# IDEA: Internet Differential Equations Activities

## Some Basic Models

In this exercise we consider some naive models for the flushing of juvenile salmon and steelhead (called smolts) down the Snake River in Washington. This problem has been in the news in recent years, since salmon and steelhead (anadromous rainbow trout) populations have dwindled dramatically. There are probably many factors contributing to the decline including commercial and sport fishing, habitat degradation due to urbanization, logging and farming, and the construction of several dams on the lower Snake and Columbia Rivers that inhibit the migration of smolts to the sea. We will focus only on how the dams slow the migration of the smolts from the rivers of their birth to the ocean where the grow to maturity.

In order to get the smolts to the Pacific, various approaches have been tried, including barging them down the river, spilling water from the dams on the river to create a more powerful current and reduce the volume of the reservoir, and combinations of these ideas. We are interested in approaches that involve spilling water. Historically, the reservoirs behind the dams are filled in the spring, and the water is used for irrigation, navigation, and hydroelectric power throughout the rest of the year, particularly during the summer. On the other hand, the smolts migrate in late spring and early summer, so that if the Snake is to be a river, then water may not be stored in the customary way. Instead, the outflow from each reservoir must be as great or greater than the inflow. In actual practice, the reservoirs have sometimes been "drawn down", meaning that much of the water from the reservoirs is flushed out.

Why is it necessary to empty the reservoir? Would it not be sufficient to make the outflow equal to the inflow, so that the reservoir would behave like a wide spot in the river? Consider the Lower Granite Reservoir, near Lewiston, Idaho, formed behind the Lower Granite Dam. It is fed by two rivers: the Snake, and the Clearwater. The Snake is fed, in turn, by the Salmon River, not far from Lewiston. Adult salmon and steelhead spawn in the headwaters of these rivers and their tributaries and thus the smolts must pass Lower Granite Dam (photo) on their return journey to the Pacific Ocean.

We will construct differential equations that model the interaction between smolts traveling down-river and the dams that control the current. Let F denote the number of fish in the reservoir behind Lower Granite Dam. We wish to derive a differential equation describing the change in fish population per day, dF/dt. We can find an expression by determining the rate at which fish enter the reservoir Rin and subtracting from this the rate at which fish leave the reservoir Rout. This assumes that the fish are uniformly distributed throughout the reservoir which is probably not true. All the necessary data is available on the web at a site entitled DART.

Problem 1. Let's begin by constructing our model equation. Let Vin and Vout denote the rate at which water is entering and leaving the reservoir respectively measured in thousands of cubic feet per second. Assume that Vout = Vin. Assuming that the fish are uniformly distributed throughout the reservoir implies that dF/dt is equal to the rate at which fish enter the reservoir (Rin) minus the rate at which fish leave the reservoir (Rout). To determine Rout you will also need to know the total volume of the reservoir. We approximated the total volume V of the reservoir by finding the acreage of Lower Granite Lake using information from the Walla Walla District of the Corp of Engineers. Lower Granite Lake is approximately 8900 acres or 387.684 million square feet. Assume that the average depth of the reservoir is 20 feet. Show that the differential equation for dF/dt having units thousands of fish per day is given by

dF/dt = Rin - 0.0111 Vout F.

(Hint: be careful with time units since Rin has units of fish per day and Vin has units of thousands of cubic feet per second!).

Problem 2. The DART Smolt Index provides a way for you to retrieve all the information you will need to complete this project. You will need four pieces of data,

1. the rate at which smolts enter the reservoir Rin, and
2. the rate at which water is entering the reservoir Vin ,
3. the rate at which water leaves the reservoir Vout , and
4. the initial volume of the reservoir V.

Assuming an average depth of 20 feet we get an approximate volume of 7753.6 million cubic feet. In this exercise we will treat Vout as a parameter and will adjust it (and hence the volume) to test it's effect on smolt migration. For the other two values let's get data for one day only, say May 1, 1997. Thus we can focus on how to get the data and how to use it in our model. Later we'll get more data to improve our model.

To retrieve the rate at which smolts enter the reservoir first go to the DART Smolt Index (you can probably open a second web browser from the File menu and do this in another window).

 Click on the box next to the word Graphics to indicate the format you wish the data to be delivered in. Under the words Select Output Format you can check Graph to view a graphical output of the data or View Data to view the raw data. Since we want to use the data in our model you should check View Data. Using your left mouse button click on Smolt Index in the first column, a year in the second column (i.e. 1997), LWG-Lower Granite in the third column to designate the dam, a type of fish in the fourth column (i.e. Steelhead), and (All Rear) in the last column. This indicates both wild and hatchery raised fish will be included in the data. Finally you need to pick a range of dates. Since we only want data for May 1, 1997 type 5/1 as a start date and stop date. Finally click on the Begin Run button to retrieve the data.

The result of your query should of been similar to the following

Smolt Index

-----------

Stlhd.97 LWG

121 292515.00

The first number 121 is the date (May 1 is the 121st day of the year) and the second number is the estimated number of steelhead smolts that made it to Lower Granite Dam on this date.

To get the volume of water entering the reservoir follow the same procedure. First click on Smolt Index to unselect that piece of data and then click on Inflow to get the volume of water entering the reservoir. The units on this data is thousands of cubic feet per second (kcfs). Thus the inflow is 169.8 thousand cubic feet of water per second. (Note: In some browsers you can choose at most 2 different items in a column and you can do this only one column at a time).

Problem 3. Use the values of Rin and Vout you found in Problem 2 to solve the differential equation for dF/dt given in Problem 1. Determine the equilibrium smolt population in the reservoir.

Problem 4. One days worth of data is hardly representative of the entire migration process. Use the DART Smolt Index to get two weeks worth of data. You should choose your two week interval when migration is relatively heavy (most smolt migration takes place from late March to early June). From this data compute average values of Rin and Vout. Use the model of Problem 1 to determine the equilibrium level of smolts in Lower Granite Reservoir.

Problem 5. Using the differential equations of Problem 1 and letting Rin and Vout be arbitrary parameters show that in this model the equilibrium concentration of fish decreases as the flow rate Vout increases. You can use the applet below or the DynaSys file salmon1.dse to illustrate this phenomenon.

In Problem 4 the smolt data you collected varied considerably over the two week period you considered. This suggests that the rate of smolt migration is not constant but changes over time. In other words, the parameter Rin really should be a function of time t. We've done some very basic statistics to determine a function that approximates the 1997 steelhead smolt data at Lower Granite Dam. The graph below shows the smolt data collected from the DART Smolt Index along with the function fitted to that data (to get the Microsoft Excel spreadsheet that generated this figure click here). Admittedly this is not a perfect fit of the data, but for the purposes of this exercises it will suffice.

We've determined that the fitted pink curve is given by

Rin(t) = exp(5.02 - .002 (t-36)2) smolts/day.

Note that t = 0 is approximately the first day steelhead smolts arrive at Lower Granite Dam

Problem 6. Use the applet below or download the DynaSys file salmon2.dse to investigate how steelhead smolts pass Lower Granite Dam when we take into account the time dependence of smolt migration. What is the equilibrium fish concentration in this case?

One strategy for getting smolts downriver quickly is a drawdown. When a dam such as Lower Granite is drawn down the outflow far exceeds the inflow to significantly lower the lake level, thereby making the "lake" more like the natural river before dam construction.

We've modeled a drawdown using the basic tools we've developed in this lab. We will begin by assuming that the rate at which fish enter the reservoir is determined by the equation used in Problem 5. The reservoir is drawn down by releasing water as shown in the figure below. For s days water is released at the rate of Vin + M kcfs. Then water is released from the dam at the same rate it enters the reservoir. Hence the water level is much lower than normal but remains constant.

We can determine the volume of Lower Granite Lake at any time t by assuming that the total volume at time t=0 is given in Problem 2 and subtracting from that the integral of Vout - Vin from 0 to t. Thus

V(t) = V0 - M t if t < s and

V(t) = V0 - M s if t > s.

Putting all these pieces together gives the differential equation

dF/dt = Rin(t) - Vout(t)/V(t) F

describing the smolt population in Lower Granite Lake.

Problem 7. Use the applet below or download the DynaSys file salmon3.dse to investigate the effect of a drawdown. Use the DART Smolt Index to determine an approximate value of Vin. Describe how changing the parameters M controlling the excess water released from the dam and s controlling the duration of the drawdown effects the rate of passage of smolts.