Project 4: Population Variability Analysis
For project 4 we're going to move to the world of ecological modeling. In particular, we're going to model the change in the population of Galapagos Penguins over time given certain ecological pressures. Our goal is to model the risk of extinction over some number of years. The model we will use is fairly simple, but you can choose to add complexity to the model once the basic system is working.
One new aspect for this project is the use of random values in the simulation. Because the world doesn't exhibit strong regularity, we're going to model this uncertainty by using a random number generator. For example, El Nino, an important factor in Galapagos Penguin survival, does not occur on a perfectly regular schedule, but it does tend to occur at least once every 5-7 years. We can model this by assigning a 1.0/5.0 or 1.0/7.0 probability of an El Nino to any given year. Then, each year of the simulation, we effectively roll a 5-sided or a 7-sided die and declare an El Nino year if it comes up with a 1. If the year is an El Nino year, then the penguin population drops significantly. If it is not an El Nino year, the population sees a small amount of growth.
Because of the random variables in the model, each time we model 100 years the computer will generate a different result. In one simulation, the population may go extinct in 20 years, in another simulation it may not go extinct at all. In order to achieve a reasonable estimate of the true probability of extinction over 100 years, we have to run the simulation many, many times and aggregate the results. For example, the probability of extinction in 100 years is the number of simulations where the population goes extinct, divided by the total number of simulations. Researchers using PVA to evaluate the risk of extinction in actual populations might run the simulation 10,000 times or more in order to get a stable estimate of extinction risk. Stability, in this case, means that if you run the simulation 10,000 times, then the difference between that average and the average of a different run of 10,000 simulations will be small.
To design the simulation, we're going to use hierarchical, modular design in order to keep each function simple to write and debug. Think about the project as having layers. The top layer is the controlling function and contains three parts. The first part sets up the parameters for the simulation. The second part runs the complete simulation many times and stores the results. The third part analyzes the results and generates the desired outputs.
The next layer down handles running a single simulation over N years. The full simulation function has two main parts. First, it sets up the initial population, then it simulates N years using a loop. It returns either the year in which the population went extinct or N if the population is still viable at the end of the simulation.
The bottom layer handles simulating a single year of the simulation. It takes in the current population and the ecological parameters and determines how many individuals are in the next year's population. It then returns the new population.
By designing the simulation in this manner, each function is well-defined and simple to write and debug. The tasks below provide more details on each level and the overall task.
- If you haven't already set yourself up for working
on the project, then do so now.
- Mount your directory on the Personal server.
- Open the Terminal and navigate to your Project4 directory on the Personal server.
- Open TextWrangler. If you want to look at any of the files you have already created, then open those files.
- Create a new file penguin.py. For this week, all of the simulation functions will be in this file, though you can use the functions in your stats.py file to analyze the results, if you choose to pursue one of those extensions.
Create a function initPopulation, which takes in two parameters: the
initial population size and the probability of an individual being
female. The function should return a list of the specified size.
Each entry in the list will be either an 'f' or an 'm'.
The function should first initialize a variable (e.g. population) to the empty list. Then it should loop population size times. Each time through the loop it should generate a random number using random.random(), which generates values between 0 and 1. If the value is less than the probability of an individual being female, append an 'f' to the population list. Otherwise, append an 'm' to the population list. Finally, return the population list.
You can use the following test function to see if your initPopulation function is doing the right thing. It should print out a list of 10 individuals with an appropriate mix of 'f' and 'm' elements. Run it a few times to make sure you're getting randomized lists.
# test function for initPopulations def test(): popsize = 10 probFemale = 0.5 pop = initPopulation(popsize, probFemale) print pop return if __name__ == "__main__": test()
Create a function simulateYear with six parameters:
- pop: the population list
- elNinoProb: the probability of an El Nino
- stdRho: the growth factor in a regular year. This number is meant to allow the population to grow each year and is expected to be greater than 1.
- elNinoRho: the growth factor in an El Nino year. This number is meant to reduce the population and is therefore less than 1.
- probFemale: the probability of a new individual being female,
- maxCapacity: the max carrying capacity of the ecosystem.
The first step in the function is to determine if it is an El Nino year. Set a variable (e.g. elNinoYear) to False. Then compare the result of random.random() to the El Nino probability. If the random.random() result is less, then set elNinoYear to True.
The second part of the function is a loop over the existing population list. Before starting the loop, create a newpop list and set it to the empty list. Inside the loop, implement the following algorithm.
# for each penguin in the original population list # if the length of the new population list is greater than maxCapacity # break, since we don't want to make any more penguins # if it is an El Nino year # if random.random() is less than the El Nino growth/reduction factor # append the penguin to the new population list # else # append the penguin to the new population list # if random.random() is less than the standard growth factor - 1.0 # if random.random() is less than the probability of a female # append an 'f' to the new population list # else # append an 'm to the new population list
Finally, return the new population list.
To test your simulateYear function, add the following code to your test function.
newpop = simulateYear(pop, 1.0, 1.188, 0.4, 0.5, 2000) print "El Nino year" print newpop newpop = simulateYear(pop, 0.0, 1.188, 0.4, 0.5, 2000) print "Standard year" print newpop
You should see the population reduce to 3-6 individuals in the El Nino year and grow to 11-14 individuals in the standard year. Run it a few times to see the variation.
The next step is to write the runSimulation function. This function
takes in 8 parameters. You can use the definition below, if you wish,
but feel free to modify the parameter names to ones that might be more
clear to you.
def runSimulation( N, initPopSize, probFemale, elNinoProb, stdRho, elNinoRho, maxCapacity, minViable ):
The function should first assign to a variable (e.g. population) the result of caling initPopulation with the appropriate arguments. Then it should set another variable (e.g. endDate) to N (the number of years to run the simulation). The variable endDate will be what the function returns.
The main part of the function should be a loop that runs N times. Each time through the loop it should call simulateYear with the appropriate arguments, assigning the result to a new list variable (e.g. newPopulation). Then it should test if there are enough individuals in the new population list to be a viable population. It should also test if there are no males or no females in the new population list. If any of these tests fails, then set endDate to the loop variable value and break out of the loop. The final step in the loop is to assign to the population variable the newPopulation variable, replacing the old population with the new one.
If you want to graph or view the population numbers over time, print out the length of the population list at the end of each loop. Alternatively, you could save the numbers to a file.
Add some test code to your test function that calls runSimulation a few times with some appropriate arguments. Make sure the initPopSize argument is larger than the minViable argument.
Default values for the simulation are as follows:
N 201 Initial Population Size 500 Probability of Females 0.5 Probability of an El Nino 1.0/7.0 Growth Factor in a standard year 1.188 Growth Factor in an El Nino year 0.41 Maximum Carrying Capacity 2000 Minimum Viable Population 10
The top level function is your main function. Give it one argument,
argv, which will be the list of strings from the command line.
Start your main function by testing if there are three arguments on the command line. The first argument will be the name of the program, the second should be the number of simulations to run, and the third should be the typical number of years between an El Nino event. If there are less than three (len(argv) < 3), print out a usage statement and exit.
After the test for arguments, cast the second argument (argv) to an integer and assign it to a variable that specifies the number of simulations to run (e.g. numSim). Cast the third argument (argv) to an integer and assign it to a variable that specifies the typical number of years between an El Nino event.
In the next section of your program, create variables for each of the simulation parameters in the table above not already assigned and give them the default values. You should also create a variable to hold the results of the simulations and initialize it to the empty list.
Loop over the number of simulations. Inside the loop, append to your result list the result of calling runSimulation with the appropriate arguments.
Finally, count how many of the results in your result list are less than N (the number of years to simulate). Divide that count by the total number of simulations and print out the result. This is the overall probability that the penguins will go extinct within the next N years.
Run your program using 100 simulations and see what you get. Does it change a lot? Run it with 1000 simulations and see if the results are more stable.
The next task is to write a function that takes the list of results,
which is a set of numbers indicating the last year in which the
population was viable, and convert it to a cumulative extinction
probability distribution (CEPD). The CEPD will be a list that is as long as
the number of years in the simulation. Each entry Y in the CEPD is
the number of simulations where the population has gone extinct by
year Y divided by the total number of simulations.
The computeCEPD function should take two arguments: the list of results from runSimulation (a list of integers), and the number of years the simulation ran (N).
The computeCEPD function has three parts.
- The first part is to create a list (e.g. CEPD) with N zeros. You can do this by appending N zeros to an empty list in a loop.
- The second part is to loop over the list of results (extinction years). If the extinction year is less than N, loop from the extinction year to N and add one to each of those entries in the CEPD list.
- The third part is to loop over the CEPD list and divide each entry by the length of the extinction year results list, which is also the number of simulations. After this, return the CEPD list.
Add a call to the computeCEPD function to the end of your top level function. Then have your top level function print out every 10th entry in the CEPD list (remember how the range function works). Run your penguin simulation 1000 time using the standard parameters. Repeat the process three times and show the results in your writeup. How much variation is there in the results?
The final step is to use your simulation to compare the extinction
distribution curves for a 3-year El Nino cycle, a 5-year El Nino
cycle, and a 7-year cycle.
Include the output from your additional two runs in your write-up. In the text of the write-up, include a description of the output and a comparison of the outputs from the three different cycle lengths.
Each assignment will have a set of suggested extensions. The required tasks constitute about 85% of the assignment, and if you do only the required tasks and do them well you will earn a B+. To earn a higher grade, you need to undertake one or more extensions. The difficulty and quality of the extension or extensions will determine your final grade for the assignment. One complex extension, done well, or 2-3 simple extensions are typical.
The following are a few suggestions on things you can do as extensions to this assignment. You are free to choose other extensions.
Add other model parameters to the command line. The complex version
of this extension is to have flags for your program so that the user
can use a flag, like -E, to specify a given parameter. For example,
python penguin.py -E 5
would modify the El Nino frequency but leave all other parameters at their default values.
- Display the CEPD data in a graph. Adjust the code in penguin.py so that it prints out the year and CEPD for all years. Redirect the output to a .csv file. Use Excel or other software to plot the data and post the image on the wiki with a thorough description of its contents.
- Explore variations on the model. For example, what effect on the extinction probability do you see if you adjust the number of years in the El Nino cycle? Does changing the carrying capacity to 3000, for example, modify the 100-year extinction probability for the 5 year El Nino cycle?
- More rigorously test the variation in the simulation outcomes. Use your standard deviation function to compare the standard deviation of the probability that the population goes extinct at a particular time. For example, consider 200 years when the El Nino cycle is 5 years. Running the simulation with numSim=100 will likely yield a larger standard deviation in the values of the CPEF at t=200 (the final year of the simulation) than running the simulation with numSim=1000 will.
- Figure out the complexity of the simulation as a function of the parameters. How many times does the main loop in runSimulation execute, on average?
Write-up and Hand-in
Turn in your code by putting it into your private hand-in directory on the Courses server. All files should be organized in a folder titled "Project 4" and you should include only those files necessary to run the program. We will grade all files turned in, so please do not turn in old, non-working, versions of files.
Make a new wiki page for your assignment. Put the label cs151sf15project4 in the label field on the bottom of the page. But give the page a meaningful title (e.g. Milo's Project 4).
In general, your intended audience for your write-up is your peers not in the class. Your goal should be to be able to use it to explain to friends what you accomplished in this project and to give them a sense of how you did it. Follow the outline below.
- A brief summary of the task, in your own words. This should be no more than a few sentences. Give the reader context and identify the key purpose of the assignment.
- A description of your solution to the tasks, including any text output or images you created. This should be a description of the form and functionality of your final code. Note any unique computational solutions you developed or any insights you gained from your code's output. You may want to incorporate code snippets in your description to point out relevant features. Code snippets should be small segments of code--usually less than a whole function--that demonstrate a particular concept. If you find yourself including more than 5-10 lines of code, it's probably not a snippet.
- A description of any extensions you undertook, including text output or images demonstrating those extensions. If you added any modules, functions, or other design components, note their structure and the algorithms you used.
- A brief description (1-3 sentences) of what you learned. Think about the answer to this question in terms of the stated purpose of the project. What are some specific things you had to learn or discover in order to complete the project?
- A list of people you worked with, including TAs and professors. Include in that list anyone whose code you may have seen, such as those of friends who have taken the course in a previous semester.
- Double-check the label. When you created the page, you should have added a the label cs151f15sproject4. Make sure it is there.
Thanks to Cathy Collins in the Colby Biology department for the project idea and documentation. The original project concept and idea came from "Problem-Solving in Conservation Biology and Wildlife Management, 2nd Edition", by James P. Gibbs, Malcolm L. Hunter, Jr., Eleanor J. Sterling.