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Lichen growth
J. theor. Biol. (1980) 82, 157-165
Lichen Growth STEPHEN CHILDRESS Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, U.S.A. AND
JOSEPH B. KELLER? Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, CA 94305, U.S.A. (Received 14 March 1979, and in revised form 30 August 1979) A theory of the growth of a lichen ispresented,basedupon the production of a carbohydrate by photosynthesis,partial consumptionof it to thicken the lichen, diffusion of the remainder toward the outer edge, and its consumptionthere to increasethe lichen radius.The mathematicalformulation of the theory leadsto a free boundary problemfor the determination of the carbohydrate concentration, and of the lichen radiusasa function of time. For the realistic caseof slow growth the problem is solved approximately. The result for the growth rate is shownto fit Proctor’s data fairly well. 1. Introduction We wish to determine the growth rate, size and weight of a circular lichen as a function of time on the basisof a simple theory. It assumesthat growth is governed by the following four processes: (1) Production of some carbohydrate by photosynthesis at a constant rate per unit area; (2) Consumption of part of the carbohydrate to thicken the lichen at a rate proportional to the carbohydrate concentration; (3) Diffusion of the rest of the carbohydrate toward the outer edge of the lichen at a rate proportional to the gradient of concentration; (4) Consumption of the carbohydrate which arrives at the outer edge to increase the lichen “radius” at a rate proportional to the arrival rate. These processeshave also been proposed to account for lichen growth by Aplin & Hill (1979). We assume that these processes occur in crustose, foliose and other saxicolour lichens. As we shall see, they account for both the prelinear or t Research supported in part by the National Science Foundation, the Army Research Office, the Air Force Office of Scientific Research, and the Office of Naval Research. 157
0022-5193/80/010157+09
$02.00/O
@ 1980 Academic Press Inc. (London) Ltd.
158
S. CHILDRESS
AND
J. B. KELLER
exponential phase of growth and for the subsequent linear phase, as well as for the transition between these two phases. They yield no postlinear phase, in agreement with the observations of Armstrong (1973,1974), that no such phase is evident in the observed growth rate. Since the carbohydrate is actually produced at a rate per unit area which varies during a day or a year, the assumed constant rate of production represents an annual average. We shall begin by describing quantitatively each of these processes and equating the rates of production and consumption. This will yield one differential equation for the concentration and another for the “radius”. We shall solve them approximately. Then we shall fit the solutions to the observational data of Proctor (1977) on lichen radius versus time, to determine two constants in the theory. With these constants determined, the theory will be shown to fit the data fairly well. The theory also predicts the weight of the lichen as a function of time, but we have no data with which to compare this prediction. Aplin & Hill (1979) have also tried to construct a mathematical model of the same four processes. Although their results fit the data, their mathematics is incorrect, as we explain in the Appendix. We thank D. J. Hill for having brought this problem to our attention, and for having sent us a copy of his paper with P. S. Aplin. 2. Formulation
of the Theory
Let us begin by representing a lichen at time I as a circle of radius R(t). We denote by c(r,t) the concentration (mass per unit area) of the carbohydrate at the radial distance r from the center at time t. Let m be the constant rate of production of the carbohydrate per unit area per unit time and let (YCbe the rate of consumption, where (Y is a rate constant with the dimensions of (time)-‘. By Fick’s law, the total outward flux of carbohydrate at radius r is -2rrDc, where D is the diffusion coefficient. We now equate ct, the rate of increase of concentration, to the rate of production minus the rate of consumption minus the divergence of the flux to obtain: OzsrsR(t).
cr=m-ac+D(~,~+r-‘c,),
(1)
This diffusion equation for c results from the first three processes described in the Introduction. To describe the fourth process we note that the rate of increase of area of the lichen is 2vrR(t) dR/dt, and the flux of carbohydrate at the edge is -2~R(t)c,[R(t), t]. Since these quantities are assumed to be proportional, we have dR -= dt
-kDc,[R(t),
t].
(2)
LICHEN
159
GROWTH
Here k is a proportionality constant with the dimensions of area/mass. Its reciprocal k-’ is the amount of massneeded to produce unit area of lichen. We also assumethat all the carbohydrate at the edge is consumed, so we have c[R(t), t] = 0. (3) In addition regularity of c at r = 0 requires that cr(0, c)= 0. (4) These equations (l)-(4) are the mathematical expresssionof the theory described in the Introduction. In order to solve them we must specify the initial radius Ro: R (0) = Ro.
(5)
We must also specify the concentration c(r, 0) at t = 0. However this will not be needed for the approximate solution which we shall present in the next section. 3. Solution for SIow Growth Since lichens grow slowly, we shall assumethat the time derivative c, is negligible and omit it from (1). Then (1) becomes Bessel’sequation. The solution of it which satisfiesthe two conditions (3) and (4) is, with 10denoting the modified Besselfunction, (6)
We now use (6) in (2) to obtain dR dt=
km(D/a)
1,2GCWW2~l &K~/~)1’2Rl
’
(7)
The result (7) gives the rate of growth of the radius as a function of the radius. It is shown in Fig. 1 in which [km(D/a)“2]-’ dR/dt is given as a function of (cx/D)“~R. For small and large values of (cY/D)“~R, (7) yields dR kmR dt2 ’ g-
km(D/a)“2,
(c.Y/D)“~R (cx/D)“~R
<< 1,
03)
>> 1.
By fitting (8) to observational data on dR/dt for R small we can find km. Similarly, by (9) the asymptotic value of dR/dt for large R yields
160
S. CHILDRESS
AND J. B. KELLER
FIG. 1. Radial growth rate dR/dr as a function of radius R based upon (7). The ordinate is [km(D/c~)“~]-’ dR/dr and the abscissa is (a/D)“‘R.
km(D/a)“*. Once these two quantities are known we can obtain (a/D)l’* = km/km(D/a)“*. Then dR/dt can be found for any value of R from (7). We can solve (7) for R(t) by first using the Bessel differential equation in the form I0 = I;l + (a/D)-1’2R-11b to eliminate I,,. Then we integrate the resulting equation and use (5) to obtain RI; [(a/D)“*R] log ROIL [(a/D)“*R,,]
(10)
= kmr’
This equation yields R as a function of t as is shown in Fig. 2. There (a/D)“*R isplottedasafunctionof kmt -log{(a/D)“*R&[(a/D)“*R~]}. This way of plotting permits the curve to be used for any value of R. by just shifting the origin of t, and for any values of the other constants. When R is small, (8) or (10) yields the prelinear or exponential growth phase, R(t) - ROekm”*,
(a/D)“*R
K 1.
(11)
For R large, (9) or (10) yields the linear growth phase, R(t)-
km(D/a)“*(f-[I),
(a/D)“*R
>>1.
(12)
The constant rl can be found from (10). To calculate the mass M(t) of the lichen as a function of time, we note that the rate of production of carbohydrate equals the area wR2 times m. Thus
LICHEN
161
GROWTH
FIG. 2. Radius R as a function of time 1 based upon (10). The ordinate is (a/D)“*R abscissa is kmr - Iog{(a/D)“ZROZ~ [(o/D)“*R~]}.
and the
we have dM
(13)
dt = mrR’(t).
If the initial mass is MO then we have, upon integrating
(1 l),
J
‘R’(r)dT.
M(t)=Mo+~m
(14)
0
By using (11) in (14) we find for R small
M(t)-M,+
(a/D)“2R
<<1.
On the other hand, for R large we use (12) in (14) to get M(t) - (7rk2m3D/3a)(t - t1)3,
Thus M(t) at first grows exponentially
(cY/D)“~R >>1.
in t, and ultimately
(16)
grows like t3.
4. Exact Solution For One Dimension The approximate solution presented in section 3 is valid for slow growth. Now we shall obtain an exact solution of the free boundary problem of section 2 which is not restricted to slow growth. However it only applies to the one dimensional case. Then the term Dr-ic, is absent from (l), the
162 S. CHILDRESS AND J. B. KELLER condition (4) is omitted, and r denotes a Cartesiancoordinate. This solution applies near the outer edge in the circular case,provided that (oJ/D)“~R is large. An exact solution of (1) with DrB1cr omitted is the plane wave c(r, t) = a-‘m(l -ebALR(“-‘l), rrR(t), (17) provided that A satisfiesthe equation AdR/dt =o -Dh2.
(18)
The solution (17) satisfies (3), while (2) yields dR/dt = kDa-‘mh.
(19)
By solving (18) and (19) we find the constantsA and dR/dt to be A
= (c~/D)“~(l+ ka-1m)-1’2,
dR/dt = km(D/a)1’2(1
+ kma-‘)-“2.
(20) (21)
We see that the velocity given by (21) agrees with that given by (12) when kma-‘c 1. A single approximate solution, valid both for slow growth and for large r, can be obtained by replacing (~y/D)l’~ by A in (6) and (7), and replacing (D/a)1’2 in (7) by Da-‘A. 5. Comparison with the Result of Proctor Woolhouse (1968) introduced the relative growth rate (RGR) defined by RGR
= ln&
- 1nA I tz- t1 ’
(22)
where AI and A2 are the areas of the lichen at times tl and t2. For the rotationally symmetric casethe differential version of (22) is RGR=;g=2R-ldt.
dR
(23)
By using our result (7) in (23) we obtain (km)-lRGR
=
IL [(cx/D)~‘~R][(~/D)~“R]-~
~olk/D)
112
RI
(24)
Proctor (1977) proposed a simple formula for RGR based upon the assumption that only the carbohydrate produced within a certain distance s
LICHEN
163
GROWTH
of the edge is available to increase the radius. This leads to the formula OSRSS,
RGR = (RGR)o, R2-(R
= WGWo
R2
-s)*
SSR.
9
(25)
Comparison of (24) with (25) shows that they agree for R small if km = (RGR)o, and they agree for R large if (D/(u)“~ = S. Proctor (1977) has determined the two constants by fitting (25) to his measurements of growth of the crustose lichen Buellia canescens. In Fig. 3 we compare both (24) and (25) with his data, using his values of the constants. Both agree quite well with the data, although for small R our result (24) seems to agree better.
I
0
I
5
IO
I
15
20
k&R
FIG. 3. Relative growth rate RGR as a function of time t based upon the present result (24) (Curve 2) and upon Proctor’s formuIa (25) (Curve I). The data pointsare from Proctor [2]. The values of the constants used are km = (RGR)o = 1.3 (year)- and (D/a)“’ = s = 1.25 mm. They were found by Proctor in fitting his curve to the data. The ordinate is Z(kmR)-‘dR/dr and the abscissa in (a/D)“‘R.
Both (24) and (25) reflect a division of growth into two phases, an exponential phase for small (cx/D)“~R, and a linear phase for large (cx/D)“~R. The transition from one phase to the other occurs when (cx/D)I’~R is approximately 1, as is clear from Fig. 2. The parameter (D/a )1’2r which has the dimensions of a length, is thus seen to be the radius of a lichen whose growth is in transition from the exponential to the Iinear phase. Similarly (km)-’ is the time for increase of area by a factor e in the exponential phase. One advantage of (24) over (25) is that the role of the parameter (D/(Y)“~ has been deduced from the theory, assuming only that growth occurs sufficiently slowly. 6. Discussion
In this note we have derived the mathematical consequences of assumptions l-4 of section 1, and shown that the resulting growth rate fits Proctor’s
164
S. CHILDRESS
AND
J. B. KELLER
(1977) observations. This fit lends support to our assumptions, such as that of outward diffusion of carbohydrate, for which there is no direct proof. However the prediction of the growth rate is a crude test, so other more direct tests of the assumptions would be worthwhile. As Armstrong (1973) remarks, in large lichens the transport of material from near the center toward the edge is inhibited. Our theory shows that this is because the concentration c becomes nearly uniform throughout the interior, except near the edge, so the gradient cr is nearly zero in the interior. Therefore the outward flux from the interior is very small. In smaller lichens the flux from the interior is not small. The transition from prelinear to linear growth occurs when the radius is of the order (O/a)“2. The theory appears to be applicable to the prelinear and linear phases of growth (Armstrong, 1974), as well as to the transition between them. It does not apply to very old lichens, in which fragmenting of the center of the thallus occurs (Armstrong, 1973, 1974). But this does not affect the radial growth rate because in large thalli the growth rate at the edge is determined by an annular band of width about two or three times (O/(U)“~ adjacent to the edge. It is only in this band that our assumptions must be satisfied. APPENDIX Aplin & Hill (1979) have proposed a model of lichen growth based upon the hypotheses given in section 1. It leads to the following two equations for the carbohydrate concentration v, and the radial flux of carbohydrate i, at radius x (see their Appendix A): dv, ---
dx-
ni, 2rhx’
(Al)
Here n, h, k and 1 are constants. By an unclear argument, they arrive at the formula 2mk m + (n1)“210[(n/l)“2r]/I~[(n/f)1’2r]
=(ix)X=r*
642)
Here r is the radial position of the edge of the lichen and m is another constant. As we shall now show, (A2) is incorrect. Elimination of vx from (Al) yields xdIdi,-ti dxxdx
-0 lx-
’
(A3)
The general solution of (A3) is i, =A~I~[(n/1)“~x]+BxK~[(n/1)“*x].
(A4)
LICHEN
GROWTH
16.5
For no choice of the constants A and B does the solution (A4) agree with their formula (A2). REFERENCES APLIN, P. S. & HILL,D.J. (1979).-K theor. Biol.78,347. ARMSTRONG,R.A.(~~~~). New Phyro[. 72,1023. ARMSTRONG,R. A.(1974). NewPhytol. 73,913. PROCTOR,M. C. F.(1977). NewPhytol. 79,659. WOOLHOUSE,H. W.(1968). Lichenologist 4,32.
Lichen Growth STEPHEN CHILDRESS Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, U.S.A. AND
JOSEPH B. KELLER? Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, CA 94305, U.S.A. (Received 14 March 1979, and in revised form 30 August 1979) A theory of the growth of a lichen ispresented,basedupon the production of a carbohydrate by photosynthesis,partial consumptionof it to thicken the lichen, diffusion of the remainder toward the outer edge, and its consumptionthere to increasethe lichen radius.The mathematicalformulation of the theory leadsto a free boundary problemfor the determination of the carbohydrate concentration, and of the lichen radiusasa function of time. For the realistic caseof slow growth the problem is solved approximately. The result for the growth rate is shownto fit Proctor’s data fairly well. 1. Introduction We wish to determine the growth rate, size and weight of a circular lichen as a function of time on the basisof a simple theory. It assumesthat growth is governed by the following four processes: (1) Production of some carbohydrate by photosynthesis at a constant rate per unit area; (2) Consumption of part of the carbohydrate to thicken the lichen at a rate proportional to the carbohydrate concentration; (3) Diffusion of the rest of the carbohydrate toward the outer edge of the lichen at a rate proportional to the gradient of concentration; (4) Consumption of the carbohydrate which arrives at the outer edge to increase the lichen “radius” at a rate proportional to the arrival rate. These processeshave also been proposed to account for lichen growth by Aplin & Hill (1979). We assume that these processes occur in crustose, foliose and other saxicolour lichens. As we shall see, they account for both the prelinear or t Research supported in part by the National Science Foundation, the Army Research Office, the Air Force Office of Scientific Research, and the Office of Naval Research. 157
0022-5193/80/010157+09
$02.00/O
@ 1980 Academic Press Inc. (London) Ltd.
158
S. CHILDRESS
AND
J. B. KELLER
exponential phase of growth and for the subsequent linear phase, as well as for the transition between these two phases. They yield no postlinear phase, in agreement with the observations of Armstrong (1973,1974), that no such phase is evident in the observed growth rate. Since the carbohydrate is actually produced at a rate per unit area which varies during a day or a year, the assumed constant rate of production represents an annual average. We shall begin by describing quantitatively each of these processes and equating the rates of production and consumption. This will yield one differential equation for the concentration and another for the “radius”. We shall solve them approximately. Then we shall fit the solutions to the observational data of Proctor (1977) on lichen radius versus time, to determine two constants in the theory. With these constants determined, the theory will be shown to fit the data fairly well. The theory also predicts the weight of the lichen as a function of time, but we have no data with which to compare this prediction. Aplin & Hill (1979) have also tried to construct a mathematical model of the same four processes. Although their results fit the data, their mathematics is incorrect, as we explain in the Appendix. We thank D. J. Hill for having brought this problem to our attention, and for having sent us a copy of his paper with P. S. Aplin. 2. Formulation
of the Theory
Let us begin by representing a lichen at time I as a circle of radius R(t). We denote by c(r,t) the concentration (mass per unit area) of the carbohydrate at the radial distance r from the center at time t. Let m be the constant rate of production of the carbohydrate per unit area per unit time and let (YCbe the rate of consumption, where (Y is a rate constant with the dimensions of (time)-‘. By Fick’s law, the total outward flux of carbohydrate at radius r is -2rrDc, where D is the diffusion coefficient. We now equate ct, the rate of increase of concentration, to the rate of production minus the rate of consumption minus the divergence of the flux to obtain: OzsrsR(t).
cr=m-ac+D(~,~+r-‘c,),
(1)
This diffusion equation for c results from the first three processes described in the Introduction. To describe the fourth process we note that the rate of increase of area of the lichen is 2vrR(t) dR/dt, and the flux of carbohydrate at the edge is -2~R(t)c,[R(t), t]. Since these quantities are assumed to be proportional, we have dR -= dt
-kDc,[R(t),
t].
(2)
LICHEN
159
GROWTH
Here k is a proportionality constant with the dimensions of area/mass. Its reciprocal k-’ is the amount of massneeded to produce unit area of lichen. We also assumethat all the carbohydrate at the edge is consumed, so we have c[R(t), t] = 0. (3) In addition regularity of c at r = 0 requires that cr(0, c)= 0. (4) These equations (l)-(4) are the mathematical expresssionof the theory described in the Introduction. In order to solve them we must specify the initial radius Ro: R (0) = Ro.
(5)
We must also specify the concentration c(r, 0) at t = 0. However this will not be needed for the approximate solution which we shall present in the next section. 3. Solution for SIow Growth Since lichens grow slowly, we shall assumethat the time derivative c, is negligible and omit it from (1). Then (1) becomes Bessel’sequation. The solution of it which satisfiesthe two conditions (3) and (4) is, with 10denoting the modified Besselfunction, (6)
We now use (6) in (2) to obtain dR dt=
km(D/a)
1,2GCWW2~l &K~/~)1’2Rl
’
(7)
The result (7) gives the rate of growth of the radius as a function of the radius. It is shown in Fig. 1 in which [km(D/a)“2]-’ dR/dt is given as a function of (cx/D)“~R. For small and large values of (cY/D)“~R, (7) yields dR kmR dt2 ’ g-
km(D/a)“2,
(c.Y/D)“~R (cx/D)“~R
<< 1,
03)
>> 1.
By fitting (8) to observational data on dR/dt for R small we can find km. Similarly, by (9) the asymptotic value of dR/dt for large R yields
160
S. CHILDRESS
AND J. B. KELLER
FIG. 1. Radial growth rate dR/dr as a function of radius R based upon (7). The ordinate is [km(D/c~)“~]-’ dR/dr and the abscissa is (a/D)“‘R.
km(D/a)“*. Once these two quantities are known we can obtain (a/D)l’* = km/km(D/a)“*. Then dR/dt can be found for any value of R from (7). We can solve (7) for R(t) by first using the Bessel differential equation in the form I0 = I;l + (a/D)-1’2R-11b to eliminate I,,. Then we integrate the resulting equation and use (5) to obtain RI; [(a/D)“*R] log ROIL [(a/D)“*R,,]
(10)
= kmr’
This equation yields R as a function of t as is shown in Fig. 2. There (a/D)“*R isplottedasafunctionof kmt -log{(a/D)“*R&[(a/D)“*R~]}. This way of plotting permits the curve to be used for any value of R. by just shifting the origin of t, and for any values of the other constants. When R is small, (8) or (10) yields the prelinear or exponential growth phase, R(t) - ROekm”*,
(a/D)“*R
K 1.
(11)
For R large, (9) or (10) yields the linear growth phase, R(t)-
km(D/a)“*(f-[I),
(a/D)“*R
>>1.
(12)
The constant rl can be found from (10). To calculate the mass M(t) of the lichen as a function of time, we note that the rate of production of carbohydrate equals the area wR2 times m. Thus
LICHEN
161
GROWTH
FIG. 2. Radius R as a function of time 1 based upon (10). The ordinate is (a/D)“*R abscissa is kmr - Iog{(a/D)“ZROZ~ [(o/D)“*R~]}.
and the
we have dM
(13)
dt = mrR’(t).
If the initial mass is MO then we have, upon integrating
(1 l),
J
‘R’(r)dT.
M(t)=Mo+~m
(14)
0
By using (11) in (14) we find for R small
M(t)-M,+
(a/D)“2R
<<1.
On the other hand, for R large we use (12) in (14) to get M(t) - (7rk2m3D/3a)(t - t1)3,
Thus M(t) at first grows exponentially
(cY/D)“~R >>1.
in t, and ultimately
(16)
grows like t3.
4. Exact Solution For One Dimension The approximate solution presented in section 3 is valid for slow growth. Now we shall obtain an exact solution of the free boundary problem of section 2 which is not restricted to slow growth. However it only applies to the one dimensional case. Then the term Dr-ic, is absent from (l), the
162 S. CHILDRESS AND J. B. KELLER condition (4) is omitted, and r denotes a Cartesiancoordinate. This solution applies near the outer edge in the circular case,provided that (oJ/D)“~R is large. An exact solution of (1) with DrB1cr omitted is the plane wave c(r, t) = a-‘m(l -ebALR(“-‘l), rrR(t), (17) provided that A satisfiesthe equation AdR/dt =o -Dh2.
(18)
The solution (17) satisfies (3), while (2) yields dR/dt = kDa-‘mh.
(19)
By solving (18) and (19) we find the constantsA and dR/dt to be A
= (c~/D)“~(l+ ka-1m)-1’2,
dR/dt = km(D/a)1’2(1
+ kma-‘)-“2.
(20) (21)
We see that the velocity given by (21) agrees with that given by (12) when kma-‘c 1. A single approximate solution, valid both for slow growth and for large r, can be obtained by replacing (~y/D)l’~ by A in (6) and (7), and replacing (D/a)1’2 in (7) by Da-‘A. 5. Comparison with the Result of Proctor Woolhouse (1968) introduced the relative growth rate (RGR) defined by RGR
= ln&
- 1nA I tz- t1 ’
(22)
where AI and A2 are the areas of the lichen at times tl and t2. For the rotationally symmetric casethe differential version of (22) is RGR=;g=2R-ldt.
dR
(23)
By using our result (7) in (23) we obtain (km)-lRGR
=
IL [(cx/D)~‘~R][(~/D)~“R]-~
~olk/D)
112
RI
(24)
Proctor (1977) proposed a simple formula for RGR based upon the assumption that only the carbohydrate produced within a certain distance s
LICHEN
163
GROWTH
of the edge is available to increase the radius. This leads to the formula OSRSS,
RGR = (RGR)o, R2-(R
= WGWo
R2
-s)*
SSR.
9
(25)
Comparison of (24) with (25) shows that they agree for R small if km = (RGR)o, and they agree for R large if (D/(u)“~ = S. Proctor (1977) has determined the two constants by fitting (25) to his measurements of growth of the crustose lichen Buellia canescens. In Fig. 3 we compare both (24) and (25) with his data, using his values of the constants. Both agree quite well with the data, although for small R our result (24) seems to agree better.
I
0
I
5
IO
I
15
20
k&R
FIG. 3. Relative growth rate RGR as a function of time t based upon the present result (24) (Curve 2) and upon Proctor’s formuIa (25) (Curve I). The data pointsare from Proctor [2]. The values of the constants used are km = (RGR)o = 1.3 (year)- and (D/a)“’ = s = 1.25 mm. They were found by Proctor in fitting his curve to the data. The ordinate is Z(kmR)-‘dR/dr and the abscissa in (a/D)“‘R.
Both (24) and (25) reflect a division of growth into two phases, an exponential phase for small (cx/D)“~R, and a linear phase for large (cx/D)“~R. The transition from one phase to the other occurs when (cx/D)I’~R is approximately 1, as is clear from Fig. 2. The parameter (D/a )1’2r which has the dimensions of a length, is thus seen to be the radius of a lichen whose growth is in transition from the exponential to the Iinear phase. Similarly (km)-’ is the time for increase of area by a factor e in the exponential phase. One advantage of (24) over (25) is that the role of the parameter (D/(Y)“~ has been deduced from the theory, assuming only that growth occurs sufficiently slowly. 6. Discussion
In this note we have derived the mathematical consequences of assumptions l-4 of section 1, and shown that the resulting growth rate fits Proctor’s
164
S. CHILDRESS
AND
J. B. KELLER
(1977) observations. This fit lends support to our assumptions, such as that of outward diffusion of carbohydrate, for which there is no direct proof. However the prediction of the growth rate is a crude test, so other more direct tests of the assumptions would be worthwhile. As Armstrong (1973) remarks, in large lichens the transport of material from near the center toward the edge is inhibited. Our theory shows that this is because the concentration c becomes nearly uniform throughout the interior, except near the edge, so the gradient cr is nearly zero in the interior. Therefore the outward flux from the interior is very small. In smaller lichens the flux from the interior is not small. The transition from prelinear to linear growth occurs when the radius is of the order (O/a)“2. The theory appears to be applicable to the prelinear and linear phases of growth (Armstrong, 1974), as well as to the transition between them. It does not apply to very old lichens, in which fragmenting of the center of the thallus occurs (Armstrong, 1973, 1974). But this does not affect the radial growth rate because in large thalli the growth rate at the edge is determined by an annular band of width about two or three times (O/(U)“~ adjacent to the edge. It is only in this band that our assumptions must be satisfied. APPENDIX Aplin & Hill (1979) have proposed a model of lichen growth based upon the hypotheses given in section 1. It leads to the following two equations for the carbohydrate concentration v, and the radial flux of carbohydrate i, at radius x (see their Appendix A): dv, ---
dx-
ni, 2rhx’
(Al)
Here n, h, k and 1 are constants. By an unclear argument, they arrive at the formula 2mk m + (n1)“210[(n/l)“2r]/I~[(n/f)1’2r]
=(ix)X=r*
642)
Here r is the radial position of the edge of the lichen and m is another constant. As we shall now show, (A2) is incorrect. Elimination of vx from (Al) yields xdIdi,-ti dxxdx
-0 lx-
’
(A3)
The general solution of (A3) is i, =A~I~[(n/1)“~x]+BxK~[(n/1)“*x].
(A4)
LICHEN
GROWTH
16.5
For no choice of the constants A and B does the solution (A4) agree with their formula (A2). REFERENCES APLIN, P. S. & HILL,D.J. (1979).-K theor. Biol.78,347. ARMSTRONG,R.A.(~~~~). New Phyro[. 72,1023. ARMSTRONG,R. A.(1974). NewPhytol. 73,913. PROCTOR,M. C. F.(1977). NewPhytol. 79,659. WOOLHOUSE,H. W.(1968). Lichenologist 4,32.