Jerome J. Weis, Bradley J. Cardinale, Kenneth J. Forshay, and Anthony R. Ives. 2007. Effects of species diversity on community biomass production change over the course of succession. Ecology 88:929–939.

Appendix A. Two-staged Lotka-Volterra model for the additive and substitutive experiments.

In this appendix, we fit a modified version of the Lotka-Volterra model of competition to the data from both additive and substitutive experiments. Our objective is to demonstrate that the data can be well-fit by a simple model, and therefore that the mechanisms underlying the temporal changes in the relationship between diversity and biomass are correspondingly simple. By fitting the model to the data, we also obtain parameter estimates (with confidence intervals) for carrying capacities and competition coefficients. These reveal strong competitive asymmetries, in which the species with the lowest carrying capacity, Chlamydomonas, also was able to outcompete the other two species. This is the underlying mechanism leading to an inverse relationship between initial diversity and biomass in late succession.

The dynamics of algal species showed a rapid initial increase, possibly due to the physiological condition of the species at the initiation of the experiment, followed by more moderate growth in mid to late succession. To account for this observed “two-phase” population growth, we used a modified Lotka-Volterra model (Eq. A.1 below),

(A.1)

where b_i(t) is the biomass of species i (i = Selenastrum, Scenedesmus, or Chlamydamonas) on day t, coefficients give the competitive effect of species j on species i, and populations grow with intrinsic growth rate up to density m_i, and then grow according to standard Lotka-Volterra dynamics with intrinsic growth rate r_i and carrying capacity K_i. Thus, in the first phase, populations grow at maximum rate , and in the second phase they grow at maximum rate r_i.

We fit Eq. A.1 for each species using all of the data for that species, including all treatments in the additive and substitutive designs, and all replicates within treatments. Because the experiment was conducted under controlled conditions and involved large population sizes, we assumed that all of the observed variation within replicates was due to measurement error. Thus, we assumed that the only difference among replicates within a treatment (aside from measurement error) was due to the initial population biomass. The fitting procedure consisted of choosing initial parameter estimates and then iterating the model for 31 days. We did not include data from the first sampling date (day 2) because low densities and stochastic probabilities of observation led to high variation among replicates. Having computed the expected dynamics for a given set of parameter values, we calculated the errors between the predicted and observed log biomasses for each sample, and from these computed the sum-of-squared deviates. Repeating this for all replicates, we summed all sums-of-squares to give a global sums-of-squares (SS). The maximum likelihood (ML) parameter estimates are those parameter values that minimize the global SS. The global SS surface was not unimodal in the parameter values. Therefore, to minimize the global SS, we used simulated annealing (Kirkpatrick et al. 1983) followed by Newton-Raphson minimization to polish the estimates. Standard errors of the estimates were obtained by inverting the information matrix using standard asymptotic arguments (e.g., Judge et al. 1985, pp. 177–180). Parameter estimates are given in Table A1.

TABLE A1: Parameter estimates and approximate standard errors for the fitted model given by Eq. A 1.

Figure A1 illustrates the fit of the model to data for Selenastrum using treatments in the additive design. The two-phase population growth is visible especially for the case of Selenastrum alone (Fig. A1a), in which there is a sudden change in dynamics once the biomass exceeded log m = 10.67. (We selected Selenastrum to illustrate the fit of the model because it had especially clear two-phase dynamics.) The model was fit to each trajectory in the panels of Fig. A1 separately; therefore, the number of fitted curves equals the number of trajectories. Because there would be too many lines if we were to show both data and fitted curves in the same panel, instead we illustrate the fit of the model using a single trajectory from the fitted model starting from an average initial biomass (dashed lines). The fitted model captures the overall dynamics for Selenastrum, and the other algal species are similarly well-fit.

FIG. A1: Observed Selenastrum biomass in each of the four treatments following the additive design: (a) Selenastrum alone, (b) with Chlamydamonas(Ch), (c) with Scenedesmus (Sc), and (d) with both species. Solid lines in each panel correspond to replicates, and an example trajectory from the model given by Eq. A.1 is shown by the dashed red line.

The fitted model also captures the overall pattern of biomass vs. time. Fig. A2 gives the biomasses for 1-, 2-, and 3-species treatments through time for additive and substitutive designs simulated by the model. These are comparable to Fig. 2a,b in the main text. Both data and fitted model give a positive relationship between diversity and biomass at mid-succession, but a negative relationship at late succession.

FIG. A2: Simulated data from Eq. A.1 giving combined species biomasses for (a) additive and (b) substitutive designs. See Fig. 2 in the main text for comparison. Dotted lines give the average of the 1-species treatments (labeled 1), dashed lines give the average of the 2-species treatments (labeled 2), and solid lines give the 3-species treatment (labeled 3).

The parameter estimates in Table A1 underlie the observed changes in the diversity-biomass relationship through time. Selenastrum had a very high initial population growth rate () and grew to a high biomass (m) before it started to slow down. In contrast, Chlamydamonas grew very slowly to low densities (K), but was a very strong competitor on a per capita basis (a). Scenedesmus was intermediate between the other two species in growth rate, carrying capacity, and competitive hierarchy. The explanation for the temporal change in the diversity-biomass relationship is that Selenastrum and, to a lesser extent, Scenedesmus populations grew rapidly, so that they dominated biomass through mid-succession. Because they exerted relatively weak competition on Chlamydamonas, in mid-succession there was effectively resource partitioning among species. Therefore, at mid-succession the total biomass was relatively greater in the 3-species treatment, leading to the positive diversity-biomass relationship. However, as Chlamydamonas populations grew further in the mixed-species treatments, its strong competitive effect exerted itself, causing plateaus or declines in Selenastrum and/or Scenedesmus biomass. Since Chlamydamonas had a low carry capacity, this suppressed the total biomass in the 3-species treatments, and the 2-species treatments in which Chlamydamonas occurred. This led to the negative diversity-biomass relationship at late-succession. In summary, the temporal change in the diversity-biomass relationship was caused by differences in population growth rates, carrying capacities, and competitive abilities; initially the 3-species treatment was dominated by rapidly growing, high biomass, and weakly competitive species (Scenedesmus and Selenastrum), but these were later dominated by a slower growing, highly competitive species (Chlamydamonas) that did not reach high biomass.

LITERATURE CITED

Judge, G. G., W. E. Griffiths, R. C. Hill, H. Lutkepohl, and T.-C. Lee. 1985. The theory and practice of econometrics, Second edition. John Wiley and Sons, New York, New York, USA.

Kirkpatrick, S., C. D. Gelatt, and M. P. Vecchi. 1983. Optimization by simulated annealing. Science 220:671–680.