- Email: [email protected]
Tree height and crown shape, as results of competitive games
Z theor BioL (1984)
112, 279-297
Tree Height and Crown Shape, as Results of Competitive Games YOH IWASAt, DAN COHEN~ AND JESUS ALBERTO LEON§
Department of Biological Science, Stanford University, Stanford, California 94305, U.S.A. (Received
12
July
1984)
Games between trees are studied to explain the height and the crown shape in a community. It is assumed that each tree maximizes the net productivity under the light environment, which is determined by surrounding trees. In both models, the asymmetry of competition is important, because higher leaves shade lower leaves but the lower don't shade the higher. In the tree-height game, each tree is assumed to choose a height given its crown shape. The height at a monomorphic equilibrium, in which all the trees of a community have the same height, increases with the tree density and the amount of leaves per tree, but decreases with the cost coefficient and the crown thickness. When the crown is thin enough, a polymorphic equilibrium appears, in which a community includes trees of different heights but with the same fitness. In the crown-shape game, each tree is assumed to choose the distribution of foliage along the height axis. A lone tree should have a hemispherical crown as the optimum. The monomorphic equilibrium in which all the trees in a community have the same crown shape is calculated. As the tree density increases, the average height of a crown of each tree increases and the height on the tree with the maximum foliage increases, but each tree should have some foliage all the way down to the ground. Several reasons for foliage cutoff at the lowest layer are discussed.
1. Introduction Trees a r e c h a r a c t e r i z e d b y t h e i r l o n g a n d t h i c k t r u n k s . T h e h e i g h t o f a d o m i n a n t tree s p e c i e s a n d t h e s h a p e o f its f o l i a g e d e t e r m i n e t h e o v e r a l l l a n d s c a p e o f m a n y t e r r e s t r i a l p l a n t c o m m u n i t i e s . Trees invest c o n s i d e r a b l e a m o u n t s o f p h o t o s y n t h e t i c p r o d u c t s in the c o n s t r u c t i o n o f t r u n k s . N o t o n l y the c o n s t r u c t i o n cost b u t t h e cost for m a i n t e n a n c e a c c o m p a n y i n g a tall t r u n k is a l s o v e r y l a r g e ; w a t e r m u s t b e c a r r i e d u p f r o m u n d e r g r o u n d r o o t s
Presentaddresses: t Department of Biology, Faculty of Science, Kyushu University, Fukuoka 812, Japan. ¢ Department of Botany, The Institute of Life Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. § Instituto de Zoologia Tropical, Facultad de Ciencias U.C.V. Apartado 47058, Caracas 1041-A, Venezuela. 279 0022-5193/85/020279+ 19 $03.00/0 © 1985 Academic Press Inc. (London) Ltd
280
Y. I W A S A
ET
AL.
to high canopy, and the photosynthetic products must be transported to other parts of the body. In addition, a trunk has to be mechanically strong enough to sustain the crown against wind stress (King, 1981 ; King & Loucks, 1978). There must be substantial benefit in keeping the foliage high, which compensates for these large costs. The primary benefit of growing up in a dense forest is to improve the light condition at the crown. The photosynthetic rate increases with light intensity, which is lessened by the foliage of surrounding trees. A tree which has a longer trunk and therefore a higher crown than its neighbors, can avoid shading by competitors and will enjoy a greater productivity. In contrast, a tree which grows alone receives no benefit due to a high crown. Thus the benefit of a trunk is produced by the surrounding trees. Consider a pure forest composed only of a single species. If all the trees lacked trunks, each tree could save the cost of producing a trunk and hence each would have a higher fitness. But such an imaginary cooperative equilibrium is not evolutionarily stable because a "cheater" mutant with a longer trunk will have a higher fitness than the surrounding cooperative individuals by enjoying better light condition. In the noncooperative (or Nash) equilibrium of a crown-height game (or ESS) in which each tree optimizes the crown height under the environment produced by other trees, each individual must spend a considerable cost in competition by producing a trunk. Therefore the adaptive significance of a tree trunk, an organ for light competition, is properly understood only in a game theoretical context. Recently Givnish (1982) discussed herb height using a game theoretical model. He predicted taller herbs in dense stands, and confirmed the predicted trend by field observation. An individual tree of a given species can have a very different shape when growing alone vs living in dense forest stand. For example the crown of an isolated oak tree is of a hemispherical shape (Fig. 1(a)); in which the amount of trunk and branches appear to be the minimum necessary to keep the leaves apart from each other. In contrast, an oak of the same species growing in a dense forest has a long trunk, and the foliage is only on the top part of the trunk (Fig. 1(b)). Such a variation of canopy shape has been explained as an optimal strategy of each tree growing under different light conditions (Horn, 1971). A model based on photosynthetic efficiency explained a multi-layer crown for successional trees growing in open habitat and a monolayer crown for climax trees in a dense closed forest. However, the crown shape of each tree can also be discussed as the result of a noncooperative game between trees, because the optimal crown shape of a tree depends on the light environment affected by the crown shape of surrounding trees.
TREE
HEIGHT
(o)
,~
AND
(b)
~t
]'t"/
,,
v' ~,
CROWN
SHAPE
281
il
FIG. 1. (a) A lone tree has a hemispherical crown which covers the trunk almost down to tile ground. (b) A tree growing in a dense forest has a long trunk with the crown only on the top.
In this paper we discuss tree height and crown shape from a game theoretical viewpoint, emphasizing the one-sided nature of light competition, i.e. the fact that the higher part of the foliage shades the lower part but the lower does not affect the higher. First we investigate a static game in which each tree chooses its height. Monomorphic equilibrium, in which all trees are of the same height, is expressed as a function of the canopy thickness, the tree density of the stand, the cost for higher crown, and the photosynthetic efficiency. When the crown is thin enough, a polymorphic equilibrium appears, in which the community includes trees of different heights but with the same fitness. Second, the foliage distribution within a tree is investigated. The adaptive significance of the cutoff of the foliage at the lowest layer is hard to understand by the static competition-for-light model. We discuss additional considerations, such as herbivore avoidance and dynamic optimization. 2. A Static Game Model for Tree Height
Consider a tree with the average crown height x. The net production rate of the tree is q~(x) =f(L(x)) - c(x), (1) where f ( L ) is the photosynthetic rate as a function of light intensity, L(x) is the light intensity at height x, and c(x) is the cost o f the tree with average crown height x. In the following, we investigate the pattern in which each tree takes the optimal height which maximizes the net production rate, ~b(x). Since the light environment of a tree depends on the height o f the surrounding trees which, if taller, can shade it, the problem is not a simple optimization but a noncooperative game in which each tree is a player which optimizes its strategy given other players' strategies.
282
Y.
IWASA
ET
AL.
The photosynthetic rate f ( L ) increases with the light intensity L, but it levels off for too strong light, obeying, say, f ( L ) = a L / ( a + bL) - m,
(2a)
f ( L ) = a log L + b.
(2b)
or
The photosynthesis-light relation for each unit leaf area fits equation (2a) well (Horn, 1971). However, the photosynthetic rate of a whole tree, an integration o f photosynthesis at different heights, may be expressed better by equation (2b) (Blackman & Wilson, 1951). In the following, we use equation (2b). Because of the shading by the foliage, the light intensity at height x is L(x) = Lo exp ( - a [ l e a f area index at height x])
(3)
(see Monsi & Saeki, 1953). The leaf area index at height x is the total amount of foliage present above height x. The coefficient a is the shading effect caused by a unit leaf area. It varies with the crown shape of a tree, and is related to the density F ( y ) of trees with height y, as follows a. (Leaf area index at x) = a
=
I;
(density of foliage at height z) dz
f(y)p(y-x)
dy
(4)
where p ( y - x ) is the shading effect on the point at height x by a tree with height y, expressed as a function of the difference in heights y - x . The function p ( y - x ) depends on the crown shape, and the reflectance and transparency of leaves. For example, if the foliage of a tree with average height y and crown thickness k, is uniformly distributed from height y - k / 2 to y + k / 2 , that tree does not cast shade on points above y + k / 2 , casts full shading on points below y - k / 2 , and an intermediate amount of shading between them: p(z)=
w(z+k/2)/k
-k/2
(5)
z<-k/2.
For simplicity we here consider the shading by foliage only, neglecting that by the trunk. The cost of having the foliage at average height x, denoted by c(x) in equation (1), includes both a part that is relatively independent of height,
TREE
HEIGHT
AND
CROWN
SHAPE
283
such as canopy respiration, and the other part that greatly increases with height x, such as the cost for carrying up the water. The latter may probably be expressed by a convex function; i.e. if height doubles, the cost will increase more than twice (positive return to scale). To specify the exact form of the function c(x), we need physiological and mechanical knowledge. Here we simply assume a quadratic form: c(x) = Co+ c2x 2.
(6)
The cost c(x) also includes an amortized construction cost assigned over many years. MONOMORPHIC EQUILIBRIUM
An equilibrium in which all individuals have the same height g is expressed by a height distribution as
(7)
F ( x ) = F. ~ ( x - ~),
where F is tree density, and 8 ( x - ~ ) is a delta function indicating that the tree height distribution has a sharp peak at g. The stability condition for this monomorphic equilibrium is ~b(x) < ~b(~),
for all x # ~
(8)
which indicates that a tree with a crown height different from 2, has a lower net productivity than the optimal. Consider the case in which the foliage is of a cylindrical shape, i.e. p(z) is given by equation (5), and the photosynthetic rate is proportional to the logarithm of the light intensity, as in equation (2b). From equations (4) and (7), the light intensity, equation (3),.becomes L(x) = Lo exp ( - F . p ( , ~ - x ) ) , and therefore the photosynthetic rate as a function of tree height is a log Lo + b - aFpw f ( L ( x ) ) = a log L o + b - a F p w / k . ( ~ + k / 2 - x ) a log L0 + b
Og+k/2
(9)
as illustrated in Fig. 2. In this case, the equilibrium condition, equation (8), is equivalent to three conditions: 4)'(g)=0,
4)"(~)<0
and
~b(0)<$(g).
284
Y.
IWASA
ET
AL.
f (L (x))
V/////////////.~I////////////)
0 X
FIG. 2. The p h o t o s y n t h e t i c rate f ( L ( x ) ) a n d the cost c(x) of a tree l i v i n g in a m o n o m o r p h i c e q u i l i b r i u m , as the f u n c t i o n s o f a v e r a g e foliage h e i g h t x. The difference b e t w e e n the two is the fitness $ ( x ) w h i c h is m a x i m u m at x = 2.
These are rewritten as aFpw/k - c'(ff) = 0
(10a)
-c"(g) < 0
(10b)
alogLo+b-aFp.,-c(O)
(10c)
respectively. From equation (6), equation (10a) becomes aFp., Y"= 2c2k'
(I1)
that is /photosynthetic]( total '~ (average foliage'~ = \ efficiency / \ s h a d i n g / \ height / (cost '~( crown '~ \coefficient/\thickness] This result implies: (1) The average foliage height ~ increases with the intensity of shading Fpw, which includes not only the transparency of each leaf and the amount of leaves per tree, but also the population density of trees at the stand. When the density is very low, the tree height should also be low, because
TREE
HEIGHT
AND
CROWN
SHAPE
285
there is no competition with respect to light between trees. We therefore expect a positive correlation between crown height and tree density in a community. On the other hand, if the amount of foliage of a tree is inversely proportional to the tree density, the tree height is expected to be constant between sites with different density. (2) The tree height ~ is inversely proportional to the crown thickness k_ If the crown is thin, an additional investment in trunk growth is greatly rewarded by an improvement of the light condition at the canopy; actually the gradient o f light intensity at 2, which produces the marginal fitness for tree height, is Fpw/k, inversely proportional to the thickness /c We should note that the height discussed here is the distance from the ground to the average height of a crown, although in common usage tree height refers to the top of the tree. (3) The height g decreases with the cost coefficient, c2. Low availability of water or nutrients as well as strong wind, which require an enduring trunk with additional investment, may reduce the crown height. (4) Height increases with the coefficient of photosynthetic rate a, but is independent o f the light intensity at the top Lo; high productivity o f the stand may result in a tall community through a high density or a smaller cost coefficient, but there is no direct effect of light intensity on height. POLYMORPHIC
EQUILIBRIUM
If the canopy is not thick enough, one of the stability conditions of monomorphic equilibrium 2:
4,(~) > 4,(0) is not satisfied. Then a mutant with no trunk ( x = 0 ) , which probably corresponds to a herb or a shrub, has a higher fitness than a tree with g, and therefore succeeds in invasion. If so, an equilibrium community may be composed of undergrowth and trees with a common height X (Fig. 3(a)). We also may have a dimorphic equilibrium in which the community is composed of trees of two classes, high trees with height X~ and low trees with x2 (Fig. 3(b)). Tree height distribution of the community is the sum of two delta functions: F ( x ) = F, . 8 ( x - g , ) + F2. 8 ( x - ~ 2 )
(12)
where Fj and F2 are the tree densities of high and low crown trees per unit area. Assume that the crowns of trees belonging to the upper class are not overlapped with those of the lower class trees. If we assume cylindrical shapes for trees o f both classes with thickness k~ and k2, respectively, we
Y. IWASA ET AL.
286 (a)
(b)
f(L(x))
~1/1////i/~
o
V///////Z g
FIG. 3. (a) An equilibrium composed of trees and shrubs. (b) A dimorphic equilibrium composed of trees of two layers, in which trees in the higher layer tend to be less thick (k~ < k2) and more dense (F I > F2) than those in the lower layer.
have the relation
kl+k2 21 --22 > ) _ 2
(13)
The s h a d i n g effect by the leaves above the height x, indicated by e q u a t i o n (4), is the leaf area index at x multiplied by the s h a d i n g coefficient a :
I
01p,.,( 2, + k l / 2 - x ) / k~
[shading] = / F l p w
| F,p~,+ F2p.(22 + k 2 / 2 - x ) / k 2 [ (Fi + F2)pw
x >1~, + kl/2 xl - k l / 2 <-- x < 2 1 + k l / 2
22+ k2/2 <- x < 2, - k~/2 22 - k2/2 <- x < 22 + k J 2 O<-x < 2 2 - k 2 / 2 (14)
then we can calculate h o w the light level decreases using e q u a t i o n (3). The g a m e ' s equilibrium condition for a d i m o r p h i c solution, e q u a t i o n (12), is that the fitness, or the net p r o d u c t i o n rate, is the same for 2~ and
TREE
HEIGHT
AND
CROWN
SHAPE
287
~2, and that the trees of height other than these two should have a lower fitness: 4,(~) = 4,(~2)> 4,(x) forall x(#~,, and x2)(15) These are based on the idea that, in a evolutionary equilibrium, all the players enjoy the same fitness and all the mutants have a smaller fitness. From Fig. 3(b), these equilibrium conditions (or ESS conditions) are expressed as 4,(~,) = 4,(~2) > 4,(0)
that is
-rl = aFzpw/2c2kz
(16a)
x2 = aF2pw/2c2k2
(16b)
aF, p~ [ aF, pw'~2 ( F2) [ aF2pw, 2 2 c2k2-~2k~) = - a F , + - ~ pw-c2k2--~2k2 ) > - a ( F , + F:)p. (16c) If only ~t and x2 are variables and F~ and F2 are parameters given arbitrarily, then these three equations cannot be satisfied except in a degenerate case, which is nongeneric in the parameter space. Here we assume that the relative densities of trees in the upper and lower classes are also determined by the above equilibrium condition, assuming that the total amount of foliage is given:
F, + F2 = F,ota,.
(17)
Then four variables, x~, x2, F~, and F2, are determined by equations (16a, b and c) and (17). Here we simply note a relation derived from an inequality Xz > x : using equations (16a) and (16b):
Fl/kl > F2/k2.
(18)
This implies that, if the thickness of the upper layer trees is larger or equal to that of lower layer trees (k~ >- k2), then the density of the upper layer trees should be much greater than that of lower layer trees, in units of shading effect (FI > F2). Otherwise, the thickness of the upper trees must be smaller than that of the lower trees (k~ < k2). In a similar way we can consider an equilibrium composed of trees of three or four layers. In general, the n-morphic equilibrium is expressed by the foliage height distribution, as F ( x ) = ~ F,8(x-~,), i=1
(19)
288
v.
IWASA
ET
AL.
where Fi is the a m o u n t of foliage at the/-layer. The equilibrium conditions for this solution are ~b(X,) = ~b(X2) = . . . =
~b(X.) = A (constant), (20)
q~(x)
for a l l x ~ x l , x 2 , . . . , x , .
Not only . ~ , . . . , ~, but also F t , . • . , F, are determined by these equilibrium conditions under the constraint of a fixed total amount of foliage: Fi = Ftota,.
(21)
i=1
For the limit case in which the thickness of each crown is small enough, and the n u m b e r of layers is very large, we have an infinite layers model: expressed by F ( x ) , a foliage height distribution. The ESS conditions are now qS(x) = A for F ( x ) > 0 ~b(x) < A for
F(x) =
0.
(22a) (22b)
Consider the case in which the photosynthetic rate is proportional to the logarithm of the light intensity, equation (2b). From equation (5) with k = 0, the fitness, equation (1), becomes 4~(x) = a log
Lo-a
f;
F(y) dypw+b-co-c2x 2.
(23)
In an interval in which equation (22a) holds, we have
-
F(y) dy=
(A-alogLo-b+co+c2x2).
(24)
apw
By calculating the derivative, F(x) =
1 apw
(25)
2c2x.
The equilibrium distribution of tree height F(x) is a combination of a positive value, equation (25), and F(x)= 0. One solution is that there is a truncation height xT, above which there is no foliage and below which equation (25) holds:
F(x) {~ c2x/apw O<-x
x ~x
(26)
T.
We can reason that only this solution can satisfy the ESS conditions as follows: Suppose that there is some gap of foliage distribution, say F(x) = 0
TREE
HEIGHT
AND
CROWN
SHAPE
289
between xj and x2. Compare the highest of the trees under the gap with the lowest of those above the gap, the heights of them being x~ and x2 respectively (xt < x2). The photosynthetic gains of them are equal because the light intensity does not decrease within the foliage gap, f(L(X~))= f(L(X2)), but the cost is larger for the higher tree, c(xl) < c(x2). Therefore their net production rates are different, ~ ( x , ) > ~(x2). This is against the ESS assumption, equation (22a). By using the constraint on the total amount of foliage, Ftot,, =
Ioo F(y) dy,
(27)
we can determine the truncation height as
(28)
xT = 4 aF, o,.,pwl c2.
This result suggests a general pattern of tree height distribution in a community with high species diversity. The tree density, measured in units of shading intensity, is the highest for the tallest trees and lower for shorter trees (Fig. 4). The height of trees in the top layer increases with total tree density, but decreases with the cost coefficient, and is relatively insensitive to light intensity. f
I
(
/
IIi
,
?
4r"
II
/..-_.% ii I
H
II FIG. 4. An equilibrium of infinitely many layers. As the limit when the crown of each tree becomes very thin, an equilibrium with infinite layers appears. The tallest trees have the highest density measured by unit of shading intensity.
3. Shape o f a Crown: Height Distribution of Leaves Within a Tree
We have assumed that a fixed size and shape of the tree crown while discussing height evolution. In this section we investigate the shape of a crown, the arrangement of foliage within trees. Denote the foliage height distribution of a tree by z(x), i.e. amount of leaves at the height x. Also denote the foliage height distribution of other
290
v.
IWASA
ET
AL.
surrounding trees by w(x). Retaining, with some pertinent changes, the notation used in the last section, the photosynthesis rate of a unit amount of leaves (not a whole tree) is
f( Loexp[-p f~ (z(y)+Dw(y))dy])
(29)
where D is a factor expressing the shading effect by other trees compared to that by the same tree. D should increase with tree density at the stand. w(x) denotes the vertical foliage distribution of other trees. The cost of a unit amount of leaves includes not only the maintenance and amortized construction cost, which increases with the height of the leaves, but also the cost of branches to separate a given amount of leaves, which may be independent o f height but increases with the amount of leaves at the height z(x). Here, we assume that the cost of a unit amount of leaves at height x is c(x) + yz(x), (30) the second term may be called "lateral cost". Therefore the fitness of a whole tree with foliage distribution z(x) is
4~=foZ(X)[f(Loexp(-o f;[z(Y)+ Dw(y)]dY))-c(x)-yz(x)] dx (31) which is a functional of the function z(x). The optimal distribution of leaves along the height axis, z*(x), is the one that maximizes the fitness, equation (31), under the constraint
z(x) >-0
z(x) dx = Ztotal.
and
(32)
We assume that the photosynthetic rate of each leaf is expressed by equation (2b), f(L) = a log L + b, which is good for the photosynthesis rate of a whole tree but not completely suitable for that of a leaf. We should rather use equation (2a), but for mathematical convenience we here assume equation (2b). By using a Lagrange's multiplier A, the above problem with the constraint of a fixed total mount o f foliage, is equivalent to the maximization o f 4~:
~ = f ? z(x)( a log [ Lo exp (-P l ; [z(Y) + Dw(y)] dY) ] + b
c(x) -
yz(x) + h / dx J
(33)
TREE
HEIGHT
AND
CROWN
SHAPE
291
under the condition z(x) >-0 only. Then the Pontryagin's maximal principle (Pontryagin et ai., 1962) can be applied. Define h(x) and W ( x ) as h(x) = W(x) = -D
z(y) dy,
(34a)
w(y) dy.
(34b)
Then from equation (32) (35a)
d h / d x = z(x).
h(0) = --Ztotab
h(oo) = 0.
(35b)
Since W ( x ) expresses the shading by other trees, it should be treated in the optimization problem as a function already given. The Hamiltonian is H = z{a log Lo+ b + ;t + ap(h(x) + W ( x ) ) - c(x) - 3"z(x)} + p ( x ) z ( x ) , (36)
where the adjoint variable p(x) follows (37)
dp / dx = -OH~Oh = -apz.
Since the terminal value h(oo) is given by equation (35b), we cannot specify the terminal value of the adjoint variable p(oo). The maximal principle states that z(x) at each x is chosen so as to maximize the Hamiltonian H under the condition that all other functions are fixed. Thus we have z*(x) =
a log L o + b + A + a p ( h ( x ) + W ( x ) ) - c ( x ) + p ( x ) . , ifit is positive. 23,
(38a)
0,
(38b)
otherwise.
Now we try to find a solution that satisfies all the above conditions. First we notice that when x grows infinitely large, both h(x) and W ( x ) tend to zero and c(x) diverges to infinity. Therefore the R.H.S. of equation (38a) is negative for sufficiently large x; hence equation (38b) holds; z * ( x ) = 0 . Denote the height of the top by x*, above which there is no foliage. Then, z*(x)=0, h ( x ) = 0 , and p(x)=constant hold for x>-x *. For x < x * , equation (38a) holds. By differentiating equation (38a), and using equations (35a), (34b), and (37), we have apD 1 dc dz*/dx = ~ w(x) - 2--~ d--~"
(39)
292
Y.
IWASA
ET
AL.
The optimal distribution o f foliage o f a tree z*(x), is determined by equation (39), and it may be influenced by the foliage distribution o f surrounding trees, w(x). CROWN SHAPE OF A LONE TREE
First consider the crown shape of a tree living alone. Since all x, equation (39) is
w(x) = 0 for
1 dc
dz*/dx
- -
2y dx"
(40)
Combining this with the condition z * ( x * ) = 0, we have
(x
z*(x)=(c(x*)-c(x))/2y
(41)
Since the cost c(x) is an increasing function o f height x, z*(x) decreases with x. Moreover the convex shape o f c(x) gives a concave curve for z*(x). Therefore we can conclude that the amount of leaves o f a lone tree is greater at lower height, and therefore a tree growing isolated has a hemispherical or cone-like shape (Fig. 5a). (a)
(b)
(c)
5.0
0.0
L
0.0
30 z " (x)
O0
3"0 z'(x)
L
I
0 i0
I
5 iO
z "(x)
FIG. 5. The foliage distribution along the height axis, z*(x), at a monomorphic noncooperative equilibrium (or ESS) of the crown-shape game, (a) when a tree grows alone (fl = 0), (b) when it grows within a stand (/3 = 0.5), and (c) within a dense stand (/3 = 2-0), calculated by equation (45). The crown becomes taller with the tree density, but each tree still has some foliage all the way down to the ground.
TREE
HEIGHT
AND
CROWN
SHAPE
293
The tree top x* is determined by the condition that the sum total of leaves is equal to a given amount Ztot,~:
(c(x*) - c(x)) dx/2"y.
Ztota, =
CROWN
SHAPE
OF TREES
IN A MONOMORPHIC
(42)
EQUILIBRIUM
Next, consider tree shape in a monomorphic equilibrium, in which all the trees have the same foliage distribution. This is mathematically expressed by optimizing z ( x ) with w ( x ) fixed and putting w(x) = z(x). Equation (39) becomes
dz,/dx=apDz,(x) 2y
I dc 23/dx"
(43)
Combining this with z * ( x * ) = 0, we have
z*(x) =
x*
"~o-2 -- x- 1 dc e-t o / ~,)~y- , _ _ _ _ . dy. 23, dy
(44)
Since d c / d x is an increasing function, z*(x) may have a hump, but it is positive for all x < x*. For example, in the case c(x) = Co+ c2x 2, by putting fl = apD/2",/, we have
z*(x) =
e-°(Y-X)y dy. c2/ y
-;;[~+x-(x*+~)e-a(x*-x'],
(45)
(see Figs 5(b), and 5(c)). The top height x* is determined by equation (32) as Z,o,a~ =
=
f; Io
z*(x) dx
:* 1--e -(apD/2~')x dc apD d--x" dx
(46)
where we used equation (44) and partial integration to obtain the second equality. Equation (46) clearly shows that the top height x* increases with the total amount of foliage Ztota,.
294
Y.
IWASA
ET
AL.
4. Discussion
It is interesting to compare the tree-height game model with the resource competition scheme assumed in the niche theory by MacArthur & Levins (1967). The tree height ~, defined as the average foliage height, corresponds to the niche position along one dimensional niche axis, and, roughly speaking, the canopy thickness k corresponds to the width of the utilization function. An important feature of the game between trees is the asymmetry of the competition, i.e. a taller tree strongly shades smaller trees but a smaller tree will not affect taller ones much, as expressed by the function p(z) in equation (5). The theories of community structure shaped by evolution have been studied and applied to the body size of animals (Roughgarden, 1972, 1976; Roughgarden, Heckel & Fuentes, 1983; Case, 1982). In these theories, it is demonstrated that individuals coexisting in an evolutionary stable community must have the same fitness, and that no mutant which happens to occur may have a higher fitness than that of the resident. These assumptions are equivalent to the equilibrium conditions in the tree-height game, such as equations (8), (15), (20), and (22). Coevolution theory for the body sizes of animals predicts that many species may coexist if the width of the utilization function of each species is small compared to that of the resource supply function. A corresponding result in the tree-height game is that a polymorphic equilibrium occurs when the canopy is thin enough. Further, the number of tree types coexisting in a community increases as the crown of each tree becomes thinner. In this paper, we assumed that each individual maximizes its own photosynthetic performance. However, if a tree is surrounded by genetically related individuals, the optimal strategy should also consider the productivity of competitors. This kinship effect is very important in many herbs in which vegetative growth is prevalent, as pointed out by Givnish (1982), and may be studied by assuming that each herb tries to maximize its inclusive fitness. The relatedness between competitors seems less important in trees because the seed produced by many parent trees are mixed in the seed pool. One interesting result of the canopy-shape game model studied in the last section is that a bare trunk accompanied by no foliage will not appear in the monomorphic ESS pattern. If trees growing in a forest have the same shape, and if that shape is the optimal one maximizing the net photosynthesis of each tree under light conditions made by others, then trees should have some leaves all the way down to the ground. This result is derived also from an intuitive argument as follows: Suppose that all trees have some bare trunk in the lowest part of their stem. The light intensity is constant
TREE
HEIGHT
AND
CROWN
SHAPE
295
for this part because of the lack of foliage which causes the shading. Therefore if a tree moves some leaves from the lowest part of the c a n o p y to the ground, it can save the cost due to height without losing photosynthetic rate. This means that the original foliage distribution with bare trunk is not an ESS. Nonexistence of bare trunks seems to be a good rule for lone trees; m a n y lone trees growing in natural environments have some leaves almost down to the ground. But for trees in a dense forest we know m a n y examples of trees with long bare trunks. N o w we would like to discuss the reasons why trees have bare trunks: (1) In a p o l y m o r p h i c equilibrium of the static crown shape game model, tall trees can have a bare trunk. Therefore in a forest c o m p o s e d of several kinds of trees with different heights, trees in the highest layer m a y have a bare trunk in the lowest level occupied by smaller trees or shrubs. Even in a pure forest c o m p o s e d of a single tree species, if there are always some younger trees of the same species with low canopy, then tall trees m a y have long bare trunks in an ESS pattern. (2) Since growth is a dynamic process, the effect on future production rate should also be taken into account in determining the shape in young trees. I f a tree can suppress the growth of surrounding trees by having a taller canopy it will gain in future production rate as well as present rate. Because o f this effect with time delay, trees m a y invest in a trunk, an organ for competition, much more than predicted by the optimal allocation in the static game model, in which the " p r e s e n t " production rate is assumed to be maximized. This hypothesis predicts that a bare trunk is more c o m m o n in a dense stand o f a uniform age where competition is intense during the growth of the stand all the way from the seedling stage. (3) Another aspect of the dynamic nature of growth is that the optimal development in a young age m a y affect the shape o f trees in later stages. Suppose that a stand is cleared by some disturbance such as fire and all the seedlings of trees begin to grow almost synchronously. In this early stage o f a tree's life history, it m a y be crucial to have a thick canopy which requires a long trunk, but this m a y be not important any more in the later stages. Then we can find old trees with a small a m o u n t of leaves at the top of a long trunk, because they would discard the lower leaves rather than the upper ones. In the last steps above, we assumed that the amortized cost for trunk construction is important so that once a long trunk is produced the cost for maintaining foliage does not strongly depend on height. The effect of p o l y m o r p h i c equilibrium discussed in (1) can also be combined with this dynamic effect. For example, in the early stage of succession there may be other trees or shrubs that cover the lowest layer,
296
v . IWASA E T A L .
and in the later stages this undergrowth may have perished due to competition for light. Then we find a m o n o m o r p h i c equilibrium of trees with long bare trunks. (4) The shading may also be caused by the tree trunk itself. This had been neglected in the model above. Because of this effect, light intensity may decrease with the decrease of height even in the lowest part without foliage. The intensity of shading by the bare trunk itself, however, seems not to be strong enough as to explain trunk existence. (5) Predation by large herbivorous animals may also explain the adaptive significance of bare trunks. For example, in a savanna, where trees are not so dense as to shade each other, they have a high canopy and a long trunk without foliage. The coevoIution of giraffes and trees in the savanna seems to be the most reasonable explanation as Lamarck pointed out. If the expectation of loss due to herbivory by m a m m a l s is very high, it may be advantageous not to produce leaves in the lowest part of the trunk. We may call the lower cutoff line of the canopy the goat line (2 m to 4 m) or the giraffe line (9 m to l l m) corresponding to the main herbivores. The hypothesis that a bare trunk is for the avoidance of m a m m a l predators may be tested by comparing different communities with varying grazing pressure. In the present model this effect may be expressed by the higher cost for lowest height, i.e. c ( x ) increases again for small x, as well as for large x. In a similar way, for trees in regions with heavy snow, the nonexistence of leaves in the lowest part of the tree might be explained as the avoidance of harm by snow. (6) Developmental constraints sometimes seem to be a main reason of a tree's crown shape. For example, a palm tree must continue growing upward to keep producing new leaves. Their crown shape, thus, m a y not be optimal from the viewpoint of photosynthetic efficiency. It is difficult to know the extent to which a plant's shape is restricted by developmental constraints; one possible way is to show an ideal crown shape in ESS without developmental constraints and to compare it with the shape of real systems. We thank D. Andow, T. Givnish, D. King, K. Naganuma, and J. Rummel for their comments. REFERENCES BLACKMAN, G. E. & WILSON, G. L. (1951). Ann. Bot. 15, 64. CASE, T. (1982). Theor. Pop. Biol. 21, 69. GIVNISH,T. J. (1982). Amer. Nat. 120, 353. HORN, H. S. (1971). Adaptive geometry of trees. Princeton: Princeton University Press.
TREE
HEIGHT
AND
CROWN
SHAPE
297
KING, D. (1981). Oecologia 51, 351. KING, D. & LOUCKS, O. L. (1978). Rad. Environm. Biophys. 15, 141. MACARTHUR, R. H. & LEVINS, R. (1967). Amen Natur. 101, 377. MONSl, M. & SAEKL T. (1953). Jap. J. Bot. 14, 22. PONTRYAG1N, L. S., BOLTYANSKII, V. G., GANKRELIDZ, R. V. & MISCHENKO, E. F. (1962). The mathematical theory of optimal processes (trans. by K. N. Trirogoff). New York: Interscience Publishers, John Wiley & Sons, Inc. ROUGHGARDEN, J. (1972). Amer. Nat. 106, 683. ROUGHGARDEN, J. (1976). Theor. Pop. Biol. 9, 388. ROUGHGARDEN, J., HECKEL, D. g¢. FUENTES, E. R. (1983). In: Lizard ecology: studies o f a modelorganism (Huey, R. B., Pianka, E. R. & Schoener, T. W. eds.), pp. 371-410. Cambridge: Harvard University Press.
112, 279-297
Tree Height and Crown Shape, as Results of Competitive Games YOH IWASAt, DAN COHEN~ AND JESUS ALBERTO LEON§
Department of Biological Science, Stanford University, Stanford, California 94305, U.S.A. (Received
12
July
1984)
Games between trees are studied to explain the height and the crown shape in a community. It is assumed that each tree maximizes the net productivity under the light environment, which is determined by surrounding trees. In both models, the asymmetry of competition is important, because higher leaves shade lower leaves but the lower don't shade the higher. In the tree-height game, each tree is assumed to choose a height given its crown shape. The height at a monomorphic equilibrium, in which all the trees of a community have the same height, increases with the tree density and the amount of leaves per tree, but decreases with the cost coefficient and the crown thickness. When the crown is thin enough, a polymorphic equilibrium appears, in which a community includes trees of different heights but with the same fitness. In the crown-shape game, each tree is assumed to choose the distribution of foliage along the height axis. A lone tree should have a hemispherical crown as the optimum. The monomorphic equilibrium in which all the trees in a community have the same crown shape is calculated. As the tree density increases, the average height of a crown of each tree increases and the height on the tree with the maximum foliage increases, but each tree should have some foliage all the way down to the ground. Several reasons for foliage cutoff at the lowest layer are discussed.
1. Introduction Trees a r e c h a r a c t e r i z e d b y t h e i r l o n g a n d t h i c k t r u n k s . T h e h e i g h t o f a d o m i n a n t tree s p e c i e s a n d t h e s h a p e o f its f o l i a g e d e t e r m i n e t h e o v e r a l l l a n d s c a p e o f m a n y t e r r e s t r i a l p l a n t c o m m u n i t i e s . Trees invest c o n s i d e r a b l e a m o u n t s o f p h o t o s y n t h e t i c p r o d u c t s in the c o n s t r u c t i o n o f t r u n k s . N o t o n l y the c o n s t r u c t i o n cost b u t t h e cost for m a i n t e n a n c e a c c o m p a n y i n g a tall t r u n k is a l s o v e r y l a r g e ; w a t e r m u s t b e c a r r i e d u p f r o m u n d e r g r o u n d r o o t s
Presentaddresses: t Department of Biology, Faculty of Science, Kyushu University, Fukuoka 812, Japan. ¢ Department of Botany, The Institute of Life Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. § Instituto de Zoologia Tropical, Facultad de Ciencias U.C.V. Apartado 47058, Caracas 1041-A, Venezuela. 279 0022-5193/85/020279+ 19 $03.00/0 © 1985 Academic Press Inc. (London) Ltd
280
Y. I W A S A
ET
AL.
to high canopy, and the photosynthetic products must be transported to other parts of the body. In addition, a trunk has to be mechanically strong enough to sustain the crown against wind stress (King, 1981 ; King & Loucks, 1978). There must be substantial benefit in keeping the foliage high, which compensates for these large costs. The primary benefit of growing up in a dense forest is to improve the light condition at the crown. The photosynthetic rate increases with light intensity, which is lessened by the foliage of surrounding trees. A tree which has a longer trunk and therefore a higher crown than its neighbors, can avoid shading by competitors and will enjoy a greater productivity. In contrast, a tree which grows alone receives no benefit due to a high crown. Thus the benefit of a trunk is produced by the surrounding trees. Consider a pure forest composed only of a single species. If all the trees lacked trunks, each tree could save the cost of producing a trunk and hence each would have a higher fitness. But such an imaginary cooperative equilibrium is not evolutionarily stable because a "cheater" mutant with a longer trunk will have a higher fitness than the surrounding cooperative individuals by enjoying better light condition. In the noncooperative (or Nash) equilibrium of a crown-height game (or ESS) in which each tree optimizes the crown height under the environment produced by other trees, each individual must spend a considerable cost in competition by producing a trunk. Therefore the adaptive significance of a tree trunk, an organ for light competition, is properly understood only in a game theoretical context. Recently Givnish (1982) discussed herb height using a game theoretical model. He predicted taller herbs in dense stands, and confirmed the predicted trend by field observation. An individual tree of a given species can have a very different shape when growing alone vs living in dense forest stand. For example the crown of an isolated oak tree is of a hemispherical shape (Fig. 1(a)); in which the amount of trunk and branches appear to be the minimum necessary to keep the leaves apart from each other. In contrast, an oak of the same species growing in a dense forest has a long trunk, and the foliage is only on the top part of the trunk (Fig. 1(b)). Such a variation of canopy shape has been explained as an optimal strategy of each tree growing under different light conditions (Horn, 1971). A model based on photosynthetic efficiency explained a multi-layer crown for successional trees growing in open habitat and a monolayer crown for climax trees in a dense closed forest. However, the crown shape of each tree can also be discussed as the result of a noncooperative game between trees, because the optimal crown shape of a tree depends on the light environment affected by the crown shape of surrounding trees.
TREE
HEIGHT
(o)
,~
AND
(b)
~t
]'t"/
,,
v' ~,
CROWN
SHAPE
281
il
FIG. 1. (a) A lone tree has a hemispherical crown which covers the trunk almost down to tile ground. (b) A tree growing in a dense forest has a long trunk with the crown only on the top.
In this paper we discuss tree height and crown shape from a game theoretical viewpoint, emphasizing the one-sided nature of light competition, i.e. the fact that the higher part of the foliage shades the lower part but the lower does not affect the higher. First we investigate a static game in which each tree chooses its height. Monomorphic equilibrium, in which all trees are of the same height, is expressed as a function of the canopy thickness, the tree density of the stand, the cost for higher crown, and the photosynthetic efficiency. When the crown is thin enough, a polymorphic equilibrium appears, in which the community includes trees of different heights but with the same fitness. Second, the foliage distribution within a tree is investigated. The adaptive significance of the cutoff of the foliage at the lowest layer is hard to understand by the static competition-for-light model. We discuss additional considerations, such as herbivore avoidance and dynamic optimization. 2. A Static Game Model for Tree Height
Consider a tree with the average crown height x. The net production rate of the tree is q~(x) =f(L(x)) - c(x), (1) where f ( L ) is the photosynthetic rate as a function of light intensity, L(x) is the light intensity at height x, and c(x) is the cost o f the tree with average crown height x. In the following, we investigate the pattern in which each tree takes the optimal height which maximizes the net production rate, ~b(x). Since the light environment of a tree depends on the height o f the surrounding trees which, if taller, can shade it, the problem is not a simple optimization but a noncooperative game in which each tree is a player which optimizes its strategy given other players' strategies.
282
Y.
IWASA
ET
AL.
The photosynthetic rate f ( L ) increases with the light intensity L, but it levels off for too strong light, obeying, say, f ( L ) = a L / ( a + bL) - m,
(2a)
f ( L ) = a log L + b.
(2b)
or
The photosynthesis-light relation for each unit leaf area fits equation (2a) well (Horn, 1971). However, the photosynthetic rate of a whole tree, an integration o f photosynthesis at different heights, may be expressed better by equation (2b) (Blackman & Wilson, 1951). In the following, we use equation (2b). Because of the shading by the foliage, the light intensity at height x is L(x) = Lo exp ( - a [ l e a f area index at height x])
(3)
(see Monsi & Saeki, 1953). The leaf area index at height x is the total amount of foliage present above height x. The coefficient a is the shading effect caused by a unit leaf area. It varies with the crown shape of a tree, and is related to the density F ( y ) of trees with height y, as follows a. (Leaf area index at x) = a
=
I;
(density of foliage at height z) dz
f(y)p(y-x)
dy
(4)
where p ( y - x ) is the shading effect on the point at height x by a tree with height y, expressed as a function of the difference in heights y - x . The function p ( y - x ) depends on the crown shape, and the reflectance and transparency of leaves. For example, if the foliage of a tree with average height y and crown thickness k, is uniformly distributed from height y - k / 2 to y + k / 2 , that tree does not cast shade on points above y + k / 2 , casts full shading on points below y - k / 2 , and an intermediate amount of shading between them: p(z)=
w(z+k/2)/k
-k/2
(5)
z<-k/2.
For simplicity we here consider the shading by foliage only, neglecting that by the trunk. The cost of having the foliage at average height x, denoted by c(x) in equation (1), includes both a part that is relatively independent of height,
TREE
HEIGHT
AND
CROWN
SHAPE
283
such as canopy respiration, and the other part that greatly increases with height x, such as the cost for carrying up the water. The latter may probably be expressed by a convex function; i.e. if height doubles, the cost will increase more than twice (positive return to scale). To specify the exact form of the function c(x), we need physiological and mechanical knowledge. Here we simply assume a quadratic form: c(x) = Co+ c2x 2.
(6)
The cost c(x) also includes an amortized construction cost assigned over many years. MONOMORPHIC EQUILIBRIUM
An equilibrium in which all individuals have the same height g is expressed by a height distribution as
(7)
F ( x ) = F. ~ ( x - ~),
where F is tree density, and 8 ( x - ~ ) is a delta function indicating that the tree height distribution has a sharp peak at g. The stability condition for this monomorphic equilibrium is ~b(x) < ~b(~),
for all x # ~
(8)
which indicates that a tree with a crown height different from 2, has a lower net productivity than the optimal. Consider the case in which the foliage is of a cylindrical shape, i.e. p(z) is given by equation (5), and the photosynthetic rate is proportional to the logarithm of the light intensity, as in equation (2b). From equations (4) and (7), the light intensity, equation (3),.becomes L(x) = Lo exp ( - F . p ( , ~ - x ) ) , and therefore the photosynthetic rate as a function of tree height is a log Lo + b - aFpw f ( L ( x ) ) = a log L o + b - a F p w / k . ( ~ + k / 2 - x ) a log L0 + b
O
(9)
as illustrated in Fig. 2. In this case, the equilibrium condition, equation (8), is equivalent to three conditions: 4)'(g)=0,
4)"(~)<0
and
~b(0)<$(g).
284
Y.
IWASA
ET
AL.
f (L (x))
V/////////////.~I////////////)
0 X
FIG. 2. The p h o t o s y n t h e t i c rate f ( L ( x ) ) a n d the cost c(x) of a tree l i v i n g in a m o n o m o r p h i c e q u i l i b r i u m , as the f u n c t i o n s o f a v e r a g e foliage h e i g h t x. The difference b e t w e e n the two is the fitness $ ( x ) w h i c h is m a x i m u m at x = 2.
These are rewritten as aFpw/k - c'(ff) = 0
(10a)
-c"(g) < 0
(10b)
alogLo+b-aFp.,-c(O)
(10c)
respectively. From equation (6), equation (10a) becomes aFp., Y"= 2c2k'
(I1)
that is /photosynthetic]( total '~ (average foliage'~ = \ efficiency / \ s h a d i n g / \ height / (cost '~( crown '~ \coefficient/\thickness] This result implies: (1) The average foliage height ~ increases with the intensity of shading Fpw, which includes not only the transparency of each leaf and the amount of leaves per tree, but also the population density of trees at the stand. When the density is very low, the tree height should also be low, because
TREE
HEIGHT
AND
CROWN
SHAPE
285
there is no competition with respect to light between trees. We therefore expect a positive correlation between crown height and tree density in a community. On the other hand, if the amount of foliage of a tree is inversely proportional to the tree density, the tree height is expected to be constant between sites with different density. (2) The tree height ~ is inversely proportional to the crown thickness k_ If the crown is thin, an additional investment in trunk growth is greatly rewarded by an improvement of the light condition at the canopy; actually the gradient o f light intensity at 2, which produces the marginal fitness for tree height, is Fpw/k, inversely proportional to the thickness /c We should note that the height discussed here is the distance from the ground to the average height of a crown, although in common usage tree height refers to the top of the tree. (3) The height g decreases with the cost coefficient, c2. Low availability of water or nutrients as well as strong wind, which require an enduring trunk with additional investment, may reduce the crown height. (4) Height increases with the coefficient of photosynthetic rate a, but is independent o f the light intensity at the top Lo; high productivity o f the stand may result in a tall community through a high density or a smaller cost coefficient, but there is no direct effect of light intensity on height. POLYMORPHIC
EQUILIBRIUM
If the canopy is not thick enough, one of the stability conditions of monomorphic equilibrium 2:
4,(~) > 4,(0) is not satisfied. Then a mutant with no trunk ( x = 0 ) , which probably corresponds to a herb or a shrub, has a higher fitness than a tree with g, and therefore succeeds in invasion. If so, an equilibrium community may be composed of undergrowth and trees with a common height X (Fig. 3(a)). We also may have a dimorphic equilibrium in which the community is composed of trees of two classes, high trees with height X~ and low trees with x2 (Fig. 3(b)). Tree height distribution of the community is the sum of two delta functions: F ( x ) = F, . 8 ( x - g , ) + F2. 8 ( x - ~ 2 )
(12)
where Fj and F2 are the tree densities of high and low crown trees per unit area. Assume that the crowns of trees belonging to the upper class are not overlapped with those of the lower class trees. If we assume cylindrical shapes for trees o f both classes with thickness k~ and k2, respectively, we
Y. IWASA ET AL.
286 (a)
(b)
f(L(x))
~1/1////i/~
o
V///////Z g
FIG. 3. (a) An equilibrium composed of trees and shrubs. (b) A dimorphic equilibrium composed of trees of two layers, in which trees in the higher layer tend to be less thick (k~ < k2) and more dense (F I > F2) than those in the lower layer.
have the relation
kl+k2 21 --22 > ) _ 2
(13)
The s h a d i n g effect by the leaves above the height x, indicated by e q u a t i o n (4), is the leaf area index at x multiplied by the s h a d i n g coefficient a :
I
01p,.,( 2, + k l / 2 - x ) / k~
[shading] = / F l p w
| F,p~,+ F2p.(22 + k 2 / 2 - x ) / k 2 [ (Fi + F2)pw
x >1~, + kl/2 xl - k l / 2 <-- x < 2 1 + k l / 2
22+ k2/2 <- x < 2, - k~/2 22 - k2/2 <- x < 22 + k J 2 O<-x < 2 2 - k 2 / 2 (14)
then we can calculate h o w the light level decreases using e q u a t i o n (3). The g a m e ' s equilibrium condition for a d i m o r p h i c solution, e q u a t i o n (12), is that the fitness, or the net p r o d u c t i o n rate, is the same for 2~ and
TREE
HEIGHT
AND
CROWN
SHAPE
287
~2, and that the trees of height other than these two should have a lower fitness: 4,(~) = 4,(~2)> 4,(x) forall x(#~,, and x2)(15) These are based on the idea that, in a evolutionary equilibrium, all the players enjoy the same fitness and all the mutants have a smaller fitness. From Fig. 3(b), these equilibrium conditions (or ESS conditions) are expressed as 4,(~,) = 4,(~2) > 4,(0)
that is
-rl = aFzpw/2c2kz
(16a)
x2 = aF2pw/2c2k2
(16b)
aF, p~ [ aF, pw'~2 ( F2) [ aF2pw, 2 2 c2k2-~2k~) = - a F , + - ~ pw-c2k2--~2k2 ) > - a ( F , + F:)p. (16c) If only ~t and x2 are variables and F~ and F2 are parameters given arbitrarily, then these three equations cannot be satisfied except in a degenerate case, which is nongeneric in the parameter space. Here we assume that the relative densities of trees in the upper and lower classes are also determined by the above equilibrium condition, assuming that the total amount of foliage is given:
F, + F2 = F,ota,.
(17)
Then four variables, x~, x2, F~, and F2, are determined by equations (16a, b and c) and (17). Here we simply note a relation derived from an inequality Xz > x : using equations (16a) and (16b):
Fl/kl > F2/k2.
(18)
This implies that, if the thickness of the upper layer trees is larger or equal to that of lower layer trees (k~ >- k2), then the density of the upper layer trees should be much greater than that of lower layer trees, in units of shading effect (FI > F2). Otherwise, the thickness of the upper trees must be smaller than that of the lower trees (k~ < k2). In a similar way we can consider an equilibrium composed of trees of three or four layers. In general, the n-morphic equilibrium is expressed by the foliage height distribution, as F ( x ) = ~ F,8(x-~,), i=1
(19)
288
v.
IWASA
ET
AL.
where Fi is the a m o u n t of foliage at the/-layer. The equilibrium conditions for this solution are ~b(X,) = ~b(X2) = . . . =
~b(X.) = A (constant), (20)
q~(x)
for a l l x ~ x l , x 2 , . . . , x , .
Not only . ~ , . . . , ~, but also F t , . • . , F, are determined by these equilibrium conditions under the constraint of a fixed total amount of foliage: Fi = Ftota,.
(21)
i=1
For the limit case in which the thickness of each crown is small enough, and the n u m b e r of layers is very large, we have an infinite layers model: expressed by F ( x ) , a foliage height distribution. The ESS conditions are now qS(x) = A for F ( x ) > 0 ~b(x) < A for
F(x) =
0.
(22a) (22b)
Consider the case in which the photosynthetic rate is proportional to the logarithm of the light intensity, equation (2b). From equation (5) with k = 0, the fitness, equation (1), becomes 4~(x) = a log
Lo-a
f;
F(y) dypw+b-co-c2x 2.
(23)
In an interval in which equation (22a) holds, we have
-
F(y) dy=
(A-alogLo-b+co+c2x2).
(24)
apw
By calculating the derivative, F(x) =
1 apw
(25)
2c2x.
The equilibrium distribution of tree height F(x) is a combination of a positive value, equation (25), and F(x)= 0. One solution is that there is a truncation height xT, above which there is no foliage and below which equation (25) holds:
F(x) {~ c2x/apw O<-x
x ~x
(26)
T.
We can reason that only this solution can satisfy the ESS conditions as follows: Suppose that there is some gap of foliage distribution, say F(x) = 0
TREE
HEIGHT
AND
CROWN
SHAPE
289
between xj and x2. Compare the highest of the trees under the gap with the lowest of those above the gap, the heights of them being x~ and x2 respectively (xt < x2). The photosynthetic gains of them are equal because the light intensity does not decrease within the foliage gap, f(L(X~))= f(L(X2)), but the cost is larger for the higher tree, c(xl) < c(x2). Therefore their net production rates are different, ~ ( x , ) > ~(x2). This is against the ESS assumption, equation (22a). By using the constraint on the total amount of foliage, Ftot,, =
Ioo F(y) dy,
(27)
we can determine the truncation height as
(28)
xT = 4 aF, o,.,pwl c2.
This result suggests a general pattern of tree height distribution in a community with high species diversity. The tree density, measured in units of shading intensity, is the highest for the tallest trees and lower for shorter trees (Fig. 4). The height of trees in the top layer increases with total tree density, but decreases with the cost coefficient, and is relatively insensitive to light intensity. f
I
(
/
IIi
,
?
4r"
II
/..-_.% ii I
H
II FIG. 4. An equilibrium of infinitely many layers. As the limit when the crown of each tree becomes very thin, an equilibrium with infinite layers appears. The tallest trees have the highest density measured by unit of shading intensity.
3. Shape o f a Crown: Height Distribution of Leaves Within a Tree
We have assumed that a fixed size and shape of the tree crown while discussing height evolution. In this section we investigate the shape of a crown, the arrangement of foliage within trees. Denote the foliage height distribution of a tree by z(x), i.e. amount of leaves at the height x. Also denote the foliage height distribution of other
290
v.
IWASA
ET
AL.
surrounding trees by w(x). Retaining, with some pertinent changes, the notation used in the last section, the photosynthesis rate of a unit amount of leaves (not a whole tree) is
f( Loexp[-p f~ (z(y)+Dw(y))dy])
(29)
where D is a factor expressing the shading effect by other trees compared to that by the same tree. D should increase with tree density at the stand. w(x) denotes the vertical foliage distribution of other trees. The cost of a unit amount of leaves includes not only the maintenance and amortized construction cost, which increases with the height of the leaves, but also the cost of branches to separate a given amount of leaves, which may be independent o f height but increases with the amount of leaves at the height z(x). Here, we assume that the cost of a unit amount of leaves at height x is c(x) + yz(x), (30) the second term may be called "lateral cost". Therefore the fitness of a whole tree with foliage distribution z(x) is
4~=foZ(X)[f(Loexp(-o f;[z(Y)+ Dw(y)]dY))-c(x)-yz(x)] dx (31) which is a functional of the function z(x). The optimal distribution of leaves along the height axis, z*(x), is the one that maximizes the fitness, equation (31), under the constraint
z(x) >-0
z(x) dx = Ztotal.
and
(32)
We assume that the photosynthetic rate of each leaf is expressed by equation (2b), f(L) = a log L + b, which is good for the photosynthesis rate of a whole tree but not completely suitable for that of a leaf. We should rather use equation (2a), but for mathematical convenience we here assume equation (2b). By using a Lagrange's multiplier A, the above problem with the constraint of a fixed total mount o f foliage, is equivalent to the maximization o f 4~:
~ = f ? z(x)( a log [ Lo exp (-P l ; [z(Y) + Dw(y)] dY) ] + b
c(x) -
yz(x) + h / dx J
(33)
TREE
HEIGHT
AND
CROWN
SHAPE
291
under the condition z(x) >-0 only. Then the Pontryagin's maximal principle (Pontryagin et ai., 1962) can be applied. Define h(x) and W ( x ) as h(x) = W(x) = -D
z(y) dy,
(34a)
w(y) dy.
(34b)
Then from equation (32) (35a)
d h / d x = z(x).
h(0) = --Ztotab
h(oo) = 0.
(35b)
Since W ( x ) expresses the shading by other trees, it should be treated in the optimization problem as a function already given. The Hamiltonian is H = z{a log Lo+ b + ;t + ap(h(x) + W ( x ) ) - c(x) - 3"z(x)} + p ( x ) z ( x ) , (36)
where the adjoint variable p(x) follows (37)
dp / dx = -OH~Oh = -apz.
Since the terminal value h(oo) is given by equation (35b), we cannot specify the terminal value of the adjoint variable p(oo). The maximal principle states that z(x) at each x is chosen so as to maximize the Hamiltonian H under the condition that all other functions are fixed. Thus we have z*(x) =
a log L o + b + A + a p ( h ( x ) + W ( x ) ) - c ( x ) + p ( x ) . , ifit is positive. 23,
(38a)
0,
(38b)
otherwise.
Now we try to find a solution that satisfies all the above conditions. First we notice that when x grows infinitely large, both h(x) and W ( x ) tend to zero and c(x) diverges to infinity. Therefore the R.H.S. of equation (38a) is negative for sufficiently large x; hence equation (38b) holds; z * ( x ) = 0 . Denote the height of the top by x*, above which there is no foliage. Then, z*(x)=0, h ( x ) = 0 , and p(x)=constant hold for x>-x *. For x < x * , equation (38a) holds. By differentiating equation (38a), and using equations (35a), (34b), and (37), we have apD 1 dc dz*/dx = ~ w(x) - 2--~ d--~"
(39)
292
Y.
IWASA
ET
AL.
The optimal distribution o f foliage o f a tree z*(x), is determined by equation (39), and it may be influenced by the foliage distribution o f surrounding trees, w(x). CROWN SHAPE OF A LONE TREE
First consider the crown shape of a tree living alone. Since all x, equation (39) is
w(x) = 0 for
1 dc
dz*/dx
- -
2y dx"
(40)
Combining this with the condition z * ( x * ) = 0, we have
(x
z*(x)=(c(x*)-c(x))/2y
(41)
Since the cost c(x) is an increasing function o f height x, z*(x) decreases with x. Moreover the convex shape o f c(x) gives a concave curve for z*(x). Therefore we can conclude that the amount of leaves o f a lone tree is greater at lower height, and therefore a tree growing isolated has a hemispherical or cone-like shape (Fig. 5a). (a)
(b)
(c)
5.0
0.0
L
0.0
30 z " (x)
O0
3"0 z'(x)
L
I
0 i0
I
5 iO
z "(x)
FIG. 5. The foliage distribution along the height axis, z*(x), at a monomorphic noncooperative equilibrium (or ESS) of the crown-shape game, (a) when a tree grows alone (fl = 0), (b) when it grows within a stand (/3 = 0.5), and (c) within a dense stand (/3 = 2-0), calculated by equation (45). The crown becomes taller with the tree density, but each tree still has some foliage all the way down to the ground.
TREE
HEIGHT
AND
CROWN
SHAPE
293
The tree top x* is determined by the condition that the sum total of leaves is equal to a given amount Ztot,~:
(c(x*) - c(x)) dx/2"y.
Ztota, =
CROWN
SHAPE
OF TREES
IN A MONOMORPHIC
(42)
EQUILIBRIUM
Next, consider tree shape in a monomorphic equilibrium, in which all the trees have the same foliage distribution. This is mathematically expressed by optimizing z ( x ) with w ( x ) fixed and putting w(x) = z(x). Equation (39) becomes
dz,/dx=apDz,(x) 2y
I dc 23/dx"
(43)
Combining this with z * ( x * ) = 0, we have
z*(x) =
x*
"~o-2 -- x- 1 dc e-t o / ~,)~y- , _ _ _ _ . dy. 23, dy
(44)
Since d c / d x is an increasing function, z*(x) may have a hump, but it is positive for all x < x*. For example, in the case c(x) = Co+ c2x 2, by putting fl = apD/2",/, we have
z*(x) =
e-°(Y-X)y dy. c2/ y
-;;[~+x-(x*+~)e-a(x*-x'],
(45)
(see Figs 5(b), and 5(c)). The top height x* is determined by equation (32) as Z,o,a~ =
=
f; Io
z*(x) dx
:* 1--e -(apD/2~')x dc apD d--x" dx
(46)
where we used equation (44) and partial integration to obtain the second equality. Equation (46) clearly shows that the top height x* increases with the total amount of foliage Ztota,.
294
Y.
IWASA
ET
AL.
4. Discussion
It is interesting to compare the tree-height game model with the resource competition scheme assumed in the niche theory by MacArthur & Levins (1967). The tree height ~, defined as the average foliage height, corresponds to the niche position along one dimensional niche axis, and, roughly speaking, the canopy thickness k corresponds to the width of the utilization function. An important feature of the game between trees is the asymmetry of the competition, i.e. a taller tree strongly shades smaller trees but a smaller tree will not affect taller ones much, as expressed by the function p(z) in equation (5). The theories of community structure shaped by evolution have been studied and applied to the body size of animals (Roughgarden, 1972, 1976; Roughgarden, Heckel & Fuentes, 1983; Case, 1982). In these theories, it is demonstrated that individuals coexisting in an evolutionary stable community must have the same fitness, and that no mutant which happens to occur may have a higher fitness than that of the resident. These assumptions are equivalent to the equilibrium conditions in the tree-height game, such as equations (8), (15), (20), and (22). Coevolution theory for the body sizes of animals predicts that many species may coexist if the width of the utilization function of each species is small compared to that of the resource supply function. A corresponding result in the tree-height game is that a polymorphic equilibrium occurs when the canopy is thin enough. Further, the number of tree types coexisting in a community increases as the crown of each tree becomes thinner. In this paper, we assumed that each individual maximizes its own photosynthetic performance. However, if a tree is surrounded by genetically related individuals, the optimal strategy should also consider the productivity of competitors. This kinship effect is very important in many herbs in which vegetative growth is prevalent, as pointed out by Givnish (1982), and may be studied by assuming that each herb tries to maximize its inclusive fitness. The relatedness between competitors seems less important in trees because the seed produced by many parent trees are mixed in the seed pool. One interesting result of the canopy-shape game model studied in the last section is that a bare trunk accompanied by no foliage will not appear in the monomorphic ESS pattern. If trees growing in a forest have the same shape, and if that shape is the optimal one maximizing the net photosynthesis of each tree under light conditions made by others, then trees should have some leaves all the way down to the ground. This result is derived also from an intuitive argument as follows: Suppose that all trees have some bare trunk in the lowest part of their stem. The light intensity is constant
TREE
HEIGHT
AND
CROWN
SHAPE
295
for this part because of the lack of foliage which causes the shading. Therefore if a tree moves some leaves from the lowest part of the c a n o p y to the ground, it can save the cost due to height without losing photosynthetic rate. This means that the original foliage distribution with bare trunk is not an ESS. Nonexistence of bare trunks seems to be a good rule for lone trees; m a n y lone trees growing in natural environments have some leaves almost down to the ground. But for trees in a dense forest we know m a n y examples of trees with long bare trunks. N o w we would like to discuss the reasons why trees have bare trunks: (1) In a p o l y m o r p h i c equilibrium of the static crown shape game model, tall trees can have a bare trunk. Therefore in a forest c o m p o s e d of several kinds of trees with different heights, trees in the highest layer m a y have a bare trunk in the lowest level occupied by smaller trees or shrubs. Even in a pure forest c o m p o s e d of a single tree species, if there are always some younger trees of the same species with low canopy, then tall trees m a y have long bare trunks in an ESS pattern. (2) Since growth is a dynamic process, the effect on future production rate should also be taken into account in determining the shape in young trees. I f a tree can suppress the growth of surrounding trees by having a taller canopy it will gain in future production rate as well as present rate. Because o f this effect with time delay, trees m a y invest in a trunk, an organ for competition, much more than predicted by the optimal allocation in the static game model, in which the " p r e s e n t " production rate is assumed to be maximized. This hypothesis predicts that a bare trunk is more c o m m o n in a dense stand o f a uniform age where competition is intense during the growth of the stand all the way from the seedling stage. (3) Another aspect of the dynamic nature of growth is that the optimal development in a young age m a y affect the shape o f trees in later stages. Suppose that a stand is cleared by some disturbance such as fire and all the seedlings of trees begin to grow almost synchronously. In this early stage o f a tree's life history, it m a y be crucial to have a thick canopy which requires a long trunk, but this m a y be not important any more in the later stages. Then we can find old trees with a small a m o u n t of leaves at the top of a long trunk, because they would discard the lower leaves rather than the upper ones. In the last steps above, we assumed that the amortized cost for trunk construction is important so that once a long trunk is produced the cost for maintaining foliage does not strongly depend on height. The effect of p o l y m o r p h i c equilibrium discussed in (1) can also be combined with this dynamic effect. For example, in the early stage of succession there may be other trees or shrubs that cover the lowest layer,
296
v . IWASA E T A L .
and in the later stages this undergrowth may have perished due to competition for light. Then we find a m o n o m o r p h i c equilibrium of trees with long bare trunks. (4) The shading may also be caused by the tree trunk itself. This had been neglected in the model above. Because of this effect, light intensity may decrease with the decrease of height even in the lowest part without foliage. The intensity of shading by the bare trunk itself, however, seems not to be strong enough as to explain trunk existence. (5) Predation by large herbivorous animals may also explain the adaptive significance of bare trunks. For example, in a savanna, where trees are not so dense as to shade each other, they have a high canopy and a long trunk without foliage. The coevoIution of giraffes and trees in the savanna seems to be the most reasonable explanation as Lamarck pointed out. If the expectation of loss due to herbivory by m a m m a l s is very high, it may be advantageous not to produce leaves in the lowest part of the trunk. We may call the lower cutoff line of the canopy the goat line (2 m to 4 m) or the giraffe line (9 m to l l m) corresponding to the main herbivores. The hypothesis that a bare trunk is for the avoidance of m a m m a l predators may be tested by comparing different communities with varying grazing pressure. In the present model this effect may be expressed by the higher cost for lowest height, i.e. c ( x ) increases again for small x, as well as for large x. In a similar way, for trees in regions with heavy snow, the nonexistence of leaves in the lowest part of the tree might be explained as the avoidance of harm by snow. (6) Developmental constraints sometimes seem to be a main reason of a tree's crown shape. For example, a palm tree must continue growing upward to keep producing new leaves. Their crown shape, thus, m a y not be optimal from the viewpoint of photosynthetic efficiency. It is difficult to know the extent to which a plant's shape is restricted by developmental constraints; one possible way is to show an ideal crown shape in ESS without developmental constraints and to compare it with the shape of real systems. We thank D. Andow, T. Givnish, D. King, K. Naganuma, and J. Rummel for their comments. REFERENCES BLACKMAN, G. E. & WILSON, G. L. (1951). Ann. Bot. 15, 64. CASE, T. (1982). Theor. Pop. Biol. 21, 69. GIVNISH,T. J. (1982). Amer. Nat. 120, 353. HORN, H. S. (1971). Adaptive geometry of trees. Princeton: Princeton University Press.
TREE
HEIGHT
AND
CROWN
SHAPE
297
KING, D. (1981). Oecologia 51, 351. KING, D. & LOUCKS, O. L. (1978). Rad. Environm. Biophys. 15, 141. MACARTHUR, R. H. & LEVINS, R. (1967). Amen Natur. 101, 377. MONSl, M. & SAEKL T. (1953). Jap. J. Bot. 14, 22. PONTRYAG1N, L. S., BOLTYANSKII, V. G., GANKRELIDZ, R. V. & MISCHENKO, E. F. (1962). The mathematical theory of optimal processes (trans. by K. N. Trirogoff). New York: Interscience Publishers, John Wiley & Sons, Inc. ROUGHGARDEN, J. (1972). Amer. Nat. 106, 683. ROUGHGARDEN, J. (1976). Theor. Pop. Biol. 9, 388. ROUGHGARDEN, J., HECKEL, D. g¢. FUENTES, E. R. (1983). In: Lizard ecology: studies o f a modelorganism (Huey, R. B., Pianka, E. R. & Schoener, T. W. eds.), pp. 371-410. Cambridge: Harvard University Press.