September 20, 2017, Wednesday
University of Colorado at Boulder Search A to Z Campus Map University of Colorado at Boulder CU 
Search Links

Resource Competition Between Algae

From MathBio

Jump to: navigation, search


In this dog-eat-dog world, a species’ distribution is often impacted by its ability to compete for limited resources. In diverse communities, describing the fate of competition for individual species quickly spirals out of control. Dr. David Tilman tested the ability of 2 models – the Monod model and the Variable Internal Stores model – to predict the outcome of competitive interactions between 2 algal species in his paper “Resource competition between planktonic algae: an experimental and theoretical approach” [1]. Unlike Lotka-Volterra type competition models, these models assume that incorporating physiological information for each species can increase the ability to predict the outcome of competitive interactions resulting from nutrient limitation.

The models predicted that two species can coexist when they are limited by different resources, but one species will inevitably overtake the other when it is superior at competing for the same resource. Laboratory studies and observations of changes in algal dominance along a nutrient gradient in Lake Michigan closely matched model predictions. Thus, Tilman was able to accurately predict the outcome of competitive interactions using either model, although the Monod model did a better job and included fewer parameters.

Mathematical Model

In this study, two resource competition models were compared. Both models are mechanistic in that they explicitly define the concentrations of resources available as well as empirical data on growth and resource utilization for the competing species. Since the goal of this study was to determine the outcome of competitive interactions, Tilman was interested in steady state conditions.

Model I (Monod Model)

This model is based on the Michaelis-Menten theory of enzyme kinetics, and was applied to algal cultures in a continuous flow system. The two important parameters are N, population size, and S, nutrient concentration.

For i of n species with j of m potentially limiting resources, changes in N and S are calculated as:

dN_{i}/dt=N_{i}*\left({\begin{array}{cc}MIN\\1<j<m\end{array}}\left[{\frac  {r_{i}S_{j}}{K_{{ij}}+S_{j}}}-D\right]\right)

dS_{j}/dt=D(_{0}S_{j}-S_{j})-\sum _{{i=1}}^{{n}}{\frac  {N_{i}r_{i}S_{j}}{(K_{{ij}}+S_{j})Y_{{ij}}}}


r_{i} = maximum growth rate for species i

K_{{ij}} = half saturation constant for species i

Y_{{ij}} = yield of species i limited by resource j (in cells/unit of resource j)

N_{i} = number of cells of species i per unit volume

S_{j} = concentration of resource j external to the cells

oS_{j} = influent concentration of resource j

D = steady-state growth rate under continuous flow (a.k.a. true dilution rate): D=ln(1/(1-f)), in which f= flow rate

  • Equation 1 assumes that the growth rate of a species is determined by the most limiting nutrient. Steady state equations were used in order to determine the boundary between silica and phosphorus limitation,
  1. N_{{i(j)}}^{*}=Y_{{ij}}(_{o}S_{j}-S_{j}^{*})
  2. S_{j}^{*}={\frac  {DK_{{ij}}}{r_{i}-D}}
  • The boundary between the growth rate of species i at the transition between limitation by resources 1 and 2 is defined as,




{\frac  {Y_{{i1}}}{Y_{{i2}}}}={\frac  {(_{o}S_{2}-S_{2}^{*})}{(_{o}S_{1}-S_{1}^{*})}}

  • The boundary between silica and phosphate limitation therefore is indicated by the ratio of the concentrations of these two limiting nutrients, which can be represented graphically:
Under steady state conditions, the Monod Model predicts Si limitation for Si/P ratios below 90 and 6 for A. formosa and C. meneghiniana, respectively, and P limitation above those ratios. When the Si/P ratio is between these values, the competing species are limited by different resources, which eases competition and allows for coexistence.

Model II (Variable Internal Stores Model)

The VIS model assumes that nutrient concentrations within the cell determine growth rates. Likewise, internal nutrient concentration is controlled by uptake rate and growth:

  1. {\frac  {dN_{i}}{dt}}=N_{i}*{\begin{array}{cc}MIN\\1<j<m\end{array}}[r_{i}(1-{\frac  {g_{{ij}}}{Q_{{ij}}}})-D]
  2. {\frac  {dQ_{{ij}}}{dt}}=V_{{ij}}\left({\frac  {S_{j}}{S_{j}+k_{{ij}}}}\right)-r_{i}(Q_{{ij}}-g_{{ij}})
  3. dS_{j}/dt=D(_{0}S_{j}-S_{j})-\sum _{{i=1,n}}\left[N_{i}V_{{ij}}({\frac  {S_{j}}{S_{j}+k_{{ij}}}})\right]

with the new parameters

g_{{ij}} = internal concentration of nutrient j for species i at which growth ceases

V_{{ij}} = maximal uptake rate of nutrient j by species i (uM cell^{-1} hr^{-1})

Q_{{ij}} = concentration of nutrient j per cell of species i

Solving for steady state conditions,

  1. N_{{i(j)}}^{*}=(_{o}S_{j}-S_{j})/Q_{{ij}}^{*}
  2. Q_{{i(j)}}^{*}=g_{{ij}}r_{i}/(r_{i}-D)
  3. S_{j}^{*}={\frac  {r_{i}g_{{ij}}k_{{ij}}D}{(V_{{ij}}(r_{i}-D)-r_{i}g_{{ij}}D)}}


The steady state predictions were tested experimentally using laboratory cultures and data collected from Lake Michigan.

The time course of competition experiments performed on the bench are shown. For each experiment, the Si/P ratio was changed in order to test the model predictions. Note that under low Si/P ratios C. meneghiniana is more abundant, but A. formosa is not excluded, which is not expected based on the steady-state conditions.
The shifts in diatom dominance in Lake Michigan could also be fit with the Monod equation.

These results support the existence of competitive interactions driven by changes in the ratios of limiting nutrients, with the Monod Model explaining over 80% of the variation in the abundance of C. meneghiniana.

Model Dynamics

Tilman's model focused on the final outcome of competition by examining steady-state conditions, but how do the model dynamics match the experimental time courses? The model was simulated in R (code) for comparison.

  • When Si/P ratios are high, A. formosa quickly dominates, driving C. meneghiniana out of the system.


As expected, P is quickly depleted, indicating P limitation at high Si/P ratios.

2speciesSi100.jpg 2speciesP100.jpg

  • Co-existence was quickly established at a Si/P ratio between 90 and 6:


  • Below the threshold for Si limitation, C.meneghiniana becomes dominant. However, it does not drive A. formosa out of the system, which was predicted by the steady state conditions. This is likely due to the slower kinetics for Si uptake. If the experiment was carried out for longer than 50 days (and if nature would be stable enough for anything close to that amount of time), C. meneghiniana should eventually exclude A. formosa.


2speciesSi0.2.jpg 2speciesP0.2.jpg

While Si and P quickly become depleted at higher Si/P ratios, at low ratios it takes a long time for Si to reach a minimum. The 1/2 saturation values for silica (1.44-3.94 uM) are over 10 times larger than for phosphate (0.02-0.25 uM), indicating that the uptake rate for silica is slower.

3 species systems

Most natural systems comprise many dozens, if not hundreds, of species that could potentially interact. What happens in this simplified system if a third algal species is introduced?

F. crotonensis. Source:

A third diatom species, F. crotonensis, was added to the model (code). Values for the parameters K and r were obtained from previous studies, and the Si/P ratio at the boundary between Si and P limitation was altered by changing the Y parameter values.

Based on the Si/P ratios, three scenarios were defined:

  1. Si/P = 120 (superior competitor for P)
  2. Si/P = 50 (intermediate competitor for both Si and P)
  3. Si/P = 1 (superior competitor for Si)

Scenario 1

Boundaries between Si and P limitation for 3 species.

3species1N200.jpg 3species1N50.jpg

3species1N10.jpg 3species1N0.1.jpg

Scenario 2

Boundaries between Si and P limitation for 3 species.

3species2N200.jpg 3species2N50.jpg

3species2N10.jpg 3species2N0.1.jpg

Scenario 3

Boundaries between Si and P limitation for 3 species.

3species3N200.jpg 3species3N50.jpg

3species3N10.jpg 3species3N0.1.jpg

These results show that when a third species is introduced to the system, the outcomes become much less clear. Specifically, the growth rate becomes more important than simply the competitive ability for a particular resource. Later studies have also shown that algal consortia actually show chaotic dynamics, with steady state conditions rarely being achieved.

R code and Presentation

  • 2 species competition model can be found here
  • 3 species competition model can be found here

References and Further Reading

  1. Tilman, D. " Resource competition between planktonic algae: an experimental and theoretical approach" Ecology, 1977, 58:338-348: