


















Population growth, which is a central process of
ecology, produces changes in community structure (Krebs 1994; p.
198). A population that has been released into a favourable
environment will begin to increase in numbers, and will not normally
grow with a constant multiplication rate (Krebs 1994; p. 198). The
simplest model of population growth, and the most intuitively obvious
one, is based upon the straightforward quantities of three properties
of population change; births, deaths and migration (emigration and
immigration) (Silvertown 1982; p. 6).
Generations overlap in many populations (including Lemna) are
continuous because their offspring begin to reproduce while their
parents are still alive and reproductively active (Blundon 1994).
When such a population is established with a few individuals in a
suitable environment, the initial rate of population growth is
exponential (Blundon 1994). The population increases without limit if
the instantaneous birth rate is greater than the instantaneous death
rate or decreases to extinction if the birth rate is less than the
death rate (Blundon 1994). The growth rate, which is of considerable
biological interest, indicates the biotic potential of a population,
the ability of that population to exploit a previously uninhabited,
unlimited environment (Blundon 1994).
Under both laboratory and natural conditions, exponential growth is
eventually submitted to environmental resistance which can cause a
decrease in natality (births), an increase in mortality (deaths), or
both (Blundon 1994). The pressures of environmental resistance on
biotic potential can be reflected by two trends, the geometric trend
or the logistic trend. If the population grows in a geometric
(Jshaped curve) way, the population size will double at a constant
rate in an unlimited environment (This situation is clearly
unrealistic because with time, it will be limited by the availability
of some resource) (Silvertown 1982; p. 7). If the population grows in
a logistic (sigmoid or Sshaped curve) way, the population will
increase towards the limits of resource availability. This will
reduce the rate of increase of the population until eventually the
population ceases to grow (Krebs 1994; p. 198). This limit density,
which the births = deaths and the instantaneous rate of increase = 0,
will be called the carrying capacity (K) of the environment (Blundon
1994). This experiment was embarked upon the intention to know if
these hypotheses are verifiable and conclusive. The objective of this
experiment was to analysis the growth of a Lemna population.
January 13 to March 3, 1994, the experiment was conceived in a biology laboratory at Camosun College during a class period. The technique of collecting data and the used material were very simple. In conformity with the Table 1, the experimentation was divided in two parts, the preparation of the laboratory and the counting of the number of thalli. During the first week, a cup, containing artificial pond water and 15 thalli (leaf unit that is over 1.5 mm in diameter), was put in a growth rack with the other groups. During the six following weeks, the number of thalli was counted and compiled every week. Details of the method are given in Blundon (1994).
Obtained data were compiled in the table 1 and table 2. Table 1 represents data that were compiled by the seven groups during the six weeks. The mean total is the mean of all groups, mean G1 & G3 is the mean for the groups 1 and 3, and S is the standard deviation total. The standard deviation is:
The sum of X is the sum of the
observations, and N is the number of observation.
Table 2 represents the analyse of the data for the six weeks. In
this table, there are the mean population size, standard deviation
total, Crude population growth rate, specific population growth rate,
exponential estimate of r, exponential estimate of N_{t}, estimate of K, natural log
of the environmental resistance in a logistic equation, linear
regression equation, and the logistic estimate of N_{t}. The mean population size
(N_{t}) and the standard
deviation total (S) are similar to Table 1. The
crude population growth rate, which is often dependent on population
size, is:
where, N_{t} is the population size at the end of a particular time period, N_{t1} is the population size at the beginning of the period, and D t=1 is the length of the time period. The highest crude population growth rate is in the last sample time and the lowest is in the first sample time. The specific population growth rate, which corrects the dependence on population size, is:
Contrary to crude population growth rate, the highest specific population growth rate is in the first sample time and the lowest is in the last sample time. The exponential estimate of r_{m} , which is the rate of increase, is:
The exponential estimate of N_{t} is:
where N_{0} = 15 and r_{m} is the highest value of
rm.
According to the Fig.
4, the estimate of K (carrying capacity of the
environment), which is the upper asymptote or maximal value of N, is
2400.
The environmental resistance in a logistic equation is:
The linear regression equation is:
This equation (above) is the straight line
(Y = a  mx), in
which ln[(KN_{t})/N_{t}]
is the Yaxis, "a" is the Y intercept, "t" is the Xaxis, and
m is the
negative slope (r). According to the Fig. 5, which
represents the environmental resistance in a logistic equation in
function of the weeks number, a= 4.63, b= 1.05 and the equation is
y = 4.63  1.05x
(negative slope).
The logistic estimate of the population size (N_{t}) is:
The trend of the logistic estimate of Nt
(Fig. 6) is different to the trend of the logistic model
(Fig. 4). The logistic model is an exponential at the beginning and
a log curve at the end, while the logistic estimate of Nt is a
negative exponential curve (Fig. 6).
Fig. 1, which represents the number of thalli in function of the
weeks number, shows upward trends (exponential curve) for most
groups, and the mean total. Because the curve of the mean total is an
exponential, it is impossible to find the carrying capacity of the
environment. Then, to find "K", "K" was approximated with an original
technique (Fig.
2). This technique allows to find best
curves (sigmoid curve) for a growing population. Thus, all groups
data were plotted in this special graph (Fig. 2).
Groups that keep a constant proportion when the total grows, should
follow a sigmoid curve or a curve that is natural for them. The
average of the two best curves (groups 1 and 3) was taken to
represent the best mean total. Fig. 1,
fig. 3, and fig.
4 show that the mean of G1 & G3, which
look a sigmoid curve, is similar to the mean total (exponential
curve) to the exception of the slope of the curve. So, with the mean
of G1 & G3, which represents very well the mean total, it is
possible to find K.
Although the experiment is not perfectly in
accordance with the initial hypotheses because of the curve of the
mean total, the experiment in general, follows the rules that was
introduced in the introduction. The logistic estimate of the
population size is a negative exponential curve. The experiment was
not conclusive despite of the fact that the mean of G1 & G3 is a
logistic curve. Fig.
1 shows that two curves (Groups 2 and 6)
seem to distort the logistic trend. The technique to find the best
sigmoid curve seems to be a good method, but that is only an
approximation because maybe it represents the mean total or maybe
not. According to the Fig.
4, the experiment wasn't long enough to
reach the K limit. The approximation of K shows that it misses about
two weeks to reach the K limit. Another reason of this failure could
be the way that the thalli were counted. Because thalli were moved
from the original cup to another container when thalli were counted,
the competition between Lemna species was eliminate. Anyway,
according to Silverston (introduction), this situation about an
unlimited growth is clearly unrealistic because with time the
population with a geometric trend will be limited by the availability
of some resources and will change in a population with a logistic
trend. An interesting improvement of this method could be to count
the number of thalli on a longer period of time and count the number
of thalli without moving the thalli constantly, container after
container, or simply put the thalli in a larger container.
Consequently, Lemna will grow as long as it will be limited by the
availability of resources or by the competition with other
individuals of its own species. In nature, it seems that young plants
grow exponentially during a certain period of time, and later in
their life, they reach a carrying capacity because of the
availability of resources and the competition with other plants of
the same species or other species. According to Thomson hypothesis,
with plants maturation, the light will stimulate the growth of
younger tissues (through hastening the onset of the elongation phase)
but inhibiting the growth of older tissues (through bringing the
elongation phase to an end before the full growth potential is
expressed) (Hart 1988; p. 108). So, according to this hypothesis,
plants growth must reach a maximum growth, despite of the fact that
the living conditions are excellent. Because conditions in the nature
change constantly, it is impossible to compare a laboratory
experiment to a natural experiment. So, a carrying capacity in
laboratory can't be comparable to a carrying capacity in nature.
Probably, because Lemna generations overlap in many populations,
Lemna should have a growth rate superior to most of the other
organisms that are submitted to important environmental resistance
(death). The greatest advantage to have an high growth rate is the
easiness to compete for a resource that is limited for a short period
of time, and exploit a previously uninhabited, unlimited environment.
The greatest disadvantage should be an rapid exhaustion of the
resource (food or space), condemning it to the starvation faster than
the other organisms.







Group 1 






Group 2 






Group 3 






Group 4 






Group 5 






Group 6 






Group 7 






Mean Total 






Mean G1 & G3 






S 






Starting Density = 15 






Week 






Mean Pop. Size (N_{t}) 






Standard Deviation Total (S) 






Crude Pop. Growth Rate 






Specific Pop. Growth Rate 






Exponential Estimate of r_{m} = ln (N_{t} / N_{t1}), when N_{t} / N_{t1} is highest 






Exponential Estimate of N_{t} = N_{0}e^{rmDt} (N_{0} = 15) 






Estimate of K = 2400 






ln ((K  N_{t}) / N_{t}) 






Linear Regression Equation ln ((K  N_{t}) / N_{t}) = a  rt a = 4,63 and r = 1,05 






Logistic Estimate of 






I would like to thank my classmate Jason, who helped me to do this experiment and all students in my class that contributed to the success of this experiment. I would like particularly thank Dr, David J. Blundon for his precious help and for its laboratory manual in which I took a lot of information and ideas.
Blundon, David J. 1994. Ecology: Laboratory Manual. Camosun College, Victoria, Canada.
Hart, J. W. 1988. Light and Plant Growth. Unwin Hyman, London, England.
Krebs, Charles J. 1994. Ecology. HarperCollins College Publishers, New York, USA.
Silvertown, Jonathan W. 1982. Introduction to Plant Population Ecology. Longman Group Limited, New York, USA.