THEORETICAL
POPULATION
25, 78-105 (1984)
BIOLOGY
Shoot/Root
Balance
of a System with
of Plants: Optimal Growth
Many Vegetative
Organs
YOH IWASA AND JONATHAN ROUGHGARDEN
Department
of Biological Sciences, Stanford
Stanford, California 94305
University,
Received January 7, 1983
The optimal growth schedule of a plant with two vegetative parts is studied to
investigate the balance between shoot and root. An intuitive justification
of
optimization procedures used in Pontryagin’s maximum principle is obtained by
defining the marginal values of shoot size, root size, and reproductive activity at
various times of the season and deriving their differential equations and terminal
conditions. The optimal growth pattern which maximizes the total reproductive
activity during the season is composed of the convergence of a plant’s shape to a
balanced growth path, followed by simultaneous growth of shoot and root
(balanced growth), ending with the reproductive growth. Along the balanced growth
path, a plant has a root/shoot ratio which maximizes the daily net photosynthesis
for a given total biomass. The model also shows a simultaneous stop of shoot and
root growth when the reproduction begins, the dependence of root/shoot ratio on
age, water and light availability,
etc., the convergence of a plant’s shape to the
balanced growth after pruning or an environmental change.
The growth pattern of plants is much more flexible than that of animals.
Relative sizes of various organs greatly change with environmental
conditions. For example, the ratio of root to shoot biomass increases when
nutrient or water availability in soil is low, but it decreases when light at the
canopy is deficient (Aung, 1974; Chapin, 1980).
In this paper we investigate the optimal growth schedule of a system with
many vegetative organs. By defining the marginal value of each part of a
plant at various time point during the season and deriving their differential
of
equations and terminal conditions, we give an intuitive justification
optimization procedures given by Pontryagin’s maximal principle. The
optimal growth pattern, which maximizes total reproductive activity, is
composed of (1) rapid convergence to a balanced growth path, (2) followed
by simultaneous growth of shoot and root (balanced growth), and (3) ending
with reproductive growth only. We discuss the simultaneous stop of shoot
and root growth, the regrowth pattern after pruning or experimental
18
0040.5809/84
Copyright
All rights
$3.00
?Z 1984 by Academic
Press, Inc.
of reproduction
in any form reserved.
SHOOT/ROOTBALANCE
OF PLANTS
79
manipulation, the dependenceof root/shoot ratio on parameters such as age,
water availability, and light level, and the growth behavior if environmental
parameter changes with time.
OPTIMAL GROWTH MODEL
The model for the growth of a plant we study is that
dx,
-=u,(~)f(X,,X*),
dt
dx,
~ dt = dt)
(14
3X,)3
(lb)
dR
- dt = UOWf(Xl 3X,)3
(lc)
ml
where X,(f) is the size of shoot (aerial portion of a plant), X,(t) is the size of
root (subterranean portion), and R(t) is the accumulated amount of
reproductive activity invested until t, including maintenance and construction
of flowers and fruits.
A plant’s production rate increases with the amount of leaves, but also
depends on the function of other organs. For example, stems improve the
local light condition at the canopy by separating leaves from each other or
by raising the canopy, and roots take in water and nutrients from the soil.
The rate of water uptake depends both on the availability of water in the soil
and on the size of the root system. Since water uptake rate affects the size of
stomata1 openings which in turn control the availability of CO, used in
photosynthesis, an investment in the roots improves the photosynthetic rate.
We therefore assume that the daily net photosynthesis f(X,, X,) increases
both with the shoot size and with the root size:
(2)
The photosynthetic material produced is allocated to shoot growth, root
growth, and reproductive activity with fractions u,(t), u*(t), and u,(t),
respectively. These change with time satisfying the constraints
u, >o,
u2 zo,
uo z 0,
and
u, + 24,+ u. = 1.
(3)
In the preceding equations, the size of the organs constructed,
photosynthesis, and reproductive investment are all measured in units of
carbon. Reutilization of the material used for the construction of vegetative
80
IWASAAND ROUGHGARDEN
organs is neglected in Eq. (1) for simplicity. This simplifying assumption
seems to be more plausible concerning the balance of carbon than
concerning the balance of nitrogen, a considerable fraction of which is
known to be reutilized during reproductive growth (Mooney, 1972).
The initial sizes of shoot and root, X,(O) = x,,,, and X,(O) = xZO, are given
by the seed size and germination processes. By definition, R(0) = 0 holds.
For simplicity, an annual plant is considered here; the length of period
suitable for plant growth (denoted by T) is fixed by environmental
constraints.
The optimal allocation schedule (u:(t), u;(t), u:(t); 0 < t < T} is the one
that maximizes the total investment in reproductive activity during the
growth period:
R(T) + maximum.
PONTRYAGIN'S MAXIMUM PRINCIPLE-DYNAMICS OF MARGINAL VALUES
The dynamic optimization problem presented above is solved by a familiar
technique of control theory, Pontryagin’s maximal principle (Pontryagin et
al., 1962). We define the Hamiltonian as
HE
i
i=O
ni(c>
ui(f)f(xl(t)
x*(c>>T
(4)
by using three costate variables A,(t), A,(l), and A,(t), each corresponding to
a state variable, X,, X,, and R, respectively. The three costate variables
satisfy the differential equations
aH
+--
af
\‘ Ail+ -,
ax,
T
ax,
;+g=o,
(5b)
(5c)
and the terminal conditions
A,(T) = l,(T)
= 0,
and
L,(T) = 1.
(6)
The maximum principle claims that optimal allocation u*(t) at every time
point should be chosen to maximize the Hamiltonian given state variables
SHOOT/ROOT
BALANCE
81
OF PLANTS
(X,, X,, and R) and costate variables (A,, A,, and A,). With an additional
necessary condition for singular control subarc, we can thus determine the
optimal allocation schedule.
Frequently these equations are used formally, only as a mathematical
device by which we can routinely extract an optimal solution. However, it
seems worthwhile to offer a biologically
intuitive justification
for the
maximum principle in order to increase its usefulness.
In the following, we will show that the notion of marginal value of each
organ as a function of time and the dynamics of these marginal values are
keys to understanding the maximum principle.
Consider the marginal value of shoot and root sizes at various time points
in the following way: Suppose that the size of shoot is perturbed from a
growth path by a small amount, keeping other organs the same size (Fig. 1).
This small increase in shoot size will result in a slightly larger photosynthetic
rate after that time point, and the surplus product will be invested to
reproductive activity or in root and shoot growth, which will again enhance
the photosynthetic rate. Eventually, reproductive success will increase. The
increase in total reproductive activity caused by unit perturbation of shoot
size at time t may be called the marginal value of shoot size at time t.
Denote this by A,(t). In a similar manner, we can define the marginal value
of root size L,(t), and that of reproductive activity A,,(t).
Note that the marginal value of shoot or root depends on time t. Since an
increase in the vegetative parts of the plant contributes to the objective
function only through a larger photosynthetic rate between that time and the
end of the season T, a perturbation of shoot size early in the season affects
the reproductive success much more than a perturbation later in the season.
The marginal value of shoot size, A,(t), therefore decreases with time. Since
0
t
1
FIG. 1. The marginal value of shoot size at t, A(t). is the increase in total reproductive
activity R(T) caused by unit perturbation of the shoot size X, at time t.
82
IWASA AND ROUGHGARDEN
the carbon produced by photosynthesis at time t is invested in shoot growth,
root growth, and reproductive activity with the fractions u,(t), u,(t), and
u,(t), respectively, and the marginal values of investments to these three
targets are Al(t), A,(t), and A,(t), then the unit amount of carbon contributes
to the objective function by
L(t) = x q(t) A,(t),
i=o
(7)
which may be called the marginal value of carbon available at t.
Differential equations satisfied by these marginal values are derived as
follows: Suppose that small amount of carbon AC is invested into shoot
growth at time t, so that shoot size increases from X, to X, + AC. Because of
this perturbation, total reproductive activity increases by k,(t) AC, according
to the definition of l,(t).
The carbon invested to vegetative organs
contributes to the total reproduction
only through the increase of
photosynthetic rate. By separating this contribution into two parts; the
increase of photosynthesis during a short period At and the increase after
that period, we have a relationship:
&WAC=
[f(X,
+ Ac,X,)-f(X,,X,)]
AtI
t n,(t + At)Ac.
(8)
The first term in the rhs is the value of the additional photosynthetic product
during At caused by the increment AC of Xi, taking into account that the
marginal value of photosynthetic product is A(t). The second term is the
ultimate effect of the increment of X, by AC at t + At. Rearrange terms and
divide both sides by AC At; Eq. (8) then becomes
&(t + At) -k,(x)
At
= _ .0x,
f AC, X*) - fW, 2*2> qt>
AC
By taking a limit when At + 0 and AC + 0, we have
(9)
which is identical to the equation for a costate variable Eq. (5a) because of
Eq. (7). In a similar manner, we can derive a differential equation for l,(t),
the marginal value of root size, which is identical to Eq. (Sb). Both equations
indicate how the marginal values of vegetative organs decrease with time.
The marginal values of vegetative organs at the end of the season are zero
because the increase of their size at the end of the season will not contribute
to reproduction at all:
k,(T) = i,(T)
= 0.
(10)
SHOOT/ROOT
BALANCE
OF PLANTS
83
Since we choose the total sum of reproductive activity during the season
as the objective function, the marginal value of reproduction is unity
regardless of the time point at which it is invested,
k,(t) = 1
for all t
(11)
which is also a result of differential equation (5~) and the terminal condition
Eq. (6) for costate variable Lo.
Thus the dynamics of marginal values are identical to those of the costate
variables in Pontryagin’s maximal principle.
Once the marginal values of shoot size, and reproductive activity are
known at each time of the season, then the optimal allocation is to invest the
photosynthate only into the activity with the highest marginal value.
q(t) > 0
only if
Z+(t) = 0
if
Ii(t) = max[l,(t),
l*(t), A,(t)],
ni(t) < max[~,(t>, &(t>, Ao(t>l,
(124
(12b)
because it maximizes the contribution to the total reproductive activity made
by the photosynthate. For example, if the marginal value of shoot size is
larger than both that of root and that of reproduction, then a plant should
invest only to shoot growth. If the marginal value of root size is higher than
the other two, only roots should grow. Shoot and root can simultaneously
grow in the optimal processes, but only when the marginal values of shoot
and root sizes are exactly equal.
This intuitive argument is equivalent to the formal procedures by
Pontryagin’s maximum principle, which requires the maximization of H
defined as Eq. (4). Sincef(X, , X,) is positive, it becomes
max t
l,(t) q(t)
under
u,(t) > 0,
i=O
which is
Since
f(X,,X,)
material
5 uj(t) = 1,
.i=O
evidently realized by Eq. (12).
the amount of photosynthetic material produced during At is
At, the increase in ultimate reproductive success caused by this
is
n(t)./-@, , X,) At = ‘7 uJ,.f(X,
, X,) At
(13)
,TO
which is equal to the Hamiltonian, Eq. (4), multiplied by At. Therefore the
maximization of the Hamiltonian, according to the maximum principle, can
be interpreted as the maximization of the increment of the total reproductive
activity caused by photosynthetic products at each time, given the state
variables and their marginal values.
84
IWASAANDROUGHGARDEN
OPTIMAL GROWTH PATH
An optimal growth schedule which satisfies all the conditions required by
Pontryagin’s maximal principle is constructed as follows, by considering a
vegetative growth path and then determining when the reproductive growth
begins.
During the vegetative growth period, a balanced growth path represented
by a curve on a X,X, plane,
plays an important role. If the initial sizes of shoot and root (xIO, xZO) do not
satisfy the above relation, the first phase of optimal growth is the
convergence to the balanced growth path; i.e., if the root/shoot ratio is
higher than that indicated by the balanced growth, all photosynthetic
material is invested to shoot growth and the root size is constant. If it is
lower, root only grows. This is expressed as
u,(t) = 1,
u,(t) = 0,
U*(t) = 0,
nZ(t)=l,
if
+
> +,
1
if
2
&<+.
I
2
Once the state point (X,(t),X,(t))
representing a plant shape reaches the
balanced growth curve (at t = ti), shoot and root grow simultaneously so
that the state point moves along the balanced growth curve, which we
assume has a positive slope.
For the vegetative growth path defined in this way, a switching time point
t, at which the reproductive growth begins is calculated by an implicit
equation:
max
&(IJ,&(fJ].
(T-ft,)=
1.
[
The optimal growth path is vegetative growth before t, with no investment in
reproduction
(u, = 0) and reproductive
growth after t,, when all
photosynthetic material is used for the reproductive activity (ZQ,= 1,
u, =u*=O).
If t, <t,, a whole season is divided into three phases: the
convergence of a plant’s shape to a balanced growth path, the simultaneous
growth of shoot and root (balanced growth), and then reproductive growth,
as illustrated in Fig. 2. If 0 < t, < t,, the reproductive growth begins before a
state point reaches the balanced growth curve. If lhs of Eq. (16) is lower than
SHOOT/ROOT
BALANCE
OF PLANTS
85
unity, even at t = 0, the reproductive growth occurs during a whole period;
neither vegetative part grows.
In Appendix A we give a proof that the growth path constructed in this
way satisfies the maximum principle, but the following intuitive argument
may be helpful:
Toward the end of the season, additional growth of a vegetative organ
gives only a small benefit because of the shortage of the photosynthetic
a
,
b
FIG. 2. Optimal growth pattern: (a)
photosynthesis is given by Eq. (19) with
and W = 1. initial size are x1,, = I, xzO =
The season is devided into three phases;
state variables and (b) costate variables. The daily
parameters: a, = 2, a2 = 60, b, = 0.5, b, = 2, L = 1.
2.5, and the length of the growing season is T= 153.
I, = 17, t, = 100.
86
IWASA AND ROUGHGARDEN
period. The direct investment to reproductive activity is thus better than an
investment in vegetative growth. This is expressed-&(t) is greater than both
A,(t) and L,(t). During the last period, all the photosynthetic material is
invested only into reproductive activity.
Before the switching time t,, however, the marginal value of vegetative
growth is larger than that of reproductive activity. In a case illustrated in
Fig. 2, material is invested into both shoot and root simultaneously between
t, and t,. During that period the marginal values of shoot and root sizes
must be the same; hence A1= 1, and d,l,/dt = d&/d& from which relation
aflaX, = aflaX, is derived using Eqs. (5a) and (5b).
Since the photosynthetic rate is a function of shoot size (X,) and root size
(X2), we can visualize it on a X,X, plane by a contour map off(X, ,X,), in
which each curve connects points with the same daily photosynthesis, as
illustrated in Fig. 3. A balanced growth path is expressed by a curve
aflax, = afax, on that plane. Consider a straight line AB on which the sum
of shoot and root biomass is constant, Xi +X, = const. Point A on the
horizontal axis indicates a plant without a root; point B on the other axis a
plant without a shoot. It seems plausible to assume that photosynthesis is
f=3.63
r
FIG. 3. Optimal growth illustrated on a X,X, plane. Parameters are the same as in
Fig. 2. The state point (Xl(t), X,(t)) first converges to the balanced growth path and then
moves along it. A contour line of f(X,, X,) is tangent to a line X, +X, = const on the
balanced growth curve.
SHOOT/ROOTBALANCE
OF PLANTS
87
highest at a point with an intermediate ratio of root to shoot along the line
plane are concave,
At every point on the balanced growth path, a photosynthetic contour is
tangent to a straight line with slope minus one; hence f(X,, X,) is at its
extremum along the line. The second-order condition indicating that
f(Xi, X,) is a local maximum at a point on the balanced growth path,
instead of a minimum or an inflection, is
AB. That is why contours of f(X, , X,) on a X,X,
--a’f 2 air -a2f< 0.
ax; ax,ax, + ax;
The condition for the local maximality
of a singular subarc is
(18)
(Kelley et al., 1967). By putting ui = u and u2 = 1 -u, we can prove the
equivalence between (17) and (18). Hence, along the balanced growth path,
the daily photosynthesis is maximized for a given total biomass, and
photosynthetic product is allocated to shoot and root so as to keep a
balanced shape through time. The balanced growth path is analogous to a
“long-run expansion path” in the theory of the firm in microeconomics
(Intriligator, 1971).
The state point indicating a plant’s shape may not be on the curve because
of the initial shape limited by seed size, because of the change in environmental conditions, or because of the experimental removal of leaves or roots.
The marginal value is greater for the part which is smaller in relative size
than that indicated by the balanced growth. In Fig. 2, for example, A, is
larger than A,. All the material is allocated to one part then, so that the
plant’s shape converges to the balanced growth path as rapidly as possible.
SHOOT/ROOT BALANCE
If we know the daily net photosynthesis as a function of root and shoot
sizes, f(X, ,X,), the growth pattern will be predicted by the model assuming
that the plant grows optimally, and can be tested by comparing the
prediction with the growth behavior of a real system. Even if the precise
form of the function f(X,, X2) is not known, the following results of the
model may be checked experimentally:
(1) Once both root and shoot begin to grow simultaneously, their
balanced growth continues to the end of the vegetative growth, and both stop
growing at the same time, when reproductive growth begins. This behavior
88
IWASA AND ROUGHGARDEN
can be checked experimentally. We note, however, that this conclusion
depends on the assumption of a constant environment. If environmental
parameters change during the season, as shown later, it is possible that two
vegetative organs stop growing at different time points.
(2) By preparing plants having various sizes of root and shoot, and by
letting them grow under a common environmental condition, we can measure
the dependency of photosynthetic rate on shoot and root sizes. The theory
predicts the relation af/aX, = aflaX, during the balanced growth-the
sensitivity of net photosynthesis on shoot and root sizes should be the same.
(3) It has been reported that pruning, or an experimental removal of
some leaves, results in a stopping of root growth and a rapid recovering of
the leaves, while pruning of the roots caused the reverse to occur (Chandler,
1919; Keeble et al., 1930; Brouwer, 1963; Hart1 et al., 1964; Maggs, 1964,
1965; Aung and Kelly, 1966). The model predicts that, after pruning the
leaves, root stops growing until the shoot recovers to the same size as before
the pruning. The simultaneous growth of shoot and root restarts when the
plant returns to the original shape. Over- or undercompensation would not
be observed, according to the model.
(4) Root/shoot balance is known to change with the plant age, the
nutrient and water availability in soil, and the light intensity on the canopy
(reviewed by Mooney, 1972; Aung, 1974; and Russell, 1977). To consider
these results, let us investigate the optimal growth when the photosynthetic
rate depends not only on the shoot and root sizes but also on the resource
availability
in the environment. For example, assume that the daily
photosynthesis depends on the light level at canopy L, and water availability
in soil W,,i,, so that it is expressed as f(X,, X,, L, W,,,,). Given this
function, we can predict the optimal growth behavior of a plant for various
environmental parameters. The precise form of this function is not known in
the present situation. To illustrate that the optimal growth model can show
such a growth behavior as reported in the literature, we here study a
hypothetical functional form.
Example. Assume that
(19)
The photosynthetic rate is an increasing function of four variables, and
saturates to a constant when one variable tends to infinity with others fixed.
This form may be interpreted by saying that photosynthesis is composed of
two limiting processes; one process related to light level and shoot size, and
the other process related to water availability and root size.
SHOOT/ROOT
BALANCE
OF PLANTS
89
The balanced growth path calculated from Eq. (19) is
(20)
therefore
(root size) ci
light level
water availability
Il(b>t
I)
Ishoot size](bI+ I)!(bzt 11,
I
Russell (1977) reported that a logarithmic linear relationship, as in
Eq. (20), is usually found between the shoot and root sizes during vegetative
growth under a constant environment (e.g., Pearsall, 1927; Troughton, 1956;
1960). Second, root/shoot ratio of seedlings decreases with age (Le Clerc
and Breazeale, 19 11; Pearsall, 1923). If we choose the parameters so that
b, > b,, the balanced growth path of the above system shows such a dependency, because X,/X, decreases with total size. Third, root/shoot ratio is
known to be large when light availability is low (Crist and Stout, 1929;
Sekioka, 1962) or when water in the soil is abundant (Shank, 1945; Yanada
and Karimata 1969; Struik and Bray, 1970). This is also a result of the
optimal growth system above.
(5) So far a constant environment has been assumed. But resource
availability may greatly change with time in natural environment, and can be
changed by experimental manipulation. Assume that plants allocate material
as if the environmental resource availability remains the same from that time
t until the end of the season T. Then we can predict the following growth
behavior in a changing environment (see Fig. 4 for illustration).
Suppose that during the first part of a season a plant grows in an
environment with high water availability and low light level. The state point
indicating a plant’s shape first converges to the balanced growth path
corresponding to this environment, and afterwards moves along the balanced
growth path (A + B + C). Then suppose that the environment suddenly
changes so that water availability becomes low and light becomes highly
available. The plant’s shape is no longer balanced in this changed
environment. The balanced growth now requires a larger root to shoot ratio.
In this case the plant will initially allocate carbon only to root growth. Shoot
growth will stop (C-D).
Only after the state point reaches a new
balanced growth curve can both root and shoot again grow simultaneously
(D + E). Suppose that the environment changes one more time, returning to
the previous situation. Then only the shoot will grow (E + F). At some time
point reproductive growth begins, and the shoot stops growing as well. This
example shows that it is possible that the growth of root and shoot may stop
at different time points during the season if environmental parameters
change.
90
IWASA AND ROUGHGARDEN
100,
OO
!
50
l(
Xl
FIG. 4. Growth curve when environmental parameter changes; L = I and W= 1, for
A + C and E + F; L = 1.5 and W= 0.5, for C--t E. Other parameters are the same as in
Fig. 2.
(6) Note that not every function that increases with shoot and root
sizes, and light and water availabilities shows an optimal growth behavior as
described above. For example, consider a function
The dependence of the photosynthetic rate on each variable here is similar to
that in the previous equation, but the combination between variables is
different. In this case, the balanced growth path is
(22)
which is independent of environmental parameters, L and WSOi,. Therefore
the way the variables interact is essential to explaining the root/shoot dependency on environmental parameters. In other words, we can restrict possible
functional form of photosynthesis if we know the phenology of root/shoot
balance.
91
SHOOT/ROOT BALANCE OF PLANTS
MARGINAL
VALUE OF CARBON
So far the marginal value of carbon available at various times, A(t), was
used as a tool for calculating the optimal growth path, but it is important for
its own sake. For example, consider the cost-benefit analysis of the
production of secondary compounds used to defend the plant from
herbivores. The benefit is to save the leaf biomass grazed in the future, but
the cost is to produce the substance. Thus we need to compare carbon gain
and carbon investment at different times of the season.
In general, the marginal value L(t) is calculated by differential equations
for costate variables, Eqs. (5). As shown in Appendix B, we have
A(f>=exp
[“max
[ ‘f
&(s),-$&(s))
c
1
&J
(0 < t < t2),
(23a)
2
= 1
(t2 < t < Z-), (23b)
where aflaX, and af/~?X,, which are the same value during a balanced
growth path, indicate the additional increase of photosynthetic rate caused
by a unit increase in the size of the vegetative organs. Therefore the integral
in Eq. (23a) is the additional photosynthate produced from that time until
the end of vegetative growth.
The cumulative carbon gain, the product of photosynthetic rate and the
lifetime of the leaf, has been used to estimate the benefit of leaves (Mooney
and Gulmon, 1982). The value of a certain amount of carbon available at
some time point seems to be given as the cumulative carbon gain of the
leaves which can be made from that carbon. This procedure is equivalent to
the calculation of the integral in Eq. (23).
Nevertheless, knowing the amount of photosynthetic material produced by
the leaves equivalent to a certain investment is not sufficient information for
estimating the “value” of the investment because the photosynthetic material
may be used later for making new leaves or roots, resulting in a larger future
gain. If we take this multiplicative nature of vegetative growth processes into
account, we need an exponential symbol in Eq. (23a).
Fortunately, under a constant environment in which the photosynthetic
rate is a function of shoot and root sizes only, L(t) is easy to measure
experimentally, because it is inversely proportional to the photosynthetic rate
of a whole plant
1
n(t) cc f(xlw,
x20>>
as proved in Appendix B. We thus can calculate the marginal value of
carbon at various times in the season without knowing the precise form of
the photosynthetic rate function.
92
IWASA AND ROUGHGARDEN
OPTIMAL GROWTH IN A PREDICTABLY CHANGING ENVIRONMENT
In a previous section we discussed how a plant should grow optimally if
the environment suddenly changes during the growing season, assuming that
the plant copes with the environmental change only after it occurs. This
assumption holds if the sudden change is not predictable to the plant in
advance, as in experimental manipulations. In contrast, some environmental
changes are predictable, such as seasonal changes and some nonseasonal
changes which can be “forseen” by the plant using environmental cues. In
such a case, a plant might prepare its growth schedule much before the
actual change of the environment. The optimal growth pattern in the same
changing environment thus may depend on whether the environmental
change is predictable or not.
To illustate the difference, consider the case in which environmental
parameters suddenly change at time t = t, during the vegetative growth
period, and assume that the plant grows optimally taking the environmental
change into account from the beginning. Let f,(X, ,X,) be the daily net
photosynthesis for t < t,, and &,(X,, X,) be that for t > t,. The balanced
growth path in each period is
avia x,
-=ax, ax,
?fi 6
-=ax, ax,
(t < ts>,
(254
(t > 0
Since the two balanced growth paths are different, there must be a transition
period in which only a single vegetative part grows. For example, suppose
that the balanced growth period after ts, Eq. (25b), has a larger root/shoot
ratio than that before t,, Eq. (25a), because of a lower availability of soil
moisture. Then we can show that the following growth schedule satisfies the
conditions for the optimal growth path: (1) First, the plant follows the
balanced growth indicated by Eq. (25a). (2) At a time t,, which is before the
environmental change t,, the unbalanced growth begins, in which the root X,
solely grows with the shoot size X, stopped. (3) A time tz, which is after fs,
the plant shape reaches a new balanced growth path indicated by Eq. (25b).
Figure 5 illustrates an example of the optimal growth path.
If we choose t,, the time at which the unbalanced growth begins, we can
calculate t, when the state point reaches the new balanced growth path.
Hence the degree of freedom of such a growth schedule is 1. Since the
optimally growing plant should invest the photosynthetic material only to the
part with the larger marginal value, we have the relations
t < t, and t>t,,
(26a)
n,(t) = 4(t),
A,(t) < Mt),
t, < t < t,,
Wb)
SHOOT/ROOT
BALANCE
93
OF PLANTS
b
IlO-
IOO-
go-
80-
70
50
100
‘s
150
200
t
FIG. 5. The optimal growth path when the environment predictably changes. (a) The
shoot size (X,), the root size (X,), and (b) the marginal values of two vegetaive organs are
illustrated. The daily net photosynthesis is assumed as Eq. (19X with the parameters: a, = 2.0,
a2 = 120.0, b, = 0.5, b, = 2.0, L = 1.0. The water availability
W suddenly decreases at time
t,=12.25 (W= 3.0 for t < I,, and W= 1.0 for I > t,).
where n,(t) is the marginal value of the ith vegetative organ at time f taking
the future environmental change into account. According to the calculation
in Appendix C, we can derive a relation which determines the optimum t,
from Eqs. (26).
In contrast, if the environmental change is not predicted, the plant
continues to grow along the balanced growth path until the actual environmental change, when the unbalanced growth begins. The reproductive
94
IWASA AND ROUGHGARDEN
success of the plant, in such a case, should be lower than the one if the
environmental
change was predicted in advance. The difference of
reproductive success between the two cases is regarded as the “benefit of
information” on future environmental change.
Note that the optimal growth in a predictably changing environment
includes a period in which the plant invests the material only to a single part
but the environment has not yet changed (t, < t < t,), thus the plant shape
deviates from the balanced growth path along which the maximum daily net
productivity is realized. Thus the optimal growth behavior which realizes the
maximum reproductive activity through the season is different from the one
which realizes the maximum instantaneous (or daily) productivity.
For example, suppose that the vegetative growth period includes both wet
and dry seasons, and water availability is restricted to the deep soil layers
during the dry season. A seedling growing in the rainy season has to invest
much of its resources to produce deep roots to reach the deep moist soil.
Such plants do not maximize their growth rate during the rainy season,
where water is easily available at the top soil layers. This large investment to
root can properly understood only if we consider the effect to “future”
productivity, through which the plant realizes a larger reproductive success.
The model can handle such a situation by taking into account the given
future distribution of water availability, because the marginal values of any
change includes all the future contributions.
The same consideration may also apply to the distribution of nutrients in
the soil layers and to the predictable change of the light condition with the
season.
OPTIMAL GROWTH OF A SYSTEM WITH n VEGETATIVE ORGANS
So far we have studied the system with two vegetative organs. A similar
analysis applies the optimal growth of a system with more than two organs.
Let Xi be the size of ith vegetative organ (i = 1, 2,..., n), and R(t) be the
activity
until t. The net
accumulated investment to reproductive
X,),
is
an
increasing
function
of the sizes of the
photosynthetic rate, f(X, ,...,
n vegetative organs. The photosynthetic material produced is allocated to
these organs and to reproductive activity. Let u,(t) be the fraction of
photosynthetic material allocated to the ith vegetative organ. The growth of
a plant is described by
dX.
2 = z$(t) j-(X, )...) X,)
dt
dR
= ~&)f(X,
dt
,-.., X,).
(i = 1, 2,..., n),
Wa)
(27b)
SHOOT/ROOT
BALANCE
95
OF PLANTS
If the initial sizes of vegetative organs, X,,(O) = xiO,..., X,,,,(O)= xnO, and the
period suitable for growth, T, are given, the optimal growth schedule,
{U, ,..‘, u,, uO}, is the one that maximizes the total reproductive activity
during the season,R(T), under the constraints,
ul(t)> o,...,
u,(t)> 0, U,(l)> 0
and
f
u,(t) + u,,(f) = 1. (28)
i=l
By a similar analysis to that in previous sections it is shown that the
optimal growth schedule is again composed of three phases: (1) the
convergence to a “balanced growth path,” (2) the simultaneous growth of n
vegetative organs (balanced growth), which is defined by
-=af -=af .**=-,af
ax,
ax,
ax,
(29)
and (3) the reproductive growth. Some of these phases may not appear
depending on the initial sizes of the organs and on the length of the season.
The second-order condition for singular subarc requires that a matrix, the i-j
element of which is
(i, j= 1, 2,..., n - l),
(30)
is positive semidefinite (Robbins, 1967) during the balanced growth. This
condition is satisfied if the balanced growth point gives a local maximum of
X,,) on the plane, X, t see+X, = const. Thus we can again interpret
f (X, 3***,
the balanced growth path as a sequence of optimal shapes in which the
relative sizes of the organs are chosen to maximize the daily photosynthesis
given the total biomass.
The maximum principle gives a necessary condition for optimality, but it
is not a sufficient condition. However, if we assume an additional condition
related to the concavity off(X, ,..., X,,), for example, if the Hessian matrix of
the function is negative definite, then the resource allocation schedule that
satisfies the maximum principle is globally optimal, as shown in
Appendix D.
DISCUSSION
The growth characteristics of a plant are determined by the way it
allocates photosynthetic products into various organs; leaves, stems, roots,
flowers, or fruits. Current theoretical studies on plant growth can be
96
IWASA AND ROUGHGARDEN
classified into three categories: (1) a mechanical description of allocation
based on compartment models, (2) a cost-benefit analysis in which a single
variable is chosen for optimization,
(3) a dynamic control model of
allocation, in which an allocation schedule through the season (a function of
time) is optimally chosen.
A compartment model is useful when describing flows of materials (or
energy) among several parts of a system. Thornley (1972a,b), and Reynolds
and Thornley (1982), for example, developed this approach to study the
shoot/root allocation. Parameters in the model, such as flow coefficients,
resistances, and conductances, however, greatly change with the growing
condition and between strains. The way in which these change is often
critical to explaining a plant’s growth pattern. We may be able to describe
parameters’ changes if we use many additional compartments describing
regulartion mechanisms, but the parameters of a model with many
compartments are difficult to calculate from data.
A useful guideline for determining the parameters and their dependence on
the environmental conditions is given by an idea of optimality-a
plant
grows in a way to maximize its reproductive success. A simple optimization
model can somtimes summarize many aspects of an organism’s behavior, for
example, models of optimal foraging or the ESS sex ratio. It is amazing to us
how simple they are compared to corresponding mechanistic models
considering physiological and neural processes. Many aspects of a plant’s
vegetative growth have been discussed in reference to optimization ideas.
These aspects include the size and shape of leaves (Parkhurst and Loucks,
1972; Givnish and Vermeij, 1976; Givnish, 1979; Orians and Solbrig, 1977),
the photosynthetic enzyme content (Mooney and Gulmon, 1979), herb height
(Givnish, 1982), and root structure (Caldwell, 1979), as well as shoot/root
balance (Orians and Solbrig, 1977; Mooney and Gulmon, 1979).
Growth is, however, a dynamic process in which every variable changes
with time. An optimal strategy of material allocation is not dependent on a
single variable, but is a function of time. The resource allocation schedule
throughout the season has been studied as a life history strategy theory
(Cohen, 197 1, 1976; Paltridge and Denholm, 1974; Vincent and Pullium,
1980; King and Roughgarden, 1982a,b; Schaffer et al., 1982). Vegetative
growth can be regarded as an indirect way of maximizing reproduction via
the development of the ability to acquire resources. Since the timing of
reproduction was mainly addressed in these theories, all the vegetative
organs were described by one variable. Problems during the vegetative
growth, such as root/shoot balance, have never previously been addressed.
In this paper we studied the optimal growth schedule of a plant with both
shoot and root. The most important prediction of the model concerns
Eq. (14) during the simultaneous growth of shoot and root. This relation
indicates that the daily net production is maximized for given total biomass.
SHOOT/ROOT
BALANCE
OF PLANTS
97
This prediction, however, depends on the assumption that the efficiency of
conversion from the photosynthate (grams CO, fixed) to shoot mass (grams
dry weight) is the same as that to root mass. If they are different, the
following modification is needed: Let 8, be the dry weight of shoot made
from unit amount of photosynthesis, and 8, be that of root from the same
photosynthate. Then the relation for the balanced growth path becomes
e ‘f -eK
’ ax,
2 ax,
which indicates that the net production rate f(X,, X,) is maximized under
the constraint X,/B, + X,/B, = const. Since f(X,, X2) depends on the
environmental parameters, such as light and water availability, the growth
path may also change when the parameters change.
The convergence of a plant’s shape to the balanced growth path after an
environmental change or pruning is a plausible result. The model also claims
that shoot stops growing and only root grows while the root/shoot ratio is
lower than that of the balanced growth path, and that simultaneous growth
begins when root size recovers. This convergence may not be observed in a
real system: (1) If some part of material used for construction can be
reutilized, shoot may “shrink” during the recovery of root. (2) Since root
intakes nitrogen and shoot fixed carbon, the smaller size of root during its
recovery may result in the shortage of nitrogen when it recovers to the
original size; then the root size may temporarily be larger than the balanced
growth (overcompensation). (3) If young roots are more efficient than old
ones, a new balance between root and shoot after root’s recovery may be
attained by a smaller root size (undercompensation).
We assumethat the photosynthetic rate depends only on the size of shoot
and root and the environmental parameters at that time. If the growth history
strongly affects the productivity, plants with the same size and shape may
have different productivity. Pruning itself may also cause some harmful
effects.
In spite of these reservations in assumptions and predictions, we think the
main result of the optimal growth model, Eq. (31), is worth testing by
experiment because it connects a plant’s shape with its productivity, and
ultimately with its reproductive success, in the context of dynamic
optimization.
APPENDIX A
Along the growth path defined in the text, Xi, X2, and R clearly satisfy
the growth equations Eqs. (la) and (lb) and initial conditions. Costate
98
IWASA AND ROUGHGARDEN
variables are calculated from differential equations (5a-c) and terminal
conditions (6). We thus need only to check the maximal principle, Eqs. (12a)
and (12b).
Toward the end of the season, both L,(t) and L,(t) are smaller than
n,(t) = 1, because of their continuities and terminal conditions:
A,(T) = n,(r) = 0. During this period, only reproductive activity occurs:
u,, = 1, U, = u2 = 0; hence the maximal principle Eq. (12) is satisfied. With
the differential equation (5a) and (5b) combined with the terminal conditions
Eq. (6), we have
A,(t)= &
. (T-t).
2
The beginning of reproductive growth, t,, is determined by the condition that
the larger of the two costate variables, n,(t) and J.,(t), is equal to L,(t) = 1.
Thus we can derive the equality in (16). If both n,(O) and n,(O) are smaller
than L,(O) = 1, as indicated by the inequality in (16) with t = 0, a whole
season is devoted to reproductive growth.
To prove the maximal principle during the vegetative growth period
(before t2), we first show a relation
then J.,(t) $ n,(t)
642)
holds for all t, as follows: From Eqs. (5a), (5b), and (6), we have
/l,(t) = jr x ujllj(s) &
f i
I
(s) ds
(i= 1, 2);
therefore,
A,(t) - l,(t) = ,(I C ujAj(s) (&
t i
L
(s) - &
(s)) ds.
(-44)
2
Suppose af(t)/X, > af(t)/Z, holds at t. From the way of constructing the
growth path, we know that af(s)l
- af(s)/X, > 0 holds for all s > t, and
that the strict inequality 3f(s)/X, - af(s)l
> 0 holds for s sufficiently
close to t because of the continuity. Then from Eq. (A4), L,(t) -L,(t) > 0.
By using a similar argument, we can prove that aj(t)/LX, < af(t)l
implies n,(t) < n*(t) and that af(t)/H, = af(t)/&Y, implies n,(t) = l,(t).
Thus (A2) is proved.
SHOOT/ROOT
BALANCE
99
OF PLANTS
The vegetative part which singly grows during the first part of the season
(before tr) is the one with the higher aflaX,. Thus the maximal principle
Eq. (12) which requires that this growing vegetative part should have larger
marginal value Izi than the other, is clearly satisfied because of Eq. (A2).
During the balanced growth period (between t, and tJ, in which both
shoot and root grow simultaneously, the maximum principle requires
Lr(t) = &(t). From Eq. (A2), this is equivalent to aflaX, = aflaX,, which
holds during the balanced growth path.
APPENDIX B
From Eqs. (7) and (12), we have
A(t) = max n,(t)
(Bl)
for all t, where the maximum is attained by i = 1 or i = 2 for t < t,. From
(A3) and (7), we have
(i= 1, 2)
WI
for all t. In addition
for all
implies
s> t
(B3)
which is derived from Eq. (A2) and the way of constructing the growth path
specified in text. Combining these results, we have
l(t) = f” A(s) max g
‘t
I
(s) ds + L(t,)
(B4)
for all t < t,. By noting n(t,) = 1, we can have a solution of this integral
equation:
W)
A(t) = exp
for t < t,. Taking n(t) = n,(t) = 1 for the reproductive period into account,
we have Eqs. (23a) and (23b). Now from Eqs. (la) and (lb), we have
3x*>*
Yx*>+& u2f(X,
ff(xlJ2)=g I uLf(X,
2
WI
100
IWASAANDROUGHGARDEN
During the vegetative growth period (0 < t < t2), u1 and u2 are positive only
when aflaX, is equal to max(afl&Y, , aflaX,), because of Eqs. (12) and (A2).
Thus Eq. (B6) becomes
ff
=max ($,&j
.f;
therefore,
12
f(Xl(fd~
X&>>
af af
f@!,(t), X,(t)) = exp I. , max (ax,’ ax, 1 ds1 .
(B7)
Combining (B.5) and (B7), we have
Note that this holds after t, as well. Since the numerator is constant, the
marginal value of carbon available at t is inversely proportional to the daily
photosynthesis of a whole plant.
APPENDIX
C.
OPTIMAL GROWTH IN PREDICTABLY CHANGING ENVIRONMENT
The daily net production here explicitly
depends on the time parameter t,
as
f (X, 3x, 34 = f,(X, 3X*)
= fiGI
X2)
(f < ts>,
(Cl4
(t > 0
CClb)
The equations for the marginal values, Eqs. (5a) and (5b), are derived in the
predictably changing environment in the same way as in a constant
environment in text. Calculating the difference between Eqs. (5a) and (5b),
we have
$(A*-1,)=1,
[g&--g]
1
W)
2
during the unbalanced growth period (tI < t < t2) which includes both before
and after f,. From Eq. (26a) and the continuity of Ji, we have
A2(t2)
-
A(f2)
=
n,ct,>
- nl(td = 0.
(C3)
SHOOT/ROOT
BALANCE
101
OF PLANTS
Integrate Eq. (C2) and use Eq. (C3); we have
[$--&I
dt
(C4)
On the other hand, we can prove that the marginal value L,(t) is inversely
proportional to the daily net photosynthesis in each interval, before and after
t,, according to the same argument as in Appendix B. This does not hold to
the whole interval including t,, because the discontinuity of f(X, , X,, t) at
time t,. Therefore we have
Devide Eq. (C4) by L,(t,), and replace L,(t) in the equation by Eq. (C5); we
have
dt.
(C6)
The first term of Eq. (C6) is related to the unbalanced growth before the
environmental change at t,. During this period, root only grows and the
plant’s shape departs from the balanced growth path indicated by Eq. (25a),
but the environment has not yet been changed. Hence a root/shoot ratio is
larger than along the balanced growth path, and we have
The first term of Eq. (C6) is thus positive. In contrast, the second term of
Eq. (C6) is related to the unbalanced growth after t,, in which root/shoot
ratio is smaller than the new balanced growth, which is reached at t,. Hence
we have
holds, and the second term of Eq. (C6) is negative. Equation (C6) indicates
that these two should exactly cancel out along the optimal growth path.
As t, increases, so does t,, the time at which the plant shape reaches the
new balanced growth path. Hence the first term of Eq. (C6) decreases and
the magnitude of the second term increases with t, . There is a single t, which
satisfies Eq. (C6); thus Eq. (C6) determines the optimal growth path.
102
IWASA AND ROUGHGARDEN
APPENDIX D
We here prove that the resource allocation schedule (u:(t),..., u:(t),
u:(t)} which satisfies the maximum principle is globally optimal if the
Hessian matrix off(X, ,..., X,) is negative definite.
Let {X:‘(t),...,X,*(t), R*(t)} be the growth trajectory by the allocation
schedule {u:(t),..., u,*(t), u:(t)}. Consider any allocation schedule {u,(t),...,
u,(t), u,(t)} which satisfies the constraints Eq. (28) and the growth trajectory
enerated by this allocation schedule using Eqs. (27)
{X,WY.Y X,(t), XOWI f.4
and the initial condition.
From the assumption about the concavity of f(X,,..., X,), we have a
relation
f (x:,..., x,*> - f(X, ,a.-,X,)+
+ af *(xi-xT)>o,
Jo, t aXi 1
Pl)
for all t. Multiplied by C~~‘=ozLi~:,Eq. (Dl) becomes
i
n,u,*f(x::
i=O
)...) X,*)
-
;:
n,u,f(x,
)..., X,)
i?O
where we replace UT by ui in the second term by using a relation
Cr=r Ai@ > x1=, Aiui, which is required by the maximum principle. We
rearrange the terms in Eq. (D2), and rewrite them using the dynamics of
state variables for each growth schedule, the dynamics of costate variables,
and a relation, A,(t) = 1. Then we have
(D3)
thus can be written as
u$.f- (X,*,..., x;)
- %J-(X, ,‘**, X,) + $2
l-1
Aj(X,* - Xi) > 0.
(D4)
By integrating this inequality from 0 to T, we have
R*(T) - R(T) = Jo’ u$f(X:: ,..., X;) dt - !‘r u,f(X,
0
,..., X,) dt > 0, (D5)
SHOOT/ROOTBALANCE
103
OFPLANTS
where we used relations
+ n,(o)(x,*(o) -xi(o))
iY1
= 0,
2 n,(T)(xi”(r) -xi(r)) = 0,
because X?(O) = Xi(O) = Xio)
because A,(r) = 0
i=l
for all
i = 1, 2,..., n.
Since allocation schedule {u,(t),..., un(t), u,(f)} was arbitrarily chosen,
Eq. (D5) indicates that the schedule {u?(t),..., u:(t), u:(O)} is the global
optimum.
Inequality (Dl) is related to the “direct sufficiency theorem” for optimal
control (Leitmann, 1981). The negative-definite Hessian matrix is a stronger
assumption than Eq. (30), which is concavity on the surface X, + .+. t X, =
const, only. It is also assumed in the theory of firm in microeconomics
(Intriligator, 1971), but we think it is too restrictive to require in all the
biologically important examples.
ACKNOWLEDGMENTS
With thanks to Dan Cohen, Nelson Johnson, Harold Mooney, John Rummel, David King,
Lee Altenberg, Sherry Gulmon, David Hollinger, and Thomas Vincent for their helpful
comments. This research was partly supported by National Science Foundation Grant
DEB 81-02067.
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