Bothalia 13, 3 & 4: 5 6 9 -5 7 6 (1981)
Miscellaneous ecological notes
VARIOUS A U TH O R S
IN V E S T IG A T IO N IN T O T H E S IG N IF IC A N C E OF P L A N T D ISP E R S IO N IN A SS E S S IN G P A ST U R E C O N D IT IO N
The description o f pasture condition in terms of percent cover and species com position is undesirable as these m easures are not the sole criteria. It has been found, for example, at the Rietvlei Agricultural Re search Station, near P retoria, that heavily fertilized mown veld has a lower basal cover than unfertilized grazed veld, yet produces m ore than three times the am ount o f herbage than the unfertilized veld. The ul tim ate aim o f pasture m anagem ent is the m ainte nance o f a healthy sward and it is possible that the m easurem ent o f the uniform ity o f plant cover may be o f value in establishing an objective concept of pasture condition. It is intended, in this Note, to dis cuss various aspects o f plant dispersion and their use in interpreting certain phenom ena. In this paper the nature o f the non-random ness is term ed ‘overdis persed’ or ‘contagious’ when individuals tend to be clumped together and ‘underdispersed’ or ‘regular’ when individuals are evenly scattered. It is stressed that the term s overdispersed and underdispersed refer to the distribution curves o f d ata and not to the pattern o f plant individuals on the ground.
For the purpose o f this investigation the Bruce Levy point quad rat (Levy & M adden, 1933) was used. Three bridges were m ade with 20 points each
spaced at 1 cm, 2 cm and 5 cm intervals. The bridges,
referred to in this paper as quadrats, were laid either
100 or 200 times at random in each experimental
plot, all o f which were smaller than one hectare. A ‘t ’ test showed that sufficient samples had been taken to
ensure an accuracy o f 10 percent o f the m ean num ber
o f strikes per bridge at p = 0,05. In one analysis a bridge with 30 points at 5 cm intervals was used.
Random distributions may be described by the Binomial distribution, or by the N orm al and Poisson approxim ations to that distribution. When the frequency curves o f plants occurring in quadrats are com pared with any one o f these curves (depending on the nature o f the sample), it is noticed that there is often a considerable discrepancy between the observed and calculated values. C lapham (1936) and Black man (1942) com pared frequency distributions of species occurring in quadrats with the Poisson dis tribution and they found that many o f the species were not random ly dispersed. They expressed this deviation from the expected distribution by com par ing the observed variance o f the samples with the ex pected variance. Neyman (1939) and Thom as (1949) expressed the principle o f contagious distribution in generalized form s o f the Poisson distribution. The various frequency distributions are described below.
The Binomial distribution
The frequencies o f 0, 1, 2, . . . successes (strikes) in N sets are given in term s o f the binom ial expansion o f N (p + q)n where N is the num ber o f quadrats, p the probability o f an event occurring, q the probability o f the event not occurring and n the num ber o f trials (points per q uad rat). An easy m ethod o f calculating the successive term s in a binom ial expansion is by using the relationship:—
log (probability o f x plants
occurring in any 1 quadrat) = log n! - log
(n-x)! - log x! + ( n - x ) log q + x log p.
In the Binomial distribution the mean is np and the variance npq.
The Poisson distribution
This distribution is given by the series
probability o f x plants = mxe~ m
occurring in any 1 quadrat x!
where m is the m ean num ber o f plants per q u adrat. In the Poisson distribution the mean is equal to the variance. When m is very small the distribution is an approxim ation to a Binomial distribution.
The Neyman contagion
This distribution is a generalization o f the Poisson distribution and is given by:
p (x = 0) = e -mi (l_e m2) and subsequent term s by
p ( x = k +1) = mi m2e ^ L p ( x = k - 1)
k + 1 t = 0 t!
where the param eters m, and m2 are proportional to
the m ean num ber o f groups per quadrat and mean num ber of individuals per group respectively (Ney m an, 1939). The param eters can be estim ated from the first and second m om ents o f the distribution as
m2 = 0*2 and m, = ^t1/m2
When m2 becomes very small and m ,m : is finite the
distribution approaches the Poisson where = 1.
The Thomas distribution
This distribution is given by
p (k = 0) = e _m and subsequent terms by
k
p(k) = L m re~m (rX)k~r e~rX
r = l r! (k-r)!
The param eters m and 1 -(-X are the mean num ber o f groups per quadrat and mean num ber o f individuals per group respectively and are obtained from the first and second m om ents o f the distribution where
Hi = m (1 -i-X) and h2 = m (1 + 3X + X2)
570 M I S C E L L A N E O U S E C O L O G IC A L N OTE S
N u m b e r o f s t r i k e s N u m b e r o f s t r i k e s
N u m b e r o f s t r i k e s N u m b e r o f s t r i k e s
N u m b e r c f s t r i k e s N u m b e r o f s t r i k e s
Fi g.1.— Frequency distributions o f all plants com pared with the expected Binom ial distribution in a, plot 1; b, plot 2; c, plot 3; and d, plot 4 with 20 points per quadrat at 5 cm espacem ent and 100 quadrats; e, plot 1; and f, plot 3 with 10 points per quadrat at 10 cm espacem ent and 100 quadrats.
The frequency distribution o f all plants occurring in plot 1 is given in Fig. la. The general appearance o f the veld is that it is uniform and in excellent condi tion. The hay cuts from this heavily fertilized plot are m ore than three times those from unfertilized veld.
The veld initially was a Themeda triandra,
Hypar-rhenia hirta, Trachypogon spicatus type but, as a
result o f the treatm ent, changed to Setaria
nigrirostris, S. perennis and Eragrostis chloromelas,
with relics o f Themeda triandra and Hyparrhenia hir
ta. An exam ination o f the figure will show that the fit
o f the observed values to the expected Binomial curve
is very close (Px2 = 0,8 -0 ,9 ). The V /n p q ratio
(1,0104) is alm ost unity, signifying that the observed and expected param eters show close agreem ent. In
plot 2, in which the treatm ent is the same as for plot
1, except that it is limed, the distribution also does
V A R IO U S A U T H O R S 571
certain climax species are driven out and are replaced by others lower in the succession, and it is during this period that the productivity and vigour o f certain species is profoundly affected. The bim odality of the d istrib u tio n s in d icates th a t one is sam pling a heterogeneous population, consisting of areas of low cover and others o f high cover. The Neyman and Thom as distributions are built on premises that (a) the individuals in a cluster are random ly dispersed and (b) the clusters in a population are random ly dispersed. Because o f the close agreement between the observed and expected values obtained by A r chibald (1948, 1950), Barnes & Stanbury (1951) and in this paper, it may be accepted that these premises are valid. The bim odal curves, which cannot be ade quately described by the Neyman and Thom as series, may therefore be explained as follows: the clusters o f species which react unfavourably to high fertilizer dressings die out leaving bare areas while those species, norm ally random ly dispersed, which have responded to the fertilizer are showing increased vigour and a subsequent increase in num ber and size and are becom ing contagiously dispersed. For exam
ple, it has been observed that tussocks o f Themeda
triandra and Hyparrhenia hirta die out and only after
their death and the dissem ination o f their litter is the space which was occupied by those tussocks coloniz ed by other species.
C orby (pers. com m .) showed that the further apart the points are the closer the distribution o f species approached the Poisson values. Tidm arsh & Hav- enga (1955) showed experim entally that when the in dividuals o f a population are dispersed at random , the distance between the points must exceed the aver age diam eter o f the individual to obtain a close fit to the expected Binomial distribution. The data for Fig. la-d were obtained from points spaced at 5 cm inter vals. The distributions for plots 1 and 3 have been repeated in Figs le and If to show the effect of point espacem ent. The points of the 5 cm bridge were
num bered from 1 to 20, and by recording each strike
against the point num ber it was later, by taking alter nate points, possible to examine and com pare the dis tributions for the two espacements. The data presen ted in Fig. 2 are from a plot which has been m oder ately grazed by sheep and cattle subsequent to an accidental fire. P rior to the fire the veld had been leniently treated and used mainly as a source o f hay. A com parison o f the figures given for the 5 cm and 10 cm espacements in Fig. 1 shows that in plot 1 the closer fit to the expected values is obtained when the
points are spaced at 10 cm, while the non-random
distribution given for 5 cm points in cam p 3 becomes random when the points are spaced 10 cm apart. In Fig 2a it is seen th at the distribution changes from a
highly contagious one for points at 1 cm to a
somewhat doubtful distribution for points at 2 cm to
a random dispersion with points at 5 cm. In the distributions for the 1 cm espacement (Fig. 2b) there is close agreem ent between the observed values and the Neyman contagion, as one would expect. It is of
interest to note that in the distributions for the 2 cm
espacement (Fig. 2c), the Poisson values do not differ significantly from the observed values, whereas the Binomial series does not describe the distribution adequately. Because the Poisson distribution is an approxim ation o f the Binomial when the probability o f an event occurring is small, it is felt that in this case the Poisson values should be discarded and that it should be accepted that the plants are contagiously dispersed for th at particular espacement. The total num ber o f strikes for the three espacements show a fairly close agreem ent, being 619, 610 and 587 for the
N u m b e r o f s t r i k e s
Fig. 2 .— a, C om parison o f frequency distributions o f plants occu r ring in 200 quadrats, each with 20 variously spaced poin ts, with the expected B inom ial curve for 5 cm espacem ent; b, com p arison o f the B inom ial, P oisson and N eym an series with the actual distribution o f plants occurring in 200 quadrats each with 20 p oints spaced 1 cm apart; and c, o f 200 quadrats each with 20 poin ts spaced 2 cm apart.
1 cm, 2 cm and 5 cm espacements respectively. A ‘t ’ test showed that the differences between these figures were not significant. It is the w riter’s contention th at a cover m ade up o f evenly dispersed small individuals is m ore desirable than one o f large individuals and correspondingly large bare areas between the individ uals, which causes the soil to be more erodable. It is probable that the com petition for light and w ater is less when the dispersion o f the plants is uniform . W here clusters are present, those individuals on the periphery o f the clusters receive the greater light and w ater benefits.
572 M I S C E L L A N E O U S EC O L O G IC A L N O TE S
72
--D 50
Fig. 3 .— Frequency distributions o f individual species from plot 5 com pared with the p o isso n , N eym an and T hom as distribu tions from 100 30-point quadrats with points spaced 5 cm apart; a, Trachypogon spicatu s; b, Schyzachyrium sanguineum; c, Tristachya leucothrix', and d, Eragrostis capensis.
Ob s e r v e d
Po i s s o n
N e y m a n
T h o m a s
shown that num bers of species in a plant com m unity are not random ly dispersed and that the m ore a b u n dant a species is, the greater is the degree o f aggrega tion present. Ashby (1948) m aintains that it is possi ble that species, without a biological predisposition to overdispersion, are random ly dispersed when they first occupy an area. Barnes & Stanbury (1951) found that as the vegetation developed, through spreading o f ‘islands’ the distribution becomes highly con tagious. If it is accepted that initial colonization is random and that the random ly distributed species la te r becom e co n tag io u sly d is trib u te d , it is reasonable to suppose that subsequent development o f the vegetation will take place in, what may be term ed as a series o f waves o f random ness, contagion and a subsequent random ness as the species is re placed. This process would continue until some state o f equilibrium is reached where the climax species are contagiously dispersed while the pioneer relics are random ly distributed. Should the succession be driven back, then the reverse process would take place with the pioneer species tending tow ards co n tagion and ultimately the climax species would be random ly dispersed. From a series o f analyses m ade at the Rietvlei Research Station, there is evidence which supports the argum ent propounded above.
Fig. 3 gives four frequency distributions from plot 5 which is rotationally grazed. From observations, this plot may be regarded as climax grassland and is in good ‘health’. A bridge with 30 points spaced at 5 cm
intervals was laid 100 times at random in the camp.
The V /n pq ratio is 0,9525 and x2 probability lies be
tween 0,8 and 0,9. It will be noted th at the distribu
tions for Trachypogon spicatus, Schizachyrium
sangiunewn and Tristachya leucothrix show a closer
agreem ent to the Neyman and T hom as series than to the Poisson distribution. (W hen the probability of a plant being struck is very small (i.e. when np is small) then the Poisson series may be used to describe the distribution, however, when np becomes larger it is
better to use the Binomial distribution). Eragrostis
capensis, a relic pioneer is distributed at random , as
are many other rare species such as Brachiaria serrata
and Eragrostis racemosa (not plotted). The data for
the four species given in Fig. 4 are from plot 3. It will be remem bered th at the distribution o f all individuals in this plot was not random for points spaced at 5 cm
intervals (Fig. lc). H eteropogon con tortu s and
Setaria perennis may be regarded as species near the
climax stage and are overdispersed. E. capensis and
E. racemosa have distributions approaching ran
VA R I O U S A U T H O R S 573
Fi g. 4 .— Frequency distributions o f individual species from plot 3 com pared with the P o isso n , N eym an and T hom as distribu tions from 100 30-point quadrats with points 5 cm apart; a, Setaria perennis; b, Eragrostis capensis; c, Eragrostis racemosa; and d, H eteropogon contortus.
O b s e r v e d
P o i s s o n
N e y m a n
T h o m a s
data suggest that both climax and pioneer species are contagiously dispersed. It has been shown by the writer that some species in which the frequency distribution cannot adequately be described by the Neyman or Thom as series, a better fit can be obtain ed if the points are moved further apart. It is quite
possible that Setaria perennis and H eteropogon con
tortus would give a better fit to the contagious series
if the points were moved apart. However, with the bridge used for these analysis, it was not possible to
use 10 points at 10 cm intervals as there would be too
few points to give an adequate representation of the d isp ersio n . It is n evertheless obvio u s th a t the distribution is not random for these two species. From Figs 3 & 4 it appears as if the dispersion o f the species can m ake a considerable contribution to the interpretation o f botanical analyses, and the assess m ent o f the condition o f the sward. In plot 5, in which all plants are random ly dispersed, it is noticed
that certain climax grasses such as Schizachyrium
sanguineum, Tristachya leucothrix and Themeda
triandra are not ran d o m ly d istrib u te d , whereas
grasses near the pioneer stage such as Eragrostis
capensis, Brachiaria serrata and Eragrostis racemosa
are random ly distributed. The distributions for these last two grasses are not given, but the d ata indicate that they are random ly dispersed. In plot 3, in which the distribution o f all plants is not random , it is found that both climax species and pioneer species are overdispersed.
This paper may be regarded as a preliminary report o f certain points which came to notice during routine botanical analyses o f grazing experiments on the Rietvlei A gricultural Research Station. It is felt that a deeper insight into the mathem atical distribution o f species and o f individuals will throw light on the mechanism o f vegetation changes and the direction in which these changes are proceeding. It is possible that the following points will indicate stability: (a) the frequency distribution o f all plants is random when the points are closely spaced, the criterion of spacem ent being determ ined by the average size of the plant and the density o f the cover; (b) the climax species are contagiously and the pioneer species ra n domly dispersed. When these conditions are not ful filled, then there are indications that vegetation changes are occurring and a concept o f the m agni tude o f these changes may be obtained by com paring the observed param eters o f the population with theoretical param eters.
R EFER E N C ES
Ar c h i b a l d, E . E. A . , 1948. Plant pop u lation s. I. A new ap p lica tio n o f N ey m a n ’s con tagious distribution. Ann. Bot. 12: 2 2 1 -2 3 5 .
A r c h i b a l d , E. E. A . , 1950. Plant pop u lation s. 11. The estim ation o f the number o f individuals per unit area o f species in heterogeneous plant pop u lation s. Ann. Bot. 14: 7 - 2 1 . As h b y, E ., 194 8 . S ta tis tic a l e c o l o g y . II. A r ea ssessm en t. Bot. Rev.
574 M I S C E L L A N E O U S E C O L O G IC A L N O T E S
Ba r n e s, H . & St a n b u r y, F. A ., 1951. A statistical study o f plant distribution during the co lo n iza tio n and early develop m en t o f vegetation on china clay residues. J. Ecol. 39: 1 7 1 -1 8 1 .
Bl a c k m a n, G . E ., 1935. A s t u d y b y s t a t is t ic a l m e t h o d s o f t h e d is t r ib u t io n o f s p e c ie s in g r a s s la n d a s s o c i a t i o n s . A nn. Bot.
49: 7 4 9 -7 7 7 .
Bl a c k m a n, G . E ., 1942. Statistical and ecological studies in the distribution o f species in plant com m u n ities. I. D ispersion as a factor in the study o f changes in plant p o p u la tio n s. Ann. Bot.6: 3 5 1 -3 7 0 .
Cl a p h a m, A . R ., 1936. O ver-dispersion in grassland and c o m m unities and the use o f statistical m ethods in plant eco lo g y .
J. Ecol. 24: 2 3 2 -2 5 1 .
Le v y, E. B. & Ma d d e n, E. A ., 1933. The point m ethod o f pasture analysis. N. Z. J. Agric. 46: 2 6 7 -2 7 9 .
Ne y m a n, J ., 1939. On a new class o f ‘c o n ta g io u s’ distributions a p plicable in en tom ology and bacteriology. Ann. M ath. Stat.
10: 3 5 - 5 7 .
Pi d g e o n, I. M . & As h b y, E ., 1940. Studies in applied eco lo g y . I. A
statistical analysis o f regeneration fo llo w in g protection from grazing. Proc. Linn. Soc. N . S. W. 65: 1 2 3 -1 4 3 .
Th o m a s, Ma r j o r i e, 1949. A generalization o f P o isso n ’s Binom ial limit for use in eco lo g y . Biom etrika 36: 1 8 -2 5 .
Ti d m a r s h, C . E. M . & Ha v e n g a, C . M . , 1955. The w heel-point m ethod o f survey and m easurem ent o f sem i-open grasslands and karoo vegetation in South A frica. M em. bot. Surv. S. A fr. 29.
H . H . V O N B r o e m b s e n *
*This and the fo llo w in g ecological n ote were written by the late H . H. von Broem bsen o f the Botanical R esearch Institute, Pretoria, w ho died in 1966. The notes were prepared for publication by J. W. M orris o f the sam e Institute. Address: Private Bag. X I 16, P retoria, 0001.
A SIM P L E M E T H O D FOR D E T E R M IN IN G T H E D E N SIT Y O F P L A N T S IN A R A N D O M L Y -D IS P E R S E D P O P U L A T IO N
It is sometimes necessary to determ ine the densities o f plants in experimental plots which have been sown to some crop or other. Where the seed has been sown in rows, it is a simple m atter to count the num ber of plants growing in unit lengths o f row, and then com pute the total num ber of plants per plot. W here the seed, however, has been broadcast the m ethod in which the density o f plants, or the total num ber of plants per plot, is determ ined, is different.
When seed is broadcast over a plot, the m anner in which the seed falls to the ground is a random p ro cess, and the dispersion of the seed over the plot may be described by the Poisson distribution (Feller,
1957). The Poisson distribution is given by
P(k plants growing on unit area a) = d ake~da k!
where d is the density of plants per unit area. The probability o f zero plants growing in any unit area is then e _da. Now, if n quadrats are placed at random in a plot, and it is found that y quadrats contain no plants then
y = n e _da
a n d J = log n -lo g y ... {1)
a log e
which means than the density o f plants may be com puted from a knowledge o f the total num ber of q u ad rats which are used for sam pling and the num ber o f quadrats containing no plants. By using this relationship the necessity for counting the num ber o f individuals in the quadrats is elim inated.
W hen using the relationship (1) care must be taken to choose a quadrat size which will not be, (a) so large that it always contains plants, or, (b) so small th at the num ber o f empty quadrats is too large. It is recom m ended that a rough estim ate o f the average
density o f the individuals first be m ade over all treat ments in the experim ent. The size o f quadrat which will give the desired percentage o f em pty quadrats may then be com puted from
a ' = log 100- l o g y^
d log e
where a is the area o f the q u ad rat, y the desired percentage o f em pty quadrats and d the estim ate o f average density. It is recom m ended that y equal 50 to 60 percent. W ith y equal to 50 percent, 34 percent o f
the quadrats will contain one plant, 12 percent two
plants, and four percent m ore than two plants. In order to determ ine the num ber o f quadrats which are required to detect predeterm ined signifi cant differences in densities the following relation ship is used:
t = __^?..Z_X2 %/n ... (2)
•s/v
Since the variance (v) o f the Poisson distribution is
equal to the m ean (da), equation (2) may be re
written in the form
n =
(d0a)J
where d0 is the difference in density you wish to
regard as being significant, and t = 1,96 for P = 0,05, or 2,58 for P = 0,01.
The estim ates o f density should be transform ed, using a square root tran sfo rm atio n , before being subjected to an analysis o f variance.
R EF E R E N C E
Fe l l e r, W ., 1957. An introduction to prob a b ility theory and its applications. N ew York: W iley.
H. H . v o n B r o e m b s e n
A N O T E O N T H E E X T E N S IO N O F T H E D E G R E E R E F E R E N C E SY STE M FOR C IT IN G B IO L O G IC A L D IS T R IB U T IO N R EC O R D S TO N O R T H O F E Q U A T O R A N D W EST OF G R E E N W IC H M E R ID IA N
The Degree Reference System proposed by
Edwards & Jessop (1967) and Edwards & Leistner
(1971) has been in general use in South A frica for well over ten years. A ttention has recently been draw n to a requirem ent for extending the System for plotting the distribution o f plants for the whole o f A frica and for surrounding islands such as T ristan da
