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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 of1trajectories.  Explain the meanings of the direction arrows, the trajectories, and the equilibriumpoint (in terms of rabbits and foxes).  How can one tell from the picture which population isthe predator and which is the prey?  In other words, how can we tell whether foxes eat rabbitsor rabbits eat foxes?Step 2.  Assume that at time 0, the preserve is stocked with 200 rabbits and 50 foxes.  Fora second picture, obtain a plot of both the rabbit and fox populations versus time (using thegraph* menu of the display screen).  Discuss the behavior of the populations over time.  Yourdiscussion should include estimates of the minimums and maximums of each population andwhen they occur.  What is the period of oscillation of the rabbit and fox populations?  Is itreasonable that such populations would be cyclical?  If, at a low point in the fox population, anillness would wipe out the remaining foxes, what would then happen to the rabbit populationaccording to the model?  Would it still be cyclical?Step 3.  Assume you are managing the preserve discussed in Step 2.  You recognize that therabbit and fox populations are quite unstable, with populations swinging dramatically betweenhigh and low values, so you plan to trap and remove a number of foxes.  Based on the pictures,at what point should you take this step and about how many foxes should you remove in orderto  achieve  the  goal  of  making  the  population  somewhat  more  stable?   Discuss  options  andobserve what happen if the timing or the number is wrong.  At what point on the trajectorywould the removal of foxes have the most destabilizing effect?2Grading Rubric for ProjectsMathematical Writing assignments involve writing skills as well as mathematical skills, sothe evaluation criteria include both.  The criteria for this assignment are:1.  Comprehensiveness (30%):(a)  all aspects of the assignment are covered;(b)  all questions stated in the assignment are answered;(c)  explanations are thorough;(d)  supporting arguments are included.2.  Mathematical Correctness (30%):(a)  appropriate mathematical and computer methods are used;(b)  mathematical steps are correct;(c)  mathematical logic is clear.3.  Expository Quality (30%):(a)  organization is logical and includes an introduction and conclusion;(b)  transitions from one idea to the next are smooth;(c)  mathematical work and computer output are smoothly integrated into the text;(d)  mathematical statements are presented within complete sentences;(e)  any figures or computer output are described in the exposition;(f)  grammar, spelling and punctuation are correct.4.  Extras (10%):(a)  the overall impact is positive;(b)  creativity and innovation are apparent.

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