
Resource Competition Between AlgaeFrom MathBioContentsSummaryIn this dogeatdog 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 LotkaVolterra 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 ModelIn 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 MichaelisMenten theory of enzyme kinetics, and was applied to algal cultures in a continuous flow system. The two important parameters are , population size, and , nutrient concentration. For i of n species with j of m potentially limiting resources, changes in N and S are calculated as:
where = maximum growth rate for species i = half saturation constant for species i = yield of species i limited by resource j (in cells/unit of resource j) = number of cells of species i per unit volume = concentration of resource j external to the cells = influent concentration of resource j = steadystate growth rate under continuous flow (a.k.a. true dilution rate): , in which = flow rate
. Thus,
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: with the new parameters = internal concentration of nutrient j for species i at which growth ceases = maximal uptake rate of nutrient j by species i (uM cell^{1} hr^{1}) = concentration of nutrient j per cell of species i
ResultsThe steady state predictions were tested experimentally using laboratory cultures and data collected from Lake Michigan. 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 DynamicsTilman's model focused on the final outcome of competition by examining steadystate conditions, but how do the model dynamics match the experimental time courses? The model was simulated in R (code) for comparison.
As expected, P is quickly depleted, indicating P limitation at high Si/P ratios.
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 (1.443.94 uM) are over 10 times larger than for phosphate (0.020.25 uM), indicating that the uptake rate for silica is slower.
3 species systemsMost 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? A third diatom species, F. crotonensis, was added to the model (code). Values for the parameters and were obtained from previous studies, and the Si/P ratio at the boundary between Si and P limitation was altered by changing the parameter values. Based on the Si/P ratios, three scenarios were defined:
Scenario 1Scenario 2Scenario 3These 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
References and Further Reading
