Help! Phase plane method

What have you done / what problems are you having?

I have problems with part b,c and d

For b) you must have done an example or two about writing odes in dimensionless form on your course at uni?

I only got to do one example.
I have tried part b, just that i didnt get the final answer.
For c) i dont know how to start answering the question and I dont know what I have to find

can you post what you tried for b)?

For c) I presume they want you to do something like the diagrams in 5.2 onwards in
http://mitran-lab.amath.unc.edu/courses/MATH564/biblio/text/05.pdf
(this is just from a 1 hit google, there will be other better examples if you search/look at your notes). A stable state will correspond to the derivatives being zero, .... For stability, it may just be for you to interpret the curves or it may be they want something more quantitative (local linearization) but I suspect not from the way the question part is worded.

d) would be interpreting the steady state / stability in terms of the original plant - herbivore model.

It looks reasonable but a bit confusing with the notation. In the 4th and 3rd lines from the end, you seem to drop a q* from the second term on the right hand side from the earlier dqstar/dtstar equation. If you didnt do that it seems about right? Not carefully checked the expressions for the scaling constants though if you end up with the right form, theyre probably good and agree with a 1/2 working/scribble by me.

Oh right, I forgot to write that down.
But the value for K and alpha are still the same. However, the answer given is K=K2K4Io^3 and alpha=K3K4Io

Im getting your values. Note that with the given answer (where does it come from?), there is no dependence on K1, which would be strange.

This question is from our textbook, and some selected answer are given. I got the answer from the back page of this book.

This question is from our textbook, and some selected answer are given. I got the answer from the back page of this book.
So im assuming my answer so far is correct.
Now im left with part c and d

For c) did you look at the previous link / your textbook / notes?

Yes. The notes in the link are the same as my textbook. But I still doesn't understand.

You still dont understand what? The usual approach is to sketch the curves where each ode is zero and consider their intersection(s) as steady state points. The zero derivative cuurves also divide up the space into regions where the derivatives have the same sign and you can approximately (qualitatively argue about) the dynamical behaviour in each region, so whether steady state values are stable or ...

Will have a look at it later and as before, Im not 100% exactly sure what youve covered/what the question is expecting but there are a few predator prey models and something like
http://alun.math.ncsu.edu/wp-content/uploads/sites/2/2018/11/ma331-f18-lecture12.pdf
describes how they construct the phase portrait etc. If its from a book, then maybe refer to one of the examples in the text.

Had a quick look through the book and your work. The book looks like it pretty much has a similar worked example to what youre trying to do in 5.10 and youve not really presented the qualitative analysis which the first part of c) asks for. You may want to comment on the cubics for q and I and whether there is a single steady state solution or. Your linearized analysis seems ok though not checked carefully.