Crown depth as a result of evolutionary games: decreasing solar angle should lead to shallower, not deeper crowns

Authors

  • Peter Johannes Vermeulen

    Corresponding author
    1. Department of Biological Sciences, Macquarie University, Sydney, NSW, Australia
    2. Centre for Crop Systems Analysis, Wageningen University, AK Wageningen, the Netherlands
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Summary

  • There is a general notion in the literature that, with increasing latitude, trees have deeper crowns as a result of a lower solar elevation angle. However, these predictions are based on models that did not include the effects of competition for light between individuals. Here, I argue that there should be selection for trees to increase the height of the crown base, as this decreases shading by neighbouring trees, leading to an evolutionarily stable strategy (ESS). Because the level of between-tree shading increases with decreasing solar angle, the predicted ESS will shift to higher crown base height.
  • This argument is supported by a simulation model to check for the effects of crown shape and the change of light intensity that occurs with changing solar angle on model outcomes.
  • So, the lower solar angle at higher latitudes would tend to select for shallower, and not deeper, crowns.
  • This casts doubt on the common belief that a decreasing solar angle increases crown depth. More importantly, it shows that different assumptions about what should be optimized can lead to different predictions, not just for absolute trait values, but for the direction of selection itself.

Introduction

It has long been thought that latitude should influence the vertical structure of forests, as a consequence of its effects on the angle of light that penetrates the canopy. Terborgh (1985), for instance, hypothesized that at higher latitude there is less vertical layering, and hence less diversity, because (1) the lower light elevation angles (angles of the light with the horizontal plane) cause light to pass through more leaves of the dominant canopy trees, and (2) trees at higher latitude have deeper crowns, because this is the form most effective in capturing light when light enters the canopy from a lower elevation angle. Modelling work has indeed suggested that, at low solar elevation angles, a canopy consisting of trees with deeper crowns is more efficient in terms of light interception and carbon gain, while at higher solar angles a shorter crown length produces a more efficient canopy (Oker-blom & Kellomaki, 1983; Kuuluvainen & Pukkala, 1987, 1989; Wang & Jarvis, 1990).

Although one can assume that more factors affect crown shape evolution at different latitudes, this modelling work has led to a long-standing notion in the literature that trees are indeed flat or shallowly domed near the equator while their shapes tend to be more vertically extended and steeply inclined closer to the poles, as a consequence of the effects of the solar angle (Richards, 1952; Halle et al., 1978; Whitmore, 1975; Terborgh, 1985; Hiura, 1998; King, 2005; Tateishi et al., 2010; Bomfleur et al., 2013, amongst others). However, most studies that refer to such a possible shift in crown shape base this argument on the theoretical work of Terborgh (1985) and Kuuluvainen (1992), and hence have implicitly assumed the general applicability of these models. This in turns means there is a heavy dependence on the model outcomes being robust against changes in the underlying assumptions; otherwise such translation from model outcomes to ecosystem functioning should be done more cautiously.

However, one of the main assumptions of these crown models may not be valid in naturally evolved forests. It is implicitly assumed that the benefit of a given vertical leaf distribution can be assessed via a population- or community-level outcome for a population in isolation, that is, through a simple optimization approach. This assumption does not apply to the evolution of crown shapes, because the benefit of height is pre-emptive access to light: taller individuals shade shorter plants but not vice versa (Ford, 1975; Weiner & Thomas, 1986). Therefore, the competitive benefits of height in general are frequency-dependent or game-theoretical (Givnish, 1982; Mäkelä, 1985; King, 1990; Falster & Westoby, 2003), meaning that the performance of individuals, and not of whole canopies, should be evaluated in competition with other possible strategies (e.g. Maynard Smith & Price, 1973; Riechert & Hammerstein, 1983).

It would therefore be interesting to determine whether such a game-theoretical approach would lead to the same predictions about how the solar angle influences crown depth. An important feature of game-theoretical models is that evolution through natural selection can lead to trait values that do not optimize the performance of the population as a whole (Schieving & Poorter, 1999; Anten & Hirose, 2001; Nowak & Sigmund, 2004; McNickle & Dybzinski, 2013); therefore, they predict different trait values from optimization models. But, as Franklin et al. (2012) showed, the two approaches could yield similar predictions. Yet, as they treat the selective nature of the environmental factors differently, their underlying mechanisms are not the same, and they cannot both give the correct interpretation as to why the shown patterns emerge (Anten & During, 2011). Different approaches become even more problematic when they produce results in opposite directions from each other. It is then that the conclusions drawn will depend on the model and its underlying assumptions, and will therefore not be robust against situations in which the conditions are not met.

Here, I show that a game-theoretical approach to model the evolution of crown depth leads to a different conclusion from the one prevalent in the literature, by showing that forests that experience light with a lower solar elevation angle are predicted to consist of individuals with shallower crowns than when the solar elevation angle is high. This conclusion follows from the following arguments.

  1. A higher crown base (Cb) results in leaves that are placed higher up the canopy, resulting in higher light capture as a consequence of decreased neighbour shading.
  2. The advantage of having a higher crown base, and hence a shallower crown, decreases as self-shading becomes a progressively larger fraction of all shading. This leads to an evolutionary endpoint (an evolutionarily stable strategy (ESS); Maynard Smith, 1982) where a change in crown base is no longer beneficial.
  3. As the relative amount of neighbour shading is greater at lower solar elevation angles, the advantage of increasing the crown base in terms of decreased shading by neighbours is also greater. Hence, the ESS is predicted to shift to higher values (i.e. shallower crowns) with decreasing solar angle.

In the model description I present the logic behind these arguments, supported by Fig. 1. Then, I shortly discuss the result of a simulation model (found in more detail in Supporting Information Notes S1) that was used as a sensitivity analysis to assess the effects of crown shape per se, and reduced light intensity with lower solar angle.

Figure 1.

The effects of (a, b) increased crown base (Cb) and (c) decreased solar angle on within-tree and between-tree shading. (a) The focal tree is either a resident tree within the resident population (light green) or a mutant that may potentially invade (dark green). Bold arrows (denoted by Sd) indicate the light beam that divides the leaves of the focal tree into the fraction of leaves that are shaded only by self (Rs, area above the arrow) and the fraction of leaves that are also shaded by neighbours (Rn, area below the arrow). Fn represents the distance the light beam travels through the neighbour before reaching a focal leaf in the middle at the bottom of the crown, and Fs the distance within the focal plant itself, with _r indicating the resident, and _m the mutant. Grey dotted arrows indicate light that reaches a focal tree with Cb of the resident population, and the black dotted arrow a mutant with higher Cb than the residents. (b) The point where a further increase in Cb does not lead to a further decrease in Fn_m (representing a potential endpoint of evolution (an evolutionarily stable strategy)). (c) The effect of decreasing the solar angle when the resident population has the Cb_r value reached in graph (b): Fn_r > 0, so crowns should become even thinner. A more detailed explanation can be found in the text.

Description

The conceptual model: model assumptions

The approach essentially explores how the crown base values of adult mature trees evolve under the influence of the solar angle, rather than explaining current growth trajectories. Hence, as in the crown models mentioned in the previous section, the stage at which selection is assumed to take place in this model is implicitly defined as trees that have reached the top of the canopy and have stopped growing in size. This also means that all trees have similar maximum height (Hmax), for instance because the switch to maximal reproductive investment is made (Falster & Westoby, 2003), or because further maximum height growth is limited by the increasing cost of increasing the diameter of the trunk (West et al., 1999). Individuals are assumed to have similar stem costs per unit height.

Related to this are several choices that were made. Costs associated with growth, such as construction and turn-over costs, are not modelled. The reason for this is that, as I argue later in the discussion, the cost function itself has no effect on the direction of model outcomes, just on the absolute trait values. Hence, to understand model outcomes, it is not necessary to complicate the cost function.

The model assumes a fixed leaf area index per tree. Thus, only the leaf area density (LAD; the leaf area per unit volume; m2 leaf area m−3) changes with an increase in crown base (Cb), packing the leaves closer together. Because this alters the distance between the leaf layers, and hence decreases the chance that a light beam is not intercepted by two sequential layers, increasing the LAD by increasing Cb potentially increases the light extinction coefficient (k). Assuming that plants do not adjust their leaf angle, such an increase in k would lead to more self-shading, and can thus be seen as a cost of increasing the crown base. However, as already stated, such a cost increase does not influence the direction of model outcomes, and hence a change in k resulting from denser leaf packing is not included in the model.

Iwasa et al. (1985) did apply a game-theoretical approach to the distribution of leaves over the vertical, predicting that there should be leaves from the top to the bottom of the crown. However, they simplified their analytical model by assuming complete mixture of strategies, excluding the distinction in between-tree and within-tree shading, and did not include effects of the solar angle. Crowns of trees with similar height do not mix, and there can sometimes be gaps between individual trees in forest stands (Putz et al., 1984). This clustering of leaves in a forest canopy within distinctly separated individual crowns causes shading of leaves to be a combination of within- and between-tree shading (Oker-Blom & Kellomaki, 1982). When the light enters the vegetation under an angle, part of the light will still travel through neighbouring trees (see Fig. 1). Hence, the balance between the within-tree and between-tree shading can be expected to shift with solar elevation angle, in which a lower angle increases the path of the light through neighbouring trees (Oker-Blom & Kellomaki, 1982; Pukkala & Kuuluvainen, 1987; Lusk et al., 2011). Because the benefit of having leaves slightly higher than one's competitors (and thus the benefit of a shallower crown) arises through reduction of shading by neighbours, a shift in solar elevation angle may lead to different selection pressures along the latitudinal gradient. Therefore, a game-theoretical model of tree crown shape should include the distinction between self- and neighbour-shading.

In the simulation model, it is assumed that a tree's fitness is equal to the carbon gain of all leaves within the tree:

display math(Eqn 1)

where Pnet(Cb) is the carbon gain of a leaf with a height between Hmax and Cb. For example, for a leaf placed in the middle of a tree, the height is 0.5(Hmax + Cb).

Following game-theoretical principles for height, the evolution towards thinner crowns is predicted to stop when the benefits of a higher Cb of a new phenotype entering a resident population are offset by the costs (see Iwasa et al., 1985; Falster & Westoby, 2003):

display math(Eqn 2)

That is, the crown base value of the resident population where an increase or decrease in Cb by a focal tree does not lead to an increase in carbon gain of that tree (see also Eqn S8).

Pnet(Cb) in Eqn 1 is the difference between the gross photosynthetic rate of the individual leaf, Pleaf(Cb) (depending on the light capture of the leaf, Ileaf), and the costs of supporting that leaf, Cs:

display math(Eqn 3)

Because the photosynthetic characteristics are assumed to remain equal, what happens to Pnet(Cb) can be inferred from the changes in the light capture of the leaves, Ileaf (see Notes S1 for formulas used in the simulation model to link light capture to gross photosynthetic rate).

Light capture of a leaf can be found assuming Beer's law (Hikosaka, 2003):

display math(Eqn 4)

with k the light extinction coefficient, Io the light above the canopy, and Fleaf(Cb) the total leaf area that light has travelled through to reach the leaf as a function of the crown base, as Cb determines the height of the leaf in the canopy. Light capture of a leaf is therefore inversely related to Fleaf(Cb), that is, an increase in Fleaf(Cb) decreases the light capture. Fleaf(Cb) can be separated into shading by neighbouring trees, Fnleaf(Cb), and within-tree shading, Fsleaf(Cb), that is:

display math(Eqn 5)

Note that this is where the difference from simple optimization models occurs: simple optimization models do not distinguish between neighbour shade and self-shading. This could lead to a state in which Fleaf(Cb) = Fsleaf(Cb) is maximized at the canopy level, where an increase in Cb could still allow a mutant to invade, because the decrease in Fnleaf(Cb) is larger than the increase in Fsleaf(Cb), leading to higher fitness of the mutant compared with the resident population.

Both neighbour shading and within-tree shading depend on the distance that light has travelled through these parts, and, assuming that leaf area is equally divided over the volume of the tree, the LAD:

display math(Eqn 6)

Note that increasing the crown base while keeping total leaf area of an individual tree fixed will decrease the volume, and hence will increase LAD.

In the next section an argument follows that explains what happens to the light capture of leaves that are in part shaded by leaves of neighbouring trees, and those that are only shaded by leaves of the tree itself; that is, how a change in Cb alters the light capture in Eqn 4 through its effects on Eqns 5 and 6 Then it is discussed how the combined effect of the individual leaves affects the fitness of the tree (Eqn 1), and the ESS that is found (Eqn 2). Note that, for simplicity, the figures that are used to explain the main arguments are based on a rectangular shape. Because of the linear behaviour of the light beam, the main arguments remain valid when other basic geometric shapes are used, unless noted otherwise.

The advantage of having higher crown base decreases with higher crown base as self-shading becomes a progressively larger fraction of all shading

The reason that self-shading becomes a progressively larger fraction of all shading with increasing Cb is twofold. Firstly, assuming that trees do not overlap, a tree has a portion of leaves that only experience within-tree shading (Rs, leaves for which Fnleaf(Cb) = 0) and a portion of leaves that experience both within-tree and neighbour shading (Rn; see also Fig. 1). The ratio between the two is determined by the radius of the tree, the solar angle (which divides the area of the tree into the Rs and Rn fractions, represented by the thick arrow Sd in Fig. 1), and the height of the crown base (as maximum height is kept constant). Two things can be deduced from this: (1) the larger the radius, the more leaves only experience self-shading, as light travels a greater distance within the focal tree; (2) as Cb increases, the average height of the leaves increases, and the Rn portion decreases; that is, Rs increases (compare area of tree under Sd in Fig. 1a).

The second reason that self-shading becomes a progressively larger fraction of all shading with increasing crown base is that, for the Rn portion of the leaves, an increase in crown base reduces the distance that light travels through neighbouring trees (compare Fn_m with Fn_r in Fig. 1a). This is stronger than the decrease in the distance travelled within the tree itself (Fs_m versus Fs_r), and thus the influence of neighbouring trees also decreases for the Rn portion of the leaves within the trees.

For these two reasons, self-shading becomes a progressively larger fraction of all shading for the whole tree when the crown base increases. Consequently, the benefits in terms of lowering Fnleaf by further increasing the crown base decrease with higher Cb values (compare Fig. 1a and b).

Costs of increasing the crown base

An invader with a higher crown base than the resident population will incur more costs for several reasons. First, costs per leaf (such as for the support of leaves) will increase with leaf height. Secondly, Fsleaf tends to increase with increasing Cb. Assuming that the total leaf area remains the same with increasing crown base, the LAD increases, that is, the leaves become more densely packed. Hence, per unit distance a light beam will pass through more leaf area within the tree itself. Therefore, when the distance that light travels through self does not decrease with increasing crown base as strongly as the LAD increases, the Fsleaf will increase (see Eqn 5). In the rectangular example, this is only the case for leaves in the Rn part, but not in the Rs part (compare thin arrows in Fig. 1), where the increase in LAD is cancelled out by the same decrease in distance that light travels. However, this is a special characteristic of a rectangular shape, where LAD is linearly related to the crown radius and crown depth. As a consequence of simple geometry, in most basic geometric shapes (cones, cylinders, and ellipsoids) the LAD increases more strongly than the distance the light travels with increasing Cb as a result of a quadratic term of the way volume changes with Cb. For example, for a cone:

display math(Eqn 7a)
display math(Eqn 7b)

Hence, Fsleaf tends to increase for leaves in the Rs part of trees as well. Therefore, in order to invade and replace the resident population, the benefits of decreased Fn should outweigh the costs of increased support costs, and increased Fs.

Endpoint of crown base evolution

As a consequence of the decreasing benefits and the increasing costs, there should be a crown base at which an invader cannot invade and replace the resident population. At this point, the invader's costs will outweigh the benefits, leading to a fitness that is lower than that of the resident population (see Fig. 1b, in which Fn of a focal leaf at the bottom of the crown approaches zero). All other traits being equal, the value of this predicted Cb will be affected by the radius of the tree and the solar elevation angle. Assuming that an increase of the radius means that the number of individual trees per unit ground surface area decreases, while the leaf area index remains constant, more light will pass through the individual tree itself. Therefore, there is less benefit in increasing the Cb when the crown radius is larger, and hence a lower crown base should evolve. This crown base height would be even further constrained if lateral costs interact with height (see Anten & Schieving, 2010).

As the main focus of this paper is on the effect of solar elevation angle, its effects on the height of the crown base are discussed next.

The advantage of having a higher crown base increases with decreasing solar elevation angle, as neighbour shading becomes a progressively larger fraction of all shading

As the solar elevation angle (αs) does not affect LAD, one only has to look at its effects on the distance light travels within the tree and through its neighbours to assess the effects of solar elevation angle on crown depth. As argued previously, the solar elevation angle effectively divides the leaves within an individual tree into leaves that only experience within-tree shading (Rs), and leaves that experience both within-tree and between-tree shading (Rn). Rs will decrease (and hence Rn will increase) with decreasing solar elevation angle (compare area under Sd for resident in Fig. 1b and c). Intuitively, this follows the logic that, when the sun is directly overhead, direct light only goes through the tree itself (under the assumption that trees are separated into their own spaces, such that no leaves overlap as seen from the horizontal plane; note that, because radiation comes from the whole sky dome, a tree will always be partly shaded by neighbouring trees). With decreasing solar elevation angle, however, a larger part of the tree will be shaded by neighbouring trees.

For leaves still shaded by neighbours (Rn), a decrease in solar elevation angle also increases the distance light travels through neighbouring trees. The distance the light travels through the canopy is inversely related to the sine of the solar elevation angle (distance = height travelled/sin(αs)). Hence, when the solar elevation angle decreases, the distance light travels through the canopy increases. Because the vertical distance that a beam travels within the tree is the product of the distance of the leaf to the edge of the tree and the tangent of the solar angle, the path light travels through the tree itself decreases with decreasing solar angle. Hence, while the total distance light travels through the canopy increases with decreasing solar angle, the fraction of self-shading becomes progressively smaller with decreasing solar angle for the Rn fraction of leaves.

For these two reasons, the amount of shading by neighbour trees increases with decreasing solar angle. A resident population that is an ESS at high solar angle would therefore not be evolutionarily stable at a lower solar elevation, where it could be invaded by a mutant with a thinner crown, because neighbour shading is still high at this lower solar angle (Fig. 1c). Consequently, the ESS Cb value should increase with decreasing solar angle.

Results

Sensitivity analyses through a simulation model

The solar elevation angle influences the amount of light above the canopy (Io). In addition, changing the basic geometric shapes of the modelled trees influences the ratio between self-shading and shading by neighbours. Therefore, a simulation model was used as a sensitivity analyses. The full description of the simulation model is given in Notes S1. This includes graphs that are the result of model input given in Table S1. In short, a fixed amount of leaves was distributed evenly within a cone shape. Light interception for each leaf was calculated as the distance light had to travel through neighbours and the focal tree itself, and the consequent amount of leaf area the light had passed through. Then, the total net photosynthetic rate of the tree was calculated by summing the ensuing net photosynthetic rate of each leaf (Eqn 1), calculated from the gross photosynthetic rate of each leaf, and subtracting a height cost (Eqn 3). Next, the benefit of a slight increase in Cb for a focal mutant (in terms of net photosynthetic rate gained compared with the resident population) was calculated for each Cb value. An ESS was then determined as the Cb value at which the resident population could not be invaded by a mutant with a different Cb value (Eqn 2). This ESS was calculated for three solar elevation angles (see Fig. S1).

The model confirmed that outcomes were not affected by the decrease in Io with decreasing solar angle, even when the decrease was stronger than is usually assumed. It confirmed that costs only affected the absolute values of the ESS, but not the direction. And it showed that different shapes can have an effect on the absolute values: a cylinder shape is predicted to lead to thinner crowns (ESS Cb of 23.1, 28.8 and 28.0 for solar elevation angle = 22.5°, 45.0° and 67.5°, respectively, with Hmax set at 30 m) compared with a cone-shaped tree at the same latitude (ESS Cb = 11.3, 23.9 and 27.6, respectively). Still, the direction of selection with changing solar angle remained the same.

Discussion

Terborgh (1985) hypothesized that forest canopies at higher latitudes have fewer layers, and that this contributes to a lower species diversity. The reasoning was that lower light angles cause light to travel longer through the dominant canopy trees, and tree crowns at higher latitude are deeper, because this is the form most effective in capturing light. The former argument seems quite valid, under the assumption that all else is equal. But I show here that the validity of the latter argument is more questionable, on the basis that lower solar angles lead to more between-tree shading, and hence stronger selection for an increased crown base, reducing the crown depth.

Overall, the results of the model presented here suggest a strong increase in the relative importance of self-shading with rising crown base. There is a decrease in between-tree shading with increasing crown base. At the same time, the level of self-shading is predicted to increase. This indicates that, similar to isolated trees (Duursma & Mäkelä, 2007; Sinoquet et al., 2007; Duursma et al., 2010), trees in dense canopies that seem to compete for light may in fact be shaded to a large degree by themselves.

This means that the model predicts the crown base (Cb) to evolve away from Cb = 0, but equally to select against Cb = Hmax (maximum tree height). This is an interesting finding for two reasons. First, unless the costs of increasing Cb always outweigh the benefits (in which case trees should have leaves along the whole vertical, for example, in a lone standing tree, or in vegetation types with very low leaf area indexes, either through a low leaf area index per individual tree or through a low density of trees), it provides an alternative to Iwasa et al. (1985)'s hypotheses as to why individual trees do not have leaves all the way to the ground: such trees can be invaded and replaced by mutants with a higher crown base. Secondly, the outcome that there is an endpoint is independent of the way the costs increase with height. This is because the fraction of leaves that are shaded by neighbours, and hence the benefits of further increasing crown base, will always go to zero when the crown base nears the maximum height. This is an important point, as several cost functions have been used when modelling height growth. The model in the present paper is a pipe model, leading to a linear increase of costs with leaf height. Iwasa et al. (1985) used an exponential function with an exponent of 2 (a quadratic increase of mass with height), while others have argued that, because of the mechanical strength that is needed to support the weight of leaves and branches, the exponent should be even larger (3 or 4; Anten & Schieving, 2010). Costs in the present model may be relatively low because an increase in Cb would increase the average height of all leaves, but not the maximum height. Then again, costs such as wood turn-over that are not explicitly modelled may differ depending on height within the crown (Mäkelä & Vanninen, 2001). But whatever the true costs are, the simple prediction is that the higher the costs with increasing leaf height are, the deeper the crown will be. It should be noted, however, that the present study has not fully quantified under which conditions a different cost function would no longer lead to a monomorphic ESS.

As the level of between-tree shading increases with decreasing solar angle, the model predicts that selection for increased crown base is stronger at higher latitudes, despite it leading to decreased photosynthetic rates at the level of the whole vegetation. As pointed out by Kuuluvainen & Pukkala (1989), having a deeper, steeply inclined crown at high latitudes does hold meaning when one wants to maximize the yield of forest plantations. But, the idea that ‘maximisation of the radiation flux on leaf surfaces is beneficial to the tree’ (Kuuluvainen, 1992) should not be assessed via a population- or community-level outcome for a population in isolation (see Falster & Westoby, 2003; Anten, 2005). The main reason for this is that the argument does not hold in an evolutionary sense. The model presented here shows that new mutants with a shallower crown can invade and replace such an ‘optimal’ population, leading to lower carbon gain. This itself is not a new prediction, as other models have predicted that, when plants are competing, selection will favour plant traits that lead to lower production at the whole-canopy level (Schieving & Poorter, 1999; Anten & Hirose, 2001). But, more importantly, the model predicts that if the only thing that differs between latitudes is the average solar elevation angle, plants should evolve to have thinner, less inclined crowns at higher rather than at lower latitudes, because the level of between-tree shading is higher at the lower solar angles associated with higher latitudes.

This outcome cannot directly be translated to how crown depth varies with latitude. Even with this simple model it can be seen that differences in other traits greatly affect the predicted crown depth. For instance, with an increasing radius of individual trees (in this model inversely related to the number of individuals per unit area), crowns are predicted to be deeper as the level of self-shading increases. Decreasing the maximum photosynthetic rate (Amax), perhaps as a result of lower temperatures, leads to deeper crowns as a result of decreased benefit in terms of net photosynthetic rate per unit captured light (see Eqn S3). Avoidance of damage caused by snow, wind, drought or temperature is likely to affect crown shape, potentially selecting for a more cone-like shape in colder, more extreme environments (Petty & Worrell, 1981; Oker-blom & Kellomaki, 1983; Davies & Ashton, 1999; Petit et al., 2011), which will lead to deeper crowns compared with cylinder-shaped trees because neighbour shading decreases. Similarly, changes in leaf distribution within the crown could affect the level of self-shading. Also, differences in other climatic factors, such as cloud cover, may drive latitudinal differences in photosynthetic rates (Nemani et al., 2003). Crown shape in general is in part driven by growth trajectories triggered by the race for light, or by other important functions (see Niklas, 1994; Pearcy et al., 2005). More comprehensive models that include all these interaction factors should come closer to a better mechanistic understanding of changes in crown depth.

The results here, however, emphasize two important points. Firstly, they cast doubt on the current general notion in the literature that crowns become deeper with latitude as a result of the effect of lower solar angle (Richards, 1952; Hiura & Fujiwara, 1999; Aiba & Kohyama, 1997; King & Clark, 2011, amongst others). Most studies that argue for such a trend have only cited the theoretical work of Terborgh (1985) and Kuuluvainen (1992), and most of them without mentioning that these studies are based on modelling results. Yet, it should be noted that not much quantitative evidence exists of how crown depth changes with latitude (but see Tanabe et al., 2001 for an intra-species comparison). Therefore, with very few data to support the traditional view, and the results presented in this paper indicating that the effect of solar angle should operate opposite to the general notion, there should at least be a reassessment of how and why canopy structure changes with latitude.

Secondly, they stress the importance of incorporating evolutionary mechanisms into models that predict how canopy structure and function respond to changing environmental conditions, through either space or time. Recently, exciting examples of game-theoretical models that explicitly include the added dimension of competition between individuals, and the effects changing resources have on those interactions, have emerged (Kohyama & Takada, 2009; Dybzinski et al., 2011; Westoby et al., 2012; Farrior et al., 2013), demonstrating the potential of such models in producing new theory in more complex conditions than the conceptual model presented in this paper. To date, one of the arguments to use game theory has been that, even though other models may produce similar results, game theory better reflects the fact that canopies are the result of competing individuals (Anten & During, 2011; McNickle & Dybzinski, 2013). More forcefully, the results presented in this paper suggest that including these competitive interactions may lead to different predictions about the direction of change from current models that implicitly ignore them. While this is no proof that using a game-theoretical approach leads to better predictions, it should serve as a warning that models trying to predict how changes in abiotic factors influence plant responses may miss important drivers of change if biotic dynamics are not included.

Acknowledgements

Thanks to M. Westoby for discussions and comments and C. R. Stoof and N. P. R. Anten for editorial advice. P.J.V. acknowledges the funding of the research programme Rubicon, which is financed by the Netherlands Organisation for Scientific Research (NWO).

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