(16 µm). Then transferred to a 250 ml flask with culture medium without nitrate, and left in

Transcription

1 Oikos OIK Barreiro, A., Roy, S. and Vasconcelos, V. M Allelopathy prevents competitive exclusion and promotes phytoplankton biodiversity. Oikos doi: /oik Appendix 1 Parameterization of uptake, growth and allelopathic effect Uptake experiments were performed in batch cultures. The phytoplankton species were pre-cultured during 6 days in 50 ml flasks (one flask per species) with low nitrate concentration (16 µm). Then transferred to a 250 ml flask with culture medium without nitrate, and left in these conditions during 24 h. Then, phosphate was resupplied (20 µm) and a pulse of 10 µm of nitrate was added to the flasks. Then, nitrate concentration (three 6 ml replicates) and cell abundances were monitored every 2 h during 18 h. All these experiments were performed in a culture room where light conditions were 40 µmols photons m -2 s -1, during 24 h per day, and temperature 20 ºC. The data obtained were fitted to a Michaelis-Menten equation (see further details below). Experiments for the estimation of the growth parameters were performed in continuous cultures. Flask volume was 400 ml and dilution rate was 0.3 day -1. Initial nitrate concentration was 320 µm and phosphate 200 µm. Experiments were performed in culture room where light conditions were 40 µmols photons m -2 s -1, during 24 h per day and temperature 20 ºC. Nitrate concentrations (in three 6 ml replicates) and cell abundances were monitored every 24 h. Experiments lasted days. Data were fitted to the Monod model (see further details below). Allelopathic effect of each species was estimated by testing the effect of increasing

2 concentrations (0, 10%, 25%, 50% and 75% of final volume) of cell-free filtrate of the donor species on the growth rate of a target species during 24 h (but 72 h when Oscillatoria sp. was used as target species). Cell-free filtrate was obtained by filtering through 0.22 µm filters (Millipore Express membrane filters) medium from continuous cultures of the donor species in steady-state, grown in the same conditions as above. These filtrates were added to 3 replicated 5 ml-vials containing cell suspensions of the target species, with culture medium and saturating concentrations of nitrate (3200 µm) and phosphate (200 µm). Experiments were performed in culture room where light conditions were 40 µmols photons m -2 s -1, during 24 h per day and temperature 20 ºC. The initial abundances of each target species were set at approximately equal biovolumes. Cell abundances were estimated with Neubauer or Sedgewyck-Rafter counting chambers (depending on cell abundance) on each of the 3 replicates immediately at the initial time and after 24 or 72 h (see above). Nitrate analysis Nitrate analysis were performed in samples of medium filtered through 0.22 µm filters (Millipore Express membrane filters). Analysis was performed in a Skalar Sanplus nutrient auto-analyzer, using the method Skalar M (EPA 353.2). Model formulations In order to fit the results from the experiments shown in Figure 1, we considered a resource-competition model of two phytoplankton competing for nitrate, and included the allelopathic effects from Oscillatoria sp. by a non-liner mortality term as function of the densities of both species. The choice of the non-linear mortality is crucial to correctly represent

3 the allelopathic effects. We tested several possible options based on the literature, and also introduced new functional forms to investigate the most suitable one that can describe our experimental results. The most basic formulations for a model of this kind could be as follows (Model 1): dn dt = D N & N F ) N 1 η ) F, N 1 η, dp ) dt = F ) N γp, P ) DP ) dp, dt = F, N DP, Where N is the nitrate concentration in the culture, N 0 is the inflowing nitrate concentration, D is the dilution rate of the system, P i (being i = 1,2) are the population abundances of Ankistrodesmus falcatus and Oscillatoria sp. respectively. η i are the yield coefficients of species i. as a ratio of mass of cells to mass of nutrient, γ is the parameter denoting the allelopathic effect. The functions F i (N) are the growth functions: F / N = µμ /P / N K / + N Where µ i are the maximum growth rates of species i, and K i the half saturation constants for nitrate. A model with this formulation does not support coexistence (known from theoretical analysis), so this formulation was discarded. A second form of the model (Model 2) includes a positive relationship between the growth rate of the allelopathic species (P 2 ) and the abundance of the non-allelopathic species (P 1 ). This positive relationship could be due to nutritional exudates or alkaline phosphatases by P 1, but also to the release of nutrients from cells of P 1 that died due to allelopathy. So, the only difference in Model 2 relative to Model 1 is the growth equation of the allelopathic species.

4 The equation for the non allelopathic growth is given by: F ) N = µμ )P ) N K ) + N with the same parameters as Model 1. The growth equation for the allelopathic species is given by: F, N = µμ,p, N K, + N where: µμ, P ) = µμ,& 1 + εp ) K 4 + P ) µ 20 is the basal maximum growth rate of this species, which is affected by the abundance of the non-allelopathic species, ε is a rate associated to this process and K p the half saturation constant. This form of the model was able to fit very accurately the two extreme cases of exclusion as well, and was also able to fit coexistence. For certain parameter values and conditions, this model can fit oscillatory coexistence (S. Roy, in prep.), similar to what is shown in the Figure 1 D, E, and F (main text). However, the values of the parameters and the state variables needed to show this oscillatory coexistence differed substantially from those needed to fit the data from this manuscript. In order to be able to fit the oscillatory coexistence using a reasonable range of parameter values and initial conditions for our experiments, we needed to incorporate an additional mechanism to those previous basic model formulations. We considered the inclusion of a more biologically feasible mechanism, which is a density dependent function determining allelochemical production. Density dependent mechanisms are inherent to most of the biological rate processes. It is very reasonable as well to consider that this rate would have lower and upper activation limits. So, we incorporated a mechanism with these

5 features through two alternative functional forms. The general model formulation in both cases is as follows: dn dt = D N & N F ) N 1 η ) F, N 1 η, dp ) dt = F ) N φ ) (P, )φ, (P, )γp ) DP ) dp, dt = F, N DP, All the terms are equivalent to Model 1 except Φ 1 (P 2 ) and Φ 2 (P 2 ), which are new. In the first alternative functional form (Model 3) Φ 1 (P 2 ) is the function that determines the negative density dependent allelopathic effect with lower threshold: φ ) P, = e 9:; < + b Where a is the rate of the process and b the lower threshold. In the same Model 3, Φ 2 (P 2 ) determines the upper limit of allelopathy effect (the minimum abundance of the allelopathic species in order for the process to occur): φ, P, = max P, n & P,, 0 Where n 0 is this minimum abundance of the allelopathic species needed for the allelopatic process. This model was able to fit qualitatively and quantitatively all the outcomes we observed in our experiments. To establish that the dynamics predicted by the model is not very specific to one non-linear functional form, we further considered alternative formulations, in order to select among them, the one providing the best fits. The first of those alternatives consists in the same model structure with a different functional form for the density dependent allelopathic killing (a similar

6 function considered in Roy et al in a different context) where: Φ 1 (P 2 ) is: φ ) P, = 1 e 9D E; < e 9D <; < Where C 1 determines the upper threshold of allelochemical production, whereas C 2 affects the slope of the process. Φ 2 (P 2 ) is the same as in Model 3. We termed this new alternative model formulation Model 4. Finally, we also considered two additional model formulations, which consisted in the same basic formulations as Models 3 and 4, but including the more explicit function for the growth of the allelopathic species, as in Model 2. The first of these two models was Model 5, where, similarly as in Model 2: F ) N = µμ )P ) N K ) + N and: F, N = µμ,p, N K, + N where: µμ, P ) = µμ,& 1 + εp ) K 4 + P ) And, as in Model 3: φ ) P, φ, P, = e 9:; < + b = max P, n & P,, 0 And the second alternative formulation is Model 6, which only changes with respect to Model 5 in: φ ) P, = 1 e 9D E; < e 9D <; < Which are the same functional forms used in Model 4 for these processes.

7 There are several mechanisms that could generate oscillatory coexistence in allelopathy models. One of them consists in incorporating a fully mechanistic structure for allelopathy (including rates of allelochemical production, uptake, and degradation) and adopting a sigmoidal function for the uptake of allelochemicals (Hsu and Waltman 2004). However, this was not successful in our case, probably because the referenced model was developed for a completely different context (bacterial allelopathy via plasmids). Other features in the model formulations such as delays (Mukhopadhyay et al. 1998) and adaptation (Mougi 2013) also showed to generate oscillatory coexistence, but in Lotka-Volterra models. We discarded incorporating a nonmechanistic delay in our models such as Mukhopadhyay et al. 1998, because we were more interested in explaining the observed dynamics as much as possible with biological mechanisms. Regarding the adaptation, we considered that this is a complex process that would need more evidences to support its inclusion in our models. Model optimization and selection In order to fit a Michealis Menten equation for nitrate uptake and a Monod model of growth (data shown in Table 1, main text) we used a nonlinear least squares method implemented in the nls function from stats package in R software. The Michaelis Menten function was: V = V G:HIJ K NO M H IJK + NO M Where V maxno3 is the maximum uptake rate and H NO3 is a half saturation constant. And the Monod function: µμ G:HNO M µμ = K IJK + NO M

8 Where µ max is the maximum growth rate and K NO2 a half saturation constant. The initial parameter estimates for our tested models come from the experiments whose results are shown in Table 1, main text. From these estimates, we could set reasonable initial estimates for µ i, K i, η i, γ. Other parameters were set experimentally (D = 0.3 day -1, N 0 = 320 µm). The rest of the parameters were unknown, and their initial values were set tentatively for estimation. Parameter optimization was performed in two steps, first by applying a global optimization method, the simulated annealing, and then, with the estimates obtained from it, a local optimization method based in the Nelder Mead algorithm (Nelder and Mead 1965). The simulated annealing was implemented with the GenSA function from GenSA R package. The Nelder Mead algorithm was implemented with the optim function from the stats R package. In both global and local optimization, the objective functions were set according to a version of the Levenberg Mardquart minimization criterion (Levenberg 1944): E = YZ[ YZ) P ) 4QRS P ) TUV P ) TUV 1 CV ;E + P, 4QRS P, TUV P, TUV 1 CV ;< The parameter set providing the minimum value of E is considered the best estimate. In this formula, P ipred (being i = 1 for A. falcatus, 2 for Oscillatoria sp.) correspond to the abundances of each species predicted by the model for times t = 1 to t = n, and P iobs, the same abundances from the real data. CV Pi is the coefficient of variation of the observed population abundances. For the values of the observed population abundances, with used a smoothed series (Ellner et al. 2002). After optimization of models 3, 4, 5 and 6, we used an ad hoc version of the Akaike Information Criterion, in which the values of E obtained from the best model fits were penalized by the number of parameters in the model, according to this formula:

9 A = log10 SZ` SZ) E + 2 log10 k Where d = 1,... 9 are the experimental runs. E is the minimized value from the criterion detailed above, and k is the number of model parameters. In order to compare A between models, A models were calculated as follows: ΔA model i = A model i lowest A Another goodness of fit criterion, the relative mean error (RME) was calculated as follows: RME = YZ[ YZ) P )4QRS P )TUV P )TUV + P,4QRS P,TUV P,TUV In this formula, P ipred (being i = 1 for A. falcatus, 2 for Oscillatoria sp.) correspond to the abundances of each species predicted by the model for times t = 1 to t = n, and P iobs, the same abundances from the real data. The results from model optimization process and the values of A are shown in Table A1 considering all the experiments as a single data set and in Table A2 by using each independent experiment as a single data set. In Table A1 we can see that, for Models 3 to 6, parameters take similar values, always in the same orders of magnitude, as expected from similarly structured models, with just minor differences in some functional forms. Models 1 and 2 were not able to fit the experiments showing species coexistence (these models do not predict coexistence under any circumstances, see above). Our results showing coexistence are the main findings of this work. For this reason, they were not considered in the comparison of A and RME values with the other models. Also, these two models showed larger amplitude in the range of parameter values than models 3 to 6. Particularly for the allelopathy parameter (γ) which spans one order of magnitude. Such a large range in the value of one of the most important model parameters indicates poor performance of Models 1 and 2. This suggests that the model structure needs to include some

10 additional feature that makes it possible to modulate the effect of allelopathy without the need to vary a lot the value of this parameter. This is what models 3 to 6 do with their nonlinear functional forms. Overall, the selection criterion benefits Model 3, the simplest model (considering 3 to 6 only) together with Model 4. In our context, model selection is not a very relevant issue, since the fitting to our data could always be improved by adding more complex functional forms or additional, less relevant mechanism. Our aim here was to keep the minimum relevant mechanisms, using the simplest possible functional forms in order to generate a model that was able to fit qualitatively and quantitatively the experimental results in an acceptable way. From this point of view, we could say that models 3 or 4 are better than 5 and 6, since the latter two incorporate an additional function for growth that makes the models more complex. But, this increase in complexity does not improve the fitting substantially. However, with respect to models 3 and 4, there are no strong reasons to decide whether Model 3 is better than Model 4, despite the slightly lower value of A for Model 3. Table A2 allows us to perform the same comparison as previously for each independent experiment. This makes it easier to see the effect of each specific feature of a model for each of the outcomes of competition. Table A2 shows that the best performance (in general) of Models 3 and 4 relative to Models 5 and 6 was due to a better fit to those experiments showing exclusion of either species. If we look at the experiments showing coexistence, the performance of all the models was about the same, being 5 or 6 even slightly better than 3 or 4. The reason why, globally, Model 3 performs better (see Table A1) was because it was the most consistent model across all of the experiments.

11 Model simulation In figure 1, we simulated the models with the optimized values of the parameters and the actual initial conditions of the state variables for each individual experiment. The dilution rate of the system and the inflowing nitrate concentration were the only parameters that were not optimized, and their values were 0.3 day -1 and 320 µm respectively. References Roy, S The coevolution of two phytoplankton species on a single resource. Allelopathy as a pseudo mixotrophy. - Theor Popul Biol. 75: Roy, S Importance of allelopathy as pseudo-mixotrophy for the dynamics and diversity of phytoplankton. Biodiversity in Ecosystems - Linking Structure and Function. Eds Lo et al, InTech, UK, pp Roy, S. et al Competing effects of toxin-producing phytoplankton on overall plankton populations in the Bay of Bengal. - B Math Biol 68: Hsu, S. B. and Waltman, P A survey of mathematical models of competition with an inhibitor. - Math Biosci. 187: Mukhopadyhay, A. et al A delay differential equations model of plankton allelopathy. - Math Biosci. 149: Mougi, A Allelopathic adaptation can cause competitive coexistence. - Theor Ecol. 6: Nelder, J. A. and Mead, R A simplex method for function minimization. - Comput J. 7: Levenberg, K A Method for the Solution of Certain Non-Linear Problems in Least

12 Squares. Quart Appl Math. 2: Ellner, S.P. et al Fitting population dynamics models to time series data by gradient matching. Ecology 83:

13 Species abundance LN(cell ml 1 ) A D B E C F Time (days)

14 Scaled population abundances A(a) B(a) C(a) Time (days) Scaled A.falcatus abundance A(b) B(b) C(b) Scaled Oscillatoria sp. abundance

15

16

17 Realized peaks of P 1 within 100 days Realized peaks of P 2 within 100 days Ini$al abundance of Allelopathic species Ini$al abundance of Allelopathic species Ini$al abundance of Allelopathic species

18 Figure A1. Plots of Oscillatoria and Ankistrodesmus abundances in long term experiments shown in log-scale, which is more appropriate to see in detail when the species were excluded. A. B, C correspond to those experiments where Oscillatoria was excluded (Figure 1 A, B, C in the main text) and D, E, F correspond to those experiments where Ankistrodesmus was excluded (Figure 1 G, H, I in the main text). Figure A2. Details on the oscillatory patterns of the three experiments with intermediate Oscillatoria sp. to A. falcatus ratios: 0.11 (A(a), A(b)), 0.20 (B(a), B(b)) and 0.36 (C(a)), C(b)). Data are represented by the smoothed curves used for model fitting (see Model optimization and selection sections in this Appendix, Figure 1 in the main text). In A(a), B(a) and C(a) green color represents A. falcatus and blue Oscillatoria sp. Major period and amplitude oscillations seem to overlap with other smaller oscillations, as the model fits in Figure 1 in the main text show. Population densities in C(a) are nearly stable. Data begin in days and end in day 90. Numbers labeling the lines in plots A(b), B(b) and C(b) correspond to days. Figure A3. Results of the same model simulations from Figure 4 in the main text showing the % of occurrence of exclusion of Ankistrodesmus falcatus, P 1 (left column), coexistence (central column) and exclusion of Oscillatoria sp., P 2. The colour scale indicates increased % from blue to red. The upper row panel represents the simulations of 'Model 3' describing density-dependent allelopathic killing, and the lower row panel 'Model 4' (see Table 2). The simulations were performed with the same conditions and methods (see Methods) for Figure 4 in the main text. Spline interpolation was applied to smooth the contours of the risk of extinction and coexistence regions. Figure A4. Results of oscillation amplitudes from species population dynamics obtained from model simulations performed under the same conditions as in Figure A3 and Figure 4 in the main

19 text. Figure A5. Same as figure A3 for realized peaks shown in the dynamics of each species.

20 Table A1. Results of model optimization and criterions for model selection considering all the experimental runs together. The values of each parameter are the average of the optimized values from each individual experiment, and within brackets, the range of those values. Last three rows show the values of the different criterions for model selection (RME: relative mean error, A, and the differential with the lowest A computed). Models 1 and 2 were not considered in the model selection process due to their inability to predict either coexistence or oscillatory coexistence (see text in Appendix 1, Model optimization and selection). Parameters Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 η [ ] 9500 [ ] 8314 [ ] 7663 [ ] 7915 [ ] 9266 [ ] η [ ] 3000 [ ] 2380 [ ] 2365 [ ] 1618 [ ] 2791 [ ] µ [ ] 1.4 [ ] 0.9 [ ] 1.1[ ] 1.1 [ ] 1 [ ] µ [ ] [ ] 0.44 [ ] - - µ [ ] [ ] 0.45 [ ] K [ ] 31.2 [ ] 15.4 [ ] 15.4 [ ] 15 [9-16] 15 [15-16] K [ ] [ ] 36.2 [ ] 45.1 [20-69] 28 [22-40] 41 [40-42] ε [ ] [ ] 0.04 [ ] K p [ ] [ ] 1995 [ ] γ [ ] [ ] 549 [ ] 448 [ ] 583 [ ] 487 [ ] n [ ] [ ] [ ] [ ] a [ ] [ ] - b [ ] [ ] - C [ ] [ ] C [ ] [ ] RME A A

21 Table A2. Same model selection criterions as in Table A1, but reported for each single experiment separately. The experiments are identified with the same indexes as in Figure 1 from the main manuscript. Experiment Model RME A A Model Model A Model Model Model Model Model Model B Model Model Model Model Model Model C Model Model Model Model Model Model D Model Model Model Model Model Model E Model Model Model Model Model Model F Model Model Model Model Model Model G Model Model Model Model Model Model H Model Model Model Model Model Model I Model Model Model Model