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On the shape of trees
J. theor. Biol. (1973) 38, 111-137
On the Shape of Trees G. W. PALTRIDGE Division of Atmospheric Physics, C.S.I.R.O., Victoria, Australia
Aspendale,
(Received 24 January 1972, and in revised form 6 May 1972) Answers to questions concerning the broad-scale characteristics of trees are sought in the analytical development of a simple model in which the rate of photosynthesis is controlled by leaf water potential and by access to direct solar radiation. These concepts are introduced tirst in a simple discussion of the single isolated tree. Later they are applied to the forest-stand situation. Expressions are derived which relate growth rate (per tree and per unit area of ground) to the height, shape, spatial separation and lifetime of the trees; to the environmental conditions such as the average solar elevation, potential evaporation E, and soil moisture content; and to two main physiological factors-the proportionality factor zO relating potential drop per unit length of trunk or branch to potential evaporation, and the critical leaf water potential vO at which net photosynthesisis zero. Accepting the assumptionsin the model, the following are examples of its predictions. It is shown that at optimum tree spacingthe photosynthesisper unit area of ground may be greatestfor the shortest trees (grass?).It is shownthat for a given environmentthere may be an optimum tree spacingyielding maximum photosynthesisper unit area of ground averagedover the lifetime of the trees; that this maximum decreases with increasingultimate tree height (which in turn is determinedby E,, zOand v,,); and that this maximum and this optimum spacingdecreasewith decreasingaveragesoil moisture.It is shownthat there can be an optimum leaf distribution which in general is such that leaf density increases radially from the trunk. It is shown how the optimum shape of trees in a forest might be expected to alter with their size. In generalit appearsthat the conceptsdiscussedhere may be useful in explaining the evolutionary developmentof many of the broad features of tree growth.
1. Introduction The growth of plants depends on environmental and inherited factors in a very complex way. Because of this complexity plant ecology has suffered from a lack of mathematical models to integrate its findings and to provide
testable hypotheses. The situation is improving 111
as computers become avail-
112
G.
W.
PALTRIDGE
able because realistically complex systems of equations can now be investigated. For instance, crop growth models are becoming more accurate as they become more complex (e.g. Connor & Cartledge, 1970). Whole ecosystems can be modelled mechanistically as in the I.B.P. model described by Watts & Loucks (1969). In general, these types of system analyses adhere to the principle (attributed to Tukey) that it is better to have approximate answers to the exact problem than exact answers to an approximate problem. There is also a place for analytic solutions to approximate problems. Often these lead to useful generalizations which would not be apparent in the computational results of a complex system model. Indeed it is probably true in many cases that the simple analytic approach is an essential prerequisite for the development of the complex model. A prime example of this has been the remarkably fast and successful development of numerical weather prediction, which was guided by many decades of analysis of “approximate problems”. In this paper the analytical approach is used to study certain general features of the shape, size and spacing of trees as they have evolved under differing environmental conditions such as potential evaporation, solar elevation and soil water status. The word “trees” is perhaps something of a misnomer, since much of the analysis is applicable to plant structure in general. Both the strength and the weakness of the analytical approach to a complex problem is the necessity to simplify the system and to study only the most basic and important of its characteristics. There can be some argument as to which of the characteristics of trees are the most important, but here we consider only their ability to photosynthesize atmospheric carbon dioxide and to dispose the resultant photosynthate in the form of new leaves and limbs. Furthermore, because of the necessity for simplification, we consider only two of the many environmental and biological factors which determine the rate of photosynthesis of individual leaves. (1) Leaf water potential. The methods of water potential “control” in real plants are very complicated and include such processes as direct physical restriction of the absorption of carbon dioxide by the stomata1 apertures of the leaves. In the present work we will specify only the most general chnracteristic-namely that photosynthesis by the leaves of a tree decreases with increasing leaf water potential. (2) Available photosynthetically-active light energy. Again, the “real” process is exceedingly complicated. In the next section, which is introductory and deals only with single isolated trees, the problem will be avoided by assuming that there is always sufficient light for leaves to photosynthesize to the limit set by their water potential. In the later sections which deal with
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trees in a forest, it will be assumed that the leaf density is sufficiently high for shading by a tree to be complete. That is, leaves which are in the shade of another tree will be assumed to have insufficient light for photosynthesis. It will be seen that the two cases lead to quite different optimum tree shape. Again it is emphasized that the concepts discussed here are susceptible to more accurate but necessarily more complex development in specific situations. 2. Basic Concepts and the Single Tree Consider a tree to be an assembly of leaves attached to the ground by a number of limbs whose function is to provide support for the leaves and to provide access to the moisture in the soil. The water potential I/ of the leaves is assumed to be the only controlling factor on their consumption of carbon dioxide and evaporation of water. Furthermore, it is assumed that the control is linear, so that the net rate of photosynthesis P per unit volume (of space occupied by leaves) can be written as P=n 0
wo-II/> I
3
(1)
which states that when + reaches a critical potential lclo the net rate of photosynthesis (gross photosynthesis minus COZ respiration) is zero, and no is the net photosynthesis when $ is zero. The concept of water potential as it applies to plant growth is discussed in detail by Slatyer (1967). Very broadly, water potential at a point in the soil-plant system is a measure of the suction pressure at that point-a pressure which is brought about by the evaporation of water. In a real tree the water potential of the leaves is determined by many factors, of which three are considered here. That is, ti is determined firstly by simple gravitational potential (g per unit height), secondly by the potential drop due to the passage of water through the resistance of the limbs, and thirdly by the potential $. of the water in the soil. Thus for unit volume of leaves at height h connected to the ground by a vertical trunk and a horizontal branch of radius r we may write ti = E,zo(h+r)+gh+k, (2) where, for reasons which will be discussed shortly, Ep is the potential evaporation per unit leaf area index (say), and z. is a constant. There are several major concepts and assumptions inherent in equations (1) and (2) which need to be discussed. First, the potentials associated with the flow of water through the solidplant-atmosphere system should be negative quantities by formal definition, l-3. 8
114
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PALTRIDGE
which arbitrarily assumes the potential of pure “free” water on the ground surface to be zero. Here, the negative sign is dropped for simplicity. No confusion arises provided it is realized that reference to a potential is a reference to the absolute value of that potential. Second, although we assume for the moment that the intensity of sunlight does not enter directly into the determination of photosynthesis, solar radiation can have an effect through its control of Ep and hence leaf potential. Third, it is assumed that the potential of one unit area of leaf is independent of the potential of any other unit area of leaf except in that they may be related through their combined effect on soil water potential I/, (see later). In effect we envisage each unit area of leaf to have a separate “pipe” to the common reservoir of the soil, so that if these pipes are bundled together in the form of a trunk (say) the overall diameter of the trunk must increase towards the ground. An analytical treatment involving any other assumption is again almost impossible to handle. In fact it appears that some real trees do behave with this apparent independence of water paths, although others do not. However, since in real trees the circumference of the limbs and trunk increases towards the roots (presumably for reason of strength, but also possibly to match water potentials at junction points of the vascular system) the approximation may not be serious. Fourth, we assume that the potential of leaves is proportional to potential evaporation E, and not to actual evaporation E. This can be regarded as an approximation which apart from other considerations is necessary for the subsequent analysis to be mathematically tractable. Had $ been set as proportional to E, which on first thoughts might seem preferable, some other assumption would in any event have had to be made with regard to the proportionality constant zO and its variation with potential. For instance, we might have written II/ = Ez,(h + r) + , . . . , and made some linear approximation for E as a function of II/ such as E = E,[($, - I,@/$~]. This would be easy to picture physically since the zO could then be visualized as the actual resistance to water flow per unit length of trunk or branch. However, some dependence of zO on r+$would have to be assumed (such as letting zO be constant with Ic/), and there is no reason to suppose that this would be nearer reality than the original assumption of equation (2). At least equation (2) simulates the general characteristic of importance here-namely, that the potential of a leaf increases with increasing potential evaporation. Finally, a linear variation of P with t+Qis not in accord with experiment, which generally shows the variation of gross photosynthesis with 1,9to be highly non-linear-and of a form obviously related to the variation of stomata1 resistance with JI. However, the mathematically tractable linear relation is sufficient for the purpose in that it simulates the characteristics of
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importance to the present analysis. In addition, averaging in time (as will be done here-for instance we will deal with average solar elevations) would tend to make a linear P vs. $ relation more appropriate, since the nonlinearity of individual leaves would be smoothed. The simplest case to treat mathematically is that of a tree whose leaves are distributed uniformly throughout a hemisphere whose base is on the ground and whose supporting branches radiate from the centre point of the base. Neglecting both soil moisture and gravitational potential the total rate of photosynthesis is the simple integral of equation (1) (with substitution of equation (2)) using spherical coordinates r, 8 and r$
(3) where H is the height and therefore radius of the tree. That is
PT =
7m,
2H3 E,zoH4 - 3 wo >'
(4)
which is an equation showing that, provided the tree maintains its spherical shape, the total rate of photosynthesis will vary with height in the manner of Fig. 1. The decrease in photosynthesis after a certain height arises because rj in equation (1) (i.e. E,z,r here) can exceed tie giving an effective negative photosynthesis of those leaves at a radius greater than $,/E,z,. Physically, this simulates the case where respiration exceeds gross photosynthesis. This and the associated problem of defining the death point of a tree will be discussed later.
r
FIG.
1. Single tree total photosynthesis as function of height.
116
G. (A)
DISPOSAL
W.
PALTRIDGE OF PHOTOSYNTHATE
A major difficulty in any attempt at modelling the growth of plants is the definition of the criteria by which newly photosynthesized material is disposed about the plant. For any given species it is presumably possible to derive a set of empirical rules appropriate to that species, but in the present work we are interested in, among other things, the broader problem as to why the gross morphoIogy of a tree (say) has evolved in its particular fashion. A technique used quite often is to replace the complicated processes of natural selection with a single assumption-namely, that the overall effect of natural selection is to produce, within the constraints set by the environment, a plant of the optimum characteristics to perform a particular “function”. If one assumes further that the optimum characteristics have already been achieved then the use of this “function” as a criterion for disposal of photosynthate in a model plant should force the model to grow in a form similar to that of real plants-provided that similar environmental constraints are imposed on the model. One can be guided in the choice of a “function” only by intuition and experience, and the fact that it contributes to the survival of the species. For instance, it could be argued that trees might have evolved in such a manner as to maximize their rate of photosynthesis. Paltridge (1970) has used this assumption in a complex computer model of a pasture. Similarly, it could be argued that trees might have evolved in such a manner as to minimize their total water use. Parkhurst & Loucks (1972) have suggested a criterion whereby plants, or parts of plants, might have evolved so as to maximize the ratio R of photosynthesis over evaporation. It is worth emphasizing that all such criteria give a result dependent on the time scale over which the criteria are applied. Maximization of instantaneous photosynthesis, for instance, will generally lead to a different shape than maximization of average photosynthesis over the lifetime of the plant. Where it is appropriate in the present work the principle of maximization of photosynthesis (over a specified period) is applied. In many practical situations Man is interested in maximizing the photosynthetic return from his land. Thus a model based on maximization of photosynthesis will generate a “plant” whose characteristics may be a helpful guide to his selection of a particular species. In addition, on the present model maximization of photosynthesis is stable since the plant or tree will tend to place its new leaves in the position of minimum potential, or in general as close as possible (bearing in mind the geometrical limitations set by the form of trunk and branches) to the base of the trunk or stem. Thus in the previous very simple example of a single tree, extra leaves would be placed such that the outer leaves always formed an equipotential canopy-in other words a hemisphere. This shape would be
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maintained throughout its growth and is independent of initial conditions. If one includes the effect of gravity, it can be appreciated that the optimum shape would become more flattened since the potential would increase more quickly in the vertical than in the horizontal. On these grounds therefore, one might expect firstly that single trees would be more closely hemispherical in regions where potential evaporation is highest (the effect of gravitation would be relatively less) and secondly that in these regions the branches of such single trees would tend to radiate directly from the stump in order to minimize their length and consequent potential drop along them. [A criterion such as minimization of evaporation is unstable in that final shape would depend on initial shape. It is to be expected that actual evaporation from a leaf would decrease with increasing $ in a similar fashion to gross photosynthesis since in real plants both CO2 uptake and evaporation are controlled by a common stomata1 aperture. Thus if for some reason there was a region of the plant whose leaves were at a higher potential than elsewhere on the plant, extra leaves would tend to be added at the extremities of this region and the plant would continue to grow outwards from the initial irregularity. Again, although Parkhurst’s criterion of maximization of R is attractive, it can be shown (see Appendix A) that this also may be unstable. It is shown there, using a slightly more detailed model of photosynthesis by a leaf, that this criterion is equivalent either (a) to a maximization of growth rate, or (b) to a minimization of evaporation, depending on the relative magnitudes of mesophyll resistance and carbon dioxide respiration at the particular leaf water potential. In general (b) would apply, except towards the end of the plant’s life in our present examples when most of the outer leaves would be at very high water potential.] It is of some practical interest to consider the case of a tree which has a single vertical trunk and horizontal radial branches. The leaf potential will be a function of both h and the radial distance r as in equation (2). As in the case of the hemispherical tree, if maximization of rate of photosynthesis is assumed the outer leaves will all have the same potential, so that if H is the tree height Wh, 11)= 4W), where r, is the radius at height h. That is E,z,(h+r,)+gh
(51
= E,zOH+gH,
(6)
where 4, is again neglected. Therefore rh =
@, zo+ g)W - h) EpZO
*
(7)
118
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PALTRIDGE
Thus such trees would be expected to adopt a conical rather than shape. It is quite straightforward to calculate the photosynthesis tree as a function of height. The result is again a polynomial in H which yields a curve similar to, but of smaller amplitude than, that 3. Optimum
Leaf Distribution
spherical of such a 3 and N” of Fig. 1.
within a Tree
So far we have assumed that photosynthesis by a leaf is independent of light intensity, and have illustrated the likely form of isolated trees if photosynthesis were determined solely by leaf water potential. This may be a reasonable approximation for trees of low leaf concentration. However, light interception obviously plays a part in determining optimum tree shape, and in the subsequent treatment of a forest the effect of interception will be illustrated by assuming the opposite extreme of very high leaf concentration. The optimum shape of trees is then quite different. It should be pointed out here that considerations of light interception and leaf potential may play a part in determining the leaf distribution within a single tree. The subject is discussed in Appendices B, C, D. It is shown there that, for trees where restriction of total volume may be a desirable characteristic (e.g. the previous examples), it might be expected that leaf density should increase radially from the trunk. The outline of a possible mathematical treatment to derive the optimum form of this radial increase in leaf density is also presented. 4. Trees in a Forest Consider now a forest of trees of uniform height H in a region where the average solar elevation is 0. Let the trees be vertical cylinders of radius Ye, randomly positioned over the ground with an average centre to centre spacing of 2. Assume that each tree has sufficient leaf density to absorb (effectively) all the light falling on it, and that photosynthesis occurs only in those regions of the tree which have a view of the sun unobstructed by other trees. This means (see Fig. 2) that for two trees directly in line with the sun, photosynthesis will occur only above a height h,, where h, = H-Z tan 8. (8) In fact it can be supposed that no leaves would exist below h,. Above ho the rate of photosynthesis dP per height increment dh can be expressed dP(h) = 2nor, where here it is assumed that leaf potential
is a function only of height
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OF TREES
.z
‘.
-
FIG. 2. Schematic diagram of forest trees showing the geometry for the case of two trees in line with the sun.
(“radial” branches are neglected) and that no is the rate of photosynthesis per unit vertical area of the tree when the leaf potential is zero. Note that we make no statement about the leaf distribution within the tree. The same mathematics are applicable if we regard the trees as two-dimensional rectangular “blades” in the vertical plane, where 2ro is the average projected blade area facing the sun. As before w = (E, zo + oh WV The total rate of photosynthesis PT of a single tree in the forest is then P, = j! P(h) dh = 2n, r. ho
H-h,-
$
(H2-h;) 0
I
,
where A is a substitution for E,,z,+g. In the three-dimensional situation of a forest the value of ho is given by equation (8) only for the case of two trees in line with the sun. In general, ho will be given by this equation if 2 is replaced by an “effective” spacing 2, which can be found by a form of mean-free-path calculation where 2, corresponds to the mean-free-path. Looking down on the forest, if one “moves” any point on a tree through a horizontal unit instance then this point will collide with the number of trees contained within an area 1 by 2 ro. Therefore, if there are N trees per unit area, 2, = 1/2roN. The average area of ground 5 available to an individual tree is given by approximately 04362 2, and since N= l/t O*86Z2 z -e-
and the appropriate
2r,
’
value of ho is therefore given by h -H O*86Z2 tan 6 o2r, ’
(12)
(13)
120 Substituting
G. W.
PALTRIDGE
for k,, in equation (11)
P, which relates spacing, height factors z0 and P,/O*86Z 2, or
0.86 tan 12Z2 (14) = 0+86n,Z2 tan 8 L1 - A $0 ( H- --%--- 0 -- >I the total tree photosynthesis to the solar elevation; to tree and radius; to potential evaporation; and to the physiological $I,,. The photosynthesis per unit area of ground P, is simply 0.86 tan tIZ2 4r,
>I ’
(15)
When h,, is less than zero, that is when the sun is illuminating parts of the forest floor between the trees, P, is given by equation (11) with h, set equal to zero. Again by dividing by 0~862~ P4hocO) = ““‘o(&;!H2). @86Z2
W
Thus assuming for the moment that the trees maintain a constant radius but increase in height, a typical P, vs. height curve might look like any of the curves of Fig. 3, where the maximum rate of photosynthesis PAcmaxjoccurs when h, = 0 or when H=--- 0.86Z2 tan 8 2r,
(17)
’
From equation (15) PAcmaxjis given by Pa(mrrl=.notane(l-~H).
H = ‘i,
FIG. 3. Instantaneous
#~=*y/a
photosynthesis per unit area as a function of average tree height
for threearbitrary valuesof Z. Note that onemustassume an initial heightHI. See text.
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These equations are simply mathematical expressions of the following. Because photosynthesis depends inversely on a parameter (leaf potential here) which increases with height, and provided solar energy is not wasted on the ground as when h,, c 0, photosynthesis per unit area of ground will be increased by increasing the spacing between the trees since more and more of the lower more efficient leaves become exposed. Tracing any of the curves in Fig. 3, given initially a young forest with a spacing such that ho < 0, as the forest grows upwards the P, will increase to a maximum according to equation (16) and thereafter will decrease according to equation (15) as the region of shaded, inoperative (or non-existent) leaves extends upwards. In addition equation (18) shows that if one compares various forest or plant stands each of which has the optimum spacing calculable from equation (17), but each of which has a different height, then the stand of lowest height will have the greatest rate of photosynthesis. From this it could be argued that more fertile soils, which from the nutrient point of view can support a higher rate of photosynthesis, are more likely to support “low trees” (grass?) than a stand of tall trees of large spacing. Indeed it is generally accepted that the grasslands of the world tend to occur in relatively fertile soils. Several points should be mentioned at this stage. First, there is nothing in the analysis so far which applies specifically to trees, and no difference is made if the individual trees are regarded as single vertical sheets of leaf material such as a blade of grass. Second, the peaks of Fig. 3 would be rounded off in a real situation because the ho defined by equation (13) is only an average. Third, no mention has been made of diffuse radiation which again would have a smoothing effect. These last two points could be covered in a specific situation by detailed computer analysis. (A)
INCLUSION
OF SOIL
WATER
AVAILABILITY
In the early discussion, equation (2) contained a term tj, representing the potential of the water in the soil. In a real situation the potential drop from the main body of the soil to the root surface must be considered, and equation (9) should be dP(h) = 2n,r,
-Jlo-$1
-wg $0
1
dh ’
(19
where $r is the root surface potential at the time. Now if 8, and B are the soil water concentrations at the root surface and in the main body of the soil respectively, they will be related by an equation (see, for instance Passioura & Cowan, 1968) of the form O1 = &-QrX/2D,
122
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PALTRIDGE
where Qr is total tree evaporation rate, D is soil diffusivity, and XK l/LV, where Lis root density and Yis volume of soil available to the roots. Typically soil potential is some form of exponential function of soil water content, so we can write for instance til = Al exp (-II,
= A, exp
=
(21)
where B = A, emB1’ and we replace the exponential of Q,X/2D with the first terms of its series expansion 1 + QTX/2D. (QTX/2D is very small since in practice it can never be more than some fraction of 0). Now suppose further that total tree evaporation Qr is proportional to total tree photosynthesis PT. This is a reasonable approximation at a given potential evaporation since in real trees both evaporation and photosynthesis are controlled to a great extent by a common stomata1 aperture. Therefore we can write +r = B(l +kP,) where k is some constant, and equation (19) becomes dP(h) = 2n, r,
$,,-B(l+kP,)-Ah
(22)
J/o
If one carries through an integration (see Appendix E) similar to that used in deriving equations (15) and (16), one arrives at two equations equivalent to them but with soil moisture status taken into account. Namely A 0.86 tan 9Z2 -- A P n, tan 0 A(ho>
4ro 1+ no kB(O*86Z2 tan o/1,4,)
0) =
and
*o H I
(23)
2noro P ,4(ho
1 + 2no r. kBH/$, (B)
OPTIMUM
AVERAGE
GROWTH
RATE
Both from the philosophical and practical points of view it is of interest to enquire whether there is a spacing 2, of trees in a “forest” such that a maximum average rate of photosynthesis is achieved where the average is taken over the lifetime (or several lifetimes) of the trees. Philosophically it is of interest since, if a concept akin to a natural tendency towards maximum land use had some basis (Lotka, 1922), then one might expect natural trees to have adopted such a spacing. Practically it is of interest to a forester who is
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harvesting “trees” and who is concerned with maximizing the return from a given area of ground. Figure 3 is a graphical representation of equations (15) and (16) giving the rate of photosynthesis per unit area as a function of height for various values of Z. To answer the question posed above it is necessary to define a death point to the trees. It could be de&red as that height H,,, where total photosynthesis of the trees as defined by equation (15) (say) is zero, but this criterion leads to an H,,, which depends drastically and quite unrealistically on Z. In addition it relies on the characteristic built into the equations that photosynthesis of a leaf continues linearly to be more negative as its potential increases beyond $,,. Intuitively it is far more likely that the tree would stop growing when any new leaves could be placed only at a height such that their net photosynthesis would always be negative. Such a definition gives H
max
=-
463 E,z,+g
or in the case where soil moisture status is included H max= ~1clo-b4 E p%+i
(26)
The lower the average soil moisture content and the higher the potential evaporation, the lower is the maximum height of a tree at the end of its life. The average rate of photosynthesis per unit area of ground over the life of the trees would be
PA= j= - PA dt, T, T--T,
where Tis the time at which the height H reaches H,,, and Tl is the somewhat arbitrary time in the very early stages of growth which has to be regarded as the beginning of growth “as a tree” (since the integral does not converge for Tl = 0).
Since rO is constant presumably dH/dt = k,,P, where k, is some constant (i.e. the rate of height increase is proportional to total photosynthesis on the assumption that the volume occupied by leaves is directly proportional to available photosynthate). We can then write How., 1 dH, h = fL,x j (28) I HI f(H> where f(H) is the function of H defined by equations (15) and (16) and HI is the height at Tl. More explicitly
(29)
124
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W.
PALTRIDGE
wheref,(H) andf’(H) are the functions of H defined by equations (15) and (16), respectively, or in the case of inclusion of soil moisture status by equations (23) and (24), respectively. These integrals are worked out explicitly in Appendix F. The resultant equation for PA as a function of Z is too complicated to differentiate analytically, so it has been evaluated numerically as a function of Z for various $,/A ratios (i.e. for various H,,,) and for various values of soil moisture status. See Fig. 4 for some examples.
FIG. 4. Examples of calculated average photosynthesis over the lifetime of the trees as a function of tree spacing 2. Solid lines are for cases where soil water potential assumed to be 22 m where average soil potential B=SOO. zero. Dashed curve is for case of H,,,,,= Relevant data: r0 = 1 m; no = 1; HI = 1 m; r,u= 2000; 0 = 45”; k (see equation (27)) = 10.0.
These calculations show that (i) there is indeed an optimum spacing to give maximum P,, (ii) this optimum spacing decreases with decreasing H,,,,,, (iii) with decreasing H,,,,, the maximum P, increases (a similar conclusion to the earlier example based on optimization of maximum instantaneous rate of photosynthesis), and (iv) with decreasing soil moisture both the maximum yield at the optimum spacing and the optimum spacingitself’decreases. 5. Tree Growth-The
Optimum Shape at Various Stages
The discussion in the previous section dealt with cylindrical “trees” which were constrained to a fixed radius r,,. There was, therefore, no need to make a detailed statement about leaf distribution within the individual trees-except to say that the leaf density was sufficiently great or was distributed in such a manner that light penetration through a tree at any distance from the trunk was negligible.
THE
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TREES
We will now consider the case of cylindrical trees in a forest where the constraint on r0 is removed. This will be done in order to obtain an idea of the shape of trees in a forest if they were constrained to maximize their instantaneous rate of photosynthesis. In other words we will assume that, at any given stage of tree development, photosynthate can be disposed in the form of new leaves either at the top of the tree (thereby increasing H) or as a uniform distribution about the sides of the tree (thereby increasing r,); and that whichever of these alternatives gives the greatest increase in rate of photosynthesis will be the actual “choice” of the tree at that stage. The treatment must necessarily be very rough, and should be regarded primarily as the basis of a qualitative description. We assume that the leaf density of the trees is uniform and very high in order not to become involved in complicated discussion of leaf distribution. Thus if there is an amount dx of available photosynthate, it can be placed either (a) at the top of the tree to give an increase in tree volume d Vdefined by dV = nr:,dH
= x: dx,
(30) or (b) on the sides of the tree to give the same increase in volume defined by d V = 2m-,(H - 11,) dr, = k dx,
(31) where k here is a constant relating amount of photosynthate to the volume occupied by the leaves which it forms. Thus in case (a) the increase in rate of photosynthesis P will be c3P -=-dH
i3P
k
(32)
aH 7rri’
and in case (b) the increase in rate of photosynthesis P will be
ap dr,
ap dr, 2m,(H
k - h,)’
(33)
The problem is to evaluate the partial derivatives aP/aH and aP/&, at the various stages of tree growth. In the previous section it was assumed for simplicity that leaf potential was a function of height only. Here, since rO may vary, we must take radial variation of leaf potential into account. Thus we assume the trees to have radial branches such that the leaf potential at height 11and radius r is given by J/(h, r) = (E,zo+g)h+E,z,r, (34) which can be compared with equation (10). By following an analysis similar to that of equations (9) through (16) with the $(A) there replaced by the above @(h, r), we can obtain equations similar to equations (15) and (16) except for
126
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W.
PALTRIDGE
inclusion of a factor (1 -E,z,r,/~,). P,=n,tanB(I-E*)
Namely
[1-~(H-0’86,,nezz)(~-~~~, (35)
and P hhO
[H-&H’
(l-E,‘,o,K)].
(36)
(Actually the r. of the factor should be some constant large fraction of r. which depends on the leaf density distribution. The constant has been dropped here for simplicity.) Consider first equation (36), which gives the rate of photosynthesis of forest trees which are small enough and/or are sufficiently far apart for ho to be less than zero. The partial derivatives aP/aH and aP/ar, can be obtained, and therefore from equations (32) and (33) we can obtain the derivatives (aP/iYx)ro= C,,nStand (aP/axO)H=cons,. From these it can be shown (see Appendix G) that provided H < 2ti,i3~,
(37)
then addition of leaves to the top of the trees leads to a greater increase in rate of photosynthesis than does addition of leaves to the sides (i.e. when H < 21,b,/ So that if such trees 34 then WW),, = COnstis greater than (aP/ax),=,,,,,). were constrained to maximize instantaneous photosynthesis they would tend to grow upwards rather than expand horizontally. The physical basis for this result is that, of the leaves placed on the sides of the trees, only those which increase the vertical area of the tree as seen by the sun contribute to extra photosynthesis. Most of the leaves appear either in front of or behind leaves already in existence, and as far as total rate of photosynthesis of the tree is concerned they contribute nothing extra. This result, which effectively deals with the case of isolated trees since ho < 0, can be contrasted with that in the early examples of a single tree where interception of radiation was neglected. Consider now equation (35), which gives the rate of photosynthesis of forest trees which are large enough and/or sufficiently close for ho to be greater than zero. Again the partial derivatives aP/aH and aP/ar, can be obtained. Both ww,,=,,,,, and wax),= Constare negative in this case, showing that when ho > 0 addition of new leaves anywhere on the trees reduces the total rate of photosynthesis. Furthermore it can be shown from the expressions for these derivatives (see Appendix G) that provided ro < ; k
(H-ho), P
0
(38)
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OF TREES
So that if wwH= constis less in absolute magnitude than (CP/~x),,=,,,,,. such trees were constrained to maximize instantaneous photosynthesis they would tend to expand horizontally rather than grow upwards. Ultimately, when r0 exceeded (3/2)[A(H-h,)/EPzo], they would again grow upwards rather than expand horizontally. Thus if we wished to “design” trees which maximized the rate of photosynthesis per unit area of ground and which were to be planted at the same time, we would perhaps “design” them to follow in broad outline the growth pattern of Fig. 5. They would tend to grow upwards at first until they reached D = 2(Epzo+ g) W-ho) Epzo
j*--
1 -0 ---+I 1
H = O-86Z2tane
FIG. 5. Diagrams of the shape, at various stages of growth, of a tree which was constrained to maximize instantaneous growth rate when it is one of a forest of similar trees.
a height where their bases were shaded from the sun (i.e. where h, = 0). They would then expand in radius keeping a constant height, with ho necessarily increasing and the depth of foliage H-he decreasing. When r0 reached a value of the order of 14 times H-h, [since in many cases (EPzO +g)/ EPzO would be not much greater than unity] the trees would again tend to grow upwards while maintaining a constant depth of foliage H-h,. Whether natural trees have developed along similar lines and for similar reasons is a matter for argument. 6. Conclusion
Two of the factors which influence the photosynthesis of leaves have been considered here. These are that the net rate of photosynthesis is some decreasing function of leaf water potential, and that the net rate of photosynthesis is some increasing function of light intensity. The possible effect of these factors on the evolutionary development of the broad-scale characteristics of trees
128
G.
W.
PALTRIDGE
have been illustrated by the analytical treatment of “extreme case” examples. For instance, by considering an isolated tree it was easily seen how considerations of leaf water potential (alone) might lead to the evolution of trees of hemispherical shape. It was shown how considerations of light interception (alone) might lead to the evolution of trees whose leaf density increased radially from the trunk. Both leaf potential and light interception have been included in an “extreme case” model of a forest. Dealing first with a forest of trees whose radii were constrained to be constant, it was shown how maximum rate of photosynthesis per unit area of ground might be achieved at a particular tree spacing determined by tree height; and that the maximum possible rate of photosynthesis would be achieved by the shortest trees at their optimum spacing. As far as the theory is concerned these “shortest trees” might be grass. Similar conclusions were reached when maximum average rate of photosynthesis (i.e. the average over the lifetime of the trees) was considered. It was shown how soil water availability might influence these results. Dryer soil reduces the maximum rate of photosynthesis at optimum tree spacing, and reduces the optimum tree spacing itself. By removing the restriction on tree radius, it was shown how the shape of trees (of high leaf density) in a forest might vary if they were to achieve maximum instantaneous rate of photosynthesis at all stages of growth. That is, they would tend to grow upwards until radiation interference from adjacenl trees became significant; they would then expand horizontally while developing a crown; and they would then resume an upward growth while maintaining a constant crown volume and dupe. In summary the work here provides the analytical framework of a much simplified model of “tree” or plant behaviour. This framework has many satisfying aspects and provides a number of results which are testable in principle, since the physical and physiological parameters are capable of measurement and the detail of the geometry (say) of a particular real situation can be handled easily by computer analysis. From the most general point of view the present attempt can be regarded as an example of the many possible starting-points for the design of an overall plant growth model. For instance, here two physiological parameters zO and y?Ohave been selected and the analysis has been built about them. There is no reason to suppose, however, that our choice of parameters is superior to another set which might be suggested by other authors. Such choices can be judged only by the success of the model in predicting the morphology of trees. In addition, although zO and $O arc in one sense over-simplified real-plant characteristics, in another they are not simple enough. The question
THE
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still remains as to what have been the environmental controls on the development of their actual values. On the time scale of the evolution of plants it is unlikely that z0 and $,, could be regarded as independent variables. Thanks
are due to Dr David
Parkhurst
for his helpful criticism
and advice.
REFERENCES CO~TWOR, D. J. & CARTLEDGE, 0. (1970). J. uppl. Ecol. 7, 353. LOTKA, A. J. (1922). Proc. mtn. Acad. Sci. U.S.A. 8, 147. PALTRIDGE, G. W. (1970). Agric. Met. 7(2), 93. PARKHURST, D. & LOUCKS, 0. L. (1972). J. Ecol. 60(l). In press. PASSIOURA, J. B. & COWAN, I. R. (1968). Agric. Met. 5(2), 129. SLATYER, R. 0. (1967). Plant-Water Relationships. London: Academic Press. WATTS, D. G. & LOUCKS, 0. L. (1969). Tech. Pup. Inst. Envir. Stud. Madison,
Wisconsin:
Univ. of Wisconsin.
Appendix A
We require to show that Parkhurst’s criterion of maximization of the ratio R( = P/E) is equivalent (as far as plant development in terms of the present model is concerned) either to maximization of photosynthesis P or minimization of evaporation E. A more realistic representation of photosynthesis than equation (1) would take into account specifically mesophyll resistance r, and respiration of carbon dioxide. For instance we could write
no ’ = rm+rs~o/~~o-II/>
-noK
(Al)
where here the first term on the right-hand side represents gross photosynthesis and the second term represents respiration. There is a specific constant rs representing stomata1 resistance when J/ = 0 (fully open stomata). The no here is a constant such that when $ = 0 the first term on the right-hand side represents gross photosynthesis n,/(r,+r,). K is a constant such that n,K is the respiration rate per unit volume-on the assumption that respiration is constant and independent of $. $. here represents the potential at which the stomata are fully closed and no further uptake of CO2 is possible. Evaporation E could be written -___ bno
T.“.
E = rs+o/($o-II/)’
(A3 9
130
G.
W.
PALTRIDGE
since r,,, does not appear in the water pathway from the sub-stomata1surfaces to the atmosphere; b is a constant such that bn,/r, is the evaporation rate per unit volume in a specific environment when the stomates are fully open. If a plant were obeying maximization of photosynthesis (criterion I say) it would place its new growth in a position of minimum $. If it were obeying minimization of evaporation (criterion II say) it would place its new growth in the position of maximum $. From the above equations
and by differentiation with respect to 4’/
If dR/d$ is positive, then the plant (if it has a choice) will maximize R by placing its new growth in a position of maximum $--equivalent therefore to criterion II. If dR/d$ is negative, the reverse will be true and the placement of new material will be equivalent to criterion I. Now from equation (A4), dR/d$ is positive when the term in square brackets is positive, that is, when (A51 This condition can bc put in a more understandable format by multiplying through by n, and obtaining the condition (by referring to the definitions in equation (Al)) respiration < (gross photosynthesis)
(A5a)
or that respiration < -.-I’, grossphotosynthesis r,,, + rs,’
(A5b)
where rst is the stomata1resistancer, $0/($0 - $). This condition is likely to be satisfied for smaller (younger) trees where the leaf water potentials will be small-and where therefore stomata1 resistance will be small and gross photosynthesis will be large. Thus in the early stagesof growth, maximization of R is likely to be equivalent to the unstable criterion II. In the latter growth stagesmaximization of R may be equivalent to the “stable” condition I.
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131
Appendix B Considering a tree for which restriction of its total volume is a desirable characteristic, it seems reasonable that a further desirable characteristic should be as follows. Considering a horizontal section of a tree (presumably a circle on the assumption of radial symmetry), whatever value of absorption or “obstruction” coefficient is presented by the random leaves along a diumeter this same coefficient should be presented to a solar beam along any chord. This assumes that the obstruction by leaves along the diameter (say) is some optimum which gives maximum absorption of photosynthetically active wavelengths for minimum “wastage” of the leaves as far as their contribution to photosynthesis is concerned. [If a certain number of leaves absorb (say) 99 “/ of the radiation, doubling or trebling the number of leaves to gain the extra percent would obviously be inefficient.] We are considering here only the case of horizontal beams of light. The same principles would apply to any section of the tree parallel to the rays of the sun, but the mathematics becomes very awkward. Thus the question arises as to whether it is possible to have a distribution of leaf density n as a function only of radial distance r from the centre of the circular section such that the absorption coefficient is constant along any chord. It can be shown quite easily that there is no such distribution except for the case of infinite density at the outside radius R and zero density elsewhere. However, it is interesting that there is a distribution in two-dimensions (i.e. n(s, y) where I is the axis in the direction of the sun) which satisfies the criterion for all chords parallel to the solar beam. This distribution is derived in Appendix C and is
n(x, y) = -$--
R -y” where k is a constant depending on the actual value of the absorption coefficient, and the origin is at the centre of the circle. Unfortunately IZ --f co at the point (X = 0, .t’ = R), but apart from this n increases smoothly with both s and 1’. Bearing in mind that in general the azimuth angle of the sun varies throughout a day by up to 180”, it is reasonable that a “close-to” optimum n(rj distribution might be one which is some form of average between n(x) and n(y). It is pointed out in Appendix D that “mathematically obvious” averages require still an infinite density at r = R, so that for practical purposes it is perhaps simplest to select a radial distribution according to equation (AIO) at some “average angle”. The actual choice of such an angle would be a major investigation in itself, but here we are intending only to show that in real trees
132
G.
W.
PALTRIDGC
it is reasonable to expect that leaf density should increase radially from the trunk. If, for instance the “average angle” were 45”. then n(r)
=
@lKr’ _____
(A@
R2 -35.2
Appendix C Referring to Fig. 6 which considers one quadrant of the circle, let n(r) be the density of leavesat radius r. We require a density distribution n(x-, y) such that JR-s--q2 JU2--)J N(y) = j’ n(r) dx = 1 ,I(\/? +y’) dx = constant. (A71 0
0
Now in terms of the variable Y (A8) so that a suitable n(r) distribution such that N(y) is constant is k&.2+ n(r)
=
(A9)
---z---j
R -J’or in terms of x and y
,--- kJx2+y2-y2 n(x, y) = R2-jj2
solar _-~
FIG.
beam.-r
6. Geometry
kX
(AlO)
R’- \a”
I
associated
with
leaf
density
distribution
analysis.
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TREES
Appendix D On “average” n(r) distribution (AlO) might be
based on the n(s, y) distribution
n(r) = f $ Re2
of equation
(All)
de.
Replacing .Y with r cos 0 and y with r sin 8, and with two subsequent changes of variable
which goes to infinity
at r = R.
Appendix E From equation (22), the total evaporation is given by P,=
j2n,ro ho = 2n, r.
I-$(1+kP,)-$h 0
I[
l-
which on re-arrangement 2noro P, =
P, from a single tree in a forest
0
$-(I+&) 0
I
dh ,
1
(A131
becomes H-h,-
$ (H-h,)-
$
(II”-hi)]
(A14)
1+ 2n, z. kB(H - h,),$::
Thus for the case when ho ,< 0 (equivalent
to ho = 0)
2nor0 1-E H-IAfp p = O*86Z2 [( *o > 211/o I A 1+2n,r,kBH/~, ~’
(A15) or (24)
For the case when ho > 0, since O-86 tan BZ2 h,=H---
)
2r 0
(A16)
134
G.
one can obtain H-h, H2-h; and by substitution
W.
PALTRIDGE
and also =
(At7)
in the above equation for P, and division by 0.862 ’
(A18) or (23)
Appendix F We require to evaluate the expression (A19) where fi and f2 are the functions of H defined by equations (15) and (16), respectively, for the case where soil water status is neglected, and are functions defined by equations (23) and (24), respectively, for the case where soil water status is included. Considering the first case, the denominator (denom) is of the form denom =
(A2’3
where a = A/$o and b = 1 + (A/$,)(0*86 tan f3Z 2/4ro). When the integrals are evaluated
and pA=zEp
H denom
(Il/olA) denom’
C.422)
Considering the second case denom = ‘z2r n l ($!s2)
dH+ ;-~inme “!I’ 0
(& -tifi)
dH,
(A23)
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where here b = (1 - Bjtj,) + (A/11/,)(0*86 tan 8Z2/4r,), c = 2n, r. i&/1+5,, d = 1 + n,kB x 0.86 tan OZz/$O. e = 1 -B/$,. When the integrals are evaluated
and again
Appendix G For the case of h,, < 0 (effectively that of isolated trees) from equation (36) (A251 and using equation (30) to obtain dH/dx, (c!),o,,“,,(z!E)=~ Similarly
3?p-;!.
(A26)
from equation (36) dP ~ = 2n,Ho”r,
4n,E,z,Hr,
noAH
$0
$0
’
(A27)
and using equation (31) to obtain dr,/dx, WW Thus if 4n,AH n, n,AH 2no ~ - > r, - @,<). ro V. r.
(~29)
136
G.
W.
PALTRIDGE
then in order to achieve maximum rate of photosynthesis the trees to grow upwards. This condition simplifies to H < 2$,/3A. For the case where ho > 0 (where signjficant), from equation (35)
light obstruction
one would expect (A30) by adjacent trees is
-ap = _ XA ~-aH $0’ where X = no tan 0, and therefore
(A31)
(~),,...,,(=%3= Similarly
from equation
ap -=---
-ii&.
(A32)
(35) XE, z.
O-86 tan 9Z2 4r,2*, ’
-XAx
$0
(A33)
and therefore - XE,zo XA O-86 tan OZ’ tj0271rO(H - h,) - x 4r~27rrO(H - h, 1 -XE,zo since H-ho
_ ---XA 4nr&b,’
tA34)
I XE,zo ~~~ + -At! XA nr&/io < 2nr,t,b,(H - ho) 47-L&,’
tA35)
= $027cro(H - ho) = O-86 tan f3Z2/2ro. Thus if
then in order to achieve minimum degradation in the rate of photosynthesis the trees should grow upwards. This condition simplifies to r. > iEL
(H-h,).
(A36)
020
APPENDIX
H
List of Symbols A B D E
EP H
Epzo +g
seeequation (2 1) soil diffusivity (a function of iz and 0,) actual evaporation rate per unit volume containing leaves potential evaporation rate per unit volume containing leaves overall tree height
THE
H max HI L P PA p.4
PT QT V x Z -Z .4 h ho flo
J-0 =o
SHAPE
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137
ultimate maximum tree height initial tree height seeequation (20) net rate of photosynthesisper unit volume containing leaves net rate of photosynthesisper unit area of ground net rate of photosynthesisper unit area of ground averaged over tree lifetime total net rate of photosynthesisof singlerree total evaporation rate of a singletree seeequation (20) seeequation (20) averagecentre-to-centrehorizontal tree spacing effective centre-to-centrehorizontal tree spacing gravitational potential per unit height height above ground surface height above which leavesof a tree have an unobstructedview of sun net rate of photosynthesisper unit volume containing leaveswhen leaf water potential is zero distancealong horizontal radius of tree horizontal radius of tree proportionality factor relating leaf potential to E,, and length of trunk plus limb averagesolar elevation averagesoil moisturecontent soil moisture content at root surface leaf water potential leaf water potential at which net rate of photosynthesisis zero water potential at the root surface
On the Shape of Trees G. W. PALTRIDGE Division of Atmospheric Physics, C.S.I.R.O., Victoria, Australia
Aspendale,
(Received 24 January 1972, and in revised form 6 May 1972) Answers to questions concerning the broad-scale characteristics of trees are sought in the analytical development of a simple model in which the rate of photosynthesis is controlled by leaf water potential and by access to direct solar radiation. These concepts are introduced tirst in a simple discussion of the single isolated tree. Later they are applied to the forest-stand situation. Expressions are derived which relate growth rate (per tree and per unit area of ground) to the height, shape, spatial separation and lifetime of the trees; to the environmental conditions such as the average solar elevation, potential evaporation E, and soil moisture content; and to two main physiological factors-the proportionality factor zO relating potential drop per unit length of trunk or branch to potential evaporation, and the critical leaf water potential vO at which net photosynthesisis zero. Accepting the assumptionsin the model, the following are examples of its predictions. It is shown that at optimum tree spacingthe photosynthesisper unit area of ground may be greatestfor the shortest trees (grass?).It is shownthat for a given environmentthere may be an optimum tree spacingyielding maximum photosynthesisper unit area of ground averagedover the lifetime of the trees; that this maximum decreases with increasingultimate tree height (which in turn is determinedby E,, zOand v,,); and that this maximum and this optimum spacingdecreasewith decreasingaveragesoil moisture.It is shownthat there can be an optimum leaf distribution which in general is such that leaf density increases radially from the trunk. It is shown how the optimum shape of trees in a forest might be expected to alter with their size. In generalit appearsthat the conceptsdiscussedhere may be useful in explaining the evolutionary developmentof many of the broad features of tree growth.
1. Introduction The growth of plants depends on environmental and inherited factors in a very complex way. Because of this complexity plant ecology has suffered from a lack of mathematical models to integrate its findings and to provide
testable hypotheses. The situation is improving 111
as computers become avail-
112
G.
W.
PALTRIDGE
able because realistically complex systems of equations can now be investigated. For instance, crop growth models are becoming more accurate as they become more complex (e.g. Connor & Cartledge, 1970). Whole ecosystems can be modelled mechanistically as in the I.B.P. model described by Watts & Loucks (1969). In general, these types of system analyses adhere to the principle (attributed to Tukey) that it is better to have approximate answers to the exact problem than exact answers to an approximate problem. There is also a place for analytic solutions to approximate problems. Often these lead to useful generalizations which would not be apparent in the computational results of a complex system model. Indeed it is probably true in many cases that the simple analytic approach is an essential prerequisite for the development of the complex model. A prime example of this has been the remarkably fast and successful development of numerical weather prediction, which was guided by many decades of analysis of “approximate problems”. In this paper the analytical approach is used to study certain general features of the shape, size and spacing of trees as they have evolved under differing environmental conditions such as potential evaporation, solar elevation and soil water status. The word “trees” is perhaps something of a misnomer, since much of the analysis is applicable to plant structure in general. Both the strength and the weakness of the analytical approach to a complex problem is the necessity to simplify the system and to study only the most basic and important of its characteristics. There can be some argument as to which of the characteristics of trees are the most important, but here we consider only their ability to photosynthesize atmospheric carbon dioxide and to dispose the resultant photosynthate in the form of new leaves and limbs. Furthermore, because of the necessity for simplification, we consider only two of the many environmental and biological factors which determine the rate of photosynthesis of individual leaves. (1) Leaf water potential. The methods of water potential “control” in real plants are very complicated and include such processes as direct physical restriction of the absorption of carbon dioxide by the stomata1 apertures of the leaves. In the present work we will specify only the most general chnracteristic-namely that photosynthesis by the leaves of a tree decreases with increasing leaf water potential. (2) Available photosynthetically-active light energy. Again, the “real” process is exceedingly complicated. In the next section, which is introductory and deals only with single isolated trees, the problem will be avoided by assuming that there is always sufficient light for leaves to photosynthesize to the limit set by their water potential. In the later sections which deal with
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113
trees in a forest, it will be assumed that the leaf density is sufficiently high for shading by a tree to be complete. That is, leaves which are in the shade of another tree will be assumed to have insufficient light for photosynthesis. It will be seen that the two cases lead to quite different optimum tree shape. Again it is emphasized that the concepts discussed here are susceptible to more accurate but necessarily more complex development in specific situations. 2. Basic Concepts and the Single Tree Consider a tree to be an assembly of leaves attached to the ground by a number of limbs whose function is to provide support for the leaves and to provide access to the moisture in the soil. The water potential I/ of the leaves is assumed to be the only controlling factor on their consumption of carbon dioxide and evaporation of water. Furthermore, it is assumed that the control is linear, so that the net rate of photosynthesis P per unit volume (of space occupied by leaves) can be written as P=n 0
wo-II/> I
3
(1)
which states that when + reaches a critical potential lclo the net rate of photosynthesis (gross photosynthesis minus COZ respiration) is zero, and no is the net photosynthesis when $ is zero. The concept of water potential as it applies to plant growth is discussed in detail by Slatyer (1967). Very broadly, water potential at a point in the soil-plant system is a measure of the suction pressure at that point-a pressure which is brought about by the evaporation of water. In a real tree the water potential of the leaves is determined by many factors, of which three are considered here. That is, ti is determined firstly by simple gravitational potential (g per unit height), secondly by the potential drop due to the passage of water through the resistance of the limbs, and thirdly by the potential $. of the water in the soil. Thus for unit volume of leaves at height h connected to the ground by a vertical trunk and a horizontal branch of radius r we may write ti = E,zo(h+r)+gh+k, (2) where, for reasons which will be discussed shortly, Ep is the potential evaporation per unit leaf area index (say), and z. is a constant. There are several major concepts and assumptions inherent in equations (1) and (2) which need to be discussed. First, the potentials associated with the flow of water through the solidplant-atmosphere system should be negative quantities by formal definition, l-3. 8
114
G.
W.
PALTRIDGE
which arbitrarily assumes the potential of pure “free” water on the ground surface to be zero. Here, the negative sign is dropped for simplicity. No confusion arises provided it is realized that reference to a potential is a reference to the absolute value of that potential. Second, although we assume for the moment that the intensity of sunlight does not enter directly into the determination of photosynthesis, solar radiation can have an effect through its control of Ep and hence leaf potential. Third, it is assumed that the potential of one unit area of leaf is independent of the potential of any other unit area of leaf except in that they may be related through their combined effect on soil water potential I/, (see later). In effect we envisage each unit area of leaf to have a separate “pipe” to the common reservoir of the soil, so that if these pipes are bundled together in the form of a trunk (say) the overall diameter of the trunk must increase towards the ground. An analytical treatment involving any other assumption is again almost impossible to handle. In fact it appears that some real trees do behave with this apparent independence of water paths, although others do not. However, since in real trees the circumference of the limbs and trunk increases towards the roots (presumably for reason of strength, but also possibly to match water potentials at junction points of the vascular system) the approximation may not be serious. Fourth, we assume that the potential of leaves is proportional to potential evaporation E, and not to actual evaporation E. This can be regarded as an approximation which apart from other considerations is necessary for the subsequent analysis to be mathematically tractable. Had $ been set as proportional to E, which on first thoughts might seem preferable, some other assumption would in any event have had to be made with regard to the proportionality constant zO and its variation with potential. For instance, we might have written II/ = Ez,(h + r) + , . . . , and made some linear approximation for E as a function of II/ such as E = E,[($, - I,@/$~]. This would be easy to picture physically since the zO could then be visualized as the actual resistance to water flow per unit length of trunk or branch. However, some dependence of zO on r+$would have to be assumed (such as letting zO be constant with Ic/), and there is no reason to suppose that this would be nearer reality than the original assumption of equation (2). At least equation (2) simulates the general characteristic of importance here-namely, that the potential of a leaf increases with increasing potential evaporation. Finally, a linear variation of P with t+Qis not in accord with experiment, which generally shows the variation of gross photosynthesis with 1,9to be highly non-linear-and of a form obviously related to the variation of stomata1 resistance with JI. However, the mathematically tractable linear relation is sufficient for the purpose in that it simulates the characteristics of
THE
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115
importance to the present analysis. In addition, averaging in time (as will be done here-for instance we will deal with average solar elevations) would tend to make a linear P vs. $ relation more appropriate, since the nonlinearity of individual leaves would be smoothed. The simplest case to treat mathematically is that of a tree whose leaves are distributed uniformly throughout a hemisphere whose base is on the ground and whose supporting branches radiate from the centre point of the base. Neglecting both soil moisture and gravitational potential the total rate of photosynthesis is the simple integral of equation (1) (with substitution of equation (2)) using spherical coordinates r, 8 and r$
(3) where H is the height and therefore radius of the tree. That is
PT =
7m,
2H3 E,zoH4 - 3 wo >'
(4)
which is an equation showing that, provided the tree maintains its spherical shape, the total rate of photosynthesis will vary with height in the manner of Fig. 1. The decrease in photosynthesis after a certain height arises because rj in equation (1) (i.e. E,z,r here) can exceed tie giving an effective negative photosynthesis of those leaves at a radius greater than $,/E,z,. Physically, this simulates the case where respiration exceeds gross photosynthesis. This and the associated problem of defining the death point of a tree will be discussed later.
r
FIG.
1. Single tree total photosynthesis as function of height.
116
G. (A)
DISPOSAL
W.
PALTRIDGE OF PHOTOSYNTHATE
A major difficulty in any attempt at modelling the growth of plants is the definition of the criteria by which newly photosynthesized material is disposed about the plant. For any given species it is presumably possible to derive a set of empirical rules appropriate to that species, but in the present work we are interested in, among other things, the broader problem as to why the gross morphoIogy of a tree (say) has evolved in its particular fashion. A technique used quite often is to replace the complicated processes of natural selection with a single assumption-namely, that the overall effect of natural selection is to produce, within the constraints set by the environment, a plant of the optimum characteristics to perform a particular “function”. If one assumes further that the optimum characteristics have already been achieved then the use of this “function” as a criterion for disposal of photosynthate in a model plant should force the model to grow in a form similar to that of real plants-provided that similar environmental constraints are imposed on the model. One can be guided in the choice of a “function” only by intuition and experience, and the fact that it contributes to the survival of the species. For instance, it could be argued that trees might have evolved in such a manner as to maximize their rate of photosynthesis. Paltridge (1970) has used this assumption in a complex computer model of a pasture. Similarly, it could be argued that trees might have evolved in such a manner as to minimize their total water use. Parkhurst & Loucks (1972) have suggested a criterion whereby plants, or parts of plants, might have evolved so as to maximize the ratio R of photosynthesis over evaporation. It is worth emphasizing that all such criteria give a result dependent on the time scale over which the criteria are applied. Maximization of instantaneous photosynthesis, for instance, will generally lead to a different shape than maximization of average photosynthesis over the lifetime of the plant. Where it is appropriate in the present work the principle of maximization of photosynthesis (over a specified period) is applied. In many practical situations Man is interested in maximizing the photosynthetic return from his land. Thus a model based on maximization of photosynthesis will generate a “plant” whose characteristics may be a helpful guide to his selection of a particular species. In addition, on the present model maximization of photosynthesis is stable since the plant or tree will tend to place its new leaves in the position of minimum potential, or in general as close as possible (bearing in mind the geometrical limitations set by the form of trunk and branches) to the base of the trunk or stem. Thus in the previous very simple example of a single tree, extra leaves would be placed such that the outer leaves always formed an equipotential canopy-in other words a hemisphere. This shape would be
THE
SHAPE
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117
maintained throughout its growth and is independent of initial conditions. If one includes the effect of gravity, it can be appreciated that the optimum shape would become more flattened since the potential would increase more quickly in the vertical than in the horizontal. On these grounds therefore, one might expect firstly that single trees would be more closely hemispherical in regions where potential evaporation is highest (the effect of gravitation would be relatively less) and secondly that in these regions the branches of such single trees would tend to radiate directly from the stump in order to minimize their length and consequent potential drop along them. [A criterion such as minimization of evaporation is unstable in that final shape would depend on initial shape. It is to be expected that actual evaporation from a leaf would decrease with increasing $ in a similar fashion to gross photosynthesis since in real plants both CO2 uptake and evaporation are controlled by a common stomata1 aperture. Thus if for some reason there was a region of the plant whose leaves were at a higher potential than elsewhere on the plant, extra leaves would tend to be added at the extremities of this region and the plant would continue to grow outwards from the initial irregularity. Again, although Parkhurst’s criterion of maximization of R is attractive, it can be shown (see Appendix A) that this also may be unstable. It is shown there, using a slightly more detailed model of photosynthesis by a leaf, that this criterion is equivalent either (a) to a maximization of growth rate, or (b) to a minimization of evaporation, depending on the relative magnitudes of mesophyll resistance and carbon dioxide respiration at the particular leaf water potential. In general (b) would apply, except towards the end of the plant’s life in our present examples when most of the outer leaves would be at very high water potential.] It is of some practical interest to consider the case of a tree which has a single vertical trunk and horizontal radial branches. The leaf potential will be a function of both h and the radial distance r as in equation (2). As in the case of the hemispherical tree, if maximization of rate of photosynthesis is assumed the outer leaves will all have the same potential, so that if H is the tree height Wh, 11)= 4W), where r, is the radius at height h. That is E,z,(h+r,)+gh
(51
= E,zOH+gH,
(6)
where 4, is again neglected. Therefore rh =
@, zo+ g)W - h) EpZO
*
(7)
118
G.
W.
PALTRIDGE
Thus such trees would be expected to adopt a conical rather than shape. It is quite straightforward to calculate the photosynthesis tree as a function of height. The result is again a polynomial in H which yields a curve similar to, but of smaller amplitude than, that 3. Optimum
Leaf Distribution
spherical of such a 3 and N” of Fig. 1.
within a Tree
So far we have assumed that photosynthesis by a leaf is independent of light intensity, and have illustrated the likely form of isolated trees if photosynthesis were determined solely by leaf water potential. This may be a reasonable approximation for trees of low leaf concentration. However, light interception obviously plays a part in determining optimum tree shape, and in the subsequent treatment of a forest the effect of interception will be illustrated by assuming the opposite extreme of very high leaf concentration. The optimum shape of trees is then quite different. It should be pointed out here that considerations of light interception and leaf potential may play a part in determining the leaf distribution within a single tree. The subject is discussed in Appendices B, C, D. It is shown there that, for trees where restriction of total volume may be a desirable characteristic (e.g. the previous examples), it might be expected that leaf density should increase radially from the trunk. The outline of a possible mathematical treatment to derive the optimum form of this radial increase in leaf density is also presented. 4. Trees in a Forest Consider now a forest of trees of uniform height H in a region where the average solar elevation is 0. Let the trees be vertical cylinders of radius Ye, randomly positioned over the ground with an average centre to centre spacing of 2. Assume that each tree has sufficient leaf density to absorb (effectively) all the light falling on it, and that photosynthesis occurs only in those regions of the tree which have a view of the sun unobstructed by other trees. This means (see Fig. 2) that for two trees directly in line with the sun, photosynthesis will occur only above a height h,, where h, = H-Z tan 8. (8) In fact it can be supposed that no leaves would exist below h,. Above ho the rate of photosynthesis dP per height increment dh can be expressed dP(h) = 2nor, where here it is assumed that leaf potential
is a function only of height
THE
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119
OF TREES
.z
‘.
-
FIG. 2. Schematic diagram of forest trees showing the geometry for the case of two trees in line with the sun.
(“radial” branches are neglected) and that no is the rate of photosynthesis per unit vertical area of the tree when the leaf potential is zero. Note that we make no statement about the leaf distribution within the tree. The same mathematics are applicable if we regard the trees as two-dimensional rectangular “blades” in the vertical plane, where 2ro is the average projected blade area facing the sun. As before w = (E, zo + oh WV The total rate of photosynthesis PT of a single tree in the forest is then P, = j! P(h) dh = 2n, r. ho
H-h,-
$
(H2-h;) 0
I
,
where A is a substitution for E,,z,+g. In the three-dimensional situation of a forest the value of ho is given by equation (8) only for the case of two trees in line with the sun. In general, ho will be given by this equation if 2 is replaced by an “effective” spacing 2, which can be found by a form of mean-free-path calculation where 2, corresponds to the mean-free-path. Looking down on the forest, if one “moves” any point on a tree through a horizontal unit instance then this point will collide with the number of trees contained within an area 1 by 2 ro. Therefore, if there are N trees per unit area, 2, = 1/2roN. The average area of ground 5 available to an individual tree is given by approximately 04362 2, and since N= l/t O*86Z2 z -e-
and the appropriate
2r,
’
value of ho is therefore given by h -H O*86Z2 tan 6 o2r, ’
(12)
(13)
120 Substituting
G. W.
PALTRIDGE
for k,, in equation (11)
P, which relates spacing, height factors z0 and P,/O*86Z 2, or
0.86 tan 12Z2 (14) = 0+86n,Z2 tan 8 L1 - A $0 ( H- --%--- 0 -- >I the total tree photosynthesis to the solar elevation; to tree and radius; to potential evaporation; and to the physiological $I,,. The photosynthesis per unit area of ground P, is simply 0.86 tan tIZ2 4r,
>I ’
(15)
When h,, is less than zero, that is when the sun is illuminating parts of the forest floor between the trees, P, is given by equation (11) with h, set equal to zero. Again by dividing by 0~862~ P4hocO) = ““‘o(&;!H2). @86Z2
W
Thus assuming for the moment that the trees maintain a constant radius but increase in height, a typical P, vs. height curve might look like any of the curves of Fig. 3, where the maximum rate of photosynthesis PAcmaxjoccurs when h, = 0 or when H=--- 0.86Z2 tan 8 2r,
(17)
’
From equation (15) PAcmaxjis given by Pa(mrrl=.notane(l-~H).
H = ‘i,
FIG. 3. Instantaneous
#~=*y/a
photosynthesis per unit area as a function of average tree height
for threearbitrary valuesof Z. Note that onemustassume an initial heightHI. See text.
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TREES
These equations are simply mathematical expressions of the following. Because photosynthesis depends inversely on a parameter (leaf potential here) which increases with height, and provided solar energy is not wasted on the ground as when h,, c 0, photosynthesis per unit area of ground will be increased by increasing the spacing between the trees since more and more of the lower more efficient leaves become exposed. Tracing any of the curves in Fig. 3, given initially a young forest with a spacing such that ho < 0, as the forest grows upwards the P, will increase to a maximum according to equation (16) and thereafter will decrease according to equation (15) as the region of shaded, inoperative (or non-existent) leaves extends upwards. In addition equation (18) shows that if one compares various forest or plant stands each of which has the optimum spacing calculable from equation (17), but each of which has a different height, then the stand of lowest height will have the greatest rate of photosynthesis. From this it could be argued that more fertile soils, which from the nutrient point of view can support a higher rate of photosynthesis, are more likely to support “low trees” (grass?) than a stand of tall trees of large spacing. Indeed it is generally accepted that the grasslands of the world tend to occur in relatively fertile soils. Several points should be mentioned at this stage. First, there is nothing in the analysis so far which applies specifically to trees, and no difference is made if the individual trees are regarded as single vertical sheets of leaf material such as a blade of grass. Second, the peaks of Fig. 3 would be rounded off in a real situation because the ho defined by equation (13) is only an average. Third, no mention has been made of diffuse radiation which again would have a smoothing effect. These last two points could be covered in a specific situation by detailed computer analysis. (A)
INCLUSION
OF SOIL
WATER
AVAILABILITY
In the early discussion, equation (2) contained a term tj, representing the potential of the water in the soil. In a real situation the potential drop from the main body of the soil to the root surface must be considered, and equation (9) should be dP(h) = 2n,r,
-Jlo-$1
-wg $0
1
dh ’
(19
where $r is the root surface potential at the time. Now if 8, and B are the soil water concentrations at the root surface and in the main body of the soil respectively, they will be related by an equation (see, for instance Passioura & Cowan, 1968) of the form O1 = &-QrX/2D,
122
G.
W.
PALTRIDGE
where Qr is total tree evaporation rate, D is soil diffusivity, and XK l/LV, where Lis root density and Yis volume of soil available to the roots. Typically soil potential is some form of exponential function of soil water content, so we can write for instance til = Al exp (-II,
= A, exp
=
(21)
where B = A, emB1’ and we replace the exponential of Q,X/2D with the first terms of its series expansion 1 + QTX/2D. (QTX/2D is very small since in practice it can never be more than some fraction of 0). Now suppose further that total tree evaporation Qr is proportional to total tree photosynthesis PT. This is a reasonable approximation at a given potential evaporation since in real trees both evaporation and photosynthesis are controlled to a great extent by a common stomata1 aperture. Therefore we can write +r = B(l +kP,) where k is some constant, and equation (19) becomes dP(h) = 2n, r,
$,,-B(l+kP,)-Ah
(22)
J/o
If one carries through an integration (see Appendix E) similar to that used in deriving equations (15) and (16), one arrives at two equations equivalent to them but with soil moisture status taken into account. Namely A 0.86 tan 9Z2 -- A P n, tan 0 A(ho>
4ro 1+ no kB(O*86Z2 tan o/1,4,)
0) =
and
*o H I
(23)
2noro P ,4(ho
1 + 2no r. kBH/$, (B)
OPTIMUM
AVERAGE
GROWTH
RATE
Both from the philosophical and practical points of view it is of interest to enquire whether there is a spacing 2, of trees in a “forest” such that a maximum average rate of photosynthesis is achieved where the average is taken over the lifetime (or several lifetimes) of the trees. Philosophically it is of interest since, if a concept akin to a natural tendency towards maximum land use had some basis (Lotka, 1922), then one might expect natural trees to have adopted such a spacing. Practically it is of interest to a forester who is
THE
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123
harvesting “trees” and who is concerned with maximizing the return from a given area of ground. Figure 3 is a graphical representation of equations (15) and (16) giving the rate of photosynthesis per unit area as a function of height for various values of Z. To answer the question posed above it is necessary to define a death point to the trees. It could be de&red as that height H,,, where total photosynthesis of the trees as defined by equation (15) (say) is zero, but this criterion leads to an H,,, which depends drastically and quite unrealistically on Z. In addition it relies on the characteristic built into the equations that photosynthesis of a leaf continues linearly to be more negative as its potential increases beyond $,,. Intuitively it is far more likely that the tree would stop growing when any new leaves could be placed only at a height such that their net photosynthesis would always be negative. Such a definition gives H
max
=-
463 E,z,+g
or in the case where soil moisture status is included H max= ~1clo-b4 E p%+i
(26)
The lower the average soil moisture content and the higher the potential evaporation, the lower is the maximum height of a tree at the end of its life. The average rate of photosynthesis per unit area of ground over the life of the trees would be
PA= j= - PA dt, T, T--T,
where Tis the time at which the height H reaches H,,, and Tl is the somewhat arbitrary time in the very early stages of growth which has to be regarded as the beginning of growth “as a tree” (since the integral does not converge for Tl = 0).
Since rO is constant presumably dH/dt = k,,P, where k, is some constant (i.e. the rate of height increase is proportional to total photosynthesis on the assumption that the volume occupied by leaves is directly proportional to available photosynthate). We can then write How., 1 dH, h = fL,x j (28) I HI f(H> where f(H) is the function of H defined by equations (15) and (16) and HI is the height at Tl. More explicitly
(29)
124
G.
W.
PALTRIDGE
wheref,(H) andf’(H) are the functions of H defined by equations (15) and (16), respectively, or in the case of inclusion of soil moisture status by equations (23) and (24), respectively. These integrals are worked out explicitly in Appendix F. The resultant equation for PA as a function of Z is too complicated to differentiate analytically, so it has been evaluated numerically as a function of Z for various $,/A ratios (i.e. for various H,,,) and for various values of soil moisture status. See Fig. 4 for some examples.
FIG. 4. Examples of calculated average photosynthesis over the lifetime of the trees as a function of tree spacing 2. Solid lines are for cases where soil water potential assumed to be 22 m where average soil potential B=SOO. zero. Dashed curve is for case of H,,,,,= Relevant data: r0 = 1 m; no = 1; HI = 1 m; r,u= 2000; 0 = 45”; k (see equation (27)) = 10.0.
These calculations show that (i) there is indeed an optimum spacing to give maximum P,, (ii) this optimum spacing decreases with decreasing H,,,,,, (iii) with decreasing H,,,,, the maximum P, increases (a similar conclusion to the earlier example based on optimization of maximum instantaneous rate of photosynthesis), and (iv) with decreasing soil moisture both the maximum yield at the optimum spacing and the optimum spacingitself’decreases. 5. Tree Growth-The
Optimum Shape at Various Stages
The discussion in the previous section dealt with cylindrical “trees” which were constrained to a fixed radius r,,. There was, therefore, no need to make a detailed statement about leaf distribution within the individual trees-except to say that the leaf density was sufficiently great or was distributed in such a manner that light penetration through a tree at any distance from the trunk was negligible.
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We will now consider the case of cylindrical trees in a forest where the constraint on r0 is removed. This will be done in order to obtain an idea of the shape of trees in a forest if they were constrained to maximize their instantaneous rate of photosynthesis. In other words we will assume that, at any given stage of tree development, photosynthate can be disposed in the form of new leaves either at the top of the tree (thereby increasing H) or as a uniform distribution about the sides of the tree (thereby increasing r,); and that whichever of these alternatives gives the greatest increase in rate of photosynthesis will be the actual “choice” of the tree at that stage. The treatment must necessarily be very rough, and should be regarded primarily as the basis of a qualitative description. We assume that the leaf density of the trees is uniform and very high in order not to become involved in complicated discussion of leaf distribution. Thus if there is an amount dx of available photosynthate, it can be placed either (a) at the top of the tree to give an increase in tree volume d Vdefined by dV = nr:,dH
= x: dx,
(30) or (b) on the sides of the tree to give the same increase in volume defined by d V = 2m-,(H - 11,) dr, = k dx,
(31) where k here is a constant relating amount of photosynthate to the volume occupied by the leaves which it forms. Thus in case (a) the increase in rate of photosynthesis P will be c3P -=-dH
i3P
k
(32)
aH 7rri’
and in case (b) the increase in rate of photosynthesis P will be
ap dr,
ap dr, 2m,(H
k - h,)’
(33)
The problem is to evaluate the partial derivatives aP/aH and aP/&, at the various stages of tree growth. In the previous section it was assumed for simplicity that leaf potential was a function of height only. Here, since rO may vary, we must take radial variation of leaf potential into account. Thus we assume the trees to have radial branches such that the leaf potential at height 11and radius r is given by J/(h, r) = (E,zo+g)h+E,z,r, (34) which can be compared with equation (10). By following an analysis similar to that of equations (9) through (16) with the $(A) there replaced by the above @(h, r), we can obtain equations similar to equations (15) and (16) except for
126
G.
W.
PALTRIDGE
inclusion of a factor (1 -E,z,r,/~,). P,=n,tanB(I-E*)
Namely
[1-~(H-0’86,,nezz)(~-~~~, (35)
and P hhO
[H-&H’
(l-E,‘,o,K)].
(36)
(Actually the r. of the factor should be some constant large fraction of r. which depends on the leaf density distribution. The constant has been dropped here for simplicity.) Consider first equation (36), which gives the rate of photosynthesis of forest trees which are small enough and/or are sufficiently far apart for ho to be less than zero. The partial derivatives aP/aH and aP/ar, can be obtained, and therefore from equations (32) and (33) we can obtain the derivatives (aP/iYx)ro= C,,nStand (aP/axO)H=cons,. From these it can be shown (see Appendix G) that provided H < 2ti,i3~,
(37)
then addition of leaves to the top of the trees leads to a greater increase in rate of photosynthesis than does addition of leaves to the sides (i.e. when H < 21,b,/ So that if such trees 34 then WW),, = COnstis greater than (aP/ax),=,,,,,). were constrained to maximize instantaneous photosynthesis they would tend to grow upwards rather than expand horizontally. The physical basis for this result is that, of the leaves placed on the sides of the trees, only those which increase the vertical area of the tree as seen by the sun contribute to extra photosynthesis. Most of the leaves appear either in front of or behind leaves already in existence, and as far as total rate of photosynthesis of the tree is concerned they contribute nothing extra. This result, which effectively deals with the case of isolated trees since ho < 0, can be contrasted with that in the early examples of a single tree where interception of radiation was neglected. Consider now equation (35), which gives the rate of photosynthesis of forest trees which are large enough and/or sufficiently close for ho to be greater than zero. Again the partial derivatives aP/aH and aP/ar, can be obtained. Both ww,,=,,,,, and wax),= Constare negative in this case, showing that when ho > 0 addition of new leaves anywhere on the trees reduces the total rate of photosynthesis. Furthermore it can be shown from the expressions for these derivatives (see Appendix G) that provided ro < ; k
(H-ho), P
0
(38)
THE
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OF TREES
So that if wwH= constis less in absolute magnitude than (CP/~x),,=,,,,,. such trees were constrained to maximize instantaneous photosynthesis they would tend to expand horizontally rather than grow upwards. Ultimately, when r0 exceeded (3/2)[A(H-h,)/EPzo], they would again grow upwards rather than expand horizontally. Thus if we wished to “design” trees which maximized the rate of photosynthesis per unit area of ground and which were to be planted at the same time, we would perhaps “design” them to follow in broad outline the growth pattern of Fig. 5. They would tend to grow upwards at first until they reached D = 2(Epzo+ g) W-ho) Epzo
j*--
1 -0 ---+I 1
H = O-86Z2tane
FIG. 5. Diagrams of the shape, at various stages of growth, of a tree which was constrained to maximize instantaneous growth rate when it is one of a forest of similar trees.
a height where their bases were shaded from the sun (i.e. where h, = 0). They would then expand in radius keeping a constant height, with ho necessarily increasing and the depth of foliage H-he decreasing. When r0 reached a value of the order of 14 times H-h, [since in many cases (EPzO +g)/ EPzO would be not much greater than unity] the trees would again tend to grow upwards while maintaining a constant depth of foliage H-h,. Whether natural trees have developed along similar lines and for similar reasons is a matter for argument. 6. Conclusion
Two of the factors which influence the photosynthesis of leaves have been considered here. These are that the net rate of photosynthesis is some decreasing function of leaf water potential, and that the net rate of photosynthesis is some increasing function of light intensity. The possible effect of these factors on the evolutionary development of the broad-scale characteristics of trees
128
G.
W.
PALTRIDGE
have been illustrated by the analytical treatment of “extreme case” examples. For instance, by considering an isolated tree it was easily seen how considerations of leaf water potential (alone) might lead to the evolution of trees of hemispherical shape. It was shown how considerations of light interception (alone) might lead to the evolution of trees whose leaf density increased radially from the trunk. Both leaf potential and light interception have been included in an “extreme case” model of a forest. Dealing first with a forest of trees whose radii were constrained to be constant, it was shown how maximum rate of photosynthesis per unit area of ground might be achieved at a particular tree spacing determined by tree height; and that the maximum possible rate of photosynthesis would be achieved by the shortest trees at their optimum spacing. As far as the theory is concerned these “shortest trees” might be grass. Similar conclusions were reached when maximum average rate of photosynthesis (i.e. the average over the lifetime of the trees) was considered. It was shown how soil water availability might influence these results. Dryer soil reduces the maximum rate of photosynthesis at optimum tree spacing, and reduces the optimum tree spacing itself. By removing the restriction on tree radius, it was shown how the shape of trees (of high leaf density) in a forest might vary if they were to achieve maximum instantaneous rate of photosynthesis at all stages of growth. That is, they would tend to grow upwards until radiation interference from adjacenl trees became significant; they would then expand horizontally while developing a crown; and they would then resume an upward growth while maintaining a constant crown volume and dupe. In summary the work here provides the analytical framework of a much simplified model of “tree” or plant behaviour. This framework has many satisfying aspects and provides a number of results which are testable in principle, since the physical and physiological parameters are capable of measurement and the detail of the geometry (say) of a particular real situation can be handled easily by computer analysis. From the most general point of view the present attempt can be regarded as an example of the many possible starting-points for the design of an overall plant growth model. For instance, here two physiological parameters zO and y?Ohave been selected and the analysis has been built about them. There is no reason to suppose, however, that our choice of parameters is superior to another set which might be suggested by other authors. Such choices can be judged only by the success of the model in predicting the morphology of trees. In addition, although zO and $O arc in one sense over-simplified real-plant characteristics, in another they are not simple enough. The question
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still remains as to what have been the environmental controls on the development of their actual values. On the time scale of the evolution of plants it is unlikely that z0 and $,, could be regarded as independent variables. Thanks
are due to Dr David
Parkhurst
for his helpful criticism
and advice.
REFERENCES CO~TWOR, D. J. & CARTLEDGE, 0. (1970). J. uppl. Ecol. 7, 353. LOTKA, A. J. (1922). Proc. mtn. Acad. Sci. U.S.A. 8, 147. PALTRIDGE, G. W. (1970). Agric. Met. 7(2), 93. PARKHURST, D. & LOUCKS, 0. L. (1972). J. Ecol. 60(l). In press. PASSIOURA, J. B. & COWAN, I. R. (1968). Agric. Met. 5(2), 129. SLATYER, R. 0. (1967). Plant-Water Relationships. London: Academic Press. WATTS, D. G. & LOUCKS, 0. L. (1969). Tech. Pup. Inst. Envir. Stud. Madison,
Wisconsin:
Univ. of Wisconsin.
Appendix A
We require to show that Parkhurst’s criterion of maximization of the ratio R( = P/E) is equivalent (as far as plant development in terms of the present model is concerned) either to maximization of photosynthesis P or minimization of evaporation E. A more realistic representation of photosynthesis than equation (1) would take into account specifically mesophyll resistance r, and respiration of carbon dioxide. For instance we could write
no ’ = rm+rs~o/~~o-II/>
-noK
(Al)
where here the first term on the right-hand side represents gross photosynthesis and the second term represents respiration. There is a specific constant rs representing stomata1 resistance when J/ = 0 (fully open stomata). The no here is a constant such that when $ = 0 the first term on the right-hand side represents gross photosynthesis n,/(r,+r,). K is a constant such that n,K is the respiration rate per unit volume-on the assumption that respiration is constant and independent of $. $. here represents the potential at which the stomata are fully closed and no further uptake of CO2 is possible. Evaporation E could be written -___ bno
T.“.
E = rs+o/($o-II/)’
(A3 9
130
G.
W.
PALTRIDGE
since r,,, does not appear in the water pathway from the sub-stomata1surfaces to the atmosphere; b is a constant such that bn,/r, is the evaporation rate per unit volume in a specific environment when the stomates are fully open. If a plant were obeying maximization of photosynthesis (criterion I say) it would place its new growth in a position of minimum $. If it were obeying minimization of evaporation (criterion II say) it would place its new growth in the position of maximum $. From the above equations
and by differentiation with respect to 4’/
If dR/d$ is positive, then the plant (if it has a choice) will maximize R by placing its new growth in a position of maximum $--equivalent therefore to criterion II. If dR/d$ is negative, the reverse will be true and the placement of new material will be equivalent to criterion I. Now from equation (A4), dR/d$ is positive when the term in square brackets is positive, that is, when (A51 This condition can bc put in a more understandable format by multiplying through by n, and obtaining the condition (by referring to the definitions in equation (Al)) respiration < (gross photosynthesis)
(A5a)
or that respiration < -.-I’, grossphotosynthesis r,,, + rs,’
(A5b)
where rst is the stomata1resistancer, $0/($0 - $). This condition is likely to be satisfied for smaller (younger) trees where the leaf water potentials will be small-and where therefore stomata1 resistance will be small and gross photosynthesis will be large. Thus in the early stagesof growth, maximization of R is likely to be equivalent to the unstable criterion II. In the latter growth stagesmaximization of R may be equivalent to the “stable” condition I.
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131
Appendix B Considering a tree for which restriction of its total volume is a desirable characteristic, it seems reasonable that a further desirable characteristic should be as follows. Considering a horizontal section of a tree (presumably a circle on the assumption of radial symmetry), whatever value of absorption or “obstruction” coefficient is presented by the random leaves along a diumeter this same coefficient should be presented to a solar beam along any chord. This assumes that the obstruction by leaves along the diameter (say) is some optimum which gives maximum absorption of photosynthetically active wavelengths for minimum “wastage” of the leaves as far as their contribution to photosynthesis is concerned. [If a certain number of leaves absorb (say) 99 “/ of the radiation, doubling or trebling the number of leaves to gain the extra percent would obviously be inefficient.] We are considering here only the case of horizontal beams of light. The same principles would apply to any section of the tree parallel to the rays of the sun, but the mathematics becomes very awkward. Thus the question arises as to whether it is possible to have a distribution of leaf density n as a function only of radial distance r from the centre of the circular section such that the absorption coefficient is constant along any chord. It can be shown quite easily that there is no such distribution except for the case of infinite density at the outside radius R and zero density elsewhere. However, it is interesting that there is a distribution in two-dimensions (i.e. n(s, y) where I is the axis in the direction of the sun) which satisfies the criterion for all chords parallel to the solar beam. This distribution is derived in Appendix C and is
n(x, y) = -$--
R -y” where k is a constant depending on the actual value of the absorption coefficient, and the origin is at the centre of the circle. Unfortunately IZ --f co at the point (X = 0, .t’ = R), but apart from this n increases smoothly with both s and 1’. Bearing in mind that in general the azimuth angle of the sun varies throughout a day by up to 180”, it is reasonable that a “close-to” optimum n(rj distribution might be one which is some form of average between n(x) and n(y). It is pointed out in Appendix D that “mathematically obvious” averages require still an infinite density at r = R, so that for practical purposes it is perhaps simplest to select a radial distribution according to equation (AIO) at some “average angle”. The actual choice of such an angle would be a major investigation in itself, but here we are intending only to show that in real trees
132
G.
W.
PALTRIDGC
it is reasonable to expect that leaf density should increase radially from the trunk. If, for instance the “average angle” were 45”. then n(r)
=
@lKr’ _____
(A@
R2 -35.2
Appendix C Referring to Fig. 6 which considers one quadrant of the circle, let n(r) be the density of leavesat radius r. We require a density distribution n(x-, y) such that JR-s--q2 JU2--)J N(y) = j’ n(r) dx = 1 ,I(\/? +y’) dx = constant. (A71 0
0
Now in terms of the variable Y (A8) so that a suitable n(r) distribution such that N(y) is constant is k&.2+ n(r)
=
(A9)
---z---j
R -J’or in terms of x and y
,--- kJx2+y2-y2 n(x, y) = R2-jj2
solar _-~
FIG.
beam.-r
6. Geometry
kX
(AlO)
R’- \a”
I
associated
with
leaf
density
distribution
analysis.
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Appendix D On “average” n(r) distribution (AlO) might be
based on the n(s, y) distribution
n(r) = f $ Re2
of equation
(All)
de.
Replacing .Y with r cos 0 and y with r sin 8, and with two subsequent changes of variable
which goes to infinity
at r = R.
Appendix E From equation (22), the total evaporation is given by P,=
j2n,ro ho = 2n, r.
I-$(1+kP,)-$h 0
I[
l-
which on re-arrangement 2noro P, =
P, from a single tree in a forest
0
$-(I+&) 0
I
dh ,
1
(A131
becomes H-h,-
$ (H-h,)-
$
(II”-hi)]
(A14)
1+ 2n, z. kB(H - h,),$::
Thus for the case when ho ,< 0 (equivalent
to ho = 0)
2nor0 1-E H-IAfp p = O*86Z2 [( *o > 211/o I A 1+2n,r,kBH/~, ~’
(A15) or (24)
For the case when ho > 0, since O-86 tan BZ2 h,=H---
)
2r 0
(A16)
134
G.
one can obtain H-h, H2-h; and by substitution
W.
PALTRIDGE
and also =
(At7)
in the above equation for P, and division by 0.862 ’
(A18) or (23)
Appendix F We require to evaluate the expression (A19) where fi and f2 are the functions of H defined by equations (15) and (16), respectively, for the case where soil water status is neglected, and are functions defined by equations (23) and (24), respectively, for the case where soil water status is included. Considering the first case, the denominator (denom) is of the form denom =
(A2’3
where a = A/$o and b = 1 + (A/$,)(0*86 tan f3Z 2/4ro). When the integrals are evaluated
and pA=zEp
H denom
(Il/olA) denom’
C.422)
Considering the second case denom = ‘z2r n l ($!s2)
dH+ ;-~inme “!I’ 0
(& -tifi)
dH,
(A23)
THE
SHAPE
OF
135
TREES
where here b = (1 - Bjtj,) + (A/11/,)(0*86 tan 8Z2/4r,), c = 2n, r. i&/1+5,, d = 1 + n,kB x 0.86 tan OZz/$O. e = 1 -B/$,. When the integrals are evaluated
and again
Appendix G For the case of h,, < 0 (effectively that of isolated trees) from equation (36) (A251 and using equation (30) to obtain dH/dx, (c!),o,,“,,(z!E)=~ Similarly
3?p-;!.
(A26)
from equation (36) dP ~ = 2n,Ho”r,
4n,E,z,Hr,
noAH
$0
$0
’
(A27)
and using equation (31) to obtain dr,/dx, WW Thus if 4n,AH n, n,AH 2no ~ - > r, - @,<). ro V. r.
(~29)
136
G.
W.
PALTRIDGE
then in order to achieve maximum rate of photosynthesis the trees to grow upwards. This condition simplifies to H < 2$,/3A. For the case where ho > 0 (where signjficant), from equation (35)
light obstruction
one would expect (A30) by adjacent trees is
-ap = _ XA ~-aH $0’ where X = no tan 0, and therefore
(A31)
(~),,...,,(=%3= Similarly
from equation
ap -=---
-ii&.
(A32)
(35) XE, z.
O-86 tan 9Z2 4r,2*, ’
-XAx
$0
(A33)
and therefore - XE,zo XA O-86 tan OZ’ tj0271rO(H - h,) - x 4r~27rrO(H - h, 1 -XE,zo since H-ho
_ ---XA 4nr&b,’
tA34)
I XE,zo ~~~ + -At! XA nr&/io < 2nr,t,b,(H - ho) 47-L&,’
tA35)
= $027cro(H - ho) = O-86 tan f3Z2/2ro. Thus if
then in order to achieve minimum degradation in the rate of photosynthesis the trees should grow upwards. This condition simplifies to r. > iEL
(H-h,).
(A36)
020
APPENDIX
H
List of Symbols A B D E
EP H
Epzo +g
seeequation (2 1) soil diffusivity (a function of iz and 0,) actual evaporation rate per unit volume containing leaves potential evaporation rate per unit volume containing leaves overall tree height
THE
H max HI L P PA p.4
PT QT V x Z -Z .4 h ho flo
J-0 =o
SHAPE
OF
TREES
137
ultimate maximum tree height initial tree height seeequation (20) net rate of photosynthesisper unit volume containing leaves net rate of photosynthesisper unit area of ground net rate of photosynthesisper unit area of ground averaged over tree lifetime total net rate of photosynthesisof singlerree total evaporation rate of a singletree seeequation (20) seeequation (20) averagecentre-to-centrehorizontal tree spacing effective centre-to-centrehorizontal tree spacing gravitational potential per unit height height above ground surface height above which leavesof a tree have an unobstructedview of sun net rate of photosynthesisper unit volume containing leaveswhen leaf water potential is zero distancealong horizontal radius of tree horizontal radius of tree proportionality factor relating leaf potential to E,, and length of trunk plus limb averagesolar elevation averagesoil moisturecontent soil moisture content at root surface leaf water potential leaf water potential at which net rate of photosynthesisis zero water potential at the root surface