Journal of General Microbiology (1978), 106, 55-65.
Printed in Great Britain
55
Mycelial Growth and the Initiation and Growth of Sporophores
in the Mushroom Crop: a Mathematical Model
By D. 0. C H A N T E R A N D J. H. M. T H O R N L E Y
Glasshouse Crops Research Institute, Littlehampton, West Sussex BN16 3PU
(Received 2 1 September 1977)
A single-substrate three-compartment model for the growth of a mushroom crop is constructed. The model describes the growth of the mycelium, and the initiation and growth
of sporophores, up to the end of the first flush. The growth of mycelium and sporophores
is controlled by the substrate density in the storage component of the mycelium, and initiation of sporophores is modelled by assuming the existence of a threshold substrate density,
below which initiation cannot take place. When the substrate density exceeds the threshold
density, the rate of initiation is assumed to be proportional to the difference between these
two densities.
Parameter values are given which lead to a solution of the model which agrees reasonably
well with observed data. Various aspects of the solution are examined, and the important
parameters are identified. The parameter controlling the rate of initiation of sporophores
has little effect on either the number of sporophores initiated or the duration of initiation.
INTRODUCTION
The role of mathematical modelling in biological research has been discussed by several
authors, including Maynard Smith (1968) and Thornley (1976). Whilst the technique has
been used with some success for many agricultural crops, few attempts have so far been made
to apply the method to the horticultural production of the cultivated mushroom, Agaricus
bisporus (Lange) Imbach. A brief description of the commercial mushroom-growing
procedure is given by Chanter (1977a).
In this paper we develop a model which includes mycelial growth, sporophore initiation
and growth, and respiration. Using biologically plausible values for the parameters, a
solution is given for the model up to the end of the first flush.
MATHEMATICAL M O D E L
The biological processes underlying initiation and growth in the cultivated mushroom
are far from fully understood (Flegg, 1975). Therefore several assumptions have to be made
in order to construct a model; one such assumption made in the model presented here is that
the initiation and growth of sporophores are both regulated by the density in the mycelium
of a single substrate. Four soluble carbohydrates (manni tol, trehalose, glucose and sucrose)
are known to be present in the mycelium (Hammond & Nichols, 1976), and it seems likely
that one of these might behave in a manner similar to that required of the substrate in the
model. A diagram of the compartmental system on which the model is based is shown in
Fig. 1, and a summary of the principal mathematical symbols used in the model is given in
Table 1.
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56
D . 0. C H A N T E R A N D J. H. M. T H O R N L E Y
~~
~
~~
COMPOST
MYCELIUM
~
SPOROPHORES
STR U CT URE
Dry weight w,“
Volume V,,
Volume V,
Substrate
density s,
Number n(t,t’)
Dry weight w(t,t’)
-I
Substrate density s,,
Fig. 1. A three-compartment model for the growth of a sii.,le flush of mushrooms.
Table 1. Principal symbols, with (for the parameters of the model) the values used in the
solution given. The number of the equation in which the symbol is first introduced is given in
Meaning
Constant in the numerator of the rectangular hyperbola
describing the dependence of sporophore growth rate on
mycelial substrate density (15)
Asymptote for the structural component of mycelial dry
weight (9)
Constant in the denominator of the rectangular hyperbola
describing the dependence of sporophore growth rate on
mycelial substrate density (15)
Exponential decay constant for sporophore growth rate (14)
Maintenance coefficient for the mycelium ( 5 )
Maintenance coefficient for the sporophores (19)
Number of sporophores at time t which were initiated
between times t’ and t’+ dt’ (10)
Total number of sporophores at time t (10)
Constant relating mycelial growth rate to substrate
density (8)
Mycelial respiration rate (20)
Sporophore respiration rate (21)
Substrate density in compost at time t (1)
Substrate density in mycelium at time t (3)
Critical substrate density in mycelium (1 1)
Time (1)
Time of initiation (10)
Rate of transport of substrate from compost to mycelial
storage (1)
Rate of utilization of substrate in mycelium ( 5 )
Rate of transport of substrate from mycelial storage
to sporophores (6)
Maximum attainable ratio sm/sc(4)
Volume of compost (1)
Volume of mycelium (2)
Initial weight of sporophores (12)
Dry weight of structural component of mycelium (2)
Weight at time t of a sporophore initiated at time t’ (13)
Total weight of sporophores at time t (13)
Growth efficiency of mycelium ( 5 )
Growth efficiency of sporophores (19)
Constant relating u, to substrate densities s, and s, (3)
Initiation rate (1 1)
Mycelial density (2)
* Value when t = 0
Units
Value
f3-l m3 d-l
1.07 x 10-4
g
375
g-1 m3
5.56 x 10-5
d-l
d-l
d-l
0.24
0.041
0.041
l2-l
m3d-’
g d-l
g d-l
g m-3
g m-3
g m-3
d
d
g d-l
2 . 4 10-5
~
22000*
8400*
18000
g d-l
g d-l
10
0.11
m3
m3
g
0.1
0.1*
g
g
g
8-1 m3 d-1
8-2
m3 d-1
g m-3
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0.83
0.83
1.329x
7.2 x 10-5
9oOoo
The muslzroom crop: a mathematical model
57
Compost
The compost is represented in the model by a compartment having a fixed volume V , and
containing substrate at a density sc. Substrate leaves the compost at a rate zlc. The quantities
sc and uc are related by the equation
The compartment which represents the mycelium is partitioned into a structural and a
storage component. The structural component has a dry weight w, and a volume K,
related by the equation
W'm = pmv,
(2)
where p, is a density. The storage component contains substrate at a density ,s, and the
volume of the storage component is considered to be the same as that of the structural
component, V,.
The rate at which substrate enters the storage component of the mycelium from the
compost, uc, will be influenced by the structural component of the mycelium, w,, and the
substrate densities sc and .s, A simple equation that can be used for this relationship is
uc = Z I U , ( S c - S , )
(3)
where 2 is a constant, making the rate of transport proportional to w, and to the difference
between the two substrate densities. This is equivalent to assuming that transport behaves
phenomenologically like diffusion, but the fact that substrate densities might be higher in
the mycelium than in the compost, due to active transport of substrate into the mycelium,
can be incorporated by multiplying sc by a factor U, which represents the maximum attainable ratio sm/sc.The equation controlling uc then becomes
uc = ZWm(USc-S,)
(4)
Within the mycelium, substrate is utilized at a rate 21, for maintenance and growth. The
rate urncan be written as the sum of the rates required for maintenance and growth,
In equation (5), m, is a maintenance coefficient and YGma growth efficiency, as defined by
Thornley (1976 ; chapter 6).
Substrate leaves the storage component of the mycelium at a rate usto support the growth
of sporophores. The difference between the rate uc and the sum of the rates urn and us
gives the net rate of input of substrate into the mycelial storage, which is related to the
substrate density by the equation
d
&(VmSm) = uc-zI,-us
(6)
Equation (2) can be substituted into equation (6) to give
M y celial growth
So far no data are available on the form of the growth curve for mycelium in compost,
because of the difficulty of separating mycelium from compost. In this model a logistic-type
growth curve has been used, following Koch (1975), who used a sequence of logistic curves
when modelling the way a colony of mycelium increases in size and weight.
A logistic growth curve for wm(t)can be achieved by writing
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58
D. 0. C H A N T E R A N D J. H. M. T H O R N L E Y
where q is a constant, thus making the specific growth rate dependent upon the storage
substrate density, sm(t). However, when equation (8) was used as it stands, it was found to
be difficult to obtain sufficient increase in snl.(t)to support sporophore initiation and
growth. The equation was therefore modified by introducing an arbitrary upper limit to
wm defined independently of Sm. It is perhaps reasonable to think of this upper limit as the
effect of the physical constraint provided by the wooden tray containing the compost and
mycelium. If this u p p x limit is denoted by A , it can be incorporated into the growth equation as follows:
dwm
(9)
(l/wm> dt = qs, [(Am-wm)/AmI
Sporophore initiat ion
The third compartment in Fig. 1 represents the sporophores. The number of sporophores
at time t which were initiated between times t' and t'+dt' can be represented by n(t,t')dt';
thus the total number of sporophores present at time t is given by
N(t)
rt
=
Jon(t,t')dt'
Chanter (1976, 1977a) has described a model for the initiation of mushroom sporophores
which gives reasonable fits to data on the number of new sporophores appearing each day.
This model is based on two assumptions: (i) that the initiation of sporophores occurs when
(and only when) the substrate density in the mycelium exceeds a threshold level, and (ii) that
the rate of initiation is proportional to the dry weight of the structural component of the
mycelium and to the difference between the actual and threshold substrate densities in the
mycelium. Assuming also that sporophores do not die and are not harvested, n(t,t') can
be described by the equation
where s,, is the critical value of s,,
process itself occuis when t = t'.
and h is a constant of proportionality. The initiation
Sporophore growth
The dry weight at time t of a sporophore initiated at time t' can be written as w(t,t'), so
that if the initial weight of a sporophore is assumed to be constant it can be written
wo = w(t,t)
Thus the total dry weight of the sporophores present at time t is given by
W(t)
rt
=
Jow(t,t')n(t,t') dt'
(12)
(13)
Chanter & Cooke (1978) show that a Gompertz (1825) growth curve,
dw
(l/w) - = ,ue-kt
dt
where w is the dry weight of a sporophore at time t, and u
, and k are parameters, provides a
reasonable description of the growth of mushroom sporophores. However, they also give
experimental evidence which suggests that the growth rate is dependent on the nutrient
status of the mycelium, and suggest that this effect can be modelled by making ,u dependent
on the substrate density in the mycelium, using a rectangular hyperbola,
f,b
where a and h are parameters.
=
as,(t)/[l
+ hSJt)]
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(15 )
The mushroom crop: a mathematical model
59
Substituting equation (15) in (14) we get
which, together with the initial condition in equation (12), defines the growth rate for any
sporophore.
The total weight of sporophores at time t is given by equation (13), and the rate of increase
in total dry weight is therefore
dW
dt = dt
”[ Sofw(t,t’)n(t,t’)dt’]
Equation (17) can be shown (see Appendix) to be equivalent to
dW = w(t,t)n(t,t)+ Sutn(t,t’)[8w(t,t’)/8t]dt‘
dt
(18)
The growth of sporophores is assumed to take place with an efficiency YGs,and the sporophores are assumed to have a maintenance coefficient m,.Thus a t time t the rate of transfer
of substrate from the mycelium to the sporophores necessary to support initiation and growth
will be
The use of equation (19) as it stands implies that the processes of initiation and growth both
operate at the same efficiency YGs.
Respiration
The rates of respiration of the mycelium and sporophores, Rm and R, respectively, can
be derived from the respiration model developed by Thornley (1976), and follow from his
equation (6.18). R,,and R,are given by
Respiration is the only process in the present model by which substrate can leave the
system, and in a numerical solution these equations can be used to check that the substrate
present at time 0 is equal to that present at time t plus that respired between times 0 and t.
Summary
The state variables of the model developed in this section are w,,(dry weight of mycelium),
s, (substrate density in compost), ,s (substrate density in mycelium), n(t,t’) (distribution of
number of sporophores by time of initiation) and w(t,t’) (distribution of weight of sporophores).
The model has 16 parameters, which relate to the various sections of the model as follows:
compost, V,; mycelium, pm, A,, U and 2 ; mycelial growth, q, m, and YGm;sporophore
initiation, s,,,~,wo and A ; sporophore growth, a, h, k , m, and YG,.
The model is defined by equations (l), (4), (5), (7), (9), ( l l ) , (12), (16), (18) and (19).
PARAMETER VALUES
The model developed above can be solved numerically for any reasonable set of parameter values and initial values. To obtain a solution which resembles the behaviour of a
mushroom crop, the parameter values have to be chosen with care. Estimates are available
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60
D. 0. C H A N T E R A N D J. H. M. T H O R N L E Y
for a few of the parameters, but for many of them little information is available. Parameters
for which some information is available are as follows :
(i) V,. The volume of compost can be chosen to correspond with the volume used in the
system with which comparisons are to be made. The mushroom trays currently in use at
the Glasshouse Crops Research Institute (G.C.R.I.) hold about 0.11 m3 of compost.
(ii) sc. The dry matter content of compost at filling is about 30 %, and 0.1 1 m3 of compost
weighs about 36 000 g (fresh weight), and so its density is about 100000g (dry matter) m-3.
This gives an upper limit to the initial value of sc. A lower limit can be obtained by calculating the expected yield from a crop. A good yield for a G.C.R.I. tray is 1000 g (dry weight),
and so a lower limit for sc is 9 100gm-3. This figure does not allow for growth of the mycelium,
but since no information is available on the dry weight of mycelium in a fully run tray of
compost, it is difficult to make an allowance for the mycelium.
(iii) wm(0).About 170 g (fresh weight) of spawn is used to inoculate a tray. Probably no
more than 5 % of this is dry weight of mycelium, so an upper limit on the initial value of
wm is 8.4 g.
(iv) U. It does not make biological sense for this parameter to be less than unity, and anything in excess of 100 would also seem to be physiologically unreasonable.
(v)pm. The density of mycelium is about 1 g (fresh weight) ~ m - so
~ ,the density in terms
of dry weight will be about 0.09 g ~ m (90000
- ~ g m-3), assuming a dry matter content of 9
( v i ) m,, YGm.
No direct estimates of these parameters are available, but it seems reasonable
that they will be similar to their counterparts in sporophores, and the same values can be
used [see (xi)and ( 4 1 .
(vii) smc.Substrate densities in the mycelium might be expected to vary so that the storage
component of the dry weight of the mycelium varies between about 5 and 40 % of the total
dry weight. The critical level s, would then have to be somewhere near the middle of this
range, i.e. between about 9000 and 27000 g IT^-^, (i.e. between 10 and 30 % of pvJ.
(viii)wo.This parameter has been taken as 0.1 g throughout this investigation. The value
was chosen because any sporophore which reaches this weight is fairly certain to continue
developing. In a real crop many primordia are formed and then abort, but these primordia
seldom reach a dry weight of 0.1 g.
(ix) a, h. Direct estimates of these parameters are not available, but as,(t) divided by
1+ hs,(t) should have a value similar to the values for p estimated by Chanter & Cooke
(1978), the mean value of which is about 0-95 d-l.
(x) k. Estimates of this parameter are also given by Chanter & Cooke (1978). The mean
value of these estimates is about 0.25 d-l.
(xi) m,.This parameter can be estimated from an experiment carried out by J. B. W.
Hammond (personal communication), and analysed by Chanter (1 976). The estimate
obtained by the second method of Chanter (19773) is 0.041 d-l.
(xii) YGs.Similarly, the estimate 0.83 is available.
BeaIing in mind the constraints and estimates listed above, the set of parameter values
indicated in Table 1 has been found to give a solution with features similar to those observed
in the first flush of a mushroom crop.
x.
RESULTS
The model has been solved numerically by computer, using the parameter values given
in Table 1. The solution is of course approximate, since a discrete time interval had to be
used; an interval of 1 h (0.04167 d) was chosen as a convenient unit. An investigation of the
effects of using a smaller time interval showed that the solution did not change much, but
that the computer time taken increased considerably.
The behaviour of the state variables s, ,s and w, is shown in Fig. 2. During the early
part of the solution nearly all the substrate which enters the mycelium is used for mycelial
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61
The mushroom crop: a mathematical model
I 11i t iu I ion
I
5
10
15
20
Time. t (d)
25
30
I
35
Fig. 2. Changes in the substrate densities in the compost and mycelium, s, and s, respectively
(-),
with time for the parameter values given in Table 1. The critical substrate density in the
mycelium, s, and dry weight of the mycelium, w, (---), are also shown.
growth, but when the rate of mycelial growth begins to slow down the substrate density in
the mycelium builds up rapidly until the critical density is reached. Sporophore initiation
and growth then take place and the substrate density in the mycelium falls, but begins to
increase again when the sporophore growth rate slows down.
The initial mycelial growth rate can be expressed by the time taken for the mycelium to
double its initial dry weight, which is 3.73 d. Observations on the growth rate of mushroom
mycelium in liquid culture have shown the doubling time to be about 3 d (D. A. W. Wood,
personal communication). The curve for w, shown in Fig. 2 represents the structural dry
weight only. The total dry weight of the mycelium, including storage, rises to a maximum
of 426.9 g at 28.6 d and falls to 397.2 g (a drop of 7.0 %) at 33-5d before beginning to increase
again, Comparable experimental data for A . bisporus are not available, but a similar drop
has been observed in other basidiomycetes, e.g. Flammulina velutipeb, where the decrease
is about 24 % (Kitamoto & Gruen, 1976).
The critical substrate density in the mycelium is reached after 25.4 d, and this is similar
to the time taken between filling and the appearance of first pins in a real crop. Initiation
continues for 3.50 d ; this is somewhat shorter than the 5 d estimated by fitting Chanter’s
(1977a) initiation model to data. However, this seems reasonable in view of the fact that in
a real tray initiation does not begin over the whole surface simultaneously, and heterogeneity within trays would tend to lead to an overestimate of this time.
The total number of sporophores initiated ( N ) is 162.6, a figure which also compares
favourably with the numbers observed in real crops. The subsequent growth of the sporophores is shown in Fig. 3, for the first and last mushrooms initiated. The growth curves
compare reasonably well with those shown by Chanter & Cooke (1978); the fact that a
wide range of growth curves can be obtained in practice makes it pointless to try and obtain
an exact match to any particular growth curve.
By using equations (20) and (21) it is possible to account for all the substrate present at
time t = 0 at any other (positive) value of t. Table 2 shows the distribution of substrate at
time t = 37.5 d. The three major uses of the substrate removed from the compost are for
mycelial dry matter, sporophore dry matter and mycelial maintenance. Of the dry matter
present at t = 37.5 d, 45-5 % is in the sporophore compartment and the remainder is
mycelium. There are no data for A . bisporus with which these figures can be compared, but
comparable figures for another basidiomycete, Coprinus lagopus, can be estimated from
data given by Madelin (1956). Taking average figures for the period when the sporophores
are mature (12-7 to 15.0 d after initiation), 44 % of the dry matter is in the sporophores.
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62
D. 0. CHANTER A N D J. H. M. T H O R N L E Y
2
4
6
8
1
0
Time since induction. ( t - t ’ ) ( d )
Fig. 3. Growth curves for sporophores initiated at the beginning and end of the initiation period
for the solution to the single-flush model, using the parameter values given in Table 1.
Table 2. The distribution of substrate at times t = 0 and t = 37-5 d in the solution to the
single-jlush model with the parameter values given in Table 1
Substrate (g)
I
Present in compost
Present in mycelium:
Structure
Storage
Sporophore dry weight
Respired to support:
Mycelial growth
Mycelial maintenance
Sporophore growth
Sporophore maintenance
Total
h
Att=O
2420.0
7
At t = 37.5 d
1202.3
4-0
0.4
-
371.9
31-0
337.0
2424.4
75.3
255.3
69.0
82.6
2424.4
SENSITIVITY ANALYSIS
To investigate the behaviour of the model, increments of 10 % were made to each of the
initial values s,(O) and w,(O) and parameter values (excepting V,, pm and wo) in turn, and
solutions were obtained. Table 3 shows the effects of these increments on five aspects of
the solution: (a) initial mycelial growth rate (doubling time); (b) time at which initiation
begins ; (c) duration of initiation ; ( d ) number of sporophores initiated ; (e) sporophore
growth rate. Some of the more interesting results are discussed below.
Mycelial growth rate
The parameter q, which ostensibly controls the growth rate 9f the mycelium, actually has
only a small effect on the time taken for the mycelium to double its initial dry weight. It
is the rate at which substrate is entering the mycelial storage from the compost, uc, which
limits mycelial growth. The effect of q is mainly to determine the substrate density in the
mycelium.
Time at which initiation begins
The parameter, A,, which determines the dry weight at which the mycelium stops growing, has an appreciable effect. An increase in this dry weight causes the mycelium to reach
its growth ceiling later, and hence the later availability of substrate for initiation. The
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The mushroom crop: a mathematical model
63
Table 3. The efect of incrementing initial values and parameter values by 10 % on (a) the
time taken for the mycelium to double its initial dry weight (d), (b) the time at which initiation
starts (d), (c) the duration of initiation (d), ( d ) the number of sporophores initiated, and
( e ) the average dry weight of sporophores 6 d after the onset of initiation (g)
Parameter
incremented
(4
(b)
(c)
(4
(el
3.73
3.37
3.73
3.73
3-38
3.73
3.70
3.78
3.45
25.43
22-40
26.26
22-40
22-56
24.91
25.57
25-96
23.12
26.13
-
3-50
3.70
3.45
3.66
3.65
3.50
3.38
3.46
3.59
3.32
3.40
3-20
3.67
3.60
3.49
3.60
162.6
182.6
170.6
175.7
173-6
162.7
177-5
158.3
163-5
177.9
161.5
132.1
177.0
169.1
161.9
179.3
1.100
1.086
1.103
1.088
1*092
I a099
1.088
1.102
1.097
1.093
1.1 10
1.342
0.973
1 -020
I -098
1.087
-
-
-
-
effect of the critical substrate density, smc:is not as large as might be expected because the
substrate density in the mycelium is increasing quite rapidly immediately before initiation.
Duration of initiution
Perhaps the most notable feature of the effects on duration is their small size; apart from
the parameter a, no parameter changes the duration by more than 6 %when incremented by
10 %. The exception, a, controls the growth rate of the sporophores, and the larger this
is the more quickly the substrate in the mycelium is used up and the sooner the substrate
density falls below the critical level.
Number of sporophores initiated
The number of sporophores initiated is closely related to the duration of initiation, and
similar parameter effects were found for both responses.
One parameter with an unexpectedly large effect on the number of sporophores is q,
which is associated with mycelial growth. Also, the effect of increasing q is to increase the
number of sporophores, an effect which is in the opposite direction to that which might
be expected, since an increase in q causes a small increase in the growth rate of the mycelium,
increasing the substrate requirement of the mycelium and reducing the substrate available
for initiation. The resolution of this apparent anomaly lies in the presence of the term
( A , - w,)/A, in equation (9). The effect of increasing q by 10 % on the solution is to make
w, equal to 300.9 g when initiation begins, compared with 289-8 g in the standard solution.
This leads to a reduction of 13 % in the value of the term given above, and hence the growth
rate of the mycelium during initiation is actually reduced by incrementing q.
Another parameter of interest is A, the initiation rate, which has hardly any effect on
the number of sporophores initiated. This is because an increase in h shortens the duration
of initiation as well, and the two effects almost cancel out.
Sporophore growth rate
Only four parameters affect the initial growth rate of the first sporophore to be initiated:
these are the three parameters directly concerned with sporophore growth, a, h and k , and
5
MIC
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106
64
D. 0. C H A N T E R A N D J . H. M. THORNLEY
the critical density smC.The effect of other parameters on the subsequent growth of sporophores was examined by looking at the average dry weight of the standing sporophores 6 d
after the onset of initiation. It was found that a, h and k dominate the effects on sporophore
growth, but s,,, has only a small effect. Among the remaining parameters, the initial value
of s, has the biggest effect, and a decrease in this parameter leads to an increase in the
average weight. This is because fewer sporophores are initiated and there is more substrate
available for each individual sporophore.
DISCUSSION
The model developed in this paper describes the growth of mycelium, and the initiation
and growth of sporophores, up to the time at which the sporophores would normally be
harvested. In its qualitative behaviour, the model agrees reasonably well with real crop
behaviour, but any serious attempt at this stage to validate the model is not possible. Of the
19 parameters listed in Table 1 only two (the initial values of s, and w,) can be directly
controlled. Changing s, experimentally (by using more or less compost) also involves
changing either V, or the density to which the compost is compressed, and both these
alternatives alter the physical properties of the compost in a manner likely to affect other
parameters of the model in an unknown way. Similarly, one effect of reducing w, experimentally (by using less spawn) is that other competing organisms present in the compost
have a better chance of becoming established, an effect which cannot be allowed for in the
present model.
There are many other aspects of mushroom growing which are not represented in the
model, for example, the role of the casing layer is not considered at all. The importance of
some of the factors which have not been included is discussed by Chanter (1976).
Because the model does not include harvesting, the results presented in this paper are of
limited use in research aimed at increasing the yield of mushroom crops. However, these
results do provide a basis for a model which includes harvesting, and which exhibits several
flushes. This model will be described in a subsequent paper, and it also provides another
parameter which can be varied experimentally, so that an attempt can be made to validate
the model, at least in respect of its response to different harvesting regimes.
APPENDIX
The derivation of equation (18)
Equation (17) involves the differentiation of an integral with variable limits. The formula
for this operation is (Whittaker & Watson, 1969)
Comparison of equations ( A l ) and (17) yields 8 = t, x = t‘,f(0,x) = w(t,t’)n(t,t’), b(8) = t.
and a(0) = 0. Noting that n(t,t’) is constant with respect to t within the range of integration,
the three terms on the right hand side of (Al) are therefore
da
f(8,a)- = 0
d8
Substituting these expressions in equation ( A l ) leads to equation (18).
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The mushroom crop: a mathematical model
65
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