MAS108 Project

MAS108 Project#

This assessment is worth 20% of your grade for MAS108. You should hand it in on Blackboard, via the ‘Assessment Information / Written Project’ folder, by 2pm Thursday 23rd January.

This project examines a model of predator-prey dynamics in discrete time. Let \(U(t)\) and \(V(t)\) be the number of prey and predators at time \(t\), respectively. We model their dynamics as follows

(1)#\[\begin{align} \label{prey} U(t+1)&=U(t)+rU(t)-pU(t)V(t), \\ \label{pred} V(t+1)&=V(t)+pcU(t)V(t)-dV(t), \end{align}\]

where \(r\), \(p\), \(c\), and \(d\) are positive constants.

Your tasks are as follows:

  1. Explain how to interpret the constants \(r\), \(p\), \(c\), and \(d\) biologically.

  2. Set \(r=0.5\), \(p=0.0001\), \(c=0.5\), \(d=0.1\). Set \(U(0)=4000+N\) where \(N\) is the last three digits of your registration number. Set \(V(0)=4000+M\) where \(M\) is the number given by the 5th, 6th, and 7th digits of your registration number. (For example, if your registration number is 160153427 then \(U(0)=4427\) and \(V(0)=4534\).) Calculate \(U(t)\) and \(V(t)\) for \(t<40\) and plot the resulting curves (of \(U(t)\) and \(V(t)\) against \(t\)) on a single graph.

  3. Explore the dynamics of the equations by repeating Task 2 for different parameter values, maximum times, and/or initial conditions.

  4. Write up what you have found by doing tasks 1-3 in a single report, including a biological interpretation of your findings, using between 800 and 1000 words, plus diagrams. Hand this report in according to the instructions at the top of this page.

This will be marked on a pass/resubmit basis first time around. To pass, your report must meet the following criteria:

  • All the above tasks must be attempted.

  • A reasonable attempt at interpreting \(r\), \(p\), \(c\), and \(d\) must be given, even if your interpretations miss the mark.

  • Your report must contain an explanation of the dynamics for at least three different sets of parameter values, maximum times, and/or initial conditions. (You can, of course, do more than three, or many more than three, if you want!)

  • The report must contain the plot from Task 2 (but it may also contain further plots).

  • There must be a reasonable attempt at interpreting your findings biologically, even if your interpretations miss the mark.

Notes

In spirit, I want to see people having a go, without worrying about whether what they produce is right or wrong. A pass can be given even if all the analysis and interpretation is incorrect, as long as your report demonstrates that you have given a decent attempt at all the tasks. The hope is that you can put your worries aside about marks and enjoy exploring the interesting dynamics!

If you pass first time around, you will receive 20 out of the 20 available marks. All of you should be able to achieve this. However, if you hand in something that does not pass first time around, I will give you some feedback as to where you fell short. Then you will have the opportunity to submit one more time (the deadline for submission will be given with my feedback). If you pass second time around, you will receive 15 out of the 20 available marks. If you do not pass second time around (hopefully no-one will manage this!), you will receive either 10 (for some sign of engagement that nevertheless falls short of the above criteria) or 0 (for nothing creditable).

Please be aware that, should you fail to hand your assignment in on time, the above marks may be subject to a late hand-in penalty in line with University regulations. At the time of writing, this reads: ‘Late submissions without an extension or beyond an agreed extension date will incur a 5% penalty for each working date they are submitted late. After 5 working days, work will not be accepted and will be an automatic fail.’ So failure to hand in within 5 working days of the deadline, either first time around or second time around, will lead to an automatic mark of 0/20 and no chance for a resubmission.

I recommend that you do not use Generative AI technologies (e.g. Gemini / Chat GPT) in producing this assignment. I doubt they would help you either acquire modelling skills or meet the criteria for passing. However, if you do choose to use them (for example to improve spelling/grammar), make sure you abide by the University’s policy on unfair means in relation to Generative AI.