BISC 111/113:Lab 11: Population Growth 2

Objectives

1. To finish the Tribolium experiment by conducting final counts and start processing data
2. To explore simple models of population growth and interactions in Excel

Lab 11 Overview

I. Complete the Tribolium population growth experiment

II. Model changes in population size using Excel

a. Geometric Growth

b. Exponential Growth

c. Linear Growth

d. Logistic Growth

III. Assignment

Completing the Tribolium population growth experiment

Today you will complete the final counts of your Tribolium populations. Your samples have been placed in the freezer for at least 48 hours prior to the lab. You can isolate the beetles/pupae by sifting some of the culture through the screen over the brown paper on your bench. Try to keep all the flour away from the computers. As you isolate the beetles, put them in a labeled petri dish. There are tools available on your lab bench to help with the counting.

Enter your data on the class spreadsheet.

Refer to your lab instructor for information about the oral presentations next week. Sign up for a topic before leaving lab today.

Population Growth and Interactions: Background

Population Structure
A population is a group of individuals of the same species in a given area with the potential to interbreed. Each population has a structure that includes features such as density, spacing, and movement of individuals over time and space.

Spatial structure addresses the dispersal of individuals in space and can be classified as evenly spaced, clumped or totally random. Genetic structure of a population describes the distribution of genetic variation within that group. There is genetic variation among individuals within the population. This variation is the basis of the population’s ability to respond to environmental changes through adaptive evolution.

The study of population dynamics examines the ways that populations grow or shrink over time. Reproduction and immigration account for population increases, and death and emigration account for decreases. While some populations may reproduce continually, others experience discrete periods of reproductive growth. Because some populations (e.g. human) have no distinct reproductive season, the population grows more or less continuously.

Geometric Growth
Mathematical models offer a way to describe and predict population growth over time and under a variety of conditions. Generally, populations grow by multiplication, if all individuals have the potential to contribute equally to the population by reproducing. It follows then that the amount of growth that occurs is proportional to population size. As an example, consider the measurements of a population’s size below. How fast is this population growing?

Year measured	Population size
2006	100
2007	160
2008	256
2009	410

To calculate the per capita growth rate (the average contribution each individual makes to the population’s growth) from 2006 to 2007, we simply compute 160/100 = 1.6. This population has grown by 60% during the year (a factor of 1.6). Doing this for all subsequent 1yr intervals (256/160 = 1.6, 410/256 = 1.6) we see that the population is growing at a constant rate of 1.6 from year to year. When calculated for discrete time intervals such as from year to year, this is called the geometric growth rate of the population, and is symbolized by λ (the Greek letter “lambda”):

λ = N_t / N_t-1

where N_t is population size at some time interval “t”, and
N_t-1 is population size at the previous time interval “t-1”.

With a little algebra, we can use this simple equation as a powerful tool to make predictions about the population's growth:

N_t = N_t-1 * λ

which is the same as:

N_t+1 = N_t * λ

If we followed a population’s growth from its initial size (N₀) at “time zero” (t = 0), its size after one year (or any specified time unit) would be N₁ = N₀ * λ. At year two, N₂ = N₁ * λ, but because N₁ = N₀ * λ, N₂ is also = N₀ * λ * λ, or N₀ * λ².

To generalize this expression:

N_t = N₀ * λ^t.

Exponential Growth
We can use this expression to predict the population size at any future time “t” from some initial size (N₀), if the population growth rate (λ) is constant. In our example above where N₀ = 100 individuals, after 15 years the population would be predicted to be quite large under a constant population growth rate of 1.6 (N₁₅ = 100 * 1.6¹⁵ , or > 115 * 10³ individuals).

Calculations involving population growth can be more easily handled if we express the population growth rate as an exponential constant, r, rather than the geometric constant, λ. We can do this by substituting e^r, for λ in our computations. “e” is the base of Naperian, or natural, logarithms, and is equal to approximately 2.718. The equation for exponential population growth then, is

N_t = N₀ * e^rt

Why use r instead of λ? The geometric constant depends on the interval of time specified; λ= 1.6 over 1 yr, in the previous example, but 2.56 over 2 yr. The exponential constant r is an instantaneous measure of relative population growth at every moment in time, and is hence more generally useful. Exponential growth is like a compound interest savings account, where the interest is continually added to the principal. Geometric growth is akin to simple interest, where the account balance is updated once a year. Exponential growth is especially applicable to continuously reproducing populations, such as the human population.

During exponential growth, the population grows by an amount dN, during a time interval dt (this is the population growth rate) and the amount of growth in this interval is proportional to the size of the population (N_t) by the constant r:

dN/dt = rN_t

Where
N_t = population size (at time “t”), and
r = the instantaneous rate of population growth.

You can also see that r = (dN/dt)/N_t . And so during exponential growth, r is the per capita rate of increase of the population.

Exponential population growth displays a J-shaped curve of N over time. Compare exponential growth to a simple linear model:

Exponential growth: N_t = N₀ * e^rt
Linear growth: N_t = N₀ + b*t

Where:
N₀ = the initial (starting) population size at time t = 0
t = time elapsed from time = 0 to time = t
b = the population linear growth rate, and is a constant that is not proportional to N; it is the slope of the N vs. t linear relationship. Here, “dN/dt” = b. (You will see that this relationship is described by a linear regression of N on time.)

Linear Growth.
Linear models may be appropriate over short time intervals, but seldom describe long-term population growth because they fail to include an increasing number of reproductive individuals as the population grows. However, while exponential growth models are more realistic than the linear models, neither reflects the dynamics of most natural populations over the long term. There are several reasons for this, as we will see below. Nonetheless, “r” is a very useful parameter in modeling (describing & predicting) population growth as an integrator of life history traits (e.g., when & how much to reproduce) and growth potential.

Logistic Growth.
Exponential growth, if unchecked, could lead to unimaginable numbers of everything from bacteria to elephants. After all, the exponential rate of population increase is proportional to the population size; the bigger the population gets, the faster it grows! Of course, most natural populations never achieve their potential for unlimited size; Darwin was keenly aware of this fact. A third model, the logistic population growth model, tends to be more realistic because it takes into account “environmental resistance” (K) or carrying capacity. K is the maximum, or equilibrium, population size that can be sustained theoretically by the environment. When K is reached the population growth rate is zero due to a balance of births and deaths. K is influenced by resource availability, waste accumulation, and other density-dependent factors (see below).

In logistic population growth, dN/dt = r N_t *[(K- N_t)/K]

Here, K = the carrying capacity, and other terms are as defined above.

Note that the bracketed term, [(K- N_t)/K] acts as a “braking” factor to slow the rate of population growth as the population nears its carrying capacity. See that when N is very small, the entire expression collapses to unity (K/K = 1), and the population grows exponentially. However, as N increases, the bracketed term approaches zero, and thus dN/dt, the population growth rate, approaches zero. This braking effect results in an S-shaped population growth curve.

The logistic equation for predicting N at some time “t” looks forbidding, but is not too difficult. The terms in this equation are as defined above.

N_t = (N₀K)/(N₀+[K- N₀]e^-rt)

Population growth in nature
The models we have presented are more often representative of laboratory populations growing under controlled conditions, rather than the patterns seen in natural populations. In the logistic model, as long as the population size (N) is less than the carrying capacity (K) the population will increase, and the rate of increase slows as N approaches K. However, even this model is often too simple to capture the dynamics of natural populations. If you read any of the original articles by Park about Tribolium beetles, you see that the growth curves are complex and not usually predictable. It is important to understand that K in the logistic equation is fixed, but that in reality K will continually change as environmental characteristics (e.g., availability of food & space) change.

Models of natural populations must account for the complexity of multiple interacting environmental factors. For example, in many populations, there are distinct breeding seasons and periods of growth so it becomes important to measure the population at the same time each year to take into account when the majority of births and deaths occur. The resources supporting populations are dynamic quantities, as are habitat characteristics, interspecific competitors, predators, and pathogens. And now, natural populations are facing the challenges of rapid local and global climate change.

Population Regulation
Density-dependent factors such as predators, quality of cover, parasites, diseases, and amount of food determine K, and therefore, may regulate populations by maintaining or restoring them to some equilibrium size.

Density-independent factors, in particular, environmental factors such as temperature, precipitation, and disturbances (e.g., fire, floods, earthquakes, tsunamis) can also alter the rate of population growth. These factors, however, are density independent because they affect a certain proportion of the population independently of population size (and largely independent of individual fitness).

Density-dependence is difficult to demonstrate in nature, in part because of frequent changes in environmental conditions. As a result, some ecologists refute the concept of “equilibrium” entirely and are currently interested in models based on chaos theory. Chaotic systems behave in unpredictable ways, even though the factors influencing them may be quite deterministic. As the number of such interacting factors increases, the ultimate “trajectory” a system takes, such as how a population grows, depends strongly on the starting conditions, and can be highly unpredictable. Chaos typifies many such “complex systems”.

Modeling Population Growth using Excel

In Excel, you will produce figures that demonstrate:

a. geometric growth

b. exponential growth

c. linear growth

d. logistic growth

Figures should cover a 12-month time period with an estimated population size for each month. Don’t worry about creating a fancy graph -- just plot the data to make sure you have set up the model correctly.

a) Geometric Growth typifies a population that grows at a fixed proportion (λ) of population size over equal time intervals. Assume that this pattern of growth applies here. First, calculate λ from the first two time intervals at t = 0 and t = 1 month. Then, calculate N_t for the remaining months, and plot against time using a line scatter graph.

Geometric model: N_t = N₀ * λ^t

Time (t, months)	Number of Individuals (Nt)
0	10 (= N0)
1	45
2	...
3	...
4, etc.	...

b) During Exponential Growth, the rate of population growth depends on the population size; the larger the population, the faster it grows! This is similar to geometric growth, except that we apply an instantaneous measure of growth, r. In the following case, let’s assume that r = 1; this “intrinsic rate” of population growth seems small, but watch what happens in a very short amount of time.

Exponential model: N_t = N₀e^rt
In Excel, the command for “e” is EXP. The right hand of the above expression would be calculated as N₀*EXP(r*t)

Time (t, months)	Number of Individuals (Nt)
0	10 (= N0)
1	10 * EXP(1*1) = 27
2	...
3	...
4, etc.	...

Plot the data using a scatter plot without lines. After you plot your data, click on the series of data points, go to Chart/Add Trendline, choose exponential for type, then in options choose “Display equation” on chart. The equation should exactly match your model of exponential population growth with time.

c) Linear population growth is characterized by a fixed amount of growth per time interval; a constant number of individuals accrues within a given time period. In the following case the population adds 20 beetles each month (b = 20) to an initial population of 10.

Linear model: N_t = N₀ + b*t

Time (t, months)	Number of Individuals (Nt)
0	10 (= N0)
1	10 + 20*1 = 30
2	...
3	...
4, etc.	...

Plot the data using a scatter plot without lines. After you plot your data, click on the series of data points, go to the "Chart/Add Trendline", choose "linear" for type, then in options choose “Display equation” on chart. You should see that the equation exactly matches your model of population growth with time.

Compare exponential and linear model outputs. Explain why the exponential model produces a J-shaped curve rather than a linear straight line. (Hint: Identify the impact of N on population growth and what happens as N increases).

d) Logistic Growth. Now, on your own, try to set up a third spreadsheet to model logistic growth.

Logistic model: N_t = (N₀K)/(N₀+[K- N₀]e^-rt)

Start at N₀ = 10, and set r = 1; let’s set K = 10,000.

Your Excel formula for N_t would be: =(10*10000)/(10+((10000-10)*EXP(-1*t))).

Excel will not plot a logistic equation through the data points, so just link the points using a line scatter graph or smooth line plot to see the overall pattern of population change with time.

You should see an S-shaped curve. Can you explain why the population increases more rapidly at intermediate size than at relatively large or small sizes?

Assignment

1. Prepare for a workshop style presentation and discussion as described by your lab instructor. The oral presentations are designed to expand your knowledge of population growth and interactions and to reflect on our Tribolium beetle experiment. The oral presentations topics and format will be described by your lab instructor. Be sure to sign up before leaving lab today.

2. Write a research paper on the Tribolium experiment. A full scientific paper in the Biological Sciences consists of the following sections: Abstract, Introduction, Methods, Results, Discussion, and Literature Cited. Follow the guidelines of your instructor regarding the sections to be included in your report.

Please refer to the chapter Science Writing Guidelines of the Wiki for a reminder of what should be covered in each of these sections. Procedures for constructing graphs and running statistical tests are available at Statistics and Graphing.

The research paper is due on the last day of classes.