MATH36031 A Fox and A R abbit

MATH36031 Project 2 - deadline 22nd November 2024, time 1100hrs.

In this project, the dynamics between a fox and a rabbit will be investigated, by solvingdifferential equations modelling their positions at different times. The initial configurationis shown in Figure 1, where the fox starts chasing the rabbit while the rabbit tries to escapefrom its predator and moves towards its burrow instantaneously.The fox, initially located1at (300, −550), pursues the rabbit with the initial speed sf0 = 15m/s in one of the followingtwo possible ways:

Warehouse R(0,0)B(600,600)NW(200,0)SW(200,−400)F(300,−550)Figure 1: Coordinates (in metres) of the fox (F), the rabbit (R) and its burrow (B) and thtwo corners (SW,NW) of the warehouse.❼ if the rabbit is in sight, the fox’s attack path points directly towards the rabbit (thedirection of the velocity vector of the fox is exact from the fox to the rabbit);

❼ if the view of the rabbit is blocked by the corner SW of an impenetrable warehouse(assuming that the warehouse is extended indefinitely to the east), then the fox runs1The subscripts f and r denote the fox and rabbit respectively. The units of the coordinates are metres.1directly towards this corner. If the rabbit is still not in sight when the corner is reached,then the fox moves parallel to the NW-SW perimeter of the warehouse until it sees therabbit.The rabbit, initially located at the origin (0, 0), runs towards its burrow at (600, 600) ina straight line with initial speed sr0 = 11m/s.

Question 1: Constant speeds.. Assuming that both the fox and the rabbit run withconstant speeds sf0 = 15m/s and sr0 = 11m/s respectively, determine whether the rabbit

can be captured before it reaches its burrow. The rabbit is considered to be captured by thefox, if the distance between them is smaller than or equal to 0.1 meter.Question 2: Diminishing speeds. Let us consider a more realistic scenario, whenthe hungry fox meets the tired rabbit. Because neither the fox nor the rabbit are in theirbest conditions, their chasing/escaping speeds diminish in time, according to the amount ofdistance (starting from the time they find each other and start running) they have travelledso far. More precisely, their speeds at time t are given bysf (t) = sf0e −µf df (t), sr(t) = sr0e −µrdr(t) , where sf0 = 15m/s and sr0 = 11m/s are the same initial speeds as above, µf = 0.0002m−1and µr = 0.0008m−1 are the rates of the diminishing speeds, df (t) and dr(t) are the distancethey have travelled up to time t(> 0). Determine whether the rabbit can be captured beforeit reaches its burrow. (You may assume that this diminishing speed starts from t = 0).Outputs required You are required to submit a report (maximum 8 pages includingany appendices) in pdf form via the Turnitin submission box on Blackboard. Additionallyyou need to submit your m-files or mlx-files used for the MATLAB codes via theBlackboardSubmission Box also on Blackboard. As part of the required output, in yourreport give(i) the time T and the location of the fox when either the rabbit is captured or the rabbitescapes to the burrow; (ii) the distance travelled by the fox in time T . Additionally providea plot showing the paths taken by both animals.

Additional information and guidelines

1. All coding must be done in MATLAB.
2. Treat both the fox and the rabbit as points, without worrying about their finite sizes(as in most models).

3. The questions can be answered with different approaches, but you need to use thebuilt-in ODE solver ode45 discussed during the lectures.

4. Avoid using hard-coded numbers. Any number in your code should either be given asinitial condition, or be derived from these conditions.

5. Keep to the page length not exceeding eight A4 pages, and there is no need for a titlepage or abstract for a relative short report like this. Font sizes should be no smallethan 11 point, and page margins 代寫MATH36031 A Fox and A R abbit no smaller than 2cm.2 List the complete code of the whole function at the end of each question, or in anappendix. Make your source code more readable, by keeping the indentation andstylistic features, and can be copied from your submitted. Your published resultsshould be reproducable from the code attached.

7. Have a look at the generic rubric about how your report will be marked, and also theintended learning outcomes about what you are expected to achieve in the end.

8. Avoid copying (too many) sentences directly from the project description, and try torestate the problem with your own words or examples if possible.

9. You may use your report in the future as evidences of written work, so take it seriously.
10. Your target audience is a fellow student on your course: explain the questions so thatthe report can be understood without this project description and your approach couldbe implemented in another computer language like Python. The report shouldindicateto the reader how well you understand the problem and the approach you took.Yourgoal will be to communicate your solutions to another person rather than toshowyou’ve completed the assignment.

11. Balance the explanation of the approach and the comments in the code. Avoid undercommenting and over-commenting.

12. Aim for precision and clarity of writing.
13. Since there is no final exam, you are advised to spend at least 15 hours on each project,with additional self-study if you are less experience with computer programming. Remember for a 10 credit module like this one, you are expect to spend 100 = 10 Ö 10hours in total (including lectures, labs, self-study and coursework).

14. Please do not put any personal information on the report, only your student ID number.
15. The submission for each project will be open two weeks before the deadline. Onlyyour last submission will be marked, and anything submitted after the deadline willbe treated late and any penalty will be applied by the Teaching and Learning OfficeinJune according to the Undergraduate Student Handbook.

16. Whilst this project can be done without the use of any artificial intelligence (AI)

software tools, if you use any AI tools or software to help you with your project, you must mention this in the report. Please study the guidelines at4on how to do this correctly. The content and accuracy of the report will be your responsibility alone, and any factually incorrect statements or mathematically incorrect contentwill bepenalised.3 Your attention is also drawn to to the University’s Academic Malpractice Policy, see

We are obliged to report cases of suspected academic malpractice, and people maybe subject to an additional oral assessment on the content of the report and codessubmitted4