Population Growth in Lemna minor
A population is a group of individuals of the same species living in the same area at the same time. In stable environments, the number of individuals in a population remains nearly constant. In rapidly changing environments, however, many individuals may die within a short period, leaving only a few survivors to perpetuate the species. In either case, the survival of the population depends on its ability to increase in number.
Models of population growth help scientists understand how a population’s size changes over time. These models have applications in microbiology, wildlife management, pest management, agricultural productivity, and toxicology. Consider two simple models of population growth: exponential and logistic.
Exponential Population Growth
When resources are unlimited, the number of individuals in a population
grows exponentially: 1, 10, 100, 1000, etc., as shown in the graph to the
right. In exponential growth, the number of individuals increases rapidly
and without limit. The population’s rate of increase, dN/dt, equals a
constant, r, multiplied by the number of individuals, N. Notice the “J” shape
of the exponential curve.
Exponential population growth equation: dN/dt=rN
The d means “change,” so the equation above reads, “the change in the
number of individuals as time changes equals a constant, r, multiplied by
the number of individuals.” The constant r is the population’s intrinsic rate
of increase and varies from environment to environment. There exists, at
least in theory, an environment that is perfect in all respects for a
population and in which it attains its maximum rate of increase. This rate is
the population’s biotic potential, r. Most environments, however, limit growth, and the population’s m
intrinsic rate of increase is less than the biotic potential. Integrating the above formula yields the equation
rt N = Ne t0
Where N is the number of individuals at Time t, Nis the number of individuals at Time 0, e is Euler’s t0
constant, and t is the length of time from Time 0 to Time t. With this equation, it is possible to calculate the number of individuals in the future, the number of individuals in the past, or a population’s intrinsic rate of increase.
In this experiment, you will calculate the maximum value of r and determine the biotic potential of Lemna minor. For example, suppose there were 3 fronds on Day 0 and 9 fronds on Day 5. What is the rt population’s intrinsic rate of increase over this period? Use the formula N= Newhere N= 9, N = 3, t 0t 0
and t = 5.
rtN= Ne t 0
5r9 = 3e 5r 3 = eTake the natural log (ln) of both sides. 5r x ln 3 = ln eUse the identity ln e= x.
ln 3 = 5r
r = (1/5) (ln 3)
r = 0.220
Note that r has the same units of time as t and the quantity of rt is unitless. The frond population’s intrinsic rate of increase over this period is 0.220 per day.
Logistic Population Growth
Exponential population growth does not continue indefinitely. If it did, one population would quickly
cover the surface of the Earth. Growth is limited by the availability of resources such as light, nutrients,
and space. The abundance of resources determines the carrying capacity, K, the number of individuals
of a population the environment can support indefinitely. The logistic population growth equation
reflects the limiting effect carrying capacity has on r and N:
Logistic population growth equation: dN/dt – rN[(K-N)/K]
Compare the logistic growth equation with the exponential
growth equation. The term [(K-N)/K] decreases the value
of the growth rate as the population increases. The term
represents environmental resistance as the population
approaches the environment’s carrying capacity. The
population growth rate slows to nearly zero at this c
population density. Although individuals die and new
individuals are born, called logistic population growth and
is shown in the graph to the right:
The population increases exponentially at first (a) when
resources are abundant. When the number of individuals
equals half the carrying capacity, the growth rate is at a maximum (c). Once this number is attained, the
growth rate slows (b), and the graph approaches a horizontal line whose y value is the carrying capacity
(K). This type of curve is called an S-curve (or sigmoidal curve, after the Greek word for the letter 2), for
(a) About Lemna minor
Lemna minor, or duckweek, is a member of the family Lemnaceae. This tiny organism is ideal for population growth experiments because it reproduces quickly, requires minimal space to grow, and requires no maintenance. This freshwater plant can be identified by its buoyant, leaf-like structures, called fronds (see diagram to right). Duckweed plants have one root, and grow by vegetative budding (a). Lemna is found in still waters from temperate to tropical zones (though it prefers temperatures around 25?C). Duckweed is the smallest flowering plant.
On average, fronds live four to five weeks. Due to its rapid growth rate, duckweed is utilized in a variety of governmental and commercial practices. For example, the US EPA requires companies that make pesticides to determine how the chemicals affect aquatic plant biology. Many companies use duckweed as the model plant, measuring a pesticide’s toxicity by observing its effect on the growth rate of duckweed.
Duckweed is also used to remove nitrogen and phosphorous from wastewater. Nitrogen and phosphorous are plant nutrients that in high concentrations (as in wastewater) promote rapid plant growth. When wastewater is released into the environment untreated, new plant growth can quickly clog waterways and cause eutrophication. To remove nutrients from wastewater, Lemna plants are added to treatment tanks to act as natural filters As the plants grow, they take up nitrogen and phosphorous from the wastewater. When the duckweed plants die, they are harvested, composted, and used as mulch. The treated wastewater continues to the next stage of water purification.
Besides applications in toxicity testing and water treatment, duckweed performs well as a food resource. Lemna has high nutrient content and is a viable livestock alternative. Duckweed gives an advantage in poultry feed by naturally offering pigments that are otherwise provided through expensive supplements. Additionally, Lemna can be used by wildlife as shelter and food. Duckweed is beneficial in pond communities because it reduces the rate of evaporation and restricts algal blooms.
During this lab, you and your classmates will test Lemna minor’s population growth rate and carrying capacity in a variety of growing mediums or treatments. The treatments being studied are conditions that may be present in natural waterways. Your class will test the following conditions:
; Phosphate level
; Nitrate level
; Amount of shade
; pH level
; control (held under normal growing conditions)
Your group will test one environmental condition (of your choosing, first come, first serve) at three different treatment levels. For example, salinity is tested at relative low, medium, and high levels, determined by your group. All parameters are tested using three replicates, and frond growth is monitored over 21 to 28 days. After making observations and gathering and analyzing data, you will identify the optimal growing conditions for Lemna minor.
Materials Available for use:
; test tubes Lemna minor
; graduated cylinders microscopes
; saline solution hand lenses
; phosphate solution distilled water
; nitrate solution petri dishes
; screening (to provide shade) small plastic containers
; rubber bands grease pencils
; pH solutions 5, 6, 7, 8, 9, anything else you need…just ask!
Day One of Inquiry:
; Make your observations and sketches of Lemna minor
; Ask your question (based upon the variable you wish to study)
; Design your experiment:
o Get list of materials
o Plan first three procedures but have the rest in mind
o Design your data tables
o Gather your supplies for tomorrow
o Determine how you will handle the collection of data over the Thanksgiving
; Write your hypothesis
o Show your instructor
Day Two of Inquiry:
; Execute your experiment
; Collect data for day one
o Don’t forget photographic evidence
Days Three-Twenty Eight (or however long it lasts!)
; Collect data
1. Plot the frond count over time for each treatment level you tested. Ideally, the curves
will closely resemble each other in slope and height. Draw each line in a different color. 2. Determine the carrying capacity for each test tube (colony) using the graph. Draw a
horizontal line to indicate the value of the carrying capacity and label the line with a “K”.
Include the r value for each curve. Label each curve with its K and r value.
3. Review your group’s logistic curves for each level of treatment. Compare the growth
rate between levels of treatment.
a. What can you conclude from your evidence about the effect of the treatment
on population growth rate?
b. Label the point on a line curve where the population growth rate is
ii. Increasing at a decreasing rate
4. Look at the overall class results.
a. Which habitat treatment was most conducive to fast population growth?
b. Which habitat treatment was most conducive to high carrying capacity? 5. Based on the class results, choose the treatment levels for EACH environmental
parameter that is optimal for Lemna minor.
6. How are birth and death rates accounted for in the exponential population growth