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SUPPORTING INFORMATION
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Appendix S1 finite Pairwise Invasibility Plot
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Since we need to trace the resource dynamics of each individual tree in the forest over a
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large number of generations, we cannot choose a very large value for the number of trees N due to
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the limited computational resources. We therefore set N equal to 100 for most calculations, but the
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results of the model remain the same when we examine a larger N, though taking a lot more time
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to run. This choice for the number of trees is plausible because most of successful pollen donors
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are within a rather short range (Sork et al. 2002; Koenig & Ashley 2003). The finiteness of the
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number of individuals results in fluctuation due to demographic stochasticity. In addition, the
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reproductive success of each tree depends on the behavior of other individuals in the population.
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The model is a "game in a finite population" (Nowak et al. 2004; Taylor et al. 2004; Nowak 2006).
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The marginal fitness is no longer a good predictor of the evolutionary outcome; instead, we need
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to calculate the fixation probability of a mutant that appears in a population dominated by the
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resident value.
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Specifically, consider a population in which there is a single mutant with parameter k'
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and N − 1 resident individuals with parameter k. The mutant will either become fixed in the
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population or extinct at the evolutionary end point. By carrying out a large number of replicates,
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we can calculate the fraction of runs with successful fixation. If the fraction of successful fixation
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runs exceeds that expected for the neutral case (1/N), we conclude that the mutant is favored by
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natural selection (Taylor et al. 2004). If the fixation rate is less than that expected for neutral
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mutants, we conclude that the resident is favored by natural selection. In between these two cases,
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we conclude that the mutant is neutral.
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Tachiki & Iwasa (2008) ran these simulations for all possible combinations of the
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resident’s k and the mutant’s k', and plotted the result on a two-dimensional square. The graph is a
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finite-population version of the pairwise invasibility plot (PIP), and was called a "finite pairwise
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invasibility plot" (fPIP). This method extends the PIP for an infinitely large population (Metz,
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Nisbet & Geritz 1992; Kisdi & Meszéna, 1995; Geritz et al. 1997; Geritz, Meijden & Metz 1999)
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to a finite population.
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After1000 replicates for each pair of resident and mutant, ratio of fixation of mutant is
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calculated. Figure S1 shows two fPIPs with different seedling survivorship. In each plot,
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horizontal axis is the resident trait k, and vertical axis is the mutant trait k'. A point k, k' is
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shaded black if the mutant has a fixation probability significantly higher than the neutral case. In
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
contrast, it is left white if the mutant has a fixation probability significantly lower than the neutral
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case. Gray shading indicates no significant deviation from the neutral expectation. Note that the
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area near the diagonal line (where the mutant k' is close to the resident k) is gray.
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Using this graph, we can find evolutionary attractors to which the phenotype converges
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as a result of evolution. Since mutants are similar to the parent in phenotype, we can focus on the
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result near the diagonal line. If the area above the line is black, and if the area below the line is
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white, the mutant with a greater phenotype (k' > k) has a higher fixation probability than the
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resident and the mutant with a smaller phenotype (k' < k) cannot invade. In such a case, we
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conclude that k should increase during evolution.
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Figure S1a illustrates the case with small seedling survivorship ( ss  0.01). The
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population starts with a value of k smaller than the evolutionary attractor, k* = 1.4. Mutants with k'
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
a little larger than the resident k have a fixation probability greater than that of the neutral case
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(indicated in black). They can invade and subsequently replace the resident, and the value of k of
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the population approaches k*. In contrast, if the resident population has k greater than k*, a mutant
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with k' a little lower than k can invade and take over the population. As a result, the population
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mean k moves toward k*. k* is an evolutionary attractor because a population with different initial
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k gradually evolves toward k*. We expect that in the long run, k will converge to k *  1.4 ,
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although the stochasticity may occasionally result in fixation of a mutant that is not favored by
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natural selection.

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Figure S1b shows the fPIP for a higher survivorship of seedlings ( ss  0.5). For a wide
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range of initial values of k, the population evolves to the same end point as that obtained by direct
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simulation of evolutionary trajectories (Fig. 2).
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
The comparison of Figs. S1a and b again shows that the evolutionary outcome of k
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increases with seedling survivorship. Moreover, the neutral region tends to become wider as
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seedling survivorship increases. The width of the neutral range has an order of magnitude similar
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to the standard deviation of k, shown in Fig. 2.
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
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Fig. S1 Finite pairwise invasibility plots for different values of seedling survivorship ss . (a)
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ss  0.01, (b) ss  0.5. In each panel, the vertical axis represents the resource depletion
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
coefficient k’ of a mutant, and the horizontal axis represents the resident k in the range 0  k 10 .
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
The black region indicates that the mutant has a fixation probability significantly higher than the
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
neutral expectation judged by 1000 replicate simulations (P < 0.01). In the white region, the
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mutant has a significantly lower fixation probability than neutrality (P < 0.01). In the gray region,
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the mutant has a fixation probability not distinguishable from the expectation of the neutral case.
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Other parameter values are N = 100, = 0.04,  = 10, and β = 2.
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Appendix S2 Synchronization index
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In this paper perfect synchronization of seed production never occur, because some
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fraction of trees are immature, and even in mature trees, some have to take several years to
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synchronize.
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
To quantify the degree of synchronization, we defined the following quantity named the
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“synchronization index” (Satake & Iwasa 2002; Uriu, Morishita & Iwasa 2010):
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IS 
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where i t is the time series of the seed production of i-th site, t  is the average of i t
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over all trees, Vart t  is the temporal variance of the time series t , Vari i t  is the
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
Vart t 
Vart t 

,
Vartotal
Vart t  Mean t Vari i t 




between-tree variance at time t, and Mean t Vari i t  is the temporal average of Vari i t .





Consider the forest in which the trees are strongly synchronized in reproduction, but the
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 engaging I
year-to-year fluctuation of
the reproductive level is large, as is the case in forest
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masting. The within-year variance, Vari i t  is small but between-year variance, Vart t 
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is large. In this case IS becomes large (  1). In contrast, in a desynchronized forest, the mean
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does not change between yeas, and the forest always
 includes some
reproductive level of forest
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trees reproducing little, with
their fractions unchanged. In such forest, the within-year variance is
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large, but the between-year variance is small. As a result, IS becomes small (  0 ).
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
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