2024-02-11: Symbolic algebra and typing

2023-08-01: Population waves

2023-05-18: Math of telephone billing mystery

2023-05-05: Franklin and DNA More information…

2023-04-25: On angle and dimension

2023-02-20: On Leonardo da Vinci and Gravity

2022-04-29: Fabricating Evidence to catch Carmen Sandiego

2022-03-04: Probabilistic law of the excluded middle

2020-05-04: Archimedes and the sphere

2019-05-16: Glow worms return

2019-04-11: Original memetic sin

2019-01-31: The theory of weight

2018-11-06: Origins of telephone network theory

2018-10-24: Modern thought

2018-09-10: Feeding a controversy

2018-06-11: Glow worm distribution

2018-04-23: Outlawing risk

2017-08-22: A rebuttal on the beauty in applying math

2017-04-22: Free googles book library

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-02-17: Dependent Non-Commuting Random Variable Systems

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: Mathematical Epidemiology citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-12: A Comment on How Biased Dispersal can Preclude Competitive Exclusion

2015-02-09: Hamilton's selfish-herd paradox

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-17: Ebola comments

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-21: Kurtosis, 4th order diffusion, and wave speed

2014-06-20: Random dispersal speeds invasions

2014-05-06: Preservation of information asymetry in Academia

2014-04-16: Dual numbers are really just calculus infinitessimals

2014-04-14: More on fairer markets

2014-03-18: It's a mad mad mad mad prisoner's dilemma

2014-03-05: Integration techniques: Fourier--Laplace Commutation

2014-02-25: Fiber-bundles for root-polishing in two dimensions

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why \(1^{\infty} eq 1\)

2013-11-15: Adobe leaks of PSU data + NSA success accounting

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-10: L'Hopital's Rule for Multidimensional Systems

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

2013-11-03: Elementary mathematical theory of the health poverty trap

2013-11-02: Proof of the circle area formula using elementary methods

Blog – a page

Population waves

Cases of Smallpox in Sweden, from Ransome

It’s rather accepted knowledge these days that many natural populations rise and fall like waves – sometimes the creatures become more numerous and easy to find (the rise), and then become less numerous and harder to find (the fall). But the history of this as a scientific idea – population waves waxing and waning – is rather incompletely understood.

To begin with, even in old-testament times, variations in common-ness of plants and animals were recognized, particular in the contexts of feast-and-famine, or the irregular arrivals of plague and pestilence. The reasons for these were often mysterious and attributed to curses or acts of gods. For plagues in particular, they were more likely to be attributed to divine intervention than biological phenomena.

Perspectives on systems of living things changed quickly over the time of the second industrial revolution, with the industrialization of hunting and the advancement of infectious disease science. By the 1950’s, it was well-documented that besides human populations, natural populations underwent dramatic changes in their numbers. And although the reasons for particular changes in abundance could still be mysterious at first, it was widely accepted that changes could be explained mechanistically in terms of ecology and environment. So, how did our collective understanding morph from deus ex machina to first principles mechanisms?

In epidemiology, the recent article The shape of epidemics by Jones and Helmreich (2020) traces the idea of epidemic waves back to be in parallel with the development of the germ theory of infectious disease. In 1868, Arthur Ransom explicitly described infectious disease epidemics like measles as progressing like waves over the course of years, with data and charts to back up the description.

In ecology, this idea has not yet been traced back so far. In 1989, Sinclair reviewed the history of population regulation, and credits the work of Nicholson in 1933 as the start of our modern understanding and soundly within the field we call “ecology” today. But Nicholson himself cited others who had recognized possibilities of both smooth regulation (Like Pearl and Reed 1920,1927) and periodic variation even earlier. The earliest explicit reference to models with oscillation seems to be in Vito Volterra’s citation of changes in numbers of fish in 1926. However, Alfred Lotka had already proposed a theory for changes in population abundance in the form of a system of ordinary differential equations that we now call the Lotka–Volterra equations. He may well have been inspired by his earlier work (1912) hypothesizing oscillatory solutions for Ross’s epidemic theory — hence connecting ecology back to epidemic theory.

That part’s all well-documented today. Yet, Lotka drew part of his inspiration from an even older source than Ransome – a source that seems to have been overlooked so far. In 1860, Herbert Spencer published his 500+ page book “First Principles”, as a foundation for his philosophy. In it, he makes numerous references to the concept of rhythmic variation in population numbers over time. He even goes on to point out that these rhythmic variations need not be perfectly periodic, but may be less regular in their nature with respect to period and amplitude.

This is something that deserves more attention. Spencer is a particularly interesting transitional figure, working in scientific philosophy, but devoid of explicit mathematics and empiricism, in parallel with the developments of germ theory and biological evolution, as well as the emerging social sciences of economics, anthropology, and sociology. Unfortunately, one of his biggests effects was to aid the rise of social darwinism as a cultural movement, which would fester and take malignant form over the next 100 years.

Herbert Spencer, 1820-1903