Supplementary Information: Takehito Yoshida et al., Rapid evolution drives ecological dynamics in a predator-prey system. Nature (2003).
The single-clone model
We begin with the model of Fussmann et al. (2000) which describes the dynamics of a chemostat predator-prey (Brachionus calyciflorus - Chlorella vulgaris) system.As it does not consider the within-population clonal dynamics of Chlorella, it can be considered a single-clone model:
(1)
Model parameters are defined in Supplementary Table 1.Here n= NV is nitrogen (mole per chemostat), and c is Chlorella cell counts (109 cells per chemostat). Similarly, b and r are Brachionus (total population and fertile population, respectively) individuals per chemostat. The parameters [(1) and Table 1] are conversions between consumption and recruitment, and do not appear in Fussmann et al (2000), because there the organism state variables (c,b,r) were expressed in units of total nitrogen rather than cell counts.
Consumption rates for Chlorella and Brachionus are described by the following functional responses. Fc is moles N consumed (per day) by 109Chlorella cells:
, where N=n/V (2)
Rotifer feeding rate, Fb , is derived from the rotifer clearance rate
), where C=c/V (3)
The form of equation (3), and the estimates of the parameter values, are derived from our own measurements of rotifer feeding rate in trials at a range of algal concentrations spanning those observed in the chemostat experiments (T. Yoshida, unpublished data). Multiplying the clearance rate by the algal concentration, C, gives the ingestionrate in units of 109 cells/rotifer/day:
. (4)
The multiple clone model: tradeoff between food value and competitive fitness
We then extend the above model, introducing clonal selection by specifying a set of competing clones in terms of their food value, p,and related competitive ability. The defensive "low food value" trait comes at the cost of reduced fitness, and this relationship may be specified by a tradeoff curve. Our experiments suggest that the cost of defense is only evident when nutrients are scarce, so we assume a tradeoff between food value and the half-saturation constant for uptake by the algae of the limiting nutrient, Kc. In the interest of simplicity, we use a two-parameter tradeoff curve (shape parameter , and scale or “cost” parameter ) to describe how Kcvaries between p0 and p=1. The tradeoff curve is
where .
(for "technical typesetting reasons" appears as α' and appears as α in the published paper).
This yields, in terms of the food value of the ith clone,
(5)
Note that Kc attains its minimum value at p=1(thus algal growth rate is maximumat p=1), so reducing a clone’s food value also effectively reduces its competitive fitness. Examples of trade-off curves are found in Yoshida et al. (2003, Figure 1a and 1c). The algal functional response (2) then becomes:
(6)
When the tradeoff curve is concave down(ibid., Figure 1a), it is more likely that the surviving co-existing clones are two extremes in food value. However, where the tradeoff curve is concave up(Figure 1c), intermediate (in terms of food value) algal types are most likely to persist. Note that for our simulations we set a minimum food value, pmin, above which food values were randomly chosen, since food values of caused rotifer populations to collapse immediately (see Table 1).
Where the original model (Fussmann et al. 2000) assumes that rotifer clearance rate (3) is a function of total algal density (i.e., that rotifers have non-discriminating palates), in our extended clonal model we assume that clearance rate is affected by the relative abundances of clones with different food values. Equation (3) thus becomes:
, where , (7)
and pi is the food value of the ith clone. The functional response for rotifer feeding rate (4) then becomes:
(8)
for the rate of feeding on algae with food value pi, and consequently
(9)
for the total feeding rate on the entire Chlorella population, where is defined above.The model then becomes
(10)
Simulation details
Values assumed for , , the minimum possible pi and other model parameters are found in Supplementary Table 1 below. For our simulations, we assumedsets ofq=1,2,3,5or 7clones at the start of the simulation run. As discussed in the main text (Yoshida et al. 2003), for each choice of clone "family" (i.e., system consisting of 1, 2, 3, 5 or 7 clones), we randomly drew a food value pfor each clone from a Uniform distribution on the interval where , and computed an associated Chlorella half-saturation value. Each resulting randomly associated set of clones was then used in a simulation run. A total of 3500 simulations were run, 700 sets each with 1, 2, 3, 5 and 7 clones initially. The software used to make these simulations was written in the R programming language (version 1.6) and run on a Windows 2000 platform.
Bifurcation diagram
Fussmann et al. (2000) experimentally determined the bifurcation diagram – the points at which transitions occur between stable and cyclic dynamics – for our chemostat system as the dilution rate is varied, and compared this with the predictions of the model without algal evolution. Although the non-evolutionary model was qualitatively correct about the sequence of bifurcations, there were quantitative discrepancies. We therefore examined whether the clonal evolution model could provide a better fit. The values of at which bifurcations occur depend on the algal tradeoff curve, which we have not measured directly. For this comparison, the tradeoff curve and conversion efficiency parameters in the clonal selection model were estimated by fitting the model simultaneously to two data sets (rotifer and algal density) at which cycles occurred, at dilution rates =0.69 and 0.96. The fitting criterion was a modification of the "probe matching" criterion in Shertzer et al. (2002), i.e. we attempted to match as closely as possible the period, amplitudes, and phase relations of the rotifer and algal cycles in the two data sets. Optimization was performed in R using the Nelder-Mead simplex algorithm, provided by the optim function in R, using multiple starting values in order to search for the global optimum. The model of Fussmann et al. (2000) was also updated to include improved parameter estimates based on our subsequent results (e.g., using the rotifer functional response estimated from our own experiments, rather than values from the literature and a conventional Monod uptake equation), but this had relatively minor effects on the predicted bifurcation points. The results (Supplementary Figure 1) show that the bifurcation points predicted by the clonal evolution model are consistent with all experimental results, while the non-evolutionary model predicts cycles at low dilution rates where the actual system does not exhibit cycles.
Supplementary Table 1. Parameter values, descriptions, and sources for the model used in Yoshida et al.(2003). The two different sets of values shown for and correspond to the two different tradeoff curves used in the simulations (Figures 1a and 1c).
Parameter / Description / Value / SourceNi / Inflow concentration of limiting nutrient N (mole N /l) / 80.0 / Set
V / Chemostat volume (l) / 0.33 / Set
/ Minimum half-saturation constant for nutrient uptake by Chlorella (mole N/l) / 4.3 / F2K
Kb / Half-saturation constant for algal consumption by rotifer (109 Chlorella cells) / 0.292 / TY
/ Maximum recruitment rate, Chlorella / 3.3 / TY
/ Critical Chlorella concentration (x109) / 0.437 / TY
/ Shape parameter for algal tradeoff curve / 0.8, 1.9* / Fitted
/ Cost parameter for algal tradeoff curve / 9.5, 4.0* / Fitted
pmin / Minimum algal food value (maximum is 1.0) / 0.02 / Assumed
/ Chemostat dilution rate (l/d) / 0.69 / Set
/ Conversion efficiency by rotifers / 5400 / F2K
/ Conversion efficiency by algae / 0.05 / F2K
/ N content in 109 Chlorella cells (mole) / 20.0 / F2K
/ Assimilation efficiency / 1.0 / F2K
G / Rotifer maximum clearance rate (1/d) / 3.3x10-4 / TY
/ Rotifer senescence rate (1/d) / 0.4 / F2K
m / Rotifer mortality (1/d) / 0.055 / F2K
Set: Experimental parameter whose value was under our control.
F2K: Value is the same as in Fussmann et al. (2000) and is derived or explained in that paper.
TY: Estimated from unpublished data on our system (T. Yoshida, unpublished data)
Fitted: Derived by numerically optimizing the goodness-of-fit of the clonal selection model to data from a chemostat experiment at the dilution rate used in the simulations in which cycles occurred, using a modification of the "probe matching" criterion in Shertzer et al. (2002). We have no direct information on the shape of the tradeoff curve, and therefore used two qualitatively different curves that are capable of generating model output matching the qualitative properties of the data (cycle period, amplitude, and phase relationship).
Supplementary Figure 1. Comparison of experimental results with model predictions for stability versus cycles in the chemostat system, as a function of the dilution (flow-through) rate with the concentration of the limiting nutrient N in the supplied chemostat medium held constant at 80 moles/liter. Vertical lines show the predicted locations of Hopf bifurcations, yielding stability at low and high dilution rates and cycles at intermediate rates. Solid lines are the predictions of the clonal selection model, dashed lines the predictions of the single-clone model. Plotted values (circles and squares) are the coefficient of variation (CV=standard deviation/mean) of predator and prey populations, corrected for effects of sampling error; these are re-drawn using the same values as in Figure 3 of Fussmann et al. (2000). Two of the 7 high-CV experiments plotted in this Figure had only one cycle of data, and therefore were not used for spectral analysis (Figures 2 and 3 of the main text).
References
Fussmann, G., Ellner, S., Shertzer, K. & N. G. Hairston, Jr., Crossing the Hopf bifurcation in a live predator-prey system. Science 290, 1358-1360 (2000).
Shertzer, K.W., S.P. Ellner, G.F. Fussmann, and N.G. Hairston, Jr. 2002. Predator-prey cycles in an aquatic microcosm: testing hypotheses of mechanism. Journal of Animal Ecology 71: 802–815.
Yoshida, T., L. E. Jones, S. P. Ellner, G. F. Fussmann, N. G. Hairston Jr., Rapid evolution drives ecological dynamics in a predator-prey system. Nature(2003).
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