Evolutionary convergence to a single density-compensation strategy in the individual-based Maynard Smith and Slatkin model.

<p>In both panels, evolution is absent for the first 3⋅10⁵ generations (distinguished by a lighter background) and present thereafter. (a) Evolution of the density-compensation strategies of two coexisting species, starting from initial values that are known from Fig. 3 to enable dynamically stable coexistence (<i>b</i>₁ = 0.9, <i>b</i>₂ = 5.4). After evolution starts, the species rapidly evolve outside the coexistence region, which leads to the extinction of one species and the evolution of the other to an intermediate density-compensation strategy. (b) Evolution of the density-compensation strategies of two isolated (non-coexisting) species, starting from two different <i>b</i>-values (<i>b</i>₁ = 0.5, <i>b</i>₂ = 6). After evolution starts, the species rapidly evolve to the same <i>b</i>-value as in (a). Other parameters: <i>K</i> = 1000, , <i>m</i> = 0.3, <i>r</i> = 5, and <i>d</i> = 0.05.</p

Density-dependent reproduction ratio of the original Maynard Smith and Slatkin model (a), of the modified Maynard Smith and Slatkin model (b), and resulting pairwise invasibility plot (c).

<p>The curves plotted in (a) and (b) result from equidistant <i>b</i>-values on the logarithmic scale. The curves for <i>b</i>_low = 0.8 (red) and <i>b</i>_up = 5 (green) from eq. 9 are highlighted by dashed curves. Note that, while curves are evenly distributed for the original model, the modification creates an asymmetry between the curves above and below <i>N</i>/<i>K</i> = 1 in the modified model. The pairwise invasibility plot shows that this results in lower fitness for the intermediate strategies, as well as in the loss of evolutionary stability of the evolutionarily singular strategy.</p

Evolutionarily stable density-compensation strategies in the individual-based Maynard Smith and Slatkin model.

<p>(a) Evolutionarily stable <i>b</i>-values as a function of the intrinsic growth rate <i>r</i> and density-independent mortality <i>d</i>. Black colors indicate extinction. (b) Difference between the evolutionarily stable strategy and the critical <i>b</i>-value (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0094454#pone.0094454.e008" target="_blank">eq. 5</a>). Other parameters: <i>K</i> = 1000, , <i>m</i> = 0.1, and <i>b</i>_min = 0.17.</p

Evolutionary and dynamic stability in the individual-based Maynard Smith and Slatkin model.

<p>(a) Invasion probability, approximated by the probability that a strategy invading with one individual survives for at least 500 generations. Red shades depict areas that are mutually invasible. Dashed lines indicate the critical density-compensation strategy at <i>b</i>_cri≈2.5 (eq. 5). (b) Time until competitive exclusion (plotted in log10 units) for two species, starting with equal population sizes. The dashed curves (obtained using a kernel smoother) show combinations of <i>b</i>-values for which the two strategies have equal chances to exclude each other. The diagonal line (identical <i>b</i>-values) may be regarded as providing a reference: it shows the time until competitive exclusion under neutral drift. Moving away from the diagonal, one of the two strategies (marked by the numbers 1 and 2) tends to exclude the other, with an average time until competitive exclusion smaller than under neutral drift. More interesting, however, are the coexistence times along the other parts of the dashed curves, which are several orders of magnitude longer than along the diagonal, evidencing a non-neutral, stabilizing mechanism of coexistence. Each cell shows the results from a single simulation; hence, the variance among close-by cells provides a visual impression of the variance between simulation runs. Other parameters: <i>K</i> = 200, <i>r</i> = 5, and <i>d</i> = 0.05.</p

Supplement 1. Data and Stan code specifying the statistical model used to analyse prevalences of a phytoplasma disease in grapevines (bois noir) in the Baden region (SW-Germany) in a Bayesian framework.

<h2>File List</h2><div> <p><a href="boisnoir.txt">boisnoir.txt</a> (MD5: 9c9e10c191527b1ef0232bad81d3fbed)   Bois noir data (including grapes and environment)</p> <p><a href="RStan_boisnoir.R">RStan_boisnoir.R</a> (MD5: 10f54ea6eb0212643975b03f2c0a0f38)  Stan code</p> </div><h2>Description</h2><div> <p>The first supplement (boisnoir.txt) contains the dataset used in the paper. Edaphic data was provided by “Regierungspräsidium Freiburg, Landesamt für Geologie, Rohstoffe und Bergbau (Hrsg.) (2012): Geologische Grundflächen. – Geologische Karte 1 : 50 000, Geodaten der Integrierten geowissenschaftlichen Landesaufnahme (GeoLa). <a href="http://www.lgrbbw.de/">http://www.lgrbbw.de</a>; [26.01.2012]”. The second supplement (RStan_boisnoir.R) contains the model specification.</p> </div

Pairwise invasibility plots for different density-compensation strategies <i>b</i> in eqs. 1,2,3.

<p>The plots show the fitness (eq. 7) of an invading strategy with a density-compensation strategy along the vertical axis for residents with density-compensation strategies along the horizontal axis. Plus and minus signs indicate strategy combinations resulting in positive and negative invasion fitness, respectively; in addition, regions of negative invasion fitness are shaded. Vertical dashed lines show the critical <i>b</i>-values of the resident. Small insets in the top left of each plot show areas of mutual invasibility. Parameters: <i>r</i> = 5 (MSS), <i>r</i> = 40 (Hassell), <i>r</i> = 0.5 (Ricker), and <i>d</i> = 0.05 Different intrinsic growth rates <i>r</i> were chosen to obtain a similar growth response for similar <i>b</i>-values across the three models.</p

Effect of the density-compensation strategy <i>b</i> on the population dynamics of a single species described by the Maynard Smith and Slatkin model.

<p>(a) Reproduction ratio <i>N_t</i>₊₁/<i>N_t</i> as a function of the relative population size <i>N</i>/<i>K</i> for the analytical MSS model with intrinsic growth rate <i>r</i> = 5, density-independent mortality <i>d</i> = 0.05, and four different values of <i>b</i>. Since <i>d</i>>0, the reproduction ratio at the carrying capacity <i>K</i> is smaller than 1, which implies that the equilibrium population size remains slightly below the carrying capacity. (b) Bifurcation diagram showing population sizes at equilibrium as a function of the density-compensation strategy <i>b</i>. Cyclic population dynamics, indicating strong overcompensation, occur for <i>b</i>-values exceeding <i>b</i>_cri≈2.5 (vertical line), as predicted by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0094454#pone.0094454.e008" target="_blank">eq. 5</a>. The three insets depict the transition from stable population dynamics below the critical value to cyclic dynamics shortly above the critical value and to chaotic dynamics at even larger <i>b</i>-values.</p

Evolutionary branching (a) and time until competitive exclusion (b) for the modified Maynard Smith and Slatkin model (eq. 9).

<p>After evolution is introduced (distinguished by a darker background), evolutionary branching results in two distinct strategies that are evolutionarily stable. Analysis of the time until competitive exclusion (plotted in log10 units) indicates that these strategies are also dynamically stabilized by RNC. Other parameters: <i>K</i> = 1000, , <i>m</i> = 0.1, and <i>b</i>_min = 0.17.</p

On the Challenge of Fitting Tree Size Distributions in Ecology

<div><p>Patterns that resemble strongly skewed size distributions are frequently observed in ecology. A typical example represents tree size distributions of stem diameters. Empirical tests of ecological theories predicting their parameters have been conducted, but the results are difficult to interpret because the statistical methods that are applied to fit such decaying size distributions vary. In addition, binning of field data as well as measurement errors might potentially bias parameter estimates. Here, we compare three different methods for parameter estimation – the common maximum likelihood estimation (MLE) and two modified types of MLE correcting for binning of observations or random measurement errors. We test whether three typical frequency distributions, namely the power-law, negative exponential and Weibull distribution can be precisely identified, and how parameter estimates are biased when observations are additionally either binned or contain measurement error. We show that uncorrected MLE already loses the ability to discern functional form and parameters at relatively small levels of uncertainties. The modified MLE methods that consider such uncertainties (either binning or measurement error) are comparatively much more robust. We conclude that it is important to reduce binning of observations, if possible, and to quantify observation accuracy in empirical studies for fitting strongly skewed size distributions. In general, modified MLE methods that correct binning or measurement errors can be applied to ensure reliable results.</p> </div

Scheme of the evaluation procedure of virtual data sets.

<p>Scheme of the evaluation procedure of virtual data sets.</p