APPM 2360 Lab #1: Fish Population 1 Instructions Labs may be done in groups of 3 or less. You may use any program, but the TAs will only answer coding questions in MATLAB. One report must be turned in for each group and must be in PDF format. Make sure to read the project writing guidelines on the course webpage. Labs must include each student's: Name Student number Section number Recitation number This lab is due on Thursday, February 18, 2016 at 5pm. Each lab must be turned in through D2L on the course page. When you submit the lab please include each group members 1 information (name, student number, section number and recitation number) in the comments. This allows us to search for a student's report. Once the labs are graded you need to specifically check that your grade was entered. If you are missing a grade please bring the issue up with your TA within a week of grading. The report must be typed (including equations). Be sure to label all graph axes and curves so that, independent of the text, it is clear what is in the graph. Simply answering the lab questions will not earn you a good grade. Take time to write your report as up to 20% of your grade may be based on organization, structure, style, grammar, and spelling. 2 Introduction The goals of this lab are: Interpret a model stated in terms of a differential equation by - Model the population of fish in a suburban lake - Use some basic numerical and qualitative techniques 3 Model Suppose you live in a suburban area, with a lake stocked with fish where people may go fishing. We may assume for simplicity that humans are the only predators of the fish. We want to make sure that the people don't overfish the lake. So we model how the fish population changes over time. In many cases, the logistic model is a good model for population growth. Problem 16 in Section 2.5 of your textbook shows one way to add harvesting into the logistic model: \u0010 dy y\u0011 =r 1 y h(t) dt L Here y(t) is the size of the fish population at time t. Let the units of y be hundreds of fish, and the units of t be days. r is the natural growth rate and L is the carrying capacity (as in the regular logistic model) and h(t) is the rate of harvesting at time t. Let's change this model slightly. It's reasonable to assume that the amount of harvesting depends on the number of fish in the lake. (Because the more fish there are, the easier it is to catch them.) So let's measure the amount of harvesting in terms of y (amount of fish) instead of in terms of t. Then the model is \u0010 dy y\u0011 =r 1 y h(y). (1) dt L A reasonable harvesting function could be h(y) = py 2 , q + y2 where the parameters p and q represent how good the locals are at catching fish. Finally, notice that if we define \u0010 py 2 y\u0011 y , f (y) = r 1 L q + y2 2 then we can rewrite the differential equation (1) as y 0 (t) = f (y). It is difficult (although possible) to find the exact solution to equation (1). Also, the solution is complicated and hard to understand. So instead of using the exact solution, we might as well explore the behavior graphically. Let p = 1.2 and q = 1. Now suppose we put different types of trout in the lake. Suppose all the trout species have the same natural growth rate r = 0.65; but the carrying capacities L are different for the different species. Rainbow trout have L = 6.1, brown trout have L = 10.5, and bass have L = 20.2. Explore the behavior of the system for different values of L, starting with L = 6.1. Use Matlab (or another graphing tool if you prefer) to plot f as a function of y. Use this plot to estimate the equilibrium solutions of (1). (Remember: an equilibrium occurs where f (y) = 0.) Make sure you use a suitable scale when plotting, so that you don't miss any equilibrium values. Now remake the same plot once with L = 10.5 and again with L = 20.2. Find the equilibrium values each time. (Problem 5) Notice that some values of L have a different number of equilibria than others. By trial and error, find the approximate values of L where the number of equilibria changes. (You'll have to try values of L between the 3 you've already tried.) There are two L values in this range where the number of equilibria change. Each of these special L values is called a bifurcation value. Locate the two bifurcation values. (Problem 8) Now, plot vector fields of (1) for each of the different L values 6.1, 10.5, and 20.2. To plot the vector fields, you may use any program you like. We provide one method, in the file dirfield.m provided. Instructions for using dirfield.m are included separately. Using your direction fields, describe the long-term behavior of solutions for various initial conditions. Pay close attention to the differences in behavior for different values of L. Note that you may need to take a (very) large time interval to see the true behavior of solutions. (Problem 6) 4 Questions 1. What are the units of the parameters r and L? (Remember what the units of y and t are.) 2. Explore the harvesting function h(y). What happens to h(y) as y gets very large? What if y is close to 0? Does this make sense physically? (It may help to try graphing h(y).) 3. (a) Use separation of variables to solve (1) analytically when h(y) = 0 (i.e. no harvesting). (b) Set up the integrals that one gets when separation of variables is done on the model that includes harvesting. Suggest an analytical technique that could be used to evaluate the integrals. (However, you don't need to actually solve the integrals.) 4. (a) Describe in words what an equilibrium solution is. (b) Give a mathematical definition of an equilibrium solution. Explain how you can use this to find the equilibria from the plot of f (y). 5. For the model with harvesting, turn in two plots for each of the values L = 6.1, L = 10.5, and L = 20.2: (a) Plot f (y) as a function of y. 3 (b) Plot the direction field. Choose an appropriate range of t and y values so that you can see the relevant behavior. Indicate the equilibrium solutions in each case. Be sure to title your graphs and label the axes. 6. Interpret the function plots and vector fields from Problem 5: (a) What is happening to the population when f (y) > 0? when f (y) < 0? when f (y) = 0? (b) Use the information from part (a) to tell whether each equilibrium is stable or unstable. (c) Make sure that your function plots and vector fields are consistent with each other. Discuss what the equilibria look like on your vector fields. How can you determine stability from your vector fields? (d) What are the possible long-term behaviors? How does the behavior depend on the initial conditions (population at time t = 0) and the value of the parameter L? 7. Based on your results, which species of trout is best to stock the lake with? There may be various issues to consider- explain your decision. 8. Find the two bifurcation values for L between 6.1 and 20.2. Describe how the number of equilibria changes at each bifurcation value. 9. Discuss these questions briefly with your own thoughts: Are there any weaknesses in the model we used? How do you think could the model be improved? Do you think there are additional effects the model should account for? 4