18. Ecology

[status: barely-started]

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/an-introduction-to-population-growth-84225544/

18.2. Factors that come up in modeling population ecology

Table 18.2.1 Factors in 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

18.3.1. Unlimited

Spend some time plotting exponentials in gnuplot. Show how they dwarf linear growth, and how you need log scale to compare them.

The equation that represents population growth is:

\[\frac{dP(t)}{dt} = r P(t)\]

This is studied in detail in Section 17.5, with solutions looking like

\[P(t) = P_0 e^{r x}\]

But if you could not solve it easily then you would write it out like this:

\[\begin{split}\frac{P_{n+1} - P_n}{\Delta t} = r P_n \\ \rightarrow \\ P_{n+1} = P_n + r P_n \times \Delta t\end{split}\]

This can be demonstrated with a simple program that calculates:

P0 = 10
rate = 1.2
delta_t = 1

P = P0
print('## generation population')
for generation in range(100):
    P = P + P * (rate-1) * delta_t
    print(generation, '    ', P)

18.3.2. Resource-constrained - r-selection versus K-selection

r-selection

A strategy in which you reproduce as quickly as you can.

K-selection

A strategy in which you reproduce less and put more effort into your children.

18.3.2.1. Simulating with the logistic equation

The balance between these two is given by having large or small \(r\) or \(k\) in the logistic equation:

\[\frac{dP}{dt} = r P (1 - \frac{P}{K})\]

This can be solved with:

\[P(t) = \frac{K}{1 + A e^{-r t}}\]

where

\[A = \frac{K - P_0}{P_0}\]

This can be visualized with a simple program:

Listing 18.3.1 logistic_solution.py
#! /usr/bin/env python3

import math

def main():
    P0 = 30
    delta_t = 0.01
    print('# time   P_bigr   P_medr   P_smallr')
    for tstep in range(5000):
        t = delta_t * tstep
        P_0 = logistic(P0, t, 0.1, 500)
        P_1 = logistic(P0, t, 0.25, 500)
        P_2 = logistic(P0, t, 0.5, 500)
        P_3 = logistic(P0, t, 0.75, 500)
        P_4 = logistic(P0, t, 1.0, 500)
        print(f'{t:3}', '   ', P_0, '   ', P_1, '   ', P_2, '   ', P_3, '   ', P_4)


def logistic(P0, t, r, K):
    A = (K - P0) / P0
    result = K / (1 + A * math.exp(-r*t))
    return result

main()

$ python3 logistic_solution.py > rk.out
$ gnuplot
plot [] [0:500] 'rk.out' using 1:2 title 'r = 0.1'
replot 'rk.out' using 1:3
replot 'rk.out' using 1:4
replot 'rk.out' using 1:5
replot 'rk.out' using 1:6

18.3.2.2. Simulating with the logistic equation

https://en.wikipedia.org/wiki/Population_growth

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/world-population-growth

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/an-introduction-to-population-growth-84225544/

we can look at the table below and seek certain interesting periods in the data.

Table 18.4.1 Periods of interest in human population history

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:

https://www.pewresearch.org/fact-tank/2019/06/17/worlds-population-is-projected-to-nearly-stop-growing-by-the-end-of-the-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.

Let’s start getting to work on the equations that represent population growth

18.6. The Lotka-Volterra differential equations

18.7. Further reading

Still have to look at articles on 3-way predator-prey:

And how about simple models of a full society collapse?

Cliometrics