18. Ecology
[status: barelystarted]
18.1. Motivation, Prerequisites, Plan
As I write this, in April of 2020, it seems like a good opportunity ot get comfortable with some of the equations that come up when we talk about growth. We will look at the growth of a population, or growth of the number of infected humans (which is related to the growth of the population that carries the infection).
Let us start with mosquitoes and West Nile Virus in Texas. Watch this crash course video at:
https://www.youtube.com/watch?v=RBOsqmBQBQk&index=2&list=PL8dPuuaLjXtNdTKZkV_GiIYXpV9w4WxbX
Then have ready this nature paper on basics of ecology:
https://www.nature.com/scitable/knowledge/library/anintroductiontopopulationgrowth84225544/
18.2. Factors that come up in modeling population ecology
name 
variable 

initial pop: 
N 

birth rate: 
B 

death rate: 
D 

growth rate: 
r 
= (B  D) / N 
predation 

immigration 

emigration 

mates 

food 

space 
18.3. Exponential growth
Spend some time plotting exponentials in gnuplot. Show how they dwarf linear growth, and how you need log scale to compare them.
18.4. History of the human population on earth
Peruse the Wikipedia page on historical population estimates:
https://en.wikipedia.org/wiki/Estimates_of_historical_world_population
and the study at:
https://www.prb.org/howmanypeoplehaveeverlivedonearth/
Spend some time exploring the interactive graphs at:
https://ourworldindata.org/worldpopulationgrowth
Expand the title on “All our charts on World Population Growth”, and pick the population by country since 1500 and try to understand what areas are exponential.
Then look at the link “World population since 10,000 BCE (OurWorldInData series)”.
Download the data for this graph and zoom in on some specific periods. Look at both linear and logarithmic scales.
Following the indications shown in
https://www.nature.com/scitable/knowledge/library/anintroductiontopopulationgrowth84225544/
we can look at the table below and seek certain interesting periods in the data.
Start 
End 
What to look for 

10000 
4000 
Agricultural revolution 
4000 
600 
Early empires 
1000 
300 
Alexander and Rome 
1 
300 
Imperial Rome 
1 
1600 
Largely steady world population 
1200 
1400 
Medieval black death 
1500 
present 
Modern world 
1850 
present 
Industrial revolution 
1900 
present 
Large scale science 
18.5. The logistic function
Although the earth’s population as a whole appears to still be in an exponential growth phase, the Pew Research Center predicts that it will flatten by the end of the 21st century:
This type of function is not exponential growth anymore: it shows exponential growth, but that then slows down and we end up with what is called the Logistic Function:
https://en.wikipedia.org/wiki/Logistic_function
Think of fidget spinners.
18.6. The LotkaVolterra differential equations
18.7. Further reading
https://www.youtube.com/watch?v=NYq2078_xqc  Khan Academy video with pleasant intro to cycles and real examples, 5min.
https://www.youtube.com/watch?v=mFDiiSqGB7M  crash course on predatorprey ecology
https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations#A_simple_example
http://mcstan.org/users/documentation/casestudies/lotkavolterrapredatorprey.html
Still have to look at articles on 3way predatorprey:
chromeextension://oemmndcbldboiebfnladdacbdfmadadm/https://www.cs.unm.edu/~forrest/classes/cs365/lectures/LotkaVolterra.pdf
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.360.1552&rep=rep1&type=pdf
And how about simple models of a full society collapse?
https://en.wikipedia.org/wiki/Collapse:_How_Societies_Choose_to_Fail_or_Succeed
http://necsi.edu/projects/evolution/coevolution/predprey/coevolution_predator.html
https://www.ted.com/talks/jared_diamond_on_why_societies_collapse
https://faustusnotes.wordpress.com/2014/05/15/mathematicalmodelingofcivilizationcollapse/
Cliometrics