Functional Ecology 2012
doi: 10.1111/j.1365-2435.2012.02042.x
An evolutionary game of leaf dynamics and its
consequences for canopy structure
Kouki Hikosaka*,1,2 and Niels P. R. Anten†,3
1
Graduate School of Life Sciences, Tohoku University, Aoba, Sendai, 980-8578 Japan; 2CREST, JST, Chiyoda,
Tokyo, 102-0076 Japan; and 3Ecology Biodiversity, Institute of Environmental Biology, Utrecht University, 6 P.O. Box
800.84, 3508TB, Utrecht, The Netherlands
Summary
1. Canopy photosynthesis models combined with optimization theory have been an important
tool to understand environmental responses and interspecific variations in vegetation structure
and functioning, but their predictions are often quantitatively incorrect. Although evolutionary
game theory and the dynamic modelling of leaf turnover have been suggested useful to solve
this problem, there is no model that combines these features.
2. Here, we present such a model of leaf area dynamics that incorporates game theory.
3. Leaf area index (LAI; leaf area per unit ground area) was predicted to increase with an
increasing degree of interaction between genetically distinct neighbour plants in light interception. This implies that stands of clonal plants that consist of genetically identical daughter
ramets have different LAI from other plants. LAI was also sensitive to the assumed vertical
pattern of leaf shedding: LAI was predicted to increase with the degree to which leaves were
assumed to be shed from higher positions in the canopy. Our model provides more realistic
predictions of LAI than previous static optimization, dynamic optimization or static game
theoretical models.
4. We suggest that both leaf dynamics and game theoretical considerations of plant competition are indispensable to scale from individual leaf traits to the structure and functioning of
vegetation stands, especially in herbaceous species.
Key-words: canopy photosynthesis, evolutionarily stable strategy, game theory, leaf area
index, leaf turnover, light competition, model, nitrogen use, optimization
Introduction
The development of a theoretical framework for physicochemical and physiological functioning of plant vegetation
has been a fundamental challenge in biology, ecology, agriculture and meteorology for almost a century (Boysen
Jensen 1932; Monsi & Saeki 1953; de Wit 1965; Hirose
2005). Its importance is increasing especially in global sciences because vegetation functioning is not only sensitive
to climate change but also feedbacks to global climate
(IPCC 2007). Current models of canopy photosynthesis
are able to predict carbon uptake of vegetation if correct
values of canopy structure and leaf physiology are given
(e.g. Baldocchi & Harley 1995; Wilson, Baldocchi & Hanson 2001; Ito et al. 2006). However, both canopy structure
and leaf physiology greatly vary among stands depending
on climate, growth environment and species composition,
*Correspondence author. E-mail: [email protected]
†
Present address. Center for Crop System Analysis Wageningen
University PO Box 430 6700 AK Wageningen The Netherlands.
and such data are not available for most vegetation. Our
ability to predict plant responses to climate change is thus
still limited (Dewar et al. 2009).
Optimality theory has been a powerful tool to predict
environmental responses of canopy structure and leaf
physiology (Hirose 2005; Dewar et al. 2009; Anten & During 2011). Optimality theory is based on the concept that
some performance measure is maximized with respect to
one or more plant traits and one or more limiting factors.
For example, canopy photosynthesis per unit ground area
may be maximized if the canopy has a leaf area index
(LAI, amount of leaf area per unit ground area), whereby
the lowest leaves receive the light intensity of the compensation point for daily carbon gain (Monsi & Saeki 1953;
Saeki 1960; Ackerly 1999; Reich et al. 2009). This explains
why a canopy with vertical leaves, which allow more light
penetration into lower canopy layers, has a larger LAI
than that with horizontal leaves (Saeki 1960). Nitrogen
limitation may impose an additional constraint on leaf
area growth, as a larger LAI at fixed canopy N will lead to
© 2012 The Authors. Functional Ecology © 2012 British Ecological Society
2 K. Hikosaka & N. P. R. Anten
a dilution of leaf nitrogen content per area. This reduction
in leaf nitrogen per area in turn results in lower leaf photosynthetic capacities, since photosynthetic capacity and leaf
nitrogen per area are tightly correlated (Field & Mooney
1986; Evans 1989; Hikosaka 2004, 2010). An optimal LAI
can be derived at which whole-stand canopy photosynthesis at a given canopy nitrogen is maximized (Anten et al.
1995b). This optimal LAI increases with increasing canopy
nitrogen, which well explains the strong positive correlations that exist between LAI and nitrogen availability (e.g.
Prasertsak & Fukai 1997). Recent studies have used this
optimization approach to predict canopy traits (e.g. leaf
nitrogen content, stomatal conductance or leaf photosynthetic capacities), LAI and vegetation carbon gain under
climate change scenarios (e.g. Franklin 2007; Mäkelä,
Valentine & Helmisaari 2008; McMurtrie et al. 2008).
However, predictions by optimality models are not necessarily correct in a quantitative sense (Gersani et al. 2001;
Anten & During 2011). For example, optimal LAI values
calculated based on trait values from actual plant canopies
were always smaller than the actual ones (Anten 2002).
This discrepancy has been ascribed mainly to two assumptions in the optimality models. First, plant traits are
assumed to be optimal when they maximize whole-canopy
daily photosynthesis. This implicitly assumes that the performance of a plant is independent of the characteristics of
its neighbours (Parker & Maynard-Smith 1990). This does
not hold true in most vegetation stands where plants compete for light and soil resources. In such cases, evolutionary game theory (EGT), in which individual plant-based
maximization is considered relative to the characteristics
of neighbours, is more appropriate approach (Givnish
1982; Falster & Westoby 2003). Hikosaka & Hirose (1997)
combined EGT with a canopy photosynthesis model and
showed that a leaf angle that maximizes canopy photosynthesis is not necessarily evolutionarily stable especially
when the light competition between neighbours is strong.
It has similarly been shown that evolutionarily stable LAI
(ES-LAI) is greater than the optimal LAI at a given canopy nitrogen content (Schieving & Poorter 1999; Anten &
Hirose 2001; Anten 2002, 2005). Predicted ES-LAI values
have been found to be much closer to actual measured values than optimal LAIs (Anten 2002; Lloyd et al. 2010).
These results suggest that natural selection may lead to
plant communities with non-optimal characteristics in
terms of maximized photosynthesis at the community
level.
The second problem with classic optimization models is
that the traits underlying whole-plant photosynthetic nitrogen-use efficiency, that is, nitrogen distribution, LAI and
other traits, are treated as being static. However, leaf canopies are dynamic: new leaves are produced using photosynthates, nutrients that are absorbed by roots and resorbed
from older leaves are allocated, and old leaves are shed
with some fraction of allocated nutrients (Kikuzawa 2003;
Hikosaka 2005; Oikawa, Hikosaka & Hirose 2005; Hikosaka, Kawauchi & Kurosawa 2010). LAI and canopy
nitrogen content thus depend on various factors such as
nutrient uptake rate, leaf longevity and nutrient resorption
efficiency. Franklin & Ågren (2002) indicated that nitrogen
resorption efficiency affects optimal LAI: since plants lose
nitrogen with dead leaves, reducing LAI is not necessarily
advantageous even when the LAI is greater than the LAI
that maximizes canopy photosynthesis. They showed that
optimal LAI increases with decreasing resorption efficiency. Hikosaka (2003) developed a dynamic model of
leaf canopy. In the model, leaf area in the canopy increases
in time with the production of new leaves, which is proportional to the rate of photosynthesis in the canopy. At each
time step, uptake of nitrogen from the soil increases the
amount of nitrogen in the canopy. The optimal LAI that
maximizes canopy photosynthesis is then calculated. If leaf
area is in excess, old leaves are eliminated, and part of
nitrogen is lost with dead leaves. Consequently a new canopy having an optimal LAI with a given amount of nitrogen is obtained. Repeating this process simulates the
temporal dynamics of leaves in a growing canopy.
Recently, several models have been developed incorporating not only leaf dynamics but also processes in non-photosynthetic tissues (Franklin 2007; Mäkelä, Valentine &
Helmisaari 2008; Franklin et al. 2009).
Although the two problems, that is, competition and
canopy dynamics, have been overcome through application of the game theory and of the dynamic optimization, respectively, these two approaches have not been
combined together; that is, no one has applied EGT to
analyse leaf turnover (Hikosaka 2005; Anten & During
2011). Such an analysis would provide new insights into
the way that natural selection might have acted on leaf
turnover as a function of the degree of interaction
between neighbouring plants. Here we develop for the
first time a game theoretical model of leaf dynamics, and
thus combine two key features of plant canopies – competition and leaf turnover – that have not previously
been combined in any vegetation model. We modify the
optimality model of leaf dynamics of Hikosaka (2003)
and incorporate EGT into the model. We show that the
vertical pattern of leaf fall strongly affects the evolutionarily stable leaf area dynamics. We also compare the real
and the modelled ES-LAI of various herbaceous stands
using published trait values.
The model
INTERACTION BETWEEN NEIGHBOURS
We assume that plants compete for light with their neighbours. Each plant occupies a certain ground area and
develops leaf area within that ground area. There is no
overlap of foliage between neighbours. We can thus define
LAI at an individual plant level (note that LAI values are
the same between individual- and stand-level if the individual-level LAI is identical among individuals). Plants are
considered to influence each other’s light climate because
© 2012 The Authors. Functional Ecology © 2012 British Ecological Society, Functional Ecology
An evolutionary game of leaf dynamics
light comes not only from right above but also from
various other directions. We thus assume that at any point
in the canopy of a target plant, a fraction (g) of the radiation will have passed through canopies of neighbouring
plants and a fraction (1–g) through the canopy of the target plant itself (Hikosaka et al. 2001). Thus, g indicates
the degree of interaction with neighbours (i.e. effect of
non-self relative to total shading). The photon flux density
on a horizontal surface at a layer j around a leaf of the
target (ITj) is then described by:
ITj ¼ ð1 gÞ I0 expðKT FTj Þ þ g I0 expðKN FNJ Þ
eqn1
where the subscripts T and N indicate the target individual
and its neighbours, respectively, I0 is the I at the top of the
canopy and Fj is the cumulative LAI above layer j over the
fraction of ground area occupied by the plants.
CANOPY PHOTOSYNTHESIS AND LEAF DYNAMICS
Photosynthesis of an individual is calculated based on the
canopy photosynthesis model of Anten, Schieving & Werger (1995a). Here we explain the model briefly (see Data
S1, Supporting information). Light dependence of the photosynthetic rate is formulated with a non-rectangular
hyperbola. Both the light-saturated rate of photosynthesis
and dark respiration rate are linearly related to the nitrogen content per unit area. We divided the foliage into 100
horizontal layers and the photosynthetic rate of an individual is obtained as the sum of photosynthesis in each layer.
Nitrogen is always reallocated optimally among layers to
maximize photosynthesis.
Dynamics of leaf area is based on the model of Hikosaka (2003), with the addition of a game theoretical sensitivity analysis. Here we describe the model briefly (see
Data S1, Supporting information for detail), except for
the game theoretical part that is described in detail. We
initialize the simulation by setting the LAI and canopy N
of an individual plant to given starting values. The plant
allocates leaf nitrogen to each layer so that the canopy
photosynthesis is maximized (Anten, Schieving & Werger
1995a). The model then runs assuming time steps of
10 days. During this time step, the plant photosynthesizes
and allocates a part of the newly obtained assimilates for
construction of new leaves. We assumed this part that is
allocated to leaves as 40% of total assimilates, which is
regarded as the maximum after subtracting allocation to
other organs. New leaves are formed with a given leaf
mass per area after subtracting construction costs. The
plant simultaneously takes up nitrogen at a given rate
that is assumed to be constant during the simulation. At
the end of each time step, the LAI will have reached a
new value (termed as N-LAI). We then apply a game
theoretical sensitivity analysis (see sensitivity analysis
below) to determine the extent to which the stand can be
invaded by a ‘mutant’ individual that sheds some of its
leaves. At each time step, we thus calculate the ES-LAI.
If the value of N-LAI is greater than the ES-LAI, exces-
3
sive leaf area is eliminated after part of the nitrogen is
resorbed. The foliage with ES-LAI photosynthesizes and
produces new leaves. Repeating this process provides
growth of LAI.
GAME THEORETICAL SENSITIVITY ANALYSIS
Evolutionarily stable LAI at each time step is obtained as
follows. The target individual has the same trait values as
those of its neighbours. Both the target and neighbours
produce new leaves and then have the same amount of leaf
area (N-LAI). (i) We calculate the photosynthesis of the
target, which will evidently be the same as that of neighbours (PN). (ii) We then simulate leaf shedding by slightly
reducing the LAI of the target plant (01%). This reduction in LAI thus leads to some loss of N (nd in the model),
as not all N can be remobilized from senescing leaves
(Aerts & Chapin 2000), while the retranslocated N is
assumed to be optimally reallocated among the remaining
leaf layers (as in Anten, Schieving & Werger 1995a), resulting in N contents and associated photosynthetic capacities
of those leaves. We then calculate whole-plant photosynthesis in the new situation (PT). If PN is higher than PT,
the N-LAI is regarded as evolutionarily stable. (iii) If not,
we reduce LAI of the neighbours to the same level of the
target and calculate photosynthesis of the target (PN′). (iv)
We further reduce LAI of the target only and obtain target’s photosynthesis (PT′). The processes three and four
are repeated until we obtain PN′ > PT′. The LAI of an
individual that realizes the PN′ at PN′ > PT′ is regarded as
ES-LAI. We did not increase LAI because plants produced
maximal LAI that allowed by their assimilates. In the next
step, both target and neighbours have again the same LAI
as a result of the game. Such vegetation stands are considered to be resistant to invasion throughout the growing
season.
ASSUMPTION OF SPATIAL PATTERN OF LEAF
SHEDDING
As mentioned earlier, plants shed excessive leaf area. In
dense vegetation, light absorption of a given plant depends
strongly on the vertical distribution of leaf area relative to
that of its neighbours. It is therefore important to consider
the vertical pattern of leaf senescence; a leaf dropped from
an upper layer in the canopy may have larger positive
effect on neighbours’ light acquisition than dropping a leaf
from a low layer. That is, in terms of light competition it
is more efficient to drop leaves only from the lowest than
to drop them also from higher in the canopy. Here we consider three patterns (Fig. 1). In Case 1, leaves are dropped
from all layers equally. This spatial pattern is close to the
static EGT model of Anten (2002). In Case 2, senescent
leaves are dropped strictly from the bottom of the canopy.
Case 3 is intermediate between the two: relatively more
leaves are dropped from the lower parts of the canopy
than from higher up. It is hereby assumed that leaf area
© 2012 The Authors. Functional Ecology © 2012 British Ecological Society, Functional Ecology
4 K. Hikosaka & N. P. R. Anten
Living
leaves
Living
leaves
Senescing leaves
Case 1
Case 2
Figure 2 shows simulation results of leaf area increment as
a function of time using data of Glycine max (Anten,
Schieving & Werger 1995a). When the degree of interaction (g) is zero (no interaction with neighbours), LAI
exponentially increases at the beginning and the increment
rate gradually decreases because of leaf shedding. Finally,
LAI reaches a steady state, where leaf production rate is
equal to leaf loss rate. These results are almost independent of the assumed vertical pattern of leaf senescence
(Cases 1–3) and are quantitatively very similar to those
obtained in the dynamic optimization model (Hikosaka
2003). When g is larger than zero (i.e. neighbour plants
affect each other’s light interception), simulation patterns
differ strongly between the three cases. In Case 1, LAI
greatly increases with increasing g (Fig. 2a). When g is
very high, LAI achieves 20 and then rapidly decreases to
zero (data not shown), because canopy photosynthesis
becomes negative when LAI is very high (our model is not
designed to provide realistic response of canopy traits
when carbon gain is negative). In Case 2, LAI growth is
almost independent of g and thus the results are very similar to those of the dynamic optimization model (Fig. 2b).
This is because the alteration in leaf area at bottom layers
hardly affects light interception of neighbours. In Case 3,
LAI increases with increasing g but to a lesser extent than
in Case 1 (Fig. 2c).
Here we compare results of different models: static- vs.
dynamic-plant, and simple optimality vs. game theoretical.
Figure 3 shows the relationship between LAI and canopy
nitrogen for an g of 05 at 300 days (nearly steady state in
most situations). As mentioned earlier, results for Case 2
are almost identical to those of the dynamic optimization
model (Hikosaka 2003) where no light competition among
Senescing
leaves
Living
leaves
Results
Senescing leaves
Canopy depth
Leaf area in a layer
Case 3
Fig. 1. The three vertical pattern of leaf shedding used in the
simulations (Cases 1–3).
loss increases linearly from the top towards the bottom of
the plant.
COMPARISON OF PREDICTED AND REAL CANOPY
TRAITS
We collected data of leaf photosynthesis and canopy traits
of 21 stands of 10 species from published articles (Hirose &
Werger 1987; Schieving et al. 1992; Anten et al. 1995b;
Anten, Werger & Medina 1998; Anten 2002; Borjigidai,
Hikosaka & Hirose 2009; see Table S1 (Supporting information), including three stands grown at elevated CO2, and
simulated canopy growth with these data. Four stands are
of clonal species (indicated by ‘çlonal’ in Table S1, Supporting information) and three of them tend to form dense
mono-clonal patches. We assumed 2000 lmol m2 s1 for
noon irradiance and calculated the daily pattern of light
intensity above the canopy from this value following Hirose
& Werger (1987). Simulation started from a small canopy
with LAI = 05 and canopy nitrogen = 50 mmol m2. We
found that LAI in the steady state was independent of the
starting conditions.
5
(a)
20
(b)
4
Case 1
1·0
Case 2
3
0·75
15
2
ES-LAI (m2 m–2)
Interaction = 0·5
1
10
0
1·0 0·75
7
(c)
6
Case 3
0·5
6
5
0·75
0·5
5
4
0·25
4
0·25
3
3
Interaction = 0
Interaction = 0
2
2
1
1
0
1·0
0
50
100 150 200 250 300 350 400
0
0
50
100 150 200 250 300 350 400
Day
Fig. 2. Simulation of growth of evolutionarily stable leaf area (ES-LAI). Noon irradiance and nitrogen uptake rate are
2000 lmol m2 s1 and 5 mmol m2 day1, respectively. Values from a Glycine max canopy (Anten, Schieving & Werger 1995a) are used
for leaf and canopy traits. a, b and c shows Cases 1, 2 and 3, respectively. Note that the scale of ES-LAI changes above 8 in (a). There is
no difference among lines in Case 2.
© 2012 The Authors. Functional Ecology © 2012 British Ecological Society, Functional Ecology
An evolutionary game of leaf dynamics
8
S1
LAI (m2 m–2)
Case 1
6
S3
Case 3
Case 2
4
S2
2
0
0
200
400
600
800
Canopy nitrogen (mmol m–2)
Fig. 3. Leaf area index as a function of the canopy nitrogen per
unit ground area. g is 05 in every case. Open symbols denote
calculated results at 300 days under nitrogen uptake rates of
1–6 mmol m2 day1, and triangle, square and diamond denote
Cases 1, 2 and 3, respectively (two data points of Case 1 are outside the frame). Continuous lines are the regression using linear
(Case 1) or quadratic (Cases 2 and 3) functions. Dotted lines are
the regression for static evolutionary game theory models where
no nitrogen loss at leaf shedding is assumed (symbols are not
shown). S1, S2 and S3 assume the same leaf shedding pattern as
Cases 1, 2 and 3, respectively. Closed circle denotes data obtained
in a real stand of Glycine max (Anten, Schieving & Werger
1995a).
individuals is assumed. Results of static EGT models (S1,
S2 and S3) are also calculated with an assumption that the
nitrogen is not lost by decreasing LAI. The model S1, S2
and S3 assume the same leaf shedding pattern as in Cases
1, 2 and 3, respectively. S1 is similar to the static EGT
model of Anten (2002), and S2 is almost identical to the
static optimization model (Anten et al. 1995b). LAI shows
a linear (Case 1), convex (i.e. saturating with decreasing
slope Cases 2 and 3 and S2) or concave (i.e. with increasing slope, S1 and S3) relationship with canopy nitrogen.
The convex saturating relationship is consistent with previous experimental results, that is, mean nitrogen content
per unit leaf area (canopy N per LAI) increases with
increasing nitrogen availability (e.g. Anten et al. 1995b).
This suggests that the general pattern of the relationship
between LAI and nutrient availability is better predicted
by dynamic EGT models (Case 3) than by static EGT
models (S1 and S3).
When compared at the same canopy nitrogen, LAI is
lowest in the static optimization model (S2). LAI in
dynamic models is largest for Case 1, followed by Case 3,
and Case 2. LAI in static EGT models (S1 and S3) is
higher than that in the static optimization model especially
at lower canopy nitrogen. LAI in Case 1 is higher than
that in S1 across all canopy nitrogen values. LAI in Case 3
is higher than that in S3 at canopy nitrogen values lower
than 650 mmol m2 while the opposite holds at the higher
canopy nitrogen values. The actual LAI values obtained
from real stands are closer to the predicted values based
on Case 3 than on those based on the others.
Leaf area index values under various nitrogen uptake
rates were calculated using leaf traits obtained from 21
herbaceous stands. We obtained a regression line between
LAI and canopy nitrogen at the steady state for each species (see Fig. 3 for G. max) and calculated LAI at the canopy nitrogen content observed in the real stand. Figure 4
shows the predicted LAI values calculated based on the
Cases 1–3, as a function of the 21 measured LAI values. In
all three cases, predicted LAI was strongly correlated with
real LAI (r2 > 045) but the relationship was quantitatively
different among leaf shedding patterns. When Case 1 was
used, the predicted LAI was much greater than the real
LAI (Fig. 4a). Case 2 predicted LAIs that were slightly
smaller than real values (Fig. 4b). The regression of predicted on real LAI was very similar to the 1:1 relationship
in Case 3 (Fig. 4c). However, in the case of the three clonal grass species, which formed dense mono-clonal
patches, predicted LAI values were closer to real ones in
Case 2 than in Cases 1 and 3 (open symbols in Fig. 4).
In the simulation, an ESS is calculated every 10 days.
This time step may correspond to plastochron length,
which varies from 2 to 10 days in herbaceous plants dominating in open habitat (Hofstra, Hesketh & Myhre 1977;
Oikawa, Hikosaka & Hirose 2005). We applied various
additional time steps, however, to test whether the simulation results are robust (Fig. S1, Supporting information).
In Cases 1 and 2, LAI slightly increases with increasing
20
8
(a)
(b)
6
Case 1
r 2 = 0·45
Case 2
r 2 = 0·52
4
15
Predicted LAI (m2 m–2)
10
5
2
0
10
10
(c)
8
Case 3
r 2 = 0·53
6
5
4
2
0
0
1
2
3
4
5
6
Real LAI
0
0
1
(m2
m–2)
2
3
4
5
6
Fig. 4. Predicted and real leaf area index using data obtained from
21 stands of 10 species (see Table S1, Supporting information for
species list). a, b and c shows Cases 1, 2 and 3, respectively. Three
clonal species (H. amplexiculis, Leersia hexandra, Paspalum fasciculatum) are given as open symbols. g is 05 in every calculation.
Solid and broken lines are the regression for all data points and
the 1 : 1 relationship, respectively.
© 2012 The Authors. Functional Ecology © 2012 British Ecological Society, Functional Ecology
6 K. Hikosaka & N. P. R. Anten
time step length, but the relationship between LAI and
canopy nitrogen is not affected. In Case 3, the increase in
time step slightly decreases LAI at a given canopy nitrogen, but the difference was smaller than 10%. These results
indicate that our findings are robust and do not depend on
the chosen time step.
Discussion
The model presented in this study is the first to combine
canopy photosynthesis, leaf dynamics and EGT together.
Our results clearly show that the predicted LAI is influenced strongly by the assumptions regarding the leaf area
dynamics and degree of light competition among neighbouring plants. Both competition and leaf turnover are
dominant processes in vegetation stands, and we show that
their inclusion in canopy models is needed to make realistic predictions of LAI. The LAI predicted by the static
optimization model have been shown to be consistently
lower than real LAIs (Anten et al. 1995b, 2004; Anten
2002; Hirose et al. 1997; S2 in Fig. 3). Franklin & Ågren
(2002) suggested that incorporating dynamics of leaf and
nitrogen improves the prediction of LAI, but their model
did not consider the temporal dynamics of N uptake and
leaf turnover. We incorporated leaf dynamics using multiple time steps into the optimization model and, compared
to the static optimization model, obtained a better match
with observed LAIs (Case 2 vs. S2 in Fig. 3). Even so,
these predictions were still lower than the actual LAI in
most cases (Fig. 4b). On the basis of the view that optimal
LAIs may be evolutionarily unstable as they can be
invaded by mutants producing larger leaf areas, Anten
(2002) proposed that static EGT model should provide
better predictions of LAI than simple optimization models.
However, we show that the response of LAI to nitrogen
availability is unrealistic in the static EGT model; LAI
increases more than proportionately with canopy nitrogen
(S1 and S3 in Fig. 3). On the other hand, our Case 3
model successfully predicts realistic responses of LAI to
nitrogen availability (Fig. 3) and quantitatively valid values of LAI (Fig. 4). We thus indicate that both leaf
dynamics and competition are important factors for
determining LAI in real plants.
Predicted LAI values were higher when plant competition was taken into account as in EGT models than when
it was disregarded as in the simple optimization model
(Figs 2,3 and 4). Leaf shedding may have a positive effect
on photosynthesis because it results in concentrating nitrogen in the remaining leaves thus enhancing their photosynthetic capacity, but also has a negative effect as it reduces
area for light capture. There exists an optimum where
these two effects are balanced. However, when plants compete for light, a reduction in leaf area in one plant not only
reduces its own light acquisition but also enhances light
availability to neighbours. A delay of leaf shedding may
thus be advantageous, and by consequence, the ES-LAI
(i.e. a population with this LAI cannot be invaded by a
mutant with other leaf dynamics) is higher than the
optimal LAI (Anten 2002, 2005).
Our results show that the vertical pattern of leaf shedding
strongly affects ES-LAI. Previous static EGT models
assumed that leaf area is similarly altered in every vertical
layer (Schieving & Poorter 1999; Anten & Hirose 2001;
Anten 2002; Case 1 in Fig. 1). Anten (2002) showed that the
static EGT model provides more realistic values of LAI than
those of optimal models (S1 in Fig. 3). In the present study,
however, the Case 1 model, which used the same shedding
pattern as Anten (2002), provides unrealistically high values
of LAI (Figs 3 and 4). As the inclusion of leaf dynamics and
that of competition may both lead to an increase in predicted LAI, a combination of these aspects results in very
high LAI values. On the other hand, the Case 2 model, in
which leaves are shed only from the bottom, provides lower
LAI values than that of the Case 1 model. Its predicted LAI
values are almost identical to those by dynamic optimization
model (Hikosaka 2003). The Case 3 model, in which leaves
are shed from all layers but more from lower than from
higher layers, predicts ES-LAI values that are intermediate
between those of Cases 1 and 2. Moreover, the predictions
assuming Case 3 converge most closely to the real measured
values of LAI (Figs 3 and 4). The considerable differences
between the predictions from Cases 1 to 3 probably reflect
differences in the assumed degree of leaf area loss in the
upper layers. Because light availability is greater at upper
layers, small change in upper layers has a large influence on
light availability of neighbours and thus ES-LAI should be
high (Case 1). If the leaf area reduction occurs only at lower
layers, benefit of the delaying senescence is limited and
ES-LAI should be close to the optimal LAI (Cases 2 and 3).
Considerable differences in the predicted LAI among
leaf shedding patterns imply that ES-LAI may differ
between plants with different growth forms. For example,
erect herbaceous species develop leaves mainly from meristems located towards the top of the plant and shed them
mainly from the bottom. In some grass species that maintain their meristem near the ground surface, on the other
hand, shedding of a long leaf often results in loss of leaf
area from several canopy layers. In our data set, Carex
acutiformis develop their leaves from the ground (Hirose,
Werger & van Rheenen 1989). There was no obvious difference between C. acutiformis and other species in the
LAI–nitrogen relationship (data not shown), which is not
consistent with the hypothesis. However, our data set may
not be sufficient to test this hypothesis because of the limited number of species for each growth form. In addition,
in erect plants, leaves attached to lateral branches often
remain even when leaves attached to main stem are shed
(K. Hikosaka, personal observation); Case 3 may thus be
applicable to erect plants. Further study on the spatial
pattern of leaf senescence in combination with our modelling approach may be necessary for a better understanding
of its ecological significance in leaf dynamics.
Our results show that the predicted LAI is sensitive to the
degree of interaction with neighbours, g: the predicted LAI
© 2012 The Authors. Functional Ecology © 2012 British Ecological Society, Functional Ecology
An evolutionary game of leaf dynamics
increased with increasing g (Fig. 2). As g may be greater
when plant density is higher, our result is consistent with the
fact that plants increase their leaf area by increasing specific
leaf area when plant density is high (Nishimura et al. 2010).
Parameter g may also be influenced by other factors such as
vertical length of foliage cluster, leaf size, petiole length and
so on. For example, g is probably low in trees that tend to
have relatively broad crowns, but relatively high if the plant
has vertically long crown (e.g. erect herbaceous plants).
Some climbing plants may have particularly high g values.
g has not been estimated in real plant stands except for two
stands of an annual, Xanthium canadense, which had relatively high values (07 and 085 for low- and high-density
stands, respectively; Hikosaka et al. 2001). However,
X. canadense may have exceptionally high g values even for
herbaceous plants. When it was grown at a density where
the distance between plants was 125 cm, mean petiole
length of fully expanded leaves was 12 cm, that is, leaves
were placed in spaces occupied by neighbours (Hikosaka
et al. 2001). Such a horizontal overlap of foliage cluster is
not observed in stands of other herbaceous species such as
Chenopodium album (K. Hikosaka, personal observation),
which may have lower values of g. Evaluation of the degree
of interaction in various plant stands would be necessary.
Stands of clonal plants, that is, plants that propagate
vegetatively whereby mother ramets produce genetically
identical daughter ramets along horizontal spacers (e.g.
stolons and rhizomes), provide an interesting case for game
theoretical analyses of plant interactions. This is because
the degree of non-self/self-interaction depends on the clonal architecture, and neighbour-dependent responses such
as analysed here may have evolved differently among species depending on their clonal structure (Semchenko et al.
2007). For example, in stands of plants exhibiting the socalled phalanx growth form, which involves the production
of short spacers and associated clustering of genetically
identical ramets, the degree of self-shading may predominate, and thus g would be very low. Such plants would be
expected to exhibit more optimal leaf strategies resulting
maximum stand-level performance (Hikosaka & Hirose
1997; Anten & During 2011). In our data set, we had four
stands of clonal species (Leersia hexandra, Hymenachne
amplexicaulis, Paspalum fasciculatum and Solidago altissima). The first three tend to form large mono-genotypic
patches with very high density (>1000 plants m2; N.P.R.
Anten, personal observation). Interestingly, the real LAIs
of the stands of these three species were considerably lower
than the values that the EGT model predicted under the
assumption that g = 05 (Fig. 4). Lower g values, thus
assuming more self-shading, yielded more accurate predictions (data not shown). Their real LAI is closer to the predicted LAI in Case 2 than that in Case 3, suggesting that
they had a more optimal LAI. This result is consistent with
our expectation that clonal plants may have optimal strategies rather than evolutionarily stable ones.
In the present model, we assume that at every time step,
the plants realize ES-LAI. This implicitly assumes that
7
plants that do not realize this trait value would be eliminated from the stand. We also assumed that in each time
step, plants produce maximal leaf area at the top of the
canopy, which is allowed by their assimilates; otherwise,
plants may be overshaded by neighbours that produced
greater leaf area. These assumptions are based on the fact
that competition for light tends to be asymmetric. That is,
if, at any point during its vegetative growth, a plant fails
to develop sufficient leaves at the top of the canopy, it will
be shaded by the neighbours, resulting in a reduced growth
rate, and the growth difference is magnified over time (Nagashima, Terashima & Katoh 1995; Weiner 1990; Nagashima & Hikosaka 2011). Furthermore, such subordinate
plants have higher mortality and smaller seed production
than dominant plants (Matsumoto et al. 2008). It should
be noted that many shade tolerant species can survive and
reproduce in sub-canopy and understorey layers. Our
model is therefore most applicable to canopy species.
Conclusion
In this study, canopy photosynthesis, leaf area dynamics
and EGT are for the first time combined together. As such,
two dominant processes in vegetation stands – competition
and leaf turnover – are quantitatively integrated into a
model calculation. We focused on nitrogen and light limitation but the approach can be extended to include water
limitation (see McMurtrie et al. 2008). Our model provides
better predictions of LAI than previous static EGT or
dynamic-plant optimization models. Specifically it shows
that ES-LAI and associated stable canopy photosynthesis
is sensitive to several traits such as leaf shedding patterns
and the degree of interaction with neighbours in vegetation
stands, although further studies are necessary because values of these two traits are uncertain for most stands. We
believe that our integrated approach provides a more
mechanistic basis to analyse how plant competition and its
associated selection on leaf dynamics and leaf area growth
scale to the structure and functioning of vegetation stands.
In so doing, we believe that it may also allow a more
mechanistic scaling of plant acclimation to environmental
change to vegetation structure and functioning, which is
becoming an increasingly important issue in climate
change studies (Corlett 2011).
Acknowledgements
We thank Hendrik Poorter for valuable comments to the early draft. The
study was partly supported by KAKENHI (No. 20677001 and 21114009)
and Global COE program (J03).
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Handling Editor: Ken Thompson
9
Supporting Information
Additional Supporting Information may be found in the online
version of this article:
Data S1 Detailed explanation of the model.
Fig. S1. Effect of time step of the calculation on the leaf area
index (LAI) as a function of canopy nitrogen per ground area.
Table S1. List of species used in simulation.
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