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April 15, 2021, 07:25:49 pm

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#### keltingmeith

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##### Re: QCE Specialist Maths Questions Thread
« Reply #30 on: October 19, 2020, 11:36:59 am »
+4
Hello!
I'm currently doing a PSMT on Leslie Matrices; I had believed that I've gotten a good chunk of the assignment done but after hearing other information floating around the grade I think I'm a little lost.

So our task is to model population trends of the Tasmanian devil since the documentation of the Devil Facial Tumor Disease in 1996 up until 2030 to determine whether or not the species will go extinct.

I've calculated the initial female age distributions and now I just need to develop a Leslie matrix (which is 7x7) to model the population trends. Currently, they have provided us with the following birth and survival rates, all of which are for healthy devils, so the challenge I'm facing is determining these rates for disease-affected populations.

Survival rates for rates based on historical data for disease-free populations (where s0 = probability of surviving the 0-1 age interval):

s0 = 0.39
s1 = 0.82
s2 = 0.82
s3 = 0.82
s4 = 0.82
s5 = 0.27
s6 = 0

Breeding numbers (female per female devil):

m0 = 0
m1 = 0.03
m2 = 0.86
m3 = 1.55
m4 = 1.55
m5 = 1.55
m6 - 0.86

We've also been provided with relevant research, which I'm 99% sure we're to use for developing our survival rates. Please see the screenshots attached.

My initial guess was that we were to select appropriate survival rates from the range of 0.1-0.6, model the trends using these numbers and compare the obtained populations with actual statistics (e.g. 50% killed from 1996-2007) to establish validity in the model and change the rates where necessary to match up with these figures and thus 'refine' our model. However, recently there seems to be a stress on the sentence 'a large host population will experience a rapid decline followed by stabilization and eventual return to pre-disease numbers.'  Under the assumption that birth rates will stay the same, I'm completely unsure of how to obtain survival rates that would give us this stabilization point and subsequently model the return to pre-disease numbers. And would this all be done under one Leslie matrix? Or would we expect to have multiple to model different periods of time?

Any help on this would be greatly appreciated. Sorry for such the long question and apologies if my stress has gotten to you too

I really, really, really want to help you. I don't want your opinion of AN to be that we don't care or don't want to help with QCE. But holy shit, this is nuts. I have a university degree in mathematical statistics, and now I'm doing a PhD where I need to use statistics all the time. That's literally what all of this is - it's you, using hard maths, to answer questions based on statistical data. I had never heard of a Leslie matrix before. This is like, some real in-depth level of statistics that's only relevant to high-level ecology and zoology studies. It's incredibly niche. So I'm really sorry if I'm not able to effectively help you after this, but I'm not helping you based on something I know - I'm helping you based on my learning this on wikipedia

What I think that stressed sentence is telling you isn't that you've approached this entirely incorrectly - however, a Leslie matrix should explain how a population will grow under a specific set of circumstances. A Leslie matrix can't explain if the population numbers suddenly, massively, decrease. In fact, have you ever taken a Leslie matrix and run it to a ridiculously high number of generations? Usually, the populations then become disgustingly big. Like, orders of magnitude bigger than you'd think they'd grow to. The Leslie matrix can't predict a sudden population drop like that. Eg, in the 1820s, suddenly people were killing off devils - the Leslie matrix can't predict that, because it's designed to only consider the natural life expectancy of the devil, and how fertile the mothers of different ages are. Another example - let's say you start with a group of bee hives, and make a Leslie matrix for them. You then find after 10 years, that all of a sudden there aren't enough flowers for the bees to all get pollen from, so some hives start dying out. The Leslie matrix could not predict that this will happen, because you never gave that information. Basically - there are circumstances that the Leslie matrix CAN'T predict, and you need to account for them.

But then, consider this - what happens when those stop becoming issues? Let's say you send 90% of the hives to an entirely different area that doesn't overlap with the original 10%'s search areas. Well then, now the lack of flowers isn't an issue, because there's more than enough flowers for the bees to pollinate from, so the original Leslie matrix becomes applicable again. And similarly for the devils - when humans stop hunting those devils, they should grow according to the same level of dynamics they did before, so the same Leslie matrix SHOULD apply again. The same goes for that 2007 information - after a while, that virus is either defeated, or the devils are now immune to it, so after that point, the same original Leslie matrix SHOULD apply again.

So the question of "should I need more than one Leslie matrix" is a little complicated. I think that likely what will happen is you should only need one Leslie matrix after each population crash, and that matrix should correspond to the rise of each population. If the rise of each population doesn't match that Leslie matrix, then circumstances are sufficiently difference (and it's worth trying to figure out what those circumstances are), that the birth or survival rate of the devil has been changed - and from there, a new Leslie matrix would need to be constructed to match those changes.

Having said all of that, this is all extremely complex stuff, and I 100% feel it's worth talking to your teacher and getting any input they can on this matter.

#### Bri MT

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##### Re: QCE Specialist Maths Questions Thread
« Reply #31 on: October 19, 2020, 02:26:01 pm »
+2
Hello!
I'm currently doing a PSMT on Leslie Matrices; I had believed that I've gotten a good chunk of the assignment done but after hearing other information floating around the grade I think I'm a little lost.

So our task is to model population trends of the Tasmanian devil since the documentation of the Devil Facial Tumor Disease in 1996 up until 2030 to determine whether or not the species will go extinct.

I've calculated the initial female age distributions and now I just need to develop a Leslie matrix (which is 7x7) to model the population trends. Currently, they have provided us with the following birth and survival rates, all of which are for healthy devils, so the challenge I'm facing is determining these rates for disease-affected populations.

Survival rates for rates based on historical data for disease-free populations (where s0 = probability of surviving the 0-1 age interval):

s0 = 0.39
s1 = 0.82
s2 = 0.82
s3 = 0.82
s4 = 0.82
s5 = 0.27
s6 = 0

Breeding numbers (female per female devil):

m0 = 0
m1 = 0.03
m2 = 0.86
m3 = 1.55
m4 = 1.55
m5 = 1.55
m6 - 0.86

We've also been provided with relevant research, which I'm 99% sure we're to use for developing our survival rates. Please see the screenshots attached.

My initial guess was that we were to select appropriate survival rates from the range of 0.1-0.6, model the trends using these numbers and compare the obtained populations with actual statistics (e.g. 50% killed from 1996-2007) to establish validity in the model and change the rates where necessary to match up with these figures and thus 'refine' our model. However, recently there seems to be a stress on the sentence 'a large host population will experience a rapid decline followed by stabilization and eventual return to pre-disease numbers.'  Under the assumption that birth rates will stay the same, I'm completely unsure of how to obtain survival rates that would give us this stabilization point and subsequently model the return to pre-disease numbers. And would this all be done under one Leslie matrix? Or would we expect to have multiple to model different periods of time?

Any help on this would be greatly appreciated. Sorry for such the long question and apologies if my stress has gotten to you too

Hello!

Throwing in my 2 cents as an ecology and conservation biology major with a stats minor.

I don't think you need to worry about the whole "stabilisation thing" in terms of a levelling out as would often be seen in population ecology models. One of the potential explanations in the doc was about the devils developing immunity and so I've modelled what that might look like roughly making up some parameters based on the provided info. I've done this by adapting work from one of my labs in a final year of uni ecology subject so yeah.... you should not be expected to do this but I thought it might be interesting to show you so you can see that even a more ecologically informed model is not going to perfectly line up with your data.

The key way that this works is that the mean absolute fitness of the population is initially below 1 in the presence of the selective agent (in this case, the disease). Therefore, the population size decreases. Then, at the point of evolutionary rescue (indicated by the dotted line), the mean absolute fitness increases to above 1 and therefore the population starts growing again. In your scenario, you've been (implicitly) told that the mean absolute fitness increasing to above 1 may be because of evolutionary rescue or not but either way the same principle applies.

Density dependence (in this case you're being told low density = more population growth) can be included as part of a leslie matrix (by multiplying with a suitable diagonal matrix). You could also construct separate matrices to represent population growth and decline. My assumption given the information that you have been given is that you use the provided info to construct a "rescue/growth" matrix and need to make a separate disease/decline matrix based on the context/research, potentially multiple matrices to match the data more closely; however, I echo Keltingmeith's suggestion to check with your teacher what the expectations are as that's the safest thing to do going forwards.

I'm not sure if that helps at all but please feel free to follow up with any more questions you have

Edit: I would include how I made the above figure and talk about the maths that goes into it but you would need more ecology knowledge to understand it and that would just lead you away from the assessment criteria
« Last Edit: October 19, 2020, 02:30:02 pm by Bri MT »

#### jasmine24

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##### Re: QCE Specialist Maths Questions Thread
« Reply #32 on: February 01, 2021, 06:11:19 am »
0
Hi, i was doing Q4 from the 2016 tasc exam paper but im not sure how step one of the induction is true and wondering if anyone could explain how.
TIA

#### jasmine24

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##### Re: QCE Specialist Maths Questions Thread
« Reply #33 on: February 01, 2021, 06:20:41 am »
0
Hi, i was doing Q4 from the 2016 tasc exam paper but im not sure how step one of the induction is true and wondering if anyone could explain how.
TIA
*Edit: i dont really understand how the rest of the proof works either

#### fun_jirachi

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##### Re: QCE Specialist Maths Questions Thread
« Reply #34 on: February 01, 2021, 08:34:48 am »
0
The dots indicate multiplication - that would seem to be the most obvious point of confusion. The first step seems quite self-explanatory if you follow that. Try looking at it like that and if you're still confused with the rest of the proof let us know
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#### jasmine24

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##### Re: QCE Specialist Maths Questions Thread
« Reply #35 on: February 03, 2021, 01:24:45 pm »
0
Would anyone be able to explain what the purpose of finding eigenvalues are and how to know when to use them?
TIA

#### keltingmeith

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##### Re: QCE Specialist Maths Questions Thread
« Reply #36 on: February 03, 2021, 03:24:43 pm »
+3
Would anyone be able to explain what the purpose of finding eigenvalues are and how to know when to use them?
TIA

Eigenvalues! Man, QCE is wild.

Asking, "what's the purpose of finding eigenvalues" is kind of like asking "what's the purpose of knowing how to solve quadratics". On the surface level, there's no purpose whatsoever, and it just looks like a thing you can do/find. As you've caught on, there must be something deeper happening, or why else would we care enough to be able to find them in the first place? Annoyingly, QCAA only ask for an investigative report, which I'm assuming you're currently doing? But in general, the answer is - if you need to use eigenvalues, you'll be told you need to use them. Just like when you need to solve a quadratic, you know you need to solve one, because either you've been specifically told to solve it, or you've been told, "if a question asks you this, they're asking you to solve a quadratic".

Firstly, some good news: according to the syllabus, your exam will not cover eigenvalues - which is kinda sucky, because they're so cool!! But, that also means if you're looking into applications, we can go a bit nuts, and don't need to worry about what you might be tested on later (confirm with your teacher if they plan on testing you on these with internal tests, that'll give you an idea of what applications you need to specifically know about). First, I highly recommend the wikipedia page for a list of applications. Some of them may go over your head, and one of my favourites is diagonalizing matrices - here's a really good article on it!

Let me know if there's anything you want clarified or more information - or if I've misunderstood the question

#### jasmine24

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##### Re: QCE Specialist Maths Questions Thread
« Reply #37 on: February 03, 2021, 09:16:37 pm »
0
Eigenvalues! Man, QCE is wild.

Asking, "what's the purpose of finding eigenvalues" is kind of like asking "what's the purpose of knowing how to solve quadratics". On the surface level, there's no purpose whatsoever, and it just looks like a thing you can do/find. As you've caught on, there must be something deeper happening, or why else would we care enough to be able to find them in the first place? Annoyingly, QCAA only ask for an investigative report, which I'm assuming you're currently doing? But in general, the answer is - if you need to use eigenvalues, you'll be told you need to use them. Just like when you need to solve a quadratic, you know you need to solve one, because either you've been specifically told to solve it, or you've been told, "if a question asks you this, they're asking you to solve a quadratic".

Firstly, some good news: according to the syllabus, your exam will not cover eigenvalues - which is kinda sucky, because they're so cool!! But, that also means if you're looking into applications, we can go a bit nuts, and don't need to worry about what you might be tested on later (confirm with your teacher if they plan on testing you on these with internal tests, that'll give you an idea of what applications you need to specifically know about). First, I highly recommend the wikipedia page for a list of applications. Some of them may go over your head, and one of my favourites is diagonalizing matrices - here's a really good article on it!

Let me know if there's anything you want clarified or more information - or if I've misunderstood the question

Thank you so much! My teacher said we need to know eigenvalues to be able to apply it to Leslie matrices for our exam and also in our investigative report where we have to use markov chain and eigenvalues.

#### keltingmeith

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##### Re: QCE Specialist Maths Questions Thread
« Reply #38 on: February 06, 2021, 05:45:51 pm »
+1
Thank you so much! My teacher said we need to know eigenvalues to be able to apply it to Leslie matrices for our exam and also in our investigative report where we have to use markov chain and eigenvalues.

Sorry my response took so long - my laptop recently died and it took a while to get a new one as I live in Perth and we just went into lockdown...

Anyway, interesting point from your teacher. I question if you do need to know how to use eigenvalues for Leslie matrices, but there's still a lot of unknowns about QCE exams, so still more than happy to discuss. For Leslie matrices, there are two questions you could be asking:

1. When do I need to use eigenvalues?
2. What does the eigenvalue of a Leslie matrix tell me?

I emphasise this, because it's really important you're NOT asking the first question. The first question seems good and like it'll give you all the information you'll need, but the truth is it's a distraction. The key to doing well in maths is to look beyond the algorithms, look beyond the equations, and to figure out what they intuitively might tell you. If you can figure this out, not only will you have a better and deeper understanding for the topic at hand, but you'll also know when to use the eigenvalues - when the question is asking about what the eigenvalues will tell you.

Now, I've not worked with Leslie matrices before, so figuring out what they mean is as much a challenge for me as it is for you - thankfully, we don't need to be experts of this. There are a million people around the world and - more importantly - on the internet that have figured it out for us. Here's a handy powerpoint I found, with some very interesting results. It tells us that the first eigenvalue, and its corresponding eigenvector tell us:

a) if the population will grow or decay
b) the final proportions of each age class

I encourage you to do some more reading! There might be some other results you can find that I couldn't. It's also worth checking with your teacher what they meant when they said you need to know about them for Leslie matrices - there's probably some applications that they have in the back of their mind that they were thinking of when they told you you need to know them for Leslie matrices.

---

The next topic I have some information about - Markov chains. I love Markov chains, and I actually spent a fair amount of time at university purely devoted to learning about Markov processes. Really interesting stuff! In the simple case, for the discrete-time Markov chain (note: I highly doubt you're covering continuous time Markov chains as they're a heavily complex topic. But, if your teacher said you need to learn about them, let me know, I'll see if I can find some simple resources for you), there is one eigenvalue we care about - the one where lambda=1. From here, I'm assuming you have basic Markov chain knowledge, where I'm using T for our transition matrix, and x is our distribution of states.

We care about this one, because if a transition matrix has a solution of the form $Tx=x$, then we call this a stationary distribution. That is, if you're ever in this distribution, then your distribution will not change. Eg, if x=(0.5,0.2,0.3) is a stationary distribution, then $T^nx=(0.5,0.2,0.3)$ - it will not change, no matter HOW MANY transitions you go through. Essentially, all eigenvectors that correspond to the eigenvalue of 1 for our transition matrix are a stationary distribution. There is also an interesting theorem that shows a transition matrix can only have 0, 1, or infinitely many stationary distributions that's easy to show. Here's a hint - suppose that $x_1$ and $x_2$ are stationary distributions of $T$. In other words, $Tx_1=x_1$ and $Tx_2=x_2$. Now, consider the third vector $x_3=ax_1+(1-a)x_2$, where a is any real number between 0 and 1. Two questions:

1. Is this new vector $x_3$ a probability distribution? (do all the probabilities in it sum to 1?)
2. Does this vector also solve the equation $Tx=x$?

If both of these are yes, then $x_3$ represents an infinite amount of other stationary distributions.

So, I've spoken alot about stationary distributions, but there's one more really important bit about Markov chains - for some specific chains, you will always reach a stationary distribution after an infinite amount of time (for real-life applications, you can usually reach something close enough to the stationary distribution after a large amount of transitions that the next state is essentially the same as the one before it). Trying to define which situations they will would take ages though, sorry, and unfortunately I don't have a lot of time to be going through everything - but the wiki page is pretty good at discussing a lot of the requirements.

#### jasmine24

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##### Re: QCE Specialist Maths Questions Thread
« Reply #39 on: February 14, 2021, 07:34:47 pm »
0
In markov chains, how would I alter the transition matrix to include a new brand if i wasnt given any data on the new brand

#### jasmine24

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##### Re: QCE Specialist Maths Questions Thread
« Reply #40 on: April 06, 2021, 12:09:41 pm »
0
When sketching hyperbolas how do you know where the graphs go? like for the one I've attached how do you know the graph is horizontal, not vertical?
TIA

#### fun_jirachi

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##### Re: QCE Specialist Maths Questions Thread
« Reply #41 on: April 06, 2021, 12:16:57 pm »
+4
If you're stuck, it's always handy to plot a few points on the curve.

Otherwise, for a hyperbola in the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, this will be a horizontal hyperbola. Use the diagram in your question to determine why no point on the hyperbola can be in the upper 'triangle' and bottom 'triangle' marked by the asymptotes.

For a hyperbola in the form $-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ this will be a vertical hyperbola. In this case, it might be helpful to pose a similar question to the above by drawing the hyperbola  $-\frac{x^2}{9} + \frac{y^2}{16} = 1$ on the same diagram.

If you still can't quite get this out, feel free to query again (I've only poked you along here because I think it's helpful to develop this kind of intuition on your own).
« Last Edit: April 06, 2021, 12:21:23 pm by fun_jirachi »
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#### Specialist_maths

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##### Re: QCE Specialist Maths Questions Thread
« Reply #42 on: April 14, 2021, 05:38:41 pm »
+3
For Leslie matrices, there are two questions you could be asking:

1. When do I need to use eigenvalues?
2. What does the eigenvalue of a Leslie matrix tell me?

I emphasise this, because it's really important you're NOT asking the first question. The first question seems good and like it'll give you all the information you'll need, but the truth is it's a distraction. The key to doing well in maths is to look beyond the algorithms, look beyond the equations, and to figure out what they intuitively might tell you. If you can figure this out, not only will you have a better and deeper understanding for the topic at hand, but you'll also know when to use the eigenvalues - when the question is asking about what the eigenvalues will tell you.
Outstanding advice! I wish everyone could understand this.

Anyway, interesting point from your teacher. I question if you do need to know how to use eigenvalues for Leslie matrices, but there's still a lot of unknowns about QCE exams, so still more than happy to discuss.
FYI: Eigenvalues and eigenvectors are specifically listed in the syllabus (p29) as subject matter in the course.
There is a note that states neither will feature on the External Exam (50%).

Schools write their own Internal Assessment (50%), and can choose which descriptors are assessed (all topics must be assessed, but not all descriptors). Some may only assess in the assignment; some may only assess in the exam. I suspect most schools will not assess eigenvalues/eigenvectors at all - and instead choose other "Applications of matrices".

I encourage you to do some more reading! There might be some other results you can find that I couldn't. It's also worth checking with your teacher what they meant when they said you need to know about them for Leslie matrices - there's probably some applications that they have in the back of their mind that they were thinking of when they told you you need to know them for Leslie matrices.
Unfortunately, some schools are not as supportive as others. The assignment is not meant to be a research task. This was stated in the syllabus (p31) and again in the 2020 Subject report. The teacher should be providing guidance on how eigenvalues/eigenvectors could be used. It's then up to the student to make the decision in regards to how the given problem can be solved.

Jasmine, I hope your assessment went well. If you've already done IA1 and IA2, you probably won't have to worry about eigenvalues or markov chains again (unless you're doing maths at uni next year?)
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