Why only non-linear benefits can lead to coexistence of producers and non-producers.

<p>The fitness of producers (black curve) and non producers (grey curve) as a function of the fraction (<i>x</i>) of producers within the diffusion range of the molecule, for different steepness coefficients (<i>s</i>). The arrows show the direction of the dynamics. The shaded area shows the basin of attraction of the internal stable equilibrium (if it exists). <i>n</i> = 20, <i>h</i> = 0.5. <b>A:</b> The benefit <i>B</i>(<i>x</i>) of the molecule is an almost linear (<i>s</i> = 0.001) function of its concentration. <b>B:</b> The benefit <i>B</i>(<i>x</i>) is a sigmoid (<i>s</i> = 5) function of its concentration. <b>C:</b> The benefit function is essentially a step function (<i>s</i> = 20).</p

Realistic Hill coefficients lead to coexistence of producers and non-producers.

<p>For different benefit functions <i>B</i>(<i>x</i>) and gradients of diffusion <i>G</i>(<i>i</i>), the fraction of producers over time is show for <i>c</i> = 0.05 and <i>c</i> = 0.15. The lattices show the population after 1000 generations per cell. <b>A:</b> Linear benefit (<i>s</i> = 1, <i>h</i> = 0.5) with a diffusion gradient (<i>z</i> = 3, <i>d</i> = 0, <i>D</i> = 7). <b>B:</b> Sigmoid benefit (<i>s</i> = 20, <i>h</i> = 0.5) with no diffusion gradient (<i>z</i> = 1000, <i>d</i> = 3, <i>D</i> = 6). <b>C:</b> Sigmoid benefit (<i>s</i> = 20, <i>h</i> = 0.5) with a diffusion gradient (<i>z</i> = 3, <i>d</i> = 0, <i>D</i> = 7).</p

Why sigmoid benefits lead to different results from concave and convex benefits in spatially structured populations.

<p>The fraction of producers within the diffusion range (shown by an arrow) of molecules produced by cells at the producer/non-producer front remains approximately constant (∼0.5) even as the front moves ahead. If the benefit function is concave (<i>h</i> = 0.1) or convex (<i>h</i> = 0.9), at this fraction of producers (∼0.5) non-producers have an advantage, whereas producers have an advantage if the benefit function is sigmoid (<i>h</i> = 0.5). (<i>n</i> = 20, <i>c</i> = 0.1, <i>s</i> = 10).</p

Diffusion gradients do not affect changes in population structure.

<p>Changes in degree centrality and closeness centrality over time are shown for the producer and non-producer subgraphs. <b>A</b>: Fixed diffusion range with no diffusion gradient (<i>d</i> = 3, <i>D</i> = 6, <i>z</i> = 1000). <b>B</b>: Linear diffusion gradient (<i>d</i> = 3, <i>D</i> = 6, <i>z</i> = 1).</p

Projected rates of change in the timing of leaf coloration and leaf fall (5a and 5b; dates at which thresholds of 10%, 25%, 50%, 75% and 90% were reached), leaf color duration (5c; number of days between different leaf color duration thresholds and 90% leaf fall), and total amount of autumn colors (5d).

<p>For each species, the process-oriented (CDD/P) model, calibrated to 18 years of field data, was run forward using statistically downscaled climate projections from the GFDL CM2 model (IPCC A1fi and B1 scenarios; only A1fi scenario results shown in panels a through c). Projected rates of change (as plotted on the y-axis) were then calculated as the slope of the linear regression line between each phenological variable and year, over the period 2010–2099. Thus, for panels a through c, units are days per year, whereas for d, units are amount of color/year. ACRU: <i>Acer rubrum</i>; ACSA: <i>Acer saccharum</i>; FRAM: <i>Fraxinus americana</i>; NYSY: <i>Nyssa sylvatica</i>; PRSE: <i>Prunus serotina</i>; QUAL: <i>Quercus alba</i>; QURU: <i>Quercus rubra</i>: QUVE: <i>Quercus velutina</i>.</p

Empirical (MLR) and process-oriented (CDD/P) model fit statistics, calculated across the entire trajectory of leaf coloration (<i>c</i>₁₀ … <i>c</i>₉₀) and leaf fall (<i>f</i>₁₀ … <i>f</i>₉₀) for all eight study species.

<p>AICc  =  Akaike's Information Criterion, corrected for small samples (△AIC  =  AICc(MLR) – AICc(CDD/P)); ME  =  model efficiency; P  =  number of fit parameters. ACRU: <i>Acer rubrum</i>; ACSA: <i>Acer saccharum</i>; FRAM: <i>Fraxins americana</i>; NYSY: <i>Nyssa sylvatica</i>; PRSE: <i>Prunus serotina</i>; QUAL: <i>Quercus alba</i>; QURU: <i>Quercus rubra</i>: QUVE: <i>Quercus velutina</i>.</p

Correlation between interannual variation in temperature and interannual variation in autumn color phenology in red maple, <i>Acer rubrum</i>.

<p>Each point (<i>x</i>,<i>y</i>) in each plot represents a time window spanning the <i>y</i> weeks (vertical axis) before day <i>x</i> (horizontal axis). The color at each point (<i>x</i>,<i>y</i>) represents the correlation between the average air temperature for the time window (<i>x</i>,<i>y</i>) and the measure of autumn leaf phenology for that plot: onset of autumn colors (<i>c</i>_i), time of leaf fall (<i>f</i>_i), duration of autumn colors (<i>d</i>_i) and total amount of color (<i>A</i>). Values of <i>R</i> are shown by colors ranging from orange-red (minimum, negative) to blue-purple (maximum, positive); absolute values of <i>R</i>>0.468 (the critical value of the Pearson product-moment correlation coefficient; <i>p</i> = 0.05; <i>d.f.</i> = 16) are inside the bold lines. Here, both leaf fall and the display of red leaves were shifted significantly later in years with warmer autumn temperatures. Dates of the full display of autumn colors (<i>c</i>₇₅, <i>c</i>₉₀) were positively correlated with temperatures from spring through (especially) autumn, while warmer spring temperatures are correlated with earlier onset of color (<i>c</i>₁₀). Both the duration of autumn colors (<i>d</i>_x) and the total amount of autumn color (<i>A</i>) tended to increase in years with warmer temperatures.</p

Interannual variability of autumn senescence stages.

<p><b>2a:</b> timing of leaf coloration stages (c₁₀ = 10% of leaves colored … c₉₀ = 90% of leaves coloured) for <i>Quercus alba</i>, white oak. <b>2b:</b> timing of 50% leaf fall for four species (ACRU  =  <i>Acer rubrum</i>; FRAM  =  <i>Fraxinus americana</i>, PRSE  =  <i>Prunus serotina</i>; QURU  =  <i>Quercus rubra</i>).</p

Comparison of the empirical and process-oriented models.

<p>Comparison of goodness-of-fit (in terms of RMSE) of empirical (MLR) and process-oriented (CDD/P) models for leaf coloration (left) and leaf fall (right), in a leave-one-out cross-validation analysis. The MLR model is shown to be less robust, as its RMSE is higher (to the right of the 1∶1 line) in a majority of cases.</p

The amount of autumn colors over time for eight deciduous broadleaf species that turn red in autumn.

<p>The amount of autumn color (0–100) is calculated as <i>i</i>_n(100−<i>j</i>_n) on day <i>n</i>, where the percentage of red leaves <i>i</i>_n is multiplied by the percentage of leaves retained (100−<i>j</i>_n). Individual years (1993–2010) are shown by dotted lines, and their average by the thick curve.</p