Supporting Information Notes S1, Table S1 & Fig. S1
Notes S1: Description of the simulation model
Following Kuuluvainen (1992), a cone was used as the basic shape. Although different shapes
may affect the balance between within- and between tree shading, different shapes (i.e. ellipsoids,
cones and cylinder) lead to similar trends (see Results below).
Whole tree performance (i.e. fitness; Ptree) was calculated as the sum of the net photosynthetic
rate of its individual leaves (see Eqn 1 in the main text).
The net photosynthetic rate of an individual leaf, Pnet(Cb), in a tree with a crown base Cb was
calculated as:
Pnet(Cb)= Pleaf(Cb) – Cs(Cb)
Eqn S1
with Pleaf being the photosynthetic rate of the leaf (μmol s-1 m-2) and Cs the costs of supporting
the leaf (μmol s-1 m-2). Pleaf can be calculated from the photosynthetic rate per unit leaf area (Parea,
μmol s-1 m-2) and the leaf area (LA):
Pleaf = Parea* LA
Eqn S2
Photosynthetic rate per unit leaf area (Parea) relates to the light intercepted by the leaf following a
rectangular hyperbola model (see Medlyn et al., 2000):
Parea = θ*Amax* Ileaf (cb) / (θ*Amax+ Ileaf(cb))
Eqn S3
with θ, the curvature factor and Amax the maximum photosynthetic rate (μmol s-1 m-2)
Ileaf (Cb) follows from Eqn S4 (Eqn 4 in main article).
Ileaf(Cb) =k*Io*e-k Fleaf(Cb)
Eqn S4
with Fleaf(Cb) being the total leaf area that light has travelled through to reach the leaf as a
function of the crown base, with k the light extinction coefficient, and Io the light above the
canopy.
Ileaf (Cb) of each leaf can then be found by calculating the Fleaf(Cb) of each of these leaves, by
multiplying the distance that light has travelled (D; m) with the leaf area density (m2 leaf area m-3
tree volume; LAD), yielding the m2 leaf area m-2 area the light has travelled through the reach the
focal leaf:
Fleaf(Cb)= Ds* LADs + Dn* LADn
Eqn S5
where the subscripts s and n denote the properties of the focal tree itself and those of the
neighbouring tree, respectively.
LAD can be found by dividing the total leaf area of the tree by the volume of the tree. Note that
this assumes a crown with constant leaf area density in both the horizontal and vertical planes.
Distance depends on the position of the leaf within the tree. To assign the height and horizontal
location to the leaves, 1 million leaves were assigned x, y and z coordinates (z being the height
direction), using the Cartesian coordinate system of a cone. For this, the crown was first divided
into 100 vertical segments. The height of the leaves within a segment was taken as the average
height within that segment, and calculated relative to the top (Hmax, value=1) and bottom of the
crown (Cb, value=0). Z-coordinates were then found by multiplying these relative values with the
crown length (the difference between Hmax and Cb). Similarly, for the x and y coordinates 100
relative coordinates were assigned with a relative distance from the middle (from -1 to 1). After
calculating the radius in the x-direction of each segment using the z-coordinate and y=0, 100 xcoordinates were assigned by multiplying the relative coordinates with the x-radius of the
segment. Similarly, for each x-coordinate the length to the edge in the y direction was calculated,
after which 100 y-coordinates were assigned by multiplying this distance with the relative
coordinates. This way the x, y, z coordinates of 1 million leaves were distributed over the volume
of the tree.
It should be noted that this large number of leaves is necessary to prevent chaotic behaviour
around the zero lines. This known modelling phenomenon occurs due to the sudden transition of
a leaf from a 1 to a 0 state; in biological terms, at a certain Cb a leaf transitions from shaded by
neighbours, thereby contributing to the benefit of a further increase in Cb, to a state where it is no
longer shaded by neighbours, contributing to a cost of a further increase in Cb. This can cause
chaotic behaviour in the model. Using a lot of leaves reduces this behaviour to acceptably low
levels.
The distance that light has travelled within the tree itself before reaching the leaf was then found
by ray tracing: calculating the intersection between the cone and a ray with an angle equal to the
solar elevation angle (αs) and ending at x, y, z. It was assumed that the tree is perfectly
symmetrical, hence the calculation is simplified by assuming light comes from the same ycoordinate, i.e. that the ray reaching the leaf has only travelled in the x- and z-direction.
The distance through the next neighbouring tree was calculated using the same ray tracing
calculations. Hence, the path of a ray was traced back from the leaf through the focal tree, and
then through the neighbouring trees, until Hmax was reached. Then, these distances where
summed. Note that similar to the study of Kuuluvainen (1992), it is assumed that each tree has its
own space, i.e. trees do not overlap.
Then, for each leaf, Parea can be calculated (Eqn S3).
Following Eqn S2, Pleaf can be found using the leaf area (LA) that each leaf represents, calculated
from the area that each x, y, z coordinate represents relative to the total leaf area of the plant.
Total leaf area is the product of the leaf area index (LAI, m2 leaf m-2ground area) times the square
of the radius.
Then, only the costs (Cs) are needed to calculate the carbon gain of an individual leaf (Eqn S1). I
use a pipe model (Shinozaka et al., 1964) to model the costs, i.e. it is assumed that each extra unit
of leaf area requires one extra unit of piping. Cs can be found by the product of leaf area,
sapwood per unit leaf area (Sa, m2 wood m-2 leaf), the total length of the pipe (Lp, m) and the
respiration rate of wood (Rw, μmol m-3 wood):
Cs= LA*Sa*Lp*Rw
Eqn S6
Lp can be found as the horizontal length (Lph) plus the height of the leaf, which in turn is found
by summing the Cb and the z-coordinate. Lph is found as the shortest distance of the x, y
coordinate to the centre of the plant:
Lph2= x2+y2
Eqn S7
Then, whole tree photosynthetic rate, and therefore a tree’s fitness (Ptree), is the result of Eqn S1
summed over all leaves.
Game theoretical approach
This model is used to calculate the Evolutionarily Stable Strategy value for Cb (ESS-Cb) at
different solar elevation angles. For clarity, the results of only three angles are presented.
First, for each Cb from 0 to Hmax, the whole tree photosynthetic rate (Ptree) is calculated assuming
all trees in the population have the same Cb values (see Fig. S1a). Following game theoretical
principles, the slope of this function, Ptree’(Cb) represents whether there is selection for an
increase in Cb (in biological terms: whether a mutant with a small deviation in Cb invades and
replaces the resident population). Hence, a local maximum (Cb*) is found at the Cb value where
this selection pressure becomes 0 (a potential predicted endpoint of crown base evolution; see
also Fig. S1b):
Ptree’(Cb)= 0
Eqn S8
These Cb* points are Evolutionarily Stable Strategies as defined by Iwasa et al. (1985),
effectively testing where a small change in Cb no longer lead to invasion. However, as this is a
simulation model we can also assess whether these local maxima are resistant to invasion by all
possible combinations (ESS following Maynard Smith, 1982).
Hence the fitness of an invader competing with a resident population with the Cb* value found in
Eqn S8, is calculated as:
Fi(Cb)= Ptree(Cb vs Cb*)-Ptree(Cb* )
Eqn S9
where Ptree(Cb vs Cb*) represents the invader competing with the resident population with the
Cb* value calculated with in Eqn S8, and Ptree(Cb*) the measure of how a resident tree with value
Cb* fares within the same resident population with value Cb*.
A mono-morphic ESS-Cb is defined when for all Cb values Fi(Cb) is smaller than 0.
For Eqns 1 and S4, light above the canopy (Io, μmol s-1m-2) was calculated using the formula:
Io= sin(αs) * In
Eqn S10
where In is light above the canopy when αs is 90°. Decreasing the Io exponentially with (sin(αs))2
resulted in only slightly lower ESS-Cb values, but did not change the general trends. Initial
simulations using a simple 2-D rectangular crown shape indicate that including the change in
solar elevation angle over the day resulted similar trends. The same applied to the sensitivity
analyses for the other parameters.
Table S1 lists the parameters used for the presented results.
The results of a cylinder based model are also discussed below. The model remains the same,
only the distance of light was calculated as the intercept between a ray and a cylinder, and the
LAD found by substituting he volume of a cone with that of a cylinder.
Results simulation model:
Fig. 1(a) shows that at low crown base values, an increase in crown base (Cb) leads to lower net
photosynthetic rates due to the increase in support tissue. The decrease in photosynthetic rate
becomes stronger with increasing crown base, due to an increase in self shading (i.e. increased
Fs), until the half of the crown that is furthest away from the direction of the sun also starts to
receive direct sunlight (i.e. when the angle of the apex becomes larger than the solar elevation
angle).
At the same time, Fig. 1(b) shows that at low Cb values, there is selection for increasing
the crown base: mutants with a slightly higher Cb can invade and replace the resident population.
However, at a certain point, the benefits of increasing the Cb become lower than the costs (slope
of function becomes negative); hence at high Cb values there is selection for decreasing the
crown base.
An interesting aspect that is unique to a cone shape is that when the angle of the apex is
higher than the solar angle, the level of self-shading is reduced, because light on the other half of
the cone will not travel through the first half of the tree. This also reduces the costs of increasing
the crown base at higher Cb values. But, as this increased lateral growth also comes at a cost
because of longer pipes, there still is selection for decreasing the crown base at these higher Cb
values, leading to a single attractor point (local maximum; in Iwasa et al. 1985-terms an ESS) for
each solar elevation angle (where slope=0). Sensitivity analysis shows that only when costs of the
pipes are approaching 0, and the crown base evolves to the point where almost no single leaf is
shaded by neighbours, a second solution arises: a branching point, where a decrease in Cb leads
to evolution towards the first local maximum found, and an increase will lead to runawayselection, where Cb approaches Hmax. Since I feel that with the current settings the costs of lateral
growth are underestimated, and the point would be unique to a particular shape, I do not think the
occurrence of a second attractor would be biologically meaningful.
Note that due to the geometry of rays, increasing Hmax would simply mean adding values to the
graph on the left side of the x-axis. That is, Cb would evolve to a different value, but crown depth
would be the same as Hmax changes with the same amount.
Fig. 1(c) shows that these attractor points are mono-morphic Evolutionarily Stable Strategies
(ESS): no mutant with a different Cb value can achieve a similar fitness at this state.
The results are consistent for the different solar angles (Fig. 1a,b). However, with a decrease in
solar elevation angle the ESS value of the crown base become higher. Hence at a lower solar
elevation angle, crowns are predicted to evolve to a shallower crown.
With a cylinder as the basic shape, the results are similar. The only difference is in the
absolute values: a cylinder shape is predicted to lead to thinner crowns (ESS-Cb= 23.1, 26.8 and
28.3. for solar elevation angles of 22.5°, 45.0° and 67.5° respectively) compared to cone shaped
trees at the same latitude (ESS-Cb= 11.3, 23.9, and 27.6 respectively). This is due to the greater
level of shading between individuals with a cylinder shape, and the lower levels of self-shading.
Constricting a tree to a cone shape inherently introduced large spaces between individuals at the
higher layers of the canopy.
One exception to the globally Evolutionarily Stable Strategy that is predicted by most
simulations, however, is that if the total costs of the residents at the local maxima are high
enough, a mutant with leaves all along the vertical could invade (note the increase of mutant
fitness with decreasing Cb in Fig. 1c). For instance, costs could be higher through a much higher
Hmax, yielding a much higher total stem costs. This is more likely to occur at low solar angle,
because at its local maximum the resident trees have a higher Cb, and therefore more stem costs.
I think this is only a realistic scenario in the mathematical sense. It would mean that at lower
solar angle, there is a higher chance that the selection towards shallower crowns starts all over
again. But as the local maximum does not change, one would end up towards a trajectory of
thinner crowns over and over again (an ESS as defined by Iwasa et al., 1985). Biologically, I
think it unlikely that a tree that does not have some sort of turnover will get to this “adult” stage.
More importantly, it is mainly a consequence of sticking to the cone shape. The reason I choose
for volume to change, and not to let gaps appear between trees by keeping volume constant and
changing the radius instead, is that all model simulations showed trees should fill all possible
gaps, unless lateral costs become very high. This latter case will only arise for branches placed
high up in the canopy, when there is a stronger gravitational pull. Therefore, if a mutant with
leaves all the way down would establish itself, the space it will not occupy higher up will be
quickly filled in by the neighbouring trees, shading it, and hence preventing it from obtaining a
high Ptree, hence preventing it from invading. Thus, the potential of a mutant to invade an
attractor population is unique for the cone shape, in that cone-shaped trees allow for ample space
between individuals.
Table S1 List of parameters used for the presented results and literature sources
Abbreviation
Trait
Unit
Value
Source
Amax
Maximum rate
µmol m-2 s-1
10.0
Mäkelä (2002)
m
30.0
-
dimensionless
0.05
Anten & Werger
of
photosynthesis
of the leaf
Hmax
Maximum
height of the tree
Θ
Curvature factor
(1996)
K
Extinction
dimensionless
0.14
Mäkelä (2002)
Sapwood per
m2 wood m-2
1/4669.0
Falster et al.
unit leaf area
leaf area
Sapwood
µmol m-3 s-1
coefficient
Sa
Rw
Io
(2011)
127.24
Calculated from
respiration of
Falster et al.
wood
(2011)
Noon PPFD
µmol m-2 s-1
2000
above the
Hirose & Werger
(1987)
canopy
LAI
Leaf area index
m2 leaf m-2
6.0
of individual tree ground surface
R
Radius (m)
m
Wang & Jarvis
(1990)
3.0
-
Fig. S1 Effects of solar angle and Crown base (Cb; x-axis) on the outcome of the simulation
model.
(a) Net photosynthetic rate (y-axis) of a resident population with crown base value Cb (x-axis),
expressed per individual tree.
(b) The selection pressure to increase the crown base (y-axis) given the resident’s crown base
value (x-axis), expressed as the change in net photosynthetic rate with a small increase in crown
base compared to net photosynthetic rate of the resident population (Pnet d/dCb). A positive value
indicates benefits to a mutant, negative means a lower fitness than the resident population. Hence,
local maxima are the points where the function goes from positive to negative (circles; see Eqn
S8).
(c) Difference in fitness (y-axis) between a mutant with crown base value Cb (x-axis) and that of
a resident population that has the crown base value found in the local maxima of Fig. 1(b) (note:
three different local maxima values for the three solar angles, indicated by top circles). Values on
the y-axis represent the fitness difference (i.e. difference in net P between the invader and the
resident population (see Eqn S9); so, when this value is lower than 0 along the whole spectrum of
potential Cb values, no mutant can invade the resident population. For all angles, values are
lower than 0, unless mutant Cb value= resident Cb value, when the difference=0. This means the
attractor Cb values found in Fig. 1(b) are ESS’s, one for each solar angle.
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