The stratification theory for plant coexistence promoted by one-sided competition

Authors


*Correspondence author. E-mail: kohyama@ees.hokudai.ac.jp

Summary

  • 1It is an essential feature of plants that leaves at higher levels have better access to light than those at lower levels. Thus, larger plants generally enjoy greater success in competing for light than smaller ones. We analyse the effect of such size-asymmetry, or one-sided competition, on the successful coexistence of plant species, using an analytically tractable model for stratified populations, in which a plant in the same layer exhibits the same crowding effect as any other, irrespective of species.
  • 2A two-layer population that is reproductive in upper layer and juvenile in lower layer has a uniquely stable (plant-size-weighted) equilibrium density, as long as its fecundity is sufficient to compensate for its mortality rate. We also calculate a unique threshold lower-layer density of this layered population when there is no upper-layer plant. This threshold lower-layer density is larger than the weighted equilibrium density with upper layer, except for the case of perfect two-sided competition.
  • 3A two-layer species can stably coexist with a one-layer, understorey species as a result of one-sided, but not two-sided competition. The coexistence condition is that the equilibrium density of the one-layer species lies between the threshold lower-layer density and the equilibrium density of the two-layer species. For an understorey species to coexist successfully with a two-layer species, any advantage in demographic performance, most prominently in a sufficiently high fecundity per plant must offset the disadvantage of living in dark conditions.
  • 4Results from a model of multi-layer populations suggest that several species differing in terms of maximum layer and fecundity can coexist under conditions of one-sided competition. We demonstrate an example of the stable coexistence of eight species. The inter-specific trade-offs predicted by the model correspond to patterns observed in a rain forest.
  • 5Synthesis. We propose a stratification theory that explains the generation and maintenance of the successful coexistence of plant species. Under the condition of one-sided competition, a canopy population that takes advantage of escaping from understorey competition shows an ability to invade an understorey with a density higher than its own equilibrium density, and which offers opportunities for an understorey population with high fecundity and/or shade tolerance to coexist. The predicted coexistence of species that share maximum canopy height is most pronounced for trees of tropical rain forests.

Introduction

The ecological principle of competitive exclusion states that competing populations cannot coexist stably in spatially homogeneous environments when the system is regulated by a single limiting resource (May 1973; Hofbauer & Sigmund 1998). This principle is apparently contradicted by the fact that many plant species that require the same set of resources coexist in natural communities. A wide range of explanations has been proposed for the coexistence of plant species (Tilman & Pacala 1993; Chesson 2000a,b; Hubbell 2001; Wright 2002; Silvertown 2004), which include spatio-temporal heterogeneity, inter-specific trade-offs, and the non-equilibrium state of communities. This article addresses the contribution of autogenic vertical heterogeneity of vegetation to the coexistence of plant species.

Tall plants make better use of light resources than short plants in crowded size-structured populations. Therefore, size-asymmetry in competition for light is common in plant populations (Schwinning & Weiner 1998). The categorization of plant species into sun and shade plants, according to their adaptation to the availability of light resources, forms a general basis for plant ecophysiology (Lambers et al. 1998). It may also be expected that niche differentiation along the light gradient from the top of the canopy to the understorey promotes the coexistence of plant species. However, given that canopy species grow from seedlings that are themselves in deep shade, a simple view of niche differentiation along a gradient of light resources is not applicable. To examine how competition for light along the vertical profile of the canopy contributes to community organization, several simulation studies have been carried out. Among them, some studies address the effect of vertical biomass (leaf mass) structure in regulating competition for light resources, rather than population density, particularly with respect to clonal plants (Rees & Bergelson 1997; Perry et al. 2003; Yachi & Loreau 2007; Nevai & Vance 2007). Other studies, in contrast, describe the dynamics of size-structured plant populations, such as forests, that employ partial differential models similar to the continuity equation of fluid dynamics, and thereby address the cross-generation dynamics of populations (Kohyama 1992, 1993, 2006; Hurtt et al. 1998; Moorcroft et al. 2001; Adams et al. 2007). Although such continuous ‘size–density–time’ systems lend themselves well to numerical simulation with demographic and ecophysiological details, they are not necessarily useful for analysing general mechanisms for the coexistence of species. One possible contrastive approach is to use discrete plant-size classes rather than a continuous size distribution, as a means of simplifying the analysis. Adams et al. (2007) have proposed modelling competition in stratified populations by dividing a population into two stages, vegetative and reproductive, while describing the dynamics of continuous plant-size structure using the continuity equations. Their approach is based on the assumption that local stratification is made up of at most two stages (where the juvenile stage suffers shading by the mature stage), and that the continuous size structure of the population implicitly reflects horizontal heterogeneity in the landscape.

In this article, we make use of an alternative approach, in which we introduce simple differential equations of species competition. In our model, we aggregate the continuous distribution of plant sizes into discrete size classes and analyse the effect of a stratified structure of vegetation on the coexistence of plant populations that have overlapping generations. Models of discrete-stage structure have previously been used in population biology (Caswell 2000; Thieme 2003). Analytical studies using the Lotka–Volterra competition model exist, extended for populations with stage structure (Takada & Nakajima 1998; Mougi & Nishimura 2005; Xu et al. 2005). However, the effect of asymmetric competition among stages of life history on species coexistence remains unexamined. In this study, we specify that any individual in the same size class or layer yields the equivalent competitive effect, irrespective of species. Therefore, no assumption is made in our model of the existence of competition coefficients, or species-pair-specific factors in competition, that are essential for the coexistence of species in the Lotka–Volterra equations. We introduce a coefficient that covers a whole range of competitive size-asymmetry from one-sided competition for light to two-sided competition for nutrients, and show how competitive size-asymmetry contributes to the configuration of plant communities.

In order to investigate the effect of competition for resources on the three demographic processes of fecundity, survival and vegetative growth, our size-classified model is demographically explicit. We analyse the stability of equilibria in a two-layer two-species model and specify possible trade-offs between species required for their stable coexistence. We also demonstrate the stable coexistence of several species with a multi-layer model, and compare the simulation results with the observed pattern in an actual tree community. We compile results of the analysis into the stratification theory for coexistence and speciation of plant populations.

The model

We propose a model that represents two competing species in a vertically stratified two-layer canopy in which horizontal heterogeneity does not exist (i.e. a mean-field approximation). The two species X and Y differ in terms of their maximum attainable height. The boundary between the upper and lower layers is set just above the maximum height of the shorter species Y, so that only species X is distributed in the two layers (Fig. 1). We assume for species X that plants in the lower layer are juvenile, and those in the upper layer are reproductive. We introduce two coefficients that characterize the stratified system, namely the plant size factor W, and the coefficient of competitive size-asymmetry Q. The upper-layer plant is W times larger in leaf mass and requires W times the resources as the lower-layer plant (thus W > 1). W is kept identical between species, and unit density of plants in a given layer is assigned an equivalent resource requirement, independent of species. The coefficient of competitive size-asymmetry Q defines the degree to which a plant in the upper layer suffers reduction in available resources as a result of the lower-layer density relative to that of the upper-layer weighted density. Q is in the range 0 ≤ Q ≤ 1. If Q = 0, the system is regulated by perfectly one-sided, or size-asymmetric competition and lower-layer plants have no competitive effect on upper-layer plants. By contrast, if Q = 1, the system is under the control of perfect two-sided, or size-symmetric competition and the presence of lower-layer plants affects the performance of upper-layer plants and vice versa, as determined by competition for resources.

Figure 1.

Outline of the model of eqn 1 for a two-layer two-species system by vegetation profile (left) and population dynamics (right). Open, species X; shaded, species Y. Demographic processes (b's for fecundity, m's for mortality and g for layer-transition rate) are regulated by the layer-specific crowding intensities: C1 for the lower-layer processes, and C2 for the upper-layer processes. Q denotes the degree of competitive asymmetry from 0 (one-sided) to 1 (two-sided), and W (> 1) denotes the size factor of an upper-layer plant relative to a lower-layer plant.

We define that species X has a density of x1 in the lower layer and x2 in the upper layer. Species Y has a density of y in the lower layer. We denote the per capita fecundity of species X in the upper layer as b, the growth rate from the lower layer to the upper layer as g, and the crowding-free mortality in the lower and upper layers as m1 and m2, respectively. Fecundity b and growth rate g are suppressed by leaf mass, or weighted density, with susceptibilities of s and v, respectively. Mortality caused by crowding, or natural thinning, is proportional to leaf mass with susceptibility u. The demographic rates b′ and m′, and the susceptibility to standing leaf mass s′ and u′ of species Y are distinguished from those of species X by ‘primes’. The overall equations for competition are:

image(eqn 1)

where C1 = y + x1 + Wx2 and C2 = Q(y + x1) + Wx2 (see Fig. 1). The model of eqn 1 assumes that population densities and demographic parameters are all non-negative. Ci denotes the crowding intensity in the i-th layer. We also use x =x1 + Wx2 for the weighted density, or leaf mass of species X. Therefore, C1 = x + y represents the overall leaf mass. When Q = 0, mortality and reproduction in the upper layer, as well as the transition rate from the lower to the upper layer for species X, are determined only by the upper-layer leaf mass, C2 = Wx2, whereas lower-layer reproduction (for Y) and mortality (for both X and Y) depend on total leaf mass C1. When Q = 1, the demographic performance in both the upper and lower layers is regulated by the total leaf mass C1 = C2.

The two-layer model of eqn 1 can be extended to a multi-layer system. The extended model describes the dynamics of the density at layer i (i = 1, ... , n) of species k (k = 1, ... , n), xi,k, where species k grows up to the reproductive k-th layer through juvenile layers, as

image(eqn 2)

where inline image, gi,k > 0 for i < k and gi,k = 0 for i ≥ k.Ci defines the crowding intensity at the i-th layer, with the coefficient of size asymmetry Q (0 ≤ Q ≤ 1), and layer-specific plant size factor Wi relative to the lowest layer W1 = 1 (Wi+1 > Wi). This definition of Ci again assumes that resource consumption at a given layer i is identical between species. When n = 2, eqn 2 becomes eqn 1, where single-layer species 1 corresponds to species Y, and two-layer species 2 is species X.

In order to assess the accuracy of the results obtained from eqn 2, we examine the distribution of the 12 most abundant tree species in a warm temperate rain forest in permanent plots in the Segire Basin of Yakushima Island, southern Japan, which have been monitored since 1981. Tree density, basal area, and recruitment rate are defined using trees that have a trunk diameter of at least 2 cm at breast height. Further details are described in Kohyama (1986) and Aiba & Kohyama (1996). These species are all shade-tolerant as they are represented by abundant saplings with smooth inverse-J-shaped size distribution in closed-canopy stands. These species – with the observed maximum height (m) in parentheses – are: Distylium racemosum (21.0), Litsea acuminata (18.5), Podocarpus nagi (17.3), Neolitsea aciculata (16.7), Camellia sasanqua (16.1), Symplocos tanakae (15.0), Camellia japonica (14.8), Cleyera japonica (13.6), Symplocos glauca (12.7), Illicium anisatum (12.6), Myrsine seguinii (9.1) and Eurya japonica (8.4). They comprised 92% of the number of trees present in the plots and occupied 87% of total basal area (Kohyama 1986).

Results

two-species coexistence in the two-layer system

When species Y exists uniquely, dy/dt (eqn 1) is a simple first-order differential equation and the population has a positive, globally stable equilibrium, or carrying capacity, given by inline image = (b′ – m′)/(bs′ + u′), provided that b′ − m′ > 0, where the symbol ‘^’ represents the density at single-species equilibrium.

The population of species X also has a unique stable positive equilibrium of inline image1 and inline image2 whenever bg/m2 – g – m1 > 0; otherwise, an equilibrium value of zero is globally stable (Appendix S1 in Supporting Information). We cannot describe explicitly the carrying capacity densities inline image1 and inline image2 or a unique equilibrium leaf mass inline imageinline image is a monotonically decreasing function of Q and W, and inline image < (b – m2)/(bs + u) at Q = 1 or W = ∞ (Appendix S1). We introduce another specific density, xL, which is the threshold lower-layer density of species X given a vacant upper layer. If the equilibrium density of the lower-layer species Y is below this threshold density, species X can invade the population of species Y. xL is the solution of eqn 1 for x1 at dx1/dt = dx2/dt = 0, obtained by setting crowding intensities of C1 = x1 and C2 = Qx1 (i.e. upper-layer crowding has no effect). At Q = 0, xL = (bg/m2 – g – m1)/u and xL monotonically decreases with Q. xL is always larger than inline image whenever Q < 1, and xL = inline image when Q = 1 (Appendix S2). xL varies independently of W, and if (unrealistically) W = 0, then xL = inline image. Figure 2 illustrates the dependence of inline image and xL on Q and W.

Figure 2.

Variation of equilibrium weighted density inline image (full line) and threshold lower-layer density at vacant upper layer xL (broken line) plotted against coefficient of size asymmetry Q, for two-layer species X. The change in inline image with plant-size factor W is also shown.

When the two species are persistent (bg/m2 – g – m1 > 0 and b′ – m′ > 0), the two-species system described in eqn 1 results either in competitive exclusion or in stable coexistence. The detailed theoretical analysis of stability and uniqueness of all the equilibrium conditions is described in Appendix S2. If inline image < inline image, species X competitively excludes species Y at a unique stable equilibrium of (inline image1inline image2, 0), and alternatively if xL < inline image, Y excludes X at (0, 0, inline image). Otherwise, if neither condition is satisfied and the two species are mutually invasive, there is a unique stable positive equilibrium of inline image where asterisk indicates the density at the coexistence equilibrium. The condition of stable coexistence is therefore:

image(eqn 3)

When Q = 1 and competition is perfectly two-sided, there is no equilibrium coexistence because xL = inline image, and the dominant species is the one with the larger single-species equilibrium. Figure 3 illustrates these conditions of equilibrium stability by means of nullcline diagrams (Appendix S3). The condition dx2/dt = 0 in eqn 1 determines the density ratio x2/x1 for given values of x and y. We can therefore obtain the nullcline of the weighted density x on the (x, y) plane, from dx1/dt + dx2/dt= 0 in eqn 1, together with the specific ratio of x2/x1 under dx2/dt = 0. The nullcline of y is simply x + y = inline image, obtained using dy/dt = 0 in eqn 1. The negative slope of the x-nullcline on the (x, y) plane is invariably steeper than that of the y-nullcline for Q < 1. The sum of the weighted densities at the coexistence equilibrium, inline image, is equal to inline image, which is the single-species equilibrium density of lower-layer species Y.

Figure 3.

Nullcline diagrams of the weighted density of species X, x = x1 + Wx2 (full line) and the density y of species Y (broken line), on (x, y) coordinates. Three possible cases are shown for Q < 1: (a) X predominates over Y, (b) X and Y coexist, and (c) Y predominates over X. Filled circles indicate stable equilibrium and open circles indicate unstable equilibrium.

The range (inline image, xL) for which species Y can coexist with species X steadily increases with decreasing Q and increasing W, as shown in Fig. 2. In order to coexist with X, the understorey species Y needs to have either higher fecundity, a lower mortality, or a lower susceptibility to crowding, than species X. In other words, there must be inter-specific demographic trade-offs for coexistence to occur. Figure 4 shows the contribution to coexistence of each of these demographic parameters of species Y. Above all other factors, increasing fecundity per plant size shows the most pronounced effect on coexistence. The larger the value of g for species X, the greater is the opportunity for species Y to coexist. In particular, when Q = 0, a large value of g allows the persistence of species X, regardless of the fecundity of species Y (Fig. 4a).

Figure 4.

The range of coexistence and competitive exclusion of two-layer species X and one-layer species Y plotted against varying g for species X, for Q = 0 and Q = 0.5. The range boundary is inline image = inline image (full line) and xL = inline image (broken line) in each case. The model of eqn 1 is used with W = 4, b = 4, m1 = m2 = 0.1, s = u = v = 1. The effects of (a) changing b′, (b) changing m′, and (c) changing s′ and u′ together for species Y are shown here. Other parameters are b′ = b/W = 1, m′ = 0.1, and s′ = u′ =  1 (identical to the equivalent values for species X).

In a single-species lower-layer population that has a carrying capacity of inline image, there may emerge a mutant population that exploits the upper layer (g > 0) which stops reproduction in the lower layer. Such a mutant is invasive provided xL of the mutant population is larger than inline image. This process triggers further sympatric speciation, which provides the opportunity for a lower-layer genotype to coexist with an upper-layer genotype, as is suggested by Fig. 4.

multiple species coexistence

As for the two-layered system, if the competition for resources is perfectly two-sided (Q = 1), there is no stable coexistence in the multiple species system of eqn 2, and the species with the highest single-species equilibrium leaf mass (or weighted density) will exclude all others. It is difficult to illustrate the analytical characteristics of a multiple species system with Q < 1, and we only show a numerical example in this article.

We demonstrate the case of Q = 0 and large gi,k relative to other demographic parameters. This case provides a condition for facilitating coexistence in a two-species system (cf. Fig. 4). Figure 5 illustrates an example of the stable coexistence of eight species, each differing in terms of its maximum layer in an eight-layered system. We assume that size of a plant is four-times larger than that in the layer one beneath, such that Wi = 4i–1. We set bk identical irrespective of species k, which gives inter-specific trade-off between maximum layer k and reproductive capacity, or fecundity per plant size bk/Wk. Consequently, the difference in reproductive capacity between the bottom species 1 and the top species 8 is 1 – 0.000061. The total weighted density, or leaf mass, for all species at an equilibrium inline image with the mixture of species 1 (i.e. the lowest-layer species), is identical to the single-species equilibrium density of species 1 alone.

Figure 5.

Variation of predicted leaf mass or size-weighted density, inline image against simulation time t for eight species (k = 1, ... 8) with different maximum layers (= k), simulated using eqn 2 with Q = 0 and Wi = 4i–1. Demographic parameters are bk = 4 for any k; gi,k = 1 and 0 for i < k and i ≥ k respectively; mi,k = 0.1, sk = uk = vk = 1 for any i and k. The simulation uses initial conditions of x1,k = 1 × 10−6 and xi,k = 0 for i > 1 at t = 0 (i.e. low-density founders only at the lowest layer).

The multi-species model of eqn 2 can readily be compared with census data from actual communities that have developed stratified structures. Figure 6 compares simulation results and field observations of the studied tree community. Recruitment rate per species basal area is correlated negatively with maximum height (Fig. 6b), which is in accordance with the reproduction rate per species leaf mass at the coexistence equilibrium, inline image for the eight species in the simulation (Fig. 6a). Shorter species with high potential reproductive capacity show temporary domination during an early phase of regeneration. The ratio in terms of tree density for tree-fall gaps to closed-canopy stands is higher for shorter species (Fig. 6d). This supports the results for the model, in that lower-layer species exhibit a density maximum at an early stage of community development (Figs 5 and 6c).

Figure 6.

Comparison between model results and field observations in the maximum-size dependence of reproductive capacity (a vs. b) and of plant density at canopy gaps relative to that at closed canopy (c vs. d). The 12 most abundant tree species observed in a warm temperate rain forest on Yakushima Island were examined. (a) Simulated reproductive rate per species leaf mass at coexistence equilibrium; (b) observed recruitment rate at 2-cm trunk d.b.h. per species basal area; (c) simulated average plant density during 0 ≤ t ≤ 50 relative to that during 50 ≤ t ≤ 200 in Fig. 5; (d) observed tree density at tree-fall gaps relative to that at closed-canopy stands, for trees ≥ 2 cm trunk d.b.h.

Discussion

The present results allow us to propose a theory that explains a mechanism of species coexistence in vertically-structured communities of plants. The Stratification Theory proposed here is comprised of three components: (i) one-sided competition as the primary condition, (ii) the range of coexistence provided by a canopy species, and (iii) the demographic advantage of an understorey species. The model analysis shows that one-sided competition is essential for the stable coexistence but two-sided competition merely brings about competitive exclusion. Analysing a time-discrete model of two-layered populations regulated by two-sided competition, Takada & Nakajima (1998) also suggested that there is no coexistence equilibrium. The range of coexistence in terms of carrying-capacity densities of canopy species, (inline image, xL), exists because xL > inline image when Q < 1 (the first condition). The relative density of the reproductive upper layer x2/x1 in eqn 1 is the largest when the upper layer is vacant, and is decreased with upper-layer density x2. At the same time, the effect of overall crowding upon canopy species increases with upper-layer density. These two factors contribute to xL > inline image. The third condition (i.e. eqn 3) is that understorey species have their single-species equilibrium density (inline image) within the range of (inline image, xL). We suggest that sufficiently large fecundity per leaf mass of understorey species prominently contributes to their coexistence with canopy species (Fig. 4a), and this trade-off readily explains the coexistence of several species differing in maximum size (Fig. 5). Plot-census data of a tree community in a warm temperate rain forest (Fig. 6b), and that in a tropical rain forest (Kohyama et al. 2003) suggest the existence of this trade-off among species. The trade-off between fecundity and plant size reflects the principle of allocation between vegetative and reproductive activities under the physiological constraint (Gadgil & Bossert 1970). Another trade-off between shade tolerance and plant size can also contribute to coexistence (Fig. 4c). However, the hypothesis that tolerant understorey species coexist with intolerant canopy species may not be independently established in the stratification theory, because there is the combination effect of fecundity and tolerance on the equilibrium density of understorey species, and because intolerant canopy species tend to regenerate in canopy gaps in the patch-mosaic landscape that is not addressed in the stratification theory. Lamb et al. (2008) manipulated shoot and root competition in experimental even-aged grassland communities and showed that intensive one-sided competition for light enhances plant-size hierarchy among species and reduces community diversity. By contrast, our results suggest that one-sided competition promotes the stable coexistence. The disagreement between these results is ascribed to the time-scale difference used in the experiments. The enhanced size hierarchy at an early phase of vegetation development within a generation is not necessarily associated with the long-run competitive exclusion over generations as far as that high fecundity and/or shade tolerance offset the disadvantage of small species living in the understorey.

The present model, which assumes the discrete size classes, can be converted to the continuous-size-structure model of Kohyama (1992, 1993), which simulates the dynamics of rain-forest trees controlled by one-sided competition. In the continuous model, the intensity of size-asymmetric competition (Ci with zero Q) is expressed by the upper basal-area density specific to the size of the target tree (i). Our analytical proofs from the model of eqn 1 afford partial confirmation of the forest architecture hypothesis (Kohyama 1993), which involves a mechanism for stratification (i.e. along a vertical profile) and that is associated with shifting patch mosaic (i.e. in its horizontal structure). There are some key differences between these simulations and the model framework presented here. As a result of the limitations of the forest plot data, Kohyama models assume that the recruitment rate at a given minimum size is simply proportional to the species basal area, or i Wixi,k for species k in eqn 2 (where Wi is the individual basal area at size i), which is suppressed by the overall basal-area density, or C1. Therefore, the effect of one-sided competition is not introduced into reproductive cycles, whereas size growth and mortality are regulated in a one-sided manner. In such a partially one-sided system, coexistence is more difficult to achieve than in a fully one-sided system, as is suggested by the present model analysis. A revised simulator of Kohyama (2006) describes size-dependent fecundity that is regulated by one-sided competition (upper basal-area density Ci) at each size, and demonstrates the coexistence of twelve species that differ in their maximum attainable size (in four classes) and in shade tolerance (in three classes).

Many of the major theories that explain the coexistence of tree species in tropical rain forests are based on concepts of horizontal-spatial heterogeneity, such as gap-phase dynamics, dispersal limitation, local density dependence, heterogeneous resource availability, and stochastic turnover of species (Wright 2002). Autogenic vertical heterogeneity along the canopy profile is also an important aspect of tropical forests that can explain the diversity of the life histories of trees (Turner 2001). Tropical rain forests are characterized by a huge canopy height and a high potential tree growth rate at full light, which reflects their productive environments. Trees in tropical rain forests have relatively shallow crowns that are adapted to make the best use of direct light rays with small angles of incidence. Such crown shapes permit among-crown stratification and facilitate the competitive size-asymmetry in light competition at Q ≈ 0 (Terbough 1985). All of these characteristics of tropical forests amplify the opportunity for coexistence among tree species that differ in terms of their maximum height. As shown in the simulation of Fig. 5, several species can coexist along tall canopy profiles of tropical rain forests, solely by one-sided competition for light resources. The relationship Wi = 4i–1, employed in the Fig. 5 simulation, corresponds to the situation in which each layer is twice the height of the one beneath it (e.g. the heights are 0.5, 1, 2, 4, 8, 16, 32 and 64 m), and the leaf mass per plant that absorbs light is proportional to the square of the height. These are realistic conditions for trees in tropical rain forests because a usual threshold size of reproduction is around half of the maximum tree height (Thomas 1996; Wright et al. 2005), and because a usual crown depth is around half of the top height of rainforest trees (Kohyama et al. 2003; Poorter et al. 2006). In contrast to tropical rain forests, trees in temperate and boreal forests, even extremely tall ones, grow slowly (small g), and have deep crowns adapted to large angles of incidence of solar radiation (Q > 0). Slow growth and deep crowns prevent fine-scale stratification among species. Vertical stratification may therefore be seen as a robust component of the diversity of species of trees in tropical forests.

Competition for light resources due to crowding brings about a lowering in reproduction, vegetative growth rate and probability of survival. The present analysis allows us to investigate the effect of competition for light on each of these demographic processes. Therefore, it is an over-simplified case that crowding has an effect only on reproduction (uk = vk = 0, sk > 0) and Wi = 1 in eqn 2. These extreme assumptions cause eqn 2 to be similar to the competition equations of the successional patch-occupation model (Hastings 1980; Teramoto 1993; Tilman 1994). This model describes the coexistence of species in terms of a hierarchy of ‘dispersal–competition trade-offs’. The model assumes that a competitively superior species is established on vacant patches and, in an identical way, on patches occupied by inferior species. In fact, this assumption is too simple because even competitively superior species must grow from small seedlings, and the established adult plants of inferior species should therefore suppress the establishment of superior ones (Yu & Wilson 2001; Kisdi & Geritz 2003). It is nevertheless noteworthy that our layer-structured model, which describes asymmetric competition along the vertical gradient of light resources (without horizontal heterogeneity), shares some theoretical similarity to the patch-occupation model, which describes horizontal heterogeneity of patch occupation (without any vertical resource gradient). In either case, n types of arena are introduced for competition, where the first is exclusively for within-species competition of the tallest or latest succession species, the next is for the first and the second tallest (or latest) species, and so on, until the n-th arena is for all species together. In a competition for resources, such a species hierarchy brings about the conditions of hierarchical amensalism that allow species to coexist. In general, either vertical stratification or successional hierarchy corresponds to the coexistence of species by a mechanism of differentiating subpopulations (Chesson 2000a,b; Amarasekare & Nisbet 2001; Amarasekare et al. 2004), and to non-spatial asymmetric inter-specific competition with trade-offs between life-history traits (Adler & Mosquera 2000). However, the ecological implications of our stratification theory are different from those of the succession model. The latter suggests that shade-intolerant, early-successional species coexist with shade-tolerant, late-successional species as a result of the high capacity for colonisation of the shade-intolerant ones. In contrast, the present stratification theory implies that lower-layer species can coexist with upper-layer species when the former are characterized by high tolerance to shade in fecundity (small s′) and/or mortality (small u′).

Tall and dense-foliage species of trees that have costly supportive organs evolve as a consequence of an ‘arms race’ among plants (Falster & Westoby 2003). The stratification theory suggests that evolution towards the exploitation of higher layers facilitates sympatric speciation between understorey and canopy species. The results are in contrast to the results that are obtained when plant height is examined using game theory, where coexistence between strategies of varying maximum height is hardly observed (Maynard Smith & Brown 1986; Falster & Westoby 2003). This is not surprising, because studies that use game theory do not take into account the regeneration process of each species starting from the lowest layer, but examine a developed vertical foliage distribution within a generation of each species. Some theoretical studies that assume competitive asymmetry among species suggest the evolutionary branching of body size (Manly 1996; Kisdi 1999). The model we presented here can be applied to the examination of the conditions for the evolutionary differentiation in maximum plant size in terms of stochastic variation in demographic properties.

Acknowledgements

We thank those who gave us invaluable comments on earlier drafts of this article; Takuya Kubo, Simon Levin, Michel Loreau, Paul Moorcroft, Hisao Nakajima, Tohru Nakashizuka, Akiko Satake, Jonathan Silvertown, Akio Takenaka, Dave Tilman, Norio Yamamura and anonymous referees.

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