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:
dn
  (VN i  n)  Fc (n / V )c
dt
dc
  c Fc (n / V )c  Fb (c / V )b  c
dt
db
  b Fb (c / V )r  (  m)b
dt
dr
  b Fb (c / V )r  (  m   )r
dt
(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 109 Chlorella cells:
Fc 
1 c c N
,
c Kc  N
where N=n/V
(2)
Rotifer feeding rate, Fb , is derived from the rotifer clearance rate
g (C )  G /( K b  max( C, C*) ), 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those observed in the chemostat experiments (T. Yoshida, unpublished data).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
1
clearance rate by the algal concentration, C, gives the ingestion rate in units of 109
cells/rotifer/day:
Fb (C )  CG /( Kb  max(C, C*)) .
(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  1 , and scale or “cost”
parameter  2 ) to describe how Kc varies between p  0 and p=1. The tradeoff curve is
Kc ( pi )  Kc (1)   2 ( pi ), where 
1
1
 pi
 1.
(for "technical typesetting reasons" 1 appears as α' and  2 appears as α in the published
paper).
This yields, in terms of the food value of the ith clone,
1 1/ 1
K c ( pi )  K c ,min   2 (1  pi
)
(5)
Note that Kc attains its minimum value at p=1 (thus algal growth rate is maximum at 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:
Fc,i 
1
c  c N
 c K c ( pi )  N
(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
2
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 p  0 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:
~
~
g (C )  G /( Kb  max( C , C*)) ,
~
where C   pi Ci ,
(7)
i
and pi is the food value of the ith clone. The functional response for rotifer feeding rate (4)
then becomes:
Fb,i (ci pi )  ci piG /( Kb  max(C, C*))
(8)
for the rate of feeding on algae with food value pi, and consequently
~
~
~
(9)
Fb (C )  CG /( Kb  max( C , C*))
~
for the total feeding rate on the entire Chlorella population, where C is defined above. The
model then becomes
k
dn
  (VN i  n)   Fc ,i (n / V )ci
dt
i 1
dci
  c Fc ,i (n / V )ci  Fb ,i b   ci
dt
db
  b Fb (C )r  (  m)b
dt
dr
  b Fb (C )r  (  m   )r
dt
(10)
Simulation details
Values assumed for K c ,min , 1 ,  2 , the minimum possible pi and other model
parameters are found in Supplementary Table 1 below. For our simulations, we assumed sets
of q=1,2,3,5 or 7 clones 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 p for each clone from a Uniform distribution on the
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interval where [ pmin ,1] , 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.
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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  1 and  2 correspond to the
two different tradeoff curves used in the simulations (Figures 1a and 1c).
Parameter
Ni
V
K c ,min
Kb
c
C*
1
2
pmin

b
c
c
c
G

m

Description
Inflow concentration of limiting nutrient N
(  mole N /l)
Chemostat volume (l)
Minimum half-saturation constant for
nutrient uptake by Chlorella (  mole N/l)
Half-saturation constant for algal
consumption by rotifer (109 Chlorella cells)
Maximum recruitment rate, Chlorella
Value
80.0
Source
Set
0.33
4.3
Set
F2K
0.292
TY
3.3
TY
Critical Chlorella concentration (x10 )
Shape parameter for algal tradeoff curve
0.437
0.8, 1.9*
TY
Fitted
Cost parameter for algal tradeoff curve
9.5, 4.0*
Fitted
9
Minimum algal food value (maximum is 1.0) 0.02
Chemostat dilution rate (l/d)
0.69
Conversion efficiency by rotifers
5400
Assumed
Set
F2K
Conversion efficiency by algae
0.05
F2K
N content in 10 Chlorella cells (  mole)
20.0
F2K
Assimilation efficiency
1.0
F2K
Rotifer maximum clearance rate (1/d)
Rotifer senescence rate (1/d)
Rotifer mortality (1/d)
3.3x10-4
0.4
0.055
TY
F2K
F2K
9
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).
5
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).
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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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