Environmental Decisionmaking

Problem Set #3:

1) (10 pts) Take your two schematics from Problem Set #2 (the water in the pond and the colonization of a new island), and set them up as Excel models. For the pond problem, assume that there are 100,000 gallons of water in the pond, and that 500 gallons of water flow into the pond every hour. What must the rate coefficient be for the water flowing out of the pond, to keep the level of water from changing? Run the model with those parameters. Now, assume that the amount of incoming water increased (perhaps somebody began irrigation upstream), and so 600 gallons/hour began to flow into the pond. After about a month, would the pond reach a new steady-state (does the graph roughly level off?). For the island problem, set up the stock and flow model. Put in numbers just to make sure that it runs and makes sense, but don't worry if they aren't meaningful (though feel free to try to think about how to make them meaningful). You can e-mail me your spreadsheets.

In the pond problem, all I want you to do is to realize that the inflow is a constant (it is exogenous to the model), while the outflow is a function of the volume of water in the pond (as the water level rises, more water flows out of the outflow stream). That means that there is a rate coefficient for the outflow that you multiply by the volume of water in the pond. We know that the outflow paramter has to be equal to 0.005/hour (because 500 gallons/hour = 100,000 gallons * outflow parameter). When we increase the inflow parameter to 600 gallons/hour from 500, the level in the pond to rise until the outflow equals 600 gallons/hour as well. If we thought about the problem we would know that the final lake level will stabilize at 600 gallons/hour / 0.005/hour = 120,000 gallons. Download this example model to see how this problem might be set up.

For the new species on an island, we would essentially have a situation where the species was limited only by resources. That is a lot like the simple rabbit example which I gave in class. You can find that elsewhere on Blackboard (it is one of the examples that you asked me to post, even though really it is the answer to the question...and you guys complain that I make you work too hard...)

2) (5 pts) Set up the Rabbit and Foxes model that we went through in class (i.e. stocks of rabbits and foxes, where fox population affects rabbit deaths, and rabbit populations affect fox deaths and births). See if you can find a way to get the rabbit and fox populations relatively stable for a long period of time at some natural level (i.e. it has to be above the level that you force with a "max" function). You may not do this by simply cutting down the birth and death rates to very low values, but other than that you may use any reasonable combination of rates and you may add in any other feature in your model that you want (for example, food, disease, alternative prey species, multiple sites, different sorts of mathematical representations of the interactions as long as you can justify them, and so on). If you use explicit immigration, then I want the steady-state population to be at least twice the immigrating population (i.e., you can't just say "I have 300 rabbits immigrating every period, and my rabbit population is stable at 300!"). Explain what you have done.

Here you can download the starting point from Blackboard also (Rabbits and Foxes). The easiest way to do this is to decrease the step size (thereby limiting the overshoot, which we said was important here), and using squared terms in the death equations (i.e. fox death = constant * (foxes^2)/rabbits. This second part makes deaths rise very quickly when the populations get too high, and so keeps things from getting out of hand. You may still have to fiddle, as there are combinations of parameters where this system is still unstable. Mostly, I wanted you to fiddle and see if you learn things about your system.

3) (5 pts) Briefly explain the arguments that Meadows et al. make in the reading. What are your thoughts on the connection between exponential growth and sustainability, and how compelling do you find the discussion of this connection in the reading? Explain.

Meadows et al. believe that because of exponential trends such as increasing human consumption, economic growth, human population growth, etc., it is nearly impossible for the planet to avoid catastrophe unless explicit choices are made to limit consumption before natural limits are reached. Because we have become so good at utilizing resources and processes, we may not even realize that we are overconsuming until we are near the end of our available resources and we don't have time to develop substitutes (and in many cases the authors do not believe substitutes exist).