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January 19, 2021, 04:03:05 pm

### AuthorTopic: QCE Specialist Maths Questions Thread  (Read 5117 times)

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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.
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#### 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 »