COMPUTER LAB 1
EARTH SYSTEMS SCIENCE II
PG251, Spring 2012
Lab 1. Exponential Growth, S-shaped growth
This lab is a refresher on STELLA, and an introduction to some of the mathematical concepts that are relevant for lab this semester. This is also a refresher on Excel and some mathematical manipulations, so Part B asks that you basically work the same problem, logistic growth, using Excel and it requires that you follow the details of the problem that can be solved with this kind of math.
PART A – Using STELLA
Part 1. Exponential Growth
Remember, exponential growth occurs in many natural systems, especially biological systems, where the amount of growth per time period is a specified fraction of the amount present. For example, the reproductive rate of many populations, including unicellular organisms up to humans, is often expressed as a fraction (or percentage) per year. This is the definition of exponential growth.
Mathematically, it can be written as in equation 1.
P = r * P(1)
where P is the change in population per time (units of deer/yr, people/yr, kg-C-plankton/yr), r is the rate constant (fractional change per yr), and P is the initial population from the previous time period. Figure 1 shows the STELLA model graphical representation, and figure 2 shows a sample time series of exponential growth.
Figure 1. STELLA model for exponential growth
Figure 2. Time series of exponential growth model from figure 1, with initial value P=10 and r = 0.07/yr, with a model run length of 100 years.
Part 2. S-Shaped Growth.
In real biological systems, populations can not keep growing forever (obviously). Some limiting factor kicks in to stop the growth rate from being exponential, and eventually changing the growth rate to zero, or sometimes even a negative value (in which case the population will decline). When a population begins growing exponentially, and eventually hits a limiting factor which decreases the growth rate to zero, the trend in population change over time looks something like figure 3. This is called S-shaped growth.
Figure 3. Time series of S-shaped growth using the Logistic model, with same parameters as figure 2, and with carrying capacity K = 100.
Figure 3 was made using the Logistic model, which is the most well known mathematical representations of S-shaped growth, and one of the most well-known models in ecology. The logistic model, which is simply the exponential growth model multiplied by a “multiplier” term, is shown in equation 2.
P = r * P * M(2)
M = 1 – ( P / K)
where all the parameters have the same meaning as in equation 1, and M is the multiplier (no units), and K is the carrying capacity (same units as P). Because M gets smaller as P gets bigger, and M actually goes to zero when the population hits the carrying capacity (when P = K), the population reaches its equilibrium when the population is equal to the carrying capacity. In STELLA, the model looks like figure 4.
Figure 4. STELLA model for S-shaped Growth using the Logistic Model
The logistic model is well known because it can be solved analytically, so it was used widely before the advent of computers. However, there are some drawbacks to its use. First of all, one must specify K, carrying capacity, before you actually model the system. This can be a problem, because K is not always known. Second, this model assumes that the multiplier is a linear function, which means that the rate of growth will decrease linearly as the population increases. This is also not necessarily true. There are many ways that one can change this equation to address these problems. In the next lab we will see an example of how one can model such a thing without specifying the carrying capacity ahead of time.
ASSIGNMENT
Make a STELLA model of the logistic model. You will run two sets of experiments. These are called “sensitivity studies”, because you are investigating the sensitivity of the model’s behavior to the values of different parameters.
- Sensitivity of the model to the growth rate. Run the model for 100 years three times with P(initial)=10 and K=100: r = 0.07, 0.09, and 0.11. Hand in your STELLA file, and a word file with one graph from EXCEL containing the results of all three model runs. Explain what effect changing the growth rate has on the behavior of this model.
- Sensitivity of the model to the carrying capacity. Run the model for 100 years three times with P(initial)=10 and r = 0.07: K=100, 90, and 80. Hand in your STELLA file, and a word file with one graph from EXCEL containing the results of all three model runs. Explain what effect changing the carrying capacity has on the behavior of this model.
Part B – Chapter 13: Critical-Thinking Problem 1
Here you will calculate the change in a population of organisms (= N) as a result of births and deaths. A mathematical expression can be used to calculate next year’s population (Nt+1) on the basis of this year’s population (Nt), where t = years.
r potential growth rate = birth rate - the death rate
K carrying capacity = population size that can be supported (given constraints of food or other resource availability, competition, and so on)
(a) If the population is small relative to K, that is: if Nt < K, then the term in parentheses is essentially 1 because Nt/K < 1 so (1 - Nt/K ) approximately equal to 1. In this case, the potential growth rate is achieved. In other words,
Nt+1 Nt + rNt = Nt(1 + r)
In words: next year’s population would be simply some multiple [(1 + r) in the equation above, just a number] of this year’s population and this leads to exponential growth.
(b) If N approaches K, the population will tend to slow its growth, finally reaching the carrying capacity. In this case, Nt K, so that Nt/K is approximately equal to 1 and
(1 - Nt/K) 0 and
Nt+1 Nt
This behavior is called logistic growth.
(c) Next the fun part: behavior that has been labeled chaos!
We will use Excel to work this lab. You should be able to save your work (tables, charts) and include them in your lab report.
PART 1 (Logistic growth): Fill in the following table, and then graph Nt versus t; assume that r = 1.0 and K = 1000.
Time (years) / Nt / 1-Nt/K / rNt(1-Nt/K) / Nt+11 / 20 / 0.98 / 19.6 / 39.6
2 / 39.6
3
4
5
6
7
8
9
10
11
12
13
14
15
Describe the growth curve, and explain why it has the logistic growth shape (on the basis of the numbers you calculate).
PART 2 (Chaos) Now repeat your calculations (do at least 30 years’ worth) for the following values of r:
r = 2.0
r = 2.8
Graph your results, either on separate graphs or using different symbols or colors on the same graph (be sure the graph(s) is (are) legible). If the population growth rate goes negative, call it quits on that series of calculations; the population has gone extinct.
Describe how the behavior changes as the growth rate increases from 2.0 to 2.8. When r = 2.8, the system is described as being chaotic. A scientist who observed this population might conclude that purely random factors are controlling the size of this population. What is wrong with this conclusion?
PART 3 Leading to chaotic behavior: Let’s experiment with the values of the parameters in this problem.
(1) What do you expect will change in the results you obtained in Part 2 if you change K to (a) 500 and to (b) 1500? Perform the calculations with these values and plot your results. Explain your answer.
(2) Study the behavior of the population as r changes. Set K = 1000 as before and perform calculations for values of r as follows:
(a) 0 < r < 1 (choose at least a couple of very small values)
(b) 1 < r < 2
(c) 2 < r < 3
Explain your results.
(3) The value of Nt is the ‘initial condition’, the known population at a given time t, considered the initial time. This value was 20 in the above calculations. Investigate what happens to the results you obtained in the calculations of (2) when you change this value. Are your equilibrium solutions sensitive to changes in initial conditions? Explain your observations.
ESS II, Spring 2011, lab 1, p. 1 of 5