project : Solution Essays

All members of the group are to be involved in the computer part as well as in the write-up.You should respond to the questions posed below and write your report in the form of a paper with an introduction and conclusion.  Your report should include at least two pictures.Preliminary step.  Open MATLAB and type pplane8 in the command line.  If this program is on your computer (and in the MATLAB path) then the pplane window will open.  If not,then you need to download the program by visiting http://math.rice.edu/~dfield/ Go  to  this  site  in  a  browser  and  select  ‘pplane8.m’  and  save  the  file  as  pplane8.m  on  your computer.   As  long  as  this  file  is  in  the  MATLAB  path,  you  will  be  able  to  enter  the  word pplane8 in the command line and the program will run.  It produces menu driven windows that are somewhat self-explanatory.When you run pplane8, notice that one of the menus of the setup screen is labeled gallery.You will want to select ’predator prey’ from that menu for the problem below.  Then you enter various parameter values and hit ’proceed’ to go to the next step.  That opens a display screen with its own menus.  Under the heading solutions you will want to try ’plot several solutions’and ’find an equilibrium point’.The problem.  Investigate the non-linear, predator-prey systemx′=0.06x−0.0004yx,y′=−.08y+ 0.0002xy. The system models the populations of rabbitsxand foxesyin a hypothetical nature pre-serve, where timet is measured in months.Begin your report with an introduction of the problem.  In later paragraphs, deal with the following questions and tasks.Step 1.  Produce a picture showing the direction field and phase portrait of the system.  Plota selection of solution curves (trajectories) including the one referred to in Step 2 below.  Also,plot  the  equilibrium  point.   You  will  need  to  experiment  with  the  minimum  and  maximumpopulations in order to obtain a window that shows the equilibrium point and a good view of1