A Short History of Mathematical Population Dynamics

Frontmatter

Chapter 1. The Fibonacci sequence (1202)

Abstract

In 1202, Leonardo of Pisa, also called Fibonacci, published a book which popularized in Europe the Indian decimal number system that had also been adopted by Arab mathematicians. Among the many examples given in the book, one refers to the growth of a population of rabbits. It is one of the oldest examples of a mathematical model for the dynamics of a population.

Nicolas Bacaër

Chapter 2. Halley’s life table (1693)

Abstract

In 1693 the famous English astronomer Edmond Halley studied the birth and death records of the city of Breslau, which had been transmitted to the Royal Society by Caspar Neumann. He produced a life table showing the number of people surviving to any age from a cohort born the same year. He also used his table to compute the price of life annuities. This chapter recalls this work and puts it in the context of Halley’s life and of the early developments of “political arithmetic” and probability theory, which interested people such as Graunt, Petty, De Witt, Hudde, Huygens, Leibniz and de Moivre.

Nicolas Bacaër

Chapter 3. Euler and the geometric growth of populations (1748–1761)

Abstract

Euler wrote on several occasions on population dynamics. In his 1748 treatise, Introduction to Analysis of the Infinite, the chapter dealing with the exponential function contained four examples on the exponential growth of a population. In 1760 he published an article combining this exponential growth with an age structure for the population. This work is a forerunner of the theory of “stable” populations, which was developed in the twentieth century and plays an important role in demography. In 1761 Euler also helped Süssmilch with the second edition of his treatise on demography. He worked out an interesting model, which is a kind of variant of Fibonacci’s sequence, but did not publish his detailed analysis.

Nicolas Bacaër

Chapter 4. Daniel Bernoulli, d’Alembert and the inoculation of smallpox (1760)

Abstract

In 1760 Daniel Bernoulli wrote an article modeling smallpox. In his time there was much controversy around inoculation, a practice that could protect people but could also be deadly. He used Halley’s life table and some data concerning smallpox to show that inoculation was advantageous if the associated risk of dying was less than 11%. Inoculation could increase life expectancy at birth up to three years. D’Alembert criticized Bernoulli’s work, which was the first mathematical model in epidemiology.

Nicolas Bacaër

Chapter 5. Malthus and the obstacles to geometric growth (1798)

Abstract

In 1798 Malthus published An Essay on the Principle of Population, in which he argued that the supply of food could not follow for a long period of time the natural tendency of human populations to grow exponentially. If the population remained relatively constant, this was because a great part of mankind was suffering from food shortage. Malthus saw the “principle of population” as an argument against the writings of Godwin and Condorcet, which emphasized progress in human societies. Malthus’ essay influenced the theory of evolution of Darwin and Wallace and was criticized by Marx, but was put into practice with the Chinese one-child policy.

Nicolas Bacaër

Chapter 6. Verhulst and the logistic equation (1838)

Abstract

In 1838 the Belgian mathematician Verhulst introduced the logistic equation, which is a kind of generalization of the equation for exponential growth but with a maximum value for the population. He used data from several countries, in particular Belgium, to estimate the unknown parameters. The work of Verhulst was rediscovered only in the 1920s.

Nicolas Bacaër

Chapter 7. Bienaymé, Cournot and the extinction of family names (1845–1847)

Abstract

The French statistician Bienaymé understood in 1845 how to compute the probability of a family name becoming extinct if each male has a number of sons following a given probability distribution. If the average number of sons is less than or equal to one, the family name will become extinct. It the average is bigger than one, the extinction probability is strictly less than one. The proof of his result was published two years later in a book written by his friend Cournot. These works were rediscovered only recently.

Nicolas Bacaër

Chapter 8. Mendel and heredity (1865)

Abstract

In 1865 Mendel published the results of his pioneering experiments on the hybridization of peas. His analysis used elementary aspects of probability theory. He also considered a dynamical model for a population of self-fertilizing plants. His work, which was rediscovered only in 1900, is a milestone in the history of genetics.

Nicolas Bacaër

Chapter 9. Galton, Watson and the extinction problem (1873–1875)

Abstract

In 1873 the British statistician Galton and his compatriot mathematician Watson considered the problem of the extinction of family names without knowing the work of Bienaymé. Watson noticed that the generating function associated with the probability distribution of the number of men in each generation could be computed recursively. But he analyzed incorrectly the probability of extinction.

Nicolas Bacaër

Chapter 10. Lotka and stable population theory (1907–1911)

Abstract

In 1907 the American chemist Alfred Lotka started to study the relation between birth rate, age-specific death rates and the rate of population growth using a continuous-time model. In 1911 he published another article on the same subject with F.R. Sharpe, which also included age-specific fertility rates. The implicit equation giving the population growth rate is often called “Lotka’s equation”.

Nicolas Bacaër

Chapter 11. The Hardy–Weinberg law (1908)

Abstract

In 1908 the British mathematician Hardy and the German medical doctor Weinberg independently discovered that in an infinitely large population that mates randomly according to Mendel’s laws, the frequencies of the genotypes obtained from two alleles remain constant through generations. Their mathematical model was one of the starting points for population genetics.

Nicolas Bacaër

Chapter 12. Ross and malaria (1911)

Abstract

In 1911 the British medical doctor Ronald Ross, who had already received the 1902 Nobel prize for his work on malaria, studied a system of differential equations modelling the spread of this disease. He showed that malaria can persist only if the number of mosquitoes is above a certain threshold. Therefore it is not necessary to kill all mosquitoes to eradicate malaria – it is enough to kill just a certain fraction. Similar epidemic models were later developed by Kermack and McKendrick.

Nicolas Bacaër

Chapter 13. Lotka, Volterra and the predator–prey system (1920–1926)

Abstract

In 1920 Alfred Lotka studied a predator–prey model and showed that the populations could oscillate permanently. He developed this study in his 1925 book Elements of Physical Biology. In 1926 the Italian mathematician Vito Volterra happened to become interested in the same model to answer a question raised by the biologist Umberto d’Ancona: why were there more predator fish caught by the fishermen in the Adriatic Sea during the First World War, when the fishing effort was low?

Nicolas Bacaër

Chapter 14. Fisher and natural selection (1922)

Abstract

In 1922 the British mathematical biologist Ronald Fisher published a very influential article on population genetics. This chapter considers only one section of the article, which focuses on a variant of the Hardy–Weinberg model including natural selection. Fisher showed that if the heterozygote is favoured, then both alleles can coexist. If one of the two homozygotes is favoured, then the other allele disappears. The underlying problem is that of explaining why some genes can have several alleles.

Nicolas Bacaër

Chapter 15. Yule and evolution (1924)

Abstract

In 1924 the British statistician Yule studied a model of evolution where species can produce new species by small mutations and genera can produce new genera by large mutations. His purpose was to explain the distribution of the number of species within genera, most genera containing only one species and a few genera containing a large number of species. The stochastic “birth process” Yule introduced in his model is still a basic tool in the study of phylogenetic trees and many other areas.

Nicolas Bacaër

Chapter 16. McKendrick and Kermack on epidemic modelling (1926–1927)

Abstract

In 1926 McKendrick studied a stochastic epidemic model and found a method to compute the probability for an epidemic to reach a certain final size. He also discovered the partial differential equation governing age-structured populations in a continuous-time framework. In 1927 Kermack and McKendrick studied a deterministic epidemic model and obtained an equation for the final epidemic size, which emphasizes a certain threshold for the population density. Large epidemics can occur above but not below this threshold. These works are still very much used in contemporary epidemiology.

Nicolas Bacaër

Chapter 17. Haldane and mutations (1927)

Abstract

In another section of his 1922 article, Fisher considered the problem of a mutant gene that can be transmitted to a random number of offspring with a given probability distribution. The problem was formally the same as that of the extinction of family names but in a genetic context. Fisher showed that if the probability distribution was a Poisson distribution and if the mutant gene had no selective advantage, then the mutant gene could disappear from the population very slowly. In 1927 the British biologist Haldane pushed the study of this model further and showed that the probability of a mutant advantageous gene maintaining itself was twice its selective advantage. He also gave a more rigorous treatment of the extinction problem.

Nicolas Bacaër

Chapter 18. Erlang and Steffensen on the extinction problem (1929–1933)

Abstract

In 1929 the Danish telephone engineer Erlang considered once again the problem of extinction of family names. His compatriot statistician Steffensen worked out a complete solution of the problem. He showed in particular that the expectation of the number of offspring in each generation grows exponentially, thus making the bridge between stochastic and deterministic population models.

Nicolas Bacaër

Chapter 19. Wright and random genetic drift (1931)

Abstract

In 1931 the American biologist Sewall Wright developed the study of a stochastic model in population genetics, which is based on the same assumptions as in the Hardy–Weinberg law except that the population is not assumed infinitely large. The frequencies of the genotypes are no longer constant. One of the two alleles will in fact disappear, but maybe after a very long time. The interpretation of this model remained a subject of dispute between Wright and Fisher, the latter estimating that natural selection plays a more important role in evolution than stochasticity.

Nicolas Bacaër

Chapter 20. The diffusion of genes (1937)

Abstract

In 1937 Ronald Fisher and three Russian mathematicians, Kolmogorov, Petrovsky and Piskunov, independently studied a partial differential equation for the geographic spread of an advantageous gene. They showed that the gene frequency behaved like a wave travelling at a well-defined speed depending on the gene’s advantage and on a diffusion coefficient. Their works were the starting point for the theory of reaction–diffusion equations.

Nicolas Bacaër

Chapter 21. The Leslie matrix (1945)

Abstract

In 1945 the British ecologist P.H. Leslie analyzed a matrix model for an age-structured population of rodents, thus adapting Lotka’s work to a discrete-time framework. He emphasized that the growth rate corresponds to an eigenvalue and the “stable” age structure to an eigenvector. He also estimated numerically the “net reproduction rate” R ₀ for the brown rat.

Nicolas Bacaër

Chapter 22. Percolation and epidemics (1957)

Abstract

In 1957 Hammersley and Broadbent considered the propagation of a “fluid” in an infinite regular square network, where two neighbouring nodes are connected with a given probability. Among the possible examples, they mentioned the propagation of an epidemic in an orchard. They showed that there is critical probability below which no large epidemic can occur and above which large epidemics occur with a positive probability. Their article was the starting point of percolation theory.

Nicolas Bacaër

Chapter 23. Game theory and evolution (1973)

Abstract

In 1973 Maynard Smith and Price published an article analyzing why animals avoided using their most dangerous weapons in intraspecific conflicts. Their model used game theory and was one of those that launched the application of this mathematical theory to evolutionary problems.

Nicolas Bacaër

Chapter 24. Chaotic populations (1974)

Abstract

In 1974 Robert May, an Australian physicist turned ecologist, studied the discrete-time logistic equation as a model for population dynamics. He noticed that unexpected bifurcations occurred and that the asymptotic behaviour could even be chaotic. So long-term predictions can be impossible even with a simple deterministic model. May’s article was one of those that launched “chaos theory”.

Nicolas Bacaër

Chapter 25. China’s one-child policy (1980)

Abstract

In 1980 Song Jian and his collaborators, who had been specialists of control theory applied to airspace engineering, computed that if the birth rate in China remained at its current level, the population would reach more than two billion during the twenty-first century. Their results, based on an age-structured mathematical model, contributed to the government’s decision to turn to a one-child policy.

Nicolas Bacaër

Chapter 26. Some contemporary problems

Abstract

This chapter gives a brief overview of some contemporary problems in mathematical population dynamics: population aging in demography; emerging diseases (AIDS, SARS, vector-borne diseases…) and vaccination policy in epidemiology; fishing policies in ecology; the dispersal of genetically modified organisms in population genetics. The specialized institutions working in France on the modelling of these problems are mentioned. Various aspects of research work are also emphasized.

Nicolas Bacaër

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