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Introduction

Materials and Methods

Results

Discussion

Acknowledgements

Works Cited

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 (J-shaped 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 S-shaped 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_t-1 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 N₀ = 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[(K-N_t)/N_t] is the Y-axis, "a" is the Y intercept, "t" is the X-axis, 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.

Fig. 1: Lemna Population Growth Experiment (Groups Data)	Fig. 2: Lemna Population Growth Experiment (Groups Data)
Fig. 3: Lemna Population Growth Experiment (Mean Total and Mean for G1 & G3)	Fig. 4: Lemna Population Growth Experiment (Mean Total and Mean for G1 & G3)
Fig. 5: Logistic Growth of a Lemna Population	Fig. 6: Logistic Estimate of Nt

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.

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