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 computer simulation programs that model population growth and interactions

Lab 11 Overview

I. Completing the Triboliumn population growth experiment

II. Modeling population changes using Excel

a. Linear Growth
b. Exponential Growth
c. Logistic Growth

III. Building beetle population models in Stella

IV. 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 as well as within the entire population. This variation is the basis of the population’s ability to respond to environmental changes through 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 (human) have no distinct reproductive season, the population grows more or less continuously.

Population growth:
Exponential growth
A population that increases continuously under ideal conditions (without limitations) exhibits what is known as exponential growth. During exponential growth, the population growth rate (dN/dt) is proportional to the size of the population (N):

dN/dt = rN

Where: N = population size
r = the “intrinsic rate” of population growth. r is related to the population’s net reproductive rate R₀ and mean generation time T by the expression r = ln(R₀)/T.

You can also see that r = (dN/dt)*1/N, and during exponential growth is the theoretical maximum rate of increase of a population per individual. r under such unlimited conditions is termed r_m, or “r_max”.

Exponential population growth displays as a J-shaped curve of N over time.

The exponential model is also expressed by the equation:

Nt = N₀e^rt

Where: N₀ = the initial (starting) population size at time t = 0
N_t = population size you are trying to predict at some time “t”
e = “exponent” = the base of Naperian logarithms; e ≈ 2.718
r = the “intrinsic rate” of (exponential) population growth (usually termed rm under exponential growth)
t = time elapsed from time = 0 to time = t

Linear growth
Compare exponential growth to a simple linear model described by the equation:

N_t = N₀ + bt

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.)

While exponential growth models are more realistic than the linear models, neither reflects the dynamics of natural populations over the long term.

Logistic Population Growth Model
It turns out that 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 population size that can be sustained indefinitely by the environment. K can also be considered the equilibrium population size that can be maintained by the environment, where the population growth rate is zero due to a balance of births and deaths. It is influenced by resource availability, waste accumulation, and other density-dependent factors (see below).

In logistic population growth, dN/dt = rN*[(K-N)/K]. Here, K = the carrying capacity, and other terms are as defined above.

Note that the bracketed term, [(K-N)/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 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:

N_t = (N₀K)/(N₀+[K- N₀]e^-rt)

The terms in this equation are as defined above.

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.

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, inter-specific 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 to some equilibrium size.

Density-independent factors, in particular, environmental factors such as temperature, precipitation, and disturbances e.g. fire can also alter the rate of population growth. These factors, however, are not regulatory because they affect a certain proportion of the population independently of population size.

As it turns out, 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 Changes in Beetle Populations using Excel

In Excel, you will produce three figures that demonstrate:
a.) linear growth
b.) exponential growth
c.) 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. Based on these equations, we will spend the remainder of our time modeling our own beetle populations using the Stella modeling program.

a. Linear population growth is characterized by a fixed rate of growth; 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 + bt

Time (t, months)	Number of Individuals (Nt)
0	10 (= N0)
1	10 + 20*1 = 30
2	10 + 20*2 = 50
3	10 + 20*3 = 70
4, etc.	...

Plot the data using a scatter plot without lines. After you plot your data, click on the series of data points, go 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 on population growth with time.

b. During exponential growth, the rate of population growth depends on the population size; the larger the population, the faster it grows! 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	10*EXP(1*2) = 74
3	10*EXP(1*3) = 201
4, etc.	...

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

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).

c. 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)

N₀= 10, r = 1; let’s set K = 10000. 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 dot-to-dot scatter 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?

Building Models of Beetle Populations in Stella

Let’s build a simple exponential Stella model together that you can then adapt to a logistic model.

Table 1. A description of the functions of icons used to build a Stella model.

1. Open Stella, File, New and click on the MODEL tab.

2. Place a stock (square) and name it N beetles.

3. Place a flow connecting to the stock and name it change in beetles.

4. Place a converter (circle) and name it initial beetles.

5. Place another and name it r.

6. Use connectors in the manner below.