BSC 2011 (Periods 3 and 5; Spring 1999)

Warning: this software is designed to run under DOS on a PC (I run it on a Pentium II, 350 MHz Windows 95 machine, but can't attest to how it might run on other systems -- they have fixed a bug I reported that prevented it from running on PII's -- it was originally designed back in the archaic days of Dos and 64K machines.)

So, the first thing you need to do is download and install the software.

1. Dowload Populus. (Click here to download Populus) (To find out about Populus, click here.)

a. If you are at the CIRCA labs, download the file (named pop342p.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.

3.  Make a short-cut to populus.exe (right click on populus.exe).  This will allow the dos program to run more "nicely" under windows.

4.  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).

5. 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.  It might be easiest if you keep pop342p.exe on a personal disc (unfortunately, the entire set of program files is too large for a single floppy).

6. 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, we can use the exponential growth model: N_t=N₀ e^rt.  Because we're interested in doubling time, so we set N_t/N₀=2, or 2=e^rt; which can be solved for time: ln 2=rt; or t=ln 2 / 0.0431=16.1 yrs. The population size of Kuwait will double every 16.1 years (based on the stated assumptions) or in the year 2015.

The tripling time can be found using a similar approach: N_t/N₀=3, or 3=e^rt  ln3=rt,  t = ln 3 / 0.0431 = 25.5 yrs.  The population of Kuwait will triple in 25.5 yrs.

The tripling time can be found using a similar approach as above: N_t/N₀=2, or 2=e^rt  ln2=rt, t = ln 2 / 0.009= 77 yrs.  So the U.S. population will double in size in 77 yrs or by 2076.  With this value of r, it would take the U.S. population more time to double in size than it would take Kuwait to triple in size.

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 the U.S. and Kuwait have low death rates due to improvements in medicine, hygiene, etc. However, birth rates in Kuwait are still relatively high despite the low death rate.  Birth rates in the U.S. are lower than in Kuwait, in part, because women have relatively more social, educational, and economic independence (compared to Kuwait), and women have a greater choice about whether to have children at all. They can delay starting a family, and limit family size. Past classes have raised the issue of  polygamy that 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 subsequently, birth rates in Kuwait are quite high.

dN/dt is highest at N=50 or when the population has reached half the carrying capacity, K.

dN/Ndt =0.6 when N=5 (at the starting point or when there are the fewest number of individuals in the environment and those individuals are experiencing the least amount of competition).

deceleration in log N means per capita growth rate is decreasing with increasing time, in this case, because N is increasing with time (the population has surpassed the inflection point or K/2 in the logistic model).

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 curve with the higher carrying capacity is always above the curve with the lower carrying capacity (i.e., growth was faster initially, because competition was less severe -- i.e. the system started out even further from K and there were more resources per individual). But the most important difference is that the population stabilized at a higher number of individuals due to the higher carrying capacity (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 population to reach its carrying capacity-the curve is moved to the right.

 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 (the system is only 2 time steps behind instead of 5, so the system is not responding to conditions as different as with the time lag of 5).

a) Number of age classes = 6;
b) View age class = 0;
c) time intervals = 24.
d) Then set

lx = 1, 0.5, 0.25, 0.2, 0.1, 0
mx=0, 0, 1, 2, 7
Nx=10, 0 , 0, 0, 0, 0.

e) 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 (the growth rate does not increase at a constant rate because during the first time periods, there are all newborns who are not reproducing, and during slightly later time steps the whole population consists of reproducing adults).   This causes the growth rate to be inconsistent until the age structure is stable, where 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).

This one is up to you guys-don't be afraid to check it out.

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