monographs 76:57–72.

Appendix A. The competition model.

In this Appendix, we give a detailed description of the competition model. We first consider competition for phosphorus. Next, we consider competition for light. Finally, we combine the phosphorus-based and light-based approach to complete the full competition model.

 

Competition for phosphorus

To account for the ability of many phytoplankton species to store substantial amounts of phosphorus inside their cells, we use a variable-internal-stores model (Droop 1973, Grover 1991, Ducobu et al. 1998, Sterner and Elser 2002). More specifically, we assume that the specific growth rate of species i, mR,i(Qi), increases with its intracellular phosphorus content, Qi, according to the Droop equation (Droop 1973):

(A.1)

where mmax,i is the maximal specific growth rate of species i, and Qimin is the minimal intracellular phosphorus content required for growth of species i.

We further assume that the phosphorus uptake rate of species i, vi(R,Qi), increases with the external phosphorus concentration, R, according to Michaelis-Menten kinetics, but decreases with its intracellular phosphorus content, Qi, as described by Morel (1987) and Ducobu et al. (1998):

(A.2)

where vmax,i is the maximal phosphorus uptake rate of species i, HR,i is its half-saturation constant for phosphorus uptake, and Qimax is its maximal capacity for intracellular phosphorus storage.

To model competition for phosphorus, we consider a total number of n phytoplankton species. Let Ni denote the population density of a species i. The population dynamics of the phytoplankton species depend on their growth rates, as described by the Droop equation (Eq.1), and their losses by dilution. The dynamics of the intracellular phosphorus contents depend on phosphorus uptake (Eq.2), and the spread of consumed phosphorus through population growth. The dynamics of the phosphorus concentration in the water column depend on the phosphorus supply, losses of phosphorus from the system, and phosphorus uptake by the phytoplankton. The competition model thus reads:

i = 1,...,n
(A.3a)
   
i = 1,...,n
(A.3b)
   
(A.3c)
 

where mR,i(Qi) and vi(R,Qi) are defined by Eq.1 and Eq.2, respectively, D is the dilution rate, and Rin is the phosphorus concentration in the inflow.

To analyze the model, we first consider a monoculture of species i. In a constant environment, the monoculture develops towards a steady state. Substituting Droop’s equation (Eq.1) in Eq.3a, and solving for equilibrium, shows that the steady-state intracellular phosphorus content of species i settles at:

(A.4)

The external phosphorus concentration in a steady-state monoculture of species i can be obtained by substituting Eq.4 in Eq.3b, and solving for equilibrium. This yields the R* value of species i:

(A.5)

Finally, the steady-state population density of species i can be calculated from Eq.4 and Eq.5 as:

(A.6)

In a constant environment, the predictions of the variable-internal-stores model are similar to the predictions of classic resource competition theory (Armstrong and McGehee 1980, Tilman 1982). That is, if multiple species compete for phosphorus, the species with the lowest R* value will reduce the external phosphorus concentration below the critical phosphorus requirements of all other species. As a result, the species with the lowest R* value wins (Grover 1991, Smith and Waltman 1994).

 

Competition for light

Competition for light is mediated by shading between species. Our model of competition for light describes how changes in the vertical light gradient, caused by shading, affect the growth rates of the different phytoplankton species (Huisman and Weissing 1994, Huisman et al. 1999). We consider a well-mixed water column, in which all phytoplankton species are homogeneously distributed. Vertical positions in the water column are indicated by the depth z, where z runs from z=0 at the surface to z=zm at the bottom of the water column. Let I(z) denote the light intensity at depth z. We assume that the specific growth rate of species i under light limitation, mI,i, can be calculated as the depth average of its production rate:

(A.7)

where the nested notation pi(I(z)) indicates the specific production rate of species i as a function pi of the light intensity I at depth z. We assume that the specific production rate of species i is an increasing saturating function of light intensity, described by Monod’s (1950) equation:

,
(A.8)

where mmax,i is the maximal specific growth rate of species i, and HI,i is the half-saturation constant of light-limited growth of species i.

The vertical light gradient depends on the incident light intensity at the water surface, as well as on light absorption by the phytoplankton species and by many other substances in the water column (e.g., water itself, dissolved organic matter, clay particles). More precisely, according to Lambert-Beer’s law, the vertical light gradient is given by:

(A.9)

where Iin is the incident light intensity, Kbg is the background turbidity caused by all non-phytoplankton components, kj is the specific light extinction coefficient of phytoplankton species j, and Nj is the population density of species j. We note, from Eq.9, that the vertical light gradient is dynamic. An increase in the population densities of the phytoplankton species will result in a more turbid water column.

Using the Monod equation (Eq.8) and Lambert-Beer’s Law (Eq.9), the integral term in Eq.7 can be solved analytically. Accordingly, the specific growth rate can be written as (Huisman and Weissing 1994):

(A.10)

It is useful to interpret the specific growth rate in terms of the light intensity penetrating through the water column. That is, let Iout denote the light intensity at the bottom of the water column (at z=zm). Hence, in view of Lambert-Beer’s law, the specific growth rate can be written as a function of Iout:

(A.11)

To model competition for light, we consider again a total number of n phytoplankton species. The population dynamics of the phytoplankton species depend on their light-limited growth rates (Eq.11), and their losses by dilution. Light penetration through the water column is described by Lambert-Beer’s law. The competition model thus reads:

i = 1,...,n
(A.12a)
   
.
(A.12b)

where mI,i(Iout) is defined by Eq.11, and D is the dilution rate.

The model predicts that, in monoculture, a phytoplankton species continues to increase until it has reduced the light intensity penetrating through the water column to its own critical light intensity (Iout*). Unfortunately, there is no simple analytical expression for the critical light intensity. Instead, the critical light intensity can be calculated by numerical simulation of Eqs.12; it corresponds to the value of Iout that is reached at steady state. The steady-state population density of a species can be calculated from its critical light intensity, using Lambert-Beer’s law (Huisman 1999):

(A.13)

We note from this equation that, all else being equal, the steady-state population density is inversely proportional to the depth of the water column (Huisman 1999).

If multiple species compete for light in a constant environment, the model predicts that the species with the lowest critical light intensity will reduce the light intensity penetrating to the bottom of the water column below the critical light intensities of all other species. As a result, the species with lowest critical light intensity wins (Huisman and Weissing 1994, Weissing and Huisman 1994, Huisman et al. 1999).

 

Competition for phosphorus and light

To complete our model, we combine the model of competition for phosphorus with the model of competition for light. Because pilot experiments revealed a rather abrupt shift from phosphorus-limited to light-limited physiology in response to a slight increase in the phosphorus load, we describe the transition from phosphorus limitation to light limitation by Von Liebig’s (1840) ‘Law of the Minimum’.

Hence, the complete competition model reads:

i = 1,...,n
(A.14a)
   
i = 1,...,n
(A.14b)
   
(A.14c)
   
.
(A.14d)

where mR,i(Qi), mI,i(Iout), and vi(R,Qi) are defined by Eq.1, Eq.11, and Eq.2, respectively, and min is the minimum function.

The predictions of this competition model are analogous to the predictions of classic competition theory for two limiting resources (Tilman 1982). Each species has its own critical phosphorus requirements (R*) and its own critical light intensity (Iout*). At one extreme, if the growth rates of all species become limited by phosphorus, the species with lowest R* for phosphorus wins. At the other extreme, if the growth rates of all species become limited by light, the species with lowest critical light intensity wins. The model also predicts an intermediate region, where some species become limited by phosphorus while other species become limited by light. The exact size and location of this intermediate region in parameter space depend on the phosphorus and light requirements of the species, and also on environmental factors like the phosphorus supply, light supply, dilution rate, background turbidity, and mixing depth (Huisman and Weissing 1995).

In this intermediate region, the model predicts three contrasting hypotheses (Tilman 1982, Huisman and Weissing 1995; see Fig. 1 in the main paper):

H1: If there are no trade-offs between the competitive abilities for phosphorus and light, such that the species with the lowest R* value also has the lowest critical light intensity, the model predicts competitive exclusion. The superior competitor becomes dominant.

H2: If there are trade-offs between the competitive abilities for phosphorus and light, and the superior phosphorus competitor absorbs relatively more light while the superior light competitor consumes relatively more phosphorus, the model predicts stable coexistence. The superior phosphorus competitor and the superior light competitor share dominance.

H3: If there are trade-offs between competitive abilities for phosphorus and light, and the superior phosphorus competitor consumes relatively more phosphorus while the superior light competitor absorbs relatively more light, the model predicts two alternative stable states. Depending on the initial conditions, either the superior phosphorus competitor or the superior light competitor becomes dominant.

 

LITERATURE CITED

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