Supplementary Material
Three criteria have been used to select values for the initial conditions and parameter values in the simulations: (1) a priori empirical information, when available; (2) realistic simulation results; e.g., population extinction in the water column is expected to occur before the year finishes at t < 364, with maximum population density (A + S) around 1000 rotifers l-1; and (3) for the models anticipated to show a threshold effect for the mixis ratio in relation to population density, parameter values expected to promote this effect. A systematic, broad exploration was carried out by using a range of values for many of the parameters.
Initial conditions for variables
The initial conditions refer to values when the annual dynamics start. The R(0) is the initial amount of resources available at the onset of every growing season, and A(0) is the number of individuals founding a population by hatching from diapausing eggs. It is assumed that S(0) = P(0) = 0 (i.e., no sexual females exists at the onset, and MIP from previous years is completely degraded). Annual rotifer population dynamics start with densities much lower than the maximum achieved, which tend to be around 1000 ind. l-1. Initial resource level is expected to strongly influence the maximum population density and time until population extinction in the water column population (length of the growth season), and has to be selected to fit the restrictions stated above.
Resource consumption, birth and death rates of rotifers
Resource concentration is re-scaled, so that γ (the asymptotic limit of γ R / [R + KR]) is equal to the maximum birth rate. This means that resources are measured in numbers of rotifers that can be produced, and consequently an efficiency constant for the conversion of resources in rotifer births is not needed in eq. 1b. Studies have shown the intrinsic rate of increase (r) in rotifers is in the range of 0.2–1.2 d-1 (Wallace et al., 2006). Therefore maximum birth rate (i.e., γ) around 1.2 d-1 and mortality rate (q) around 0.5 d-1 in the wild are reasonable assumptions. Here we assumed mortality rate is larger in the wild than the usual values reported in laboratory studies. KR is the resource level at which γR/(R + KR) is half of the maximum γ (around 0.6 d-1). By combining published numerical and functional responses for the rotifer Brachionus ibericus fed Tetraselmis suecica (Ciros et al., 2001), we estimate KR 1000 Ru l-1 based on the following derivation. Half maximum increase rate is achieved at 0.2 mg C l-1 (Ciros et al., 2001). With 0.2 mg C l-1, r 0.4 d-1, and for C near 0 mg l-1, r -0.3 d-1. Hence, mortality rate is estimated to be 0.3 d-1 and it will be assumed to be resource-independent. This gives a birth rate of 0.7 d-1 at 0.2 mg C l-1. Clearance rate is 25 µl ind-1 h-1 = 6.0 X 10-4l ind-1 d-1. This means that, if there is 0.2 mg C l-1 constantly available, 1.2 X 10-4 mg C ind-1 d-1 will be consumed. This figure is in the same order of magnitude as the C filtration rate estimated in Alver et al. (2006),at 3.9 X 10-4 mg C ind-1 d-1. Following the Ciros et al. (2001) experiment, with a resource consumption by maternal females of 1.2 X 10-4 mg C ind-1 d-1, 0.7 neonates mother-1 d-1 are produced, giving 1.7 X 10-4 mg C consumption neonate-1. Hence, with C = 0.2 mg C l-1 concentration, a total of 1167 neonates could be produced per liter, so that KR is estimated to be 1167Ru l-1.
MIP degradation rates
D1/2 is the half life of the MIP in the medium. Fast loss of the activity of conditioned medium has been observed in the laboratory (Kubanek & Snell, 2008). However, no measurement of degradation rate is available. From preliminary simulations D1/2 values around 1 d produced plausible dynamics.
Hypothesis-dependent functions for mixis response to MIP and MIP production
The functions assumed for each of the four hypotheses are:
H0:(eq. S.1a)
(eq. S.1b)
HR:(eq. S.2a)
(eq. S.2b)
HP1:(eq. S.3a)
(eq. S.3b)
HP2:(eq. S.4a)
(eq. S.4b)
Parameters are listed in Table1. Notice that eqs. S.1b and S.2b state that per capita production rate is 1 MIPu ind-1 d-1 due to the scale used for MIP concentration. Consequently, eq. S.3b converges to 1 as P increases. The value for mmax is assumed to be around 0.5 (Fussmann et al., 2007). Notice that the actual maximum mixis ratio might be lower than the potential one, since the required MIP concentration for the latter may not be achieved.
Hypothesis-dependent parameters
Estimation of Km is based on the assumption that, if a threshold exists for the action of MIP on the mixis ratio (HR), it occurs around Km. In experimental works, the population density threshold (NT) is estimated (e.g., for Brachionus manjavacas it is 71 l-1 (Snell et al., 2006; see also Serra et al., 2005)). Assuming equilibrium conditions (dR/dt = dA/dt = dS/dt = dP/dt = 0), MIP production equals degradation. In terms of HR and combining eqs. 1d and S.2b, this means that A*+S* = (ln(2) / D1/2) P* (asterisk means equilibrium value). In empirical research NT is not measured at equilibrium conditions. However, taking as an approximation that NT = A*+S*, an estimation on the threshold MIP concentration is NT D1/2 /ln(2) MIPu l-1. With Km = 150 MIPu l-1 and D1/2 = 1 d, NT is ca. 100 indl-1. The occurrence of thresholds in the mixis ratio vs. population density relationship might be dependent on the Km value. The parameter a controls the slope of the effect of MIP concentration on mixis ratio (see Fig, 2, panel HR). The higher the a value, the more intense the threshold effect. A low a value could be interpreted as a gradual individual response to MIP concentration. Alternatively a low a value could be interpreted as the statistical result of variance in the females’ response to MIP, or a heterogeneous MIP distribution in the medium. Figure 2 (panel HR) was generated assuming a = 0.1 l MIPu-1 (Km = 500 MIPu l-1, MIP concentration range: 0–1000 MIPu l-1). In order to promote threshold effects, α values need to be assumed to be very low. Values for KP were selected to promote threshold effects. In HP2, in order to generate a production rate equal to 1 when averaged over asexual and sexual females, the proportion of the latter is assumed to be mmax (i.e., the same maximum proportion in the population as in the offspring). Then 1 = (1 - mmax) α + mmaxσ , which results in σ = α + (1-α)/ mmax. Hence, the value of σ is completely determined by αand mmax.
The threshold index
To analyze threshold effects, a quantitative measure of this phenomenon is needed. Measuring the magnitude of a threshold effect is difficult because the models produce continuous relationships between the mixis ratio and population density, whereas the concept of a threshold implies a discontinuity. However, genetic and microenvironmental variation likely confers some range in responsiveness to, or production of, the MIP (Aparici et al., 2001), and Fussmann et al. (2007) have shown that relative age of the maternal female when reproducing impacts the likelihood that her progeny will be mictic. Thus, the observed mixis ratio for a real population likely is continuous, reflecting a range in extrinsic or intrinsic regulatory factors. Still, it remains necessary to quantify how sharp this continuous transition from completely parthenogenetic reproduction to mixis must be to qualify as a threshold. In this paper, the following conditions for determining a threshold effect are established: (1) Threshold effects are observed in populations showing density-dependence in sexual reproduction as density increases from an initial, low population density value. (2) Threshold effects in sexual reproduction appear at densities much lower than the maximum ones. The threshold analysis is restricted to this relatively low-density phase. (3) Existence of a threshold effect implies a concave up relationship of mixis ratio and population density over the interval of analysis.
The mixis threshold index developed here (T) is computed as follows (Fig. S.1). The observed mixis ratio (mT) closest to and below half the maximum mixis ratio detected in the simulations is recorded, as well as the corresponding population density NT. Restricting consideration of the response up to this point allows analysis of the initial increase of mixis ratio. A response with no threshold is defined as a proportional increase in the mixis ratio up to mT. Under this behaviour, the mixis ratio should be m = (mT / NT) N, where N is the population density (i.e., A + S). To detect the potential for a threshold response, the deviation of the observed relationship (i.e., a curve) from this proportional response is computed. This deviation is measured as the ratio, z, between two areas: (1) the area below the observed curve and (2) the area below the proportional line, both areas being defined between the population densities N = 0 and N = NT. Then, T = 1 – z.
The value for T is 1 if the mixis ratio jumps from zero directly to half the maximum mixis ratio, showing a point discontinuity; values closer to 1 imply a sharper transition to mixis. The T index is 0 if the mixis ratio increases proportionally up to half maximum mixis ratio (i.e., follows the dashed line in Fig. S.1). Values for T tend to -1 if the mixis ratio is close to half maximum mixis ratio at very low population density. Two examples are shown in Fig. S.1.
References
Alver, M. O., J. A. Alfredsen & Y. Olsen, 2006. An individual-based population model for rotifer (Brachionus plicatilis) cultures. Hydrobiologia 560: 93–108. 561.
Aparici, E., M. J. Carmona & M. Serra, 2001. Variability for mixis initiation in Brachionus plicatilis. Hydrobiologia 446/447: 45–50.
Ciros-Pérez, J., M. J. Carmona & M. Serra, 2001. Resource competition between sympatric sibling rotifer species. Limnology and Oceanography 46: 1511–1523. 576
Fussmann, G. F., G. Kramer & M. Labib, 2007. Incomplete induction of mixis in Brachionus calyciflorus: patterns of reproduction at the individual level. Hydrobiologia 593: 111–119.
Kubanek, J. & T. W. Snell, 2008. Quorum sensing in rotifers. In Winans, S. C. & B. L. Bassler (eds), Chemical Communication Among Bacteria. ASM Press, Washington: 453–461.
Serra, M., T. W. Snell & J. J. Gilbert, 2005. Delayed mixis in rotifers: an adaptive response to the effects of density-dependentsex on population growth. Journal of Plankton Research 27: 37–45.
Snell, T. W., J. Kubanek, W. Carter, A. B. Payne, J. Kim, M. K. Hicks & C.-P. Stelzer, 2006. A protein signal triggers sexual reproduction in Brachionus plicatilis (Rotifera). Marine Biology 149: 763–773.
Wallace, R. L., T. W. Snell, C. Ricci & T. Nogrady, 2006. Rotifera: Volume 1: Biology, Ecology and Systematics, 2nd edn. In Segers, H. (ed.), Guides to the Identification of the Microinvertebrates of the Continental Waters of the World 23. Kenobi Productions Ghent and Backhuys Publishers, Leiden.
Figure S.1. A. Computing the threshold index (T; see text for explanation). B and C, examples of trajectories (see arrows) of the mixis ratio vs. population density, and the correspondingT values; population dynamics were simulated using H0(panel B) and HP1 (panel C).