I have two major items I’d like you to address in discussion this week. They both deal with population growth and will be much easier if you use some software that is available (for free) from the University of Minnesota’s web site. Of course, you can, and should, also do some of this work by hand, but this software, called Populus, is great and nicely illustrates many of the issues that we’ll cover this week (and next) in lecture and discussion.
So, the first thing you need to do is download and install the software.
1. Dowload Populus. (Click here to go to the download site) (To find out about Populus, click here.) a. If you are at the CIRCA labs, download the file (named pop34p.exe) to the USER folder (i.e., subdirectory) on the C: drive. b. If you are at home, download to a new folder (wherever you’d like). c. If you at the CIRCA lab, but want to take the program home with you, you’ll need to put pop34p.exe onto a floppy disk, take it home, copy this file to your hard disk, and then pick up with step 2.
2. Extract the files. Pop34p.exe is a self-extracting file. Simply "run" this program (e.g., double click on it from the file manager or hit the start button and choose "RUN" and then specify the path and "pop34p.exe"). When you do this, pop34p.exe will "uncompact" and stick several new files in your folder. On of these files is populus.exe, the main program file.
3. Run populus by double clicking on populus.exe or by clicking the Start button, RUN and then typing in populus.exe (along with the path).
4. If you are using the CIRCA labs, as soon as you log out, these files will all be deleted. Therefore, when you return to CIRCA to run populus again, you will again need to download and extract the files again. If you are using a personal computer, of course, you won’t have to do this.
5. There are many parts of the Populus program. You will only use a couple of them for this discussion section. However, you should feel free to explore the other parts of the program.
Questions:
1. From 1980 to 1990, the human population of Kuwait had one of the highest growth rates (r=.0431/year). If this rate persists, how long will it take Kuwait’s population to double in size (i.e., in what year)? How long will it take to triple in size? [You don’t need Populus to do this – only a calculator.]
Since we are assuming that the growth rate is constant, use the exponential growth model: Nt=N0 ert . Because we're interested in doubling time, set Nt/N0=2; or 2=ert; which can be solved for time: ln 2=rt; or t=ln 2 / .0431=16.1 yrs. I.e., the population size of Kuwait will double every 16.1 years (based on the stated assumptions).
The tripling time can be found using a similar approach: t = ln 3 / .0431 = 25.5 yrs.
2. On the other hand, the population of the US grew at a rate of r=.0090/year. At this rate, will the U.S. population ever double in size? If not, why not? If so, when (i.e., in what year)? [no need for Populus – just a calculator – but you can explore Populus’ model of Exponential Growth.]
t(doubling) = ln 2 / .009= 77 yrs.
3. Why do you think the US and Kuwait have such different rates of growth? It’s also worth thinking about some of the poorest countries in the world…despite relatively low life expectancies, these populations often have high growth rates. Why?
Both US and Kuwait have low death rates due to improvements in medicine, hygiene, etc. Birth rates in the US are low, in part, because women have relatively more social, educational, and economic independence (compared to Kuwait) and have greater choice about whether to have children at all. They can delay starting a family, and limit family size. One student who was born in Kuwait noted that polygamy is practiced in Kuwait; while this doesn't necessarily lead to high birth rates, it is part of a culture in which women have few rights or opportunities outside the home, and in which birth rates are quite high.
4. Use Populus to explore population growth under the logistic model (choose Population Growth, and then Logistic Population Growth). Use the Continuous option, and set N0=5, K=100, r=.6, and T=0. Run the program by hitting the return (enter) key. Hit the space bar to see 4 graphical displays simultaneously. At what density does dN/dt reach its highest value? At what density is dN/Ndt at its greatest value (what is this value)? What does the deceleration in Log(N) vs. time signify?
dN/dt is highest at N=50; dN/Ndt =0.6 when N=0 (or at least very close to 0); deceleration in log N means per capita growth rate is decreasing with increasing time (in this case, because N is increasing with time).
5. Now hit the F4 key (this will cause Populus to remember the location
of the results of the simulation you just ran). And now go back (hit "escape
key") and change the value of
a) K (change it to 200). What did this do to the relationships? Why?
b) now change r to .3 (i.e., K=200, r=.3, N0=5). What effect
did this have? Why?
c) now change N0 to 100, and then 200 and then 300. What’s going
on now? Please draw the general relationship between dN/dt and N (and dN/Ndt
and N) for K=200 and r=0.3.
d) Reset the model to N0=5, K=200, r=0.3 and run the simulation.
Now switch from Continuous to Lagged, and set T=5. This means that the
per capita growth rate is determined by the density 5 time units ago (i.e.
there is a delay in when density-dependence acts). What effect does this
have? Reduce the Lag to 2 time steps. What happens? Why?
a. The new is always above the old one (i.e., growth was faster initially, because competition was less severe -- i.e. the system started out even further from K). But the most important difference is that he population stabilized (i.e., stopped growing at a population size of 200 rather than 100).
b. When r is halved, it takes longer (close to twice as long) for the the population to reach its carrying capacity.
c. When N0=100, the population increases to K (as before), but because it's starting out at K/2, the growth (dN/dt) constantly decreases (i.e., we're past the inflection in the logistic function). When you start a population with N0=K=200, the population will remain at 200. When you start the population at 300, the number of individuals will decline to 200 (i.e., dN/dt and dN/Ndt are negative when N>K).
d. Incorporating a time lag of 5 causes the population to overshoot the carrying capacity. The subsequent decline (which is also an overshoot) is also the result of the time lag -- the system is always responding to the conditions that existed 5 time steps previously, even though the density has changed considerably. Note, that the oscillation becomes less extreme and N eventually approaches the carrying capacity. A smaller lag (e.g., of 2 time steps) results in a much smaller initial overshoot and is much closer in behavior to the standard logistic without a time lag.
6. Now use the Age-structured Population Growth model. Ignore the first two lines of options and jump down to a) Number of age classes = 6; b) View age class = 0; c) time intervals = 24. Then set lx = 1, 0.5, 0.25, 0.2, 0.1, 0 and mx=0, 0, 1, 2, 7, and Nx=10, 0 , 0, 0, 0, 0. Run the program. Explore the output (press the space bar to move among graphs – ignore the figure on Reproductive Value). Does this population grow exponentially? When? Why doesn’t the population grow exponentially at first? Why does the age distribution "jump" around. Does the age distribution eventually stabilize? When this happens, are the numbers in each age class increasing at the same or different rates?
This population grows exponentially after a stable age distribution is achieved; up to this time, the growth rate is variable as "blips" move through the age-structure. When the age structure is stable, the number of individuals in each age class is increasing at the same rate (i.e., the proportion of the population in each age class remains constant).
7. Play around with Populus. Explore the dynamics and see how they change when you change various parameters.
Do this. It's very instructive.