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 answercoding questions in MATLAB. One report must be turned in for each group and must be in PDFformat. Labs must include each student’s:

• Name

• Student number

• Section number

• Recitation number

This lab is due on Friday, February 20, 2015 at 5pm. Each lab must be turned in throughD2L on the course page. When you submit the lab please include each group members information(name, student number, section number and recitation number) in the comments. This allows us

1

to search for a students report. Once the labs are graded you need to specifically check that yourgrade was entered. If you are missing a grade please bring the issue up with your TA within aweek of grading.

The report must be typed (including equations). Be sure to label all graph axes and curves sothat, independent of the text, it is clear what is in the graph. Simply answering the lab questionswill not earn you a good grade. Take time to write your report as up to 20% of your grade may bebased on organization, structure, style, grammar, and spelling. Project office hours will be held inECCR 143 from Monday, February 16, 2015 to Friday, February 20, 2015.

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. Wemay assume for simplicity that humans are the only predators of the fish. We want to make surethat the people don’t overfish the lake. So we model how the fish population changes over time.

The logistic model is a good model for population growth. Problem 16 in Section 2.5 of yourtextbook shows one way to add harvesting into the logistic model:

dy

dt= r(1 − y

L)y − h(t)

Here y(t) is the size of the fish population at time t. Let the units of y be hundreds of fish, andthe units of t be days. Also, r is the natural growth rate and L is the carrying capacity, as in theregular logistic model. And h(t) is the amount of harvesting at time t.

Let’s change this model slightly. It’s reasonable to assume that the amount of harvestingdepends on the number of fish in the lake. (Because the more fish there are, the easier it is to catchthem.) So let’s measure the amount of harvesting in terms of y (amount of fish) instead of in termsof t. Then the model is

dy

dt= r(1 − y

L)y − h(y) (1)

A reasonable harvesting function could be

h(y) =py2

q + y2

where the parameters p and q represent how good the locals are at catching fish.Finally, notice that if we define

f(y) = r(1 − y

L)y − py2

q + y2,

2

then we can rewrite the differential equation (1) as y′(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 wellexplore the behavior graphically.

Let p = 1.2 and q = 1. Now suppose we put different types of trout in the lake. Suppose all thetrout species have the same natural growth rate r = .65; but the carrying capacities L are differentfor the different species. Rainbow trout have L = 5.4, brown trout have L = 8.1, and bass haveL = 16.3.

Explore the behavior of the system for different values of L, starting with L = 5.4. Use Matlab(or another graphing tool if you prefer) to plot f as a function of y. Use this plot to estimate theequilibrium solutions of (1). (Remember: an equilibrium occurs where f(y) = 0.) Make sure youuse a suitable scale when plotting, so that you don’t miss any equilibrium values.

Now remake the same plot once with L = 8.1 and again with L = 16.3. Find the equilibriumvalues each time.

Notice that some values of L have a different number of equilibria than others. By trial anderror, find the approximate values of L where the number of equilibria changes. (You’ll have to tryvalues of L between the 3 you’ve already tried.) There are two L values in this range where thenumber of equilibria change. Each of these special L values is called a bifurcation value.

Now, plot vector fields of (1) for each of the different L values 5.4, 8.1, and 16.3. To plot thevector fields, you may use any program you like. We provide one method, in the file dirfield.mprovided. Instructions for using dirfield.m are included separately.

Using your direction fields, describe the long-term behavior of solutions for various initial con-ditions. Pay close attention to the differences in behavior for different values of L. Note that youmay need to take a (very) large time interval to see the true behavior of solutions.

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 yis 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 thatincludes harvesting. Suggest an analytical technique that could be used to evaluate theintegrals. (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 thisto 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 = 5.4, L = 8.1, andL = 16.3:

(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 cansee the relevant behavior.

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 determinestability from your vector fields?

(d) What are the possible long-term behaviors? How does the behavior depend on the initialconditions (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 bevarious issues to consider- explain your decision.

8. Find the two bifurcation values for L between 5.4 and 16.3. Describe how the number ofequilibria changes at each bifurcation value.

9. Discuss these questions briefly with your own thoughts: Are there any weaknesses in themodel we used? How do you think could the model be improved? Do you think there areadditional effects the model should account for?

4