Homework and Report for Lesson 9:

Your Name:                                      Student ID:                                 Your seat row No.

Use the reverse side or additional paper if necessary.
    1. (15 points). Describe the one predator and one prey model.  Identify the independent variable and dependent variables. Explain the meanings of parameters.
 
 
 
 
 
 
    2. (5 points). Consider the following fox and rabbit model with no harvesting
                  y1' = -b y1 + c y1 y2,
                  y2' = d y2 - e y1 y2,         with variable initial conditions [ y1(0)  y2(0)]'.
Write a matlab function to define the right hand side of the differential equation; write a matlab M-file to solve the equation using Matlab built in function ode23 or ode45. Attach your Matlab code(s).  Hint: see   format_f.m  or   yprf.m,  and   lesson9.m  for examples.
 
    3. (30 points) Numerical Experiments.  Take  b=0.5, c=0.01, d=0.4, e=0.01, t0=0, tfinal=50, and y1(0)=100+10*(1+0.S ), where S is your student ID. Solve the problem use your Matlab code with y2(0)=90, 1, 40 respectively (i.e. three different cases). Plot the solutions versus time and the phase plot. Label all the plots (title, xlable, and ylable). Explain the solutions you obtained.
     
     
 
     4. (Extra Credit for 5 points.) We can use the Euler's method and improved Euler's method to solve the ODE system as well. This is useful if we do not have Matlab, we can use other computer languages. In my_euler.m and main_eluer.m, the Euler's method for solving an ODE system is given.  Modify the code to use improved Euler's method to solve the ODE system. Mark your changes. Use you code to solve the problem above with one group of parameters and plot the solutions.