## Math 5270 - Modeling with Dynamical Systems Homework Assignments

### Spring 2000 Prof. Peckham

• Set 1 Due Monday 1/31/2000:
• In Strogatz: Read Chapter 1 Overview. Read Sections 2.0 - 2.3. Do problems 2.3.2, 2.3.3.
• In Nagle and Saff (Problem numbers are for the 5th edition. 4e=4th edition, 3e=3rd edition.):
• Group projects for Chapter 2. Work on the following models for discussion in class (not to turn in): A. The snowplow problem. D. Torricelli's Law of Fluid Flow (C in 4e, not in 3e).
• Read Sections 3.1 - 3.4 on Mathematical Modeling. Concentrate on obtaining the differential equations models from the real world problems rather than on obtaining the analytical solutions of those differential equations.
• Do Sec. 3.2 problems: 6 Carbon monoxide; 19 Fish population problem (17 in 3e); 21 Snowball; 25 Carbon dating (23 in 3e);
• Mixing problem: A brine solution of saltwater with concentration 0.2 kg/l flows at a constant rate of 4 l/min into a 200 l tank which initially holds 100 l of pure water. The solution in the tank is well stirred and flows out of the tank at a rate of 2 l/min. Let s(t) be the mass of salt in the solution at time t. Write down a differential equation and initial condition which determines s(t). What is the new differential equation when the tank fills? What about the initial conditions? (You need not solve for s(t).)
• Sec. 3.3: Heating and Cooling. (2 in 5e) A cold beer initially at 35 F warms up to 40 F in 3 min while sitting in a room of temperature 70 F. Set up a differential equation and corresponding initial condition which will determine the temperature as a function of time t.
• Do Sec. 3.4 problems: 8. Parachutist. Set up the differential equations and initial equations which would need to be used both before and after the parachute opens. You need not find a formula solution or determine when the parachutist hits the ground.

• Presentation 1. Choose one of the homework problems from Set 1. Present its solution to the class. Include one change in the assumptions made for the model and how that change changes the model. Each student must choose a different problem to present. Presentations should be 5 to 10 minutes.

• Set 2 Due Monday 2/14/2000:

1. In Strogatz:
1. Numerical Solutions. Read Section 2.8. Do problems 2.8.1, 2.8.2b. You may print out the slope field using Differential Systems or any other software rather than sketching by hand.
2. Basic bifurcations. Read Sections 3.0, 3.1 through p. 48, 3.2 through Fig. 3.2.2, 3.4 through p. 56. Do problems: 3.1.1, 3.2.1, 3.4.1.
3. Insect Outbreak. Read Sec. 3.7 Do problem 3.7.1.
4. Phase planes. Skim Secs 5.0, 5.1, 5.2. Read Sec. 5.3 (Love affairs). Do problems 5.3.1, 5.3.6.

2. Coupled problems. Develop models for the following scenarios. Determine differential equations and corresponding initial conditions. You need not determine solutions, but you might guess what the behavior of solutions is. This will be more difficult for some models than others. If your model has only one or two variables (once converted to first order differential equations) exhibit some numerical solutions using software (eg., Differential Systems). Be sure to define your variables and explain any modeling assumptions you make. Cite any references used, if any.
1. A two tank model with salt water flowing in and out.
2. A two population model. Create either a model for Foxes and Rabbits, or for Deer and Rabbits (competing for the same food).
3. Any one of the following:
1. A two spring/two mass system.
2. A coupled pendulum.
3. An RLC circuit with more than one loop.
4. Two pendula coupled with a spring.

• Presentation 2. Beginning Monday Feb. 21. Choose a topic from one of the alternative references or journals. Let me know the topic so we can try to prevent topics that are too easy, too hard, or the same as other students. Present the model to the class. Concentrate on the modeling issues. If numerical solutions are possible, present them as well, but don't spend much time on this part. Presentations should be in the 15-20 minute range. Turn in a one-page written analysis of the project. A second page is allowable only for a figure or numerical results (such as Differential Systems output).
Writeup due Wed. 3/8/00.

• Logistic Model, Part 1. Due Monday, 3/13/2000.
1. 1D Differential Equation:
dx/dt = ax -bx^2
Assume a>0 and b>0. Rescale x to make the new differential equation:
dx/dt = a x (1-x).
Locate all equilibrium points and how they depend on the parameter a. Determine the stability of each equilibrium point by linearization, and determine how the stability depends on a. Could this model ever allow a positive population to exist without limiting to a constant value (a ``carrying capacity")? If so, for what parameter values?
2. 1D Map (Discrete Dynamical System):
x_(n+1) = a x_n - b (x_n)^2
Assume a>0 and b>0. Rescale x to make the new differential equation:
x_(n+1) = a x_n (1-x_n).
Locate all fixed points and how they depend on the parameter a. Determine the stability of each equilibrium point by linearization, and determine how the stability depends on a. Could this model ever allow a positive population to exist without limiting to a constant value (a ``carrying capacity")? If so, for what parameter values?
3. 1D Delay Differential Equation:
dx(t)/dt = a x(t) - b x(t)x(t-r)
Assume a>0 and b>0. Rescale x to make the new differential equation:
dx/dt = a x(t) (1-x(t-r)).
Locate all equilibrium solutions and how they depend on the parameters a and r. Determine the stability of each equilibrium point by linearization, and determine how the stability depends on a and r. Could this model ever allow a positive population to exist without limiting to a constant value (a ``carrying capacity")? If so, for what parameter values?
4. 2D Differential Equation. Assume that we keep track of our population in two different age categories: x and y. Develop a differential equations model analogous to the 1D dfferential equation model. Eliminate any parameters you can. You need not analyze this system until we agree as a class on a model.
5. Delayed discrete dynamical system. Combination of 1D discrete dynamical system and the 1D delay differential equation. Start with the 1D discrete map: x_(n+1) = a x_n - b x_n^2), but replace the second x_n with x_(n-1), analogous to the 1D delay differential equation. Convert this second order difference equation into a system of two first order difference equations. Eliminate any parameters you can. You need not analyze this system until we agree as a class on a model.

• Midterm 1 Wednesday March 22, 2000. See Midterm Topic List
• Preliminary Project Topic Description. Due Friday, March 24, 2000. Describe in a half page or less a brief outline of a project that you plan on doing. Indicate roughly where you expect to spend your effort: literature search, model development, analyzing a model, data collection, writing up results, ... .
• Logistic Model, Part 2. Due Monday, 4/10/2000.
1. (Due date pushed back to 4/17/2000, but no late ones will be accepted after the beginning of class on 4/26 since we will review all the logistic models in class on 4/17.) 2D Differential Equation. We developed several models with two age classes, but I will choose the following one to analyze:

x(t)=juvenile population at time t (unable to reproduce)
y(t)=adult population at time t (able to reproduce)

Assuming the populations behave logistically, we came up with the follwing systems of differential equations:

dx/dt = B y - D1 x - D2 x^2 - D3 x y - G x
dy/dt = G x - D4 y - D5 y^2 - D6 x y

Explain the significance of each term in the system. Assume all parameters are positive. Show that by rescaling both x and y we can only eliminate one parameter. Therefore, choose the same rescaling factor for both and show that the rescaling factor can be chosen to allow the system to be rewritten as

dx/dt = b y - d1 x - d1 x^2 - d3 x y - g x
dy/dt = g x - d4 y - d5 y^2 - d6 x y

• What is the relationship between all corresponding parameters before and after rescaling? For simplicity, assume the parameters d3 and d6 are zero.
• Determine nullclines, and how they vary with the parameters. (Why did I decide to make d3 and d6 zero?) Use the nullclines to help locate all equilibrium points and how they depend on the parameters. For which parameter values is there a positive carrying capacity?
• Determine the stability of each equilibrium point by linearization, and determine how the stability depends on the parameters.
• Could this model ever allow a positive population to exist without limiting to a constant value (a ``carrying capacity")? If so, for what parameter values? (Hint: Look at the determinant and trace of the Jacobian matrix used in the linearization.)
• Corroborate your analysis with a numerical phase portrait (for example using Differential Systems software). Label any stable and unstable manifolds of saddles. (You may draw these in by hand if you wish.) Use the phase portrait to describe the long term behavior of the population and how it depends on the starting populations.

2. (Work on this problem will not be accepted after the beginning of class on Wednesday April 26, 2000.) Delayed logistic map (discrete dynamical system). Combination of 1D discrete dynamical system and the 1D delay differential equation. Start with the 1D discrete map: x_(n+1) = a x_n - b (x_n)^2, but replace one of the x_n's in (x_n)^2 with x_(n-1), analogous to the 1D delay differential equation. Rescale to make a new discrete map: x_(n+1) = a x_n - a x_n x_(n-1). Convert this second order difference equation into a system of two first order difference equations:

y_(n+1)=x_n
x_(n+1)=a x_n (1-y_n)

Locate all nonnegative fixed points and how they depend on the parameter a. Determine the stability of each equilibrium point by linearization, and determine how the stability depends on a. Could this model ever allow a positive population to exist without limiting to a constant value (a ``carrying capacity")? If so, for what parameter values? (Hint: Locate a bifurcation where the positive fixed point changes from attracting to repelling.)

• Homotopy between two predator-prey models to locate a bifurcation. Due Monday May 1, 2000.

We saw in class that two different versions of a predator-prey model exhibited different behavior:

dR/dt=5R-R^2 - 4RF
dF/dt=-5F+4RF

had an attracting positive population for both predator and prey.

dR/dt=R-R^2 - 4RF/(1+4R)
dF/dt=-.5F+4RF/(1+4R)

had a repelling positive population for both predator and prey.

• Construct a one-parameter "straight line homotopy" between the two differential equations so that when the parameter (call it a) is zero, you have the first system of differential equations, and when a=1, you have the second system of differential equations.
• Because of the more complicated interaction term in the second system of differential equations, analytically locating all equilibrium points in terms of the parameters is difficult. But locating some of them - the ones one the axes - is not as difficult. Find all the equilibria which live on either axis in terms of the parameter a. How does the stability of these point(s) depend on a? This will help understand the phase portraits for the family, but not necessarily the bifurcation.
• Using Differential Systems, vary your parameter (trial and error) until you locate a bifurcation (find the value of a) which changes the attracting equilibrium of the first system to the repelling equilibrium of the second. What are the eigenvalues of the equilibrium point at this value of the parameter a? What is the name of this type of bifurcation? What happens to orbits that are repelled by the equilibrium for parameter values "after" the bifurcation parameter)? Include printouts of phase portraits corresponding to a=0, a=1, and a values just above and just below the bifurcation value. Sketch and label on your phase portraits any stable and unstable manifolds of saddles. Focus on the first quadrant since this is a population model.