An ecologist's mathematical exploration

Eugene Wigner's famous article "The Unbelievable Effectiveness of Mathematics in Natural Science" contains two ideas from the title alone: ​​1. The necessity of mathematics for understanding nature; 2. The importance of natural phenomena in extracting the research object to enrich mathematical ideas. The idea of ​​chaos theory originated from the diligent pursuit of mathematical descriptions of different natural phenomena by different experts in different fields, and finally reached a unified understanding. This article aims to describe the mathematical characteristics of period-doubling bifurcations discovered by ecologists when using logistic mapping to describe population dynamics.

Written by Ding Jiu (Professor of Mathematics at the University of Southern Mississippi)

In my previous Fanpu article "If I say iteration, you will understand it" (hereinafter referred to as "Iteration"), I introduced the most basic function iteration phenomenon, using the simple prop of polynomials, combined with geometric and analytical methods, to show the fixed points of attraction or repulsion or periodic points with a period of two. With these basics, we can continue to walk on the road of iteration and pick colorful flowers along the way.

This article offers readers a bunch of particularly gorgeous roses, which bloom on the iterative tree with continuous branches branching into small branches. They also provide a buffer device for the members of the "Iterative Spring Tour Group" to experience a journey from order to disorder before entering the chaotic world, so that they can take a deep breath and calm down before screaming at the chaotic monster. This whole piece of flowers blooms in the garden of a series of quadratic polynomials with one parameter.

However, the nutrients that make the "period-doubling fork" red roses bloom gloriously come from a macro-branch of biological science - ecology, which is also called population dynamics. The hard-working gardener who fertilized the rose trees was Robert May (1936-2020), who later served as the chief scientist of the British government and was knighted by Queen Elizabeth.

Baron May was born in Sydney, Australia. He was only 20 years old when he graduated from Sydney University with a bachelor's degree in chemical engineering and theoretical physics. Three years later, he obtained a doctorate in theoretical physics from the same university, and then went to Harvard University for two years as a postdoctoral fellow, studying applied mathematics. After returning to his alma mater, he returned to his original profession and began teaching as a senior lecturer, and was promoted to full professor of theoretical physics. In 1971, because he was fascinated by biology again "on a whim", he went to the United States again, stayed at the Institute for Advanced Study in Princeton for a year, and made friends with biologists at Princeton University. Two years later, he became a professor of zoology at this prestigious university.

It was here that he became a theoretical ecologist and made a name for himself in creative population dynamics research based on function iteration. This discipline belongs to a branch of biology called "community ecology", which explores the interdependence and constraints between two or more different biological populations living in the same geographical range or region, as well as the fluctuations in the number of biological populations and the rise and fall of life. The goal is to find the laws behind these ever-changing numbers, the relationship between the complexity and stability of natural communities, and the mathematical description that can explain the laws of change.

Like all other natural scientists and engineers, ecologists naturally need the help of mathematics. However, in the eyes of geometers or topologists who use profound modern mathematics to study dynamic systems, such as Fields Medal winner Stephen Smale (1930-), the mathematics used in May's world-famous research seems very simple. In fact, his most well-known work only uses quadratic polynomials, but there is only one more parameter in its expression. However, the infinite parabolic graphs of this family of functions with different curvatures threw his scientific papers into the baskets of Nature and Science like the basketball in Jordan's hands.

By the way, as early as 1947, Stanislaw Ulam (1909-1984) and John von Neumann (1903-1957), American mathematicians of Polish and Hungarian origin respectively, made precise statistical research on the longest parabola when μ = 4 in the logistic mapping family, and gave the final distribution law of the iterative point orbit that is the same for almost all initial points. This will be a flower in my future popular science article on ergodic theory.

This famous mathematical model is consistent with ecologists' intuition about population size, and it is not difficult for the general public to understand. For example, in the African savannah where the strong prey on the weak, herbivorous zebras are weak, while carnivorous lions are strong. The populations of the two restrict each other. If the weak are eaten too much by the strong, the number will decrease sharply, and the strong will in turn face a survival crisis. Therefore, experience tells us that if the population is relatively small, it will increase very quickly; when the number is moderate, the growth rate will approach zero, and when the population is large, it will drop sharply. This is a basic assumption followed by ecologists in their research, and it is also reflected in the above model: if x rises, then 1-x will fall, and conversely, if x decreases, 1-x will increase, so their product restricts the change in population size. It seems that this functional relationship can indeed reflect the law of population size change in the living environment to a certain extent.

Before Professor May entered this field, early ecologists generally agreed with the view advocated by John Smith (1920-2004), a British theoretical evolutionary biologist and geneticist, in his classic book Mathematical Ideas in Biology: the population size is often approximately constant. They all believed, more or less, that in the above logistic mapping, no matter how large or small the initial population size is, it will stabilize at a fixed number after a few years. At the same time, they all believed that only "stable solutions" were attractive. If they saw a population size that refused to stabilize, they would unthinkingly conclude that it was the error of the calculation tool that was causing the trouble.

This belief, which has not been scientifically verified, has been deeply rooted in the minds of many scientists, so they are rarely willing to sit down and do some mathematical analysis of specific models, let alone engage in boring repetitive iterative calculations with great patience. Exactly ten years ago in April, Edward Wilson (1929-2021), a famous biologist who had retired from Harvard University, wrote an article in the Wall Street Journal titled "Great Scientists ≠ Good at Mathematics", trying to use his own experience to prove that "many of the most successful scientists in the world today are only semi-illiterate in mathematics". This article and a rebuttal article by a Berkeley mathematics professor four days later, "Don't Listen to Edward Wilson", were reprinted in the Journal of the American Mathematical Society in July of that year, which aroused heated discussions among many mathematicians and scientists.

As a scientist who spans multiple fields, Professor Mei obviously disagrees with the above representative view that traditional biologists despise mathematics. At Princeton University, he began numerical iteration experiments on logistic mapping. It was Professor Mei, who is both an applied mathematician and an ecologist, who patiently applied mathematics to the logistic mapping family. By performing iterative calculations himself, he saw a new scene in the field of population dynamics that amazed him.

For the sake of convenience, we assume that the time period is one year, that is, if x represents the relative population number of this year, then μx(1-x) represents the relative population number of next year.

Mei gradually increased the reproduction rate μ, trying to figure out the relationship between the final trend of population evolution and this important parameter. He soon discovered that when μ did not exceed 3, everything was normal. For example, if the parameter μ is not greater than 1, then no matter what the population size is at the beginning, the population will decrease year by year at the latest after the second year, and will eventually die out. However, when μ is greater than 1 but not greater than 3, no matter what the initial population size is, the population size will gradually stabilize after iterations year after year, and will eventually tend to a fixed number. This fixed number increases with the increase of the parameter. For example, when μ is 2.7, the final population size will be fixed at 0.6296, and when μ = 3, the ultimate population size increases to 0.6667.

Mei continued to increase the value of the parameter.

When μ was greater than 3 but not greater than an exact value of 1 + √6, which is approximately 3.45, he discovered a new phenomenon: the population number no longer tended to a fixed number, but instead rose and fell alternately year after year, and finally jumped back and forth between two different fixed numbers, eventually tending to a period-2 orbit composed of these two periodic points, regardless of the number of populations selected at the beginning of the iteration.

If the parameter value is increased slightly from 3.45 to about 3.54, the population will fluctuate regularly every four years, eventually jumping back and forth between four fixed numbers, and eventually tending to a period-4 orbit composed of these four period points, regardless of the initial population size. In this way, the two-year cycle of the population doubles to a four-year cycle.

Similarly, when the parameter value exceeds a tiny number of 3.54, the four-year cycle will jump to an eight-year cycle. Then, as the parameter value continues to increase slightly, the sixteen-year cycle, the thirty-two-year cycle, and so on and so forth, appear one after another until infinity. Then, within a new range of μ values, there is a new period doubling phenomenon whose period is not a power of 2. When μ increases to a certain extent, the final population size no longer shows periodicity, and there are signs of some kind of "chaos".

Let us summarize the above paragraphs using mathematical language.

This limit is named after the American mathematical physicist Mitchell Feigenbaum (1944-2019).

Feigenbaum received his PhD in particle physics from MIT in 1970, then spent two years at Cornell University and Virginia Polytechnic Institute and State University, during which he published only one paper but accumulated a broad knowledge base. In fact, he only published 27 scientific papers in his lifetime, either independently or in collaboration.

After he was recruited by the Los Alamos National Laboratory in the United States in 1974, in order to deeply understand the self-similarity between large and small scales in turbulence in his own way and find the essence of the problem, he did not care whether he could squeeze out the article as soon as possible, but worked "twenty-five" hours a day, relying on his hands fiddling with the calculator and his brain thinking constantly in between calculations, he discovered a new universal constant. His immediate boss in the theoretical department later commented: "Feigenbaum has the right background, he did the right thing at the right time, and did it very well. He did not do local things, but figured out the whole thing."

What is the reason for this?

In 1974, Professor May was invited by the Department of Mathematics at the University of Maryland to give a lecture on the “Biomathematics Series”. He reported on the strange phenomena he found when iterating the logistic mapping numerically. After the lecture, a professor he invited sent him to the airport. On the way, the professor named James Yorke (1941-) handed May an article. After reading it, Professor May was shocked: a theory in the article solved his confusion.

This article, which was just a draft at the time, later became one of the most famous mathematical papers in the history of chaos. Today, its title has a fixed translation in the Chinese world: "Period three means chaos." The elementary interpretation of this eight-page mathematical paper, which has been cited more than 5,650 times by the academic community, will be the content of my mathematical popular science article in the near future.

The ecological story of the mathematical phenomenon of "period-doubling bifurcation" has been roughly told. However, knowing only "the discovery anecdotes of scientific figures" but lacking a basic understanding of scientific principles is like the truth expressed by the words that American theoretical physicist Richard Feynman (1918-1988) often recalled when his father gave his son special advice when he was a child: "If you only know the name of a bird and know nothing about its habits, then your understanding of the bird is almost zero." Feynman believed that his father's simple and wise view influenced his scientific career for a lifetime, and made him understand the fundamental difference between just knowing the name of a thing and fully understanding the essence of a thing. Therefore, in the last part of this article, I turn to mathematics, mainly using easy-to-understand elementary mathematical language to explain why, among the three period-doubling bifurcation properties of the logistic mapping family fμ mentioned above, a period-2 orbit suddenly emerges when the reproduction rate μ crosses the threshold of 3, and this orbit is attractive before the value of μ exceeds 1 + √6.

I have mentioned in Iteration that the only fixed point b/(1-a) of the linear function ax + b is attractive when |a| < 1, but b/(1-a) is repulsive when |a| > 1. The coefficient a here is also the slope of the corresponding straight line graph. For any differentiable function y = f(x), its derivative value f'(x) at a point x is the slope of the tangent line of the function's graph at the point (x, f(x)). Although the graph curve of a nonlinear function is curved, the curve and the tangent line look almost the same near the tangent point. This observation is the basic reason why calculus has been so successful for hundreds of years. Since the "attractiveness" or "repulsiveness" of a fixed point is a "local property" like the concept of "derivative of a function", why can't we use a simple straight line tangent to the function graph at a fixed point to replace the complex curve to create a mathematical tool to determine whether the fixed point x* of the function f has "attractiveness" or "repulsiveness"? This idea can at least help us "come up with" a simple and easy-to-use judgment method, that is, if the fixed point x* of f satisfies |f'(x*)| < 1, then x* must be attractive; if |f'(x*)| > 1, then x* must be repelling. This idea is absolutely true.

Why? We can easily prove the above assertion geometrically by drawing a function curve that is relatively flat near the intersection with the diagonal line y = x, using the “graph iteration method”. But the following “analytical argument” is more convincing: Assume f(x*) = x* and |f'(x*)| < 1. Since the derivative f'(x*) is the limit of the ratio of the function value difference f(x) – f(x*) to the independent variable value difference x – x* when x approaches x*, we have reason to believe that when x is near x*, |f(x) – x*| ≤ δ |x – x*|, where the positive number δ is only slightly larger than |f'(x*)|, but still less than 1 like |f'(x*)| (for example, δ can be taken as the arithmetic average of |f'(x*)| and 1, that is, δ = (|f'(x*)| + 1)/2).

The above idea can also be used to prove that if a fixed point x* of f satisfies |f'(x*)| > 1, then x* is exclusive, that is, it excludes every point x in its vicinity that is not equal to it: |f(x) – x*| ≥ Δ |x – x*|, where Δ is a number greater than 1.

Readers naturally think of the conclusion when |f'(x*)| = 1. There is no way, no general conclusion, only "specific analysis of specific problems"; just like life, mathematics also has regrets. Readers who have studied the infinite series theory in advanced mathematics will remember this fact: given a judgment method for whether a series converges or not, there will always be a series for which the judgment method fails. Therefore, there is no "master key" in the world; when I see some publishers' promotional slogans such as "The world's most recognized learning method!", I can't help but laugh, just like seeing the news of the invention of a perpetual motion machine.

The bifurcation law of this type of single-parameter function family, which was numerically simulated for the first time in history by ecologist Dr. Mei, is vividly called the pitchfork type in the field of dynamical systems, because if the bifurcation diagram above is marked with dotted lines on the repulsive periodic points, it looks like a "pitchfork" commonly used in rural areas, which is inseparable from the "period-doubling bifurcation". However, there is another bifurcation phenomenon, called the "tangent type", which comes from the fact that before and after the parameter crosses a threshold, the graph of the function goes through a process of intersecting the diagonal y = x at two points, being tangent at one point, and then separating from each other, so the number of fixed points decreases from two to one and then to zero. A typical example is the exponential function family {μex}, where the parameter μ > 0. When μ = 1/e, the graph of the function is tangent to the diagonal at the fixed point 1, when μ < 1/e, the graph intersects the diagonal at two points, and when μ > 1/e, the graph of the function never intersects the diagonal. As a self-test after reading this article, I invite readers to use the "graph iteration method" for this family of exponential functions for the three cases 0 < μ < 1/e, μ = 1/e and μ > 1/e, and predict the final direction of the iterative point orbits corresponding to all initial points.

So far, our function iteration has only encountered periodic points, including fixed points, whose period is a non-negative integer power of 2. In the completely ordered natural numbers, 3 follows 2, which is an extraordinary natural number. It will set off huge waves in the sea of ​​function iteration, making the future route of the sailing ship unpredictable!

Acknowledgements: The author would like to thank scholar Yang Yunyang for reading the first draft and providing revision suggestions.

Produced by: Science Popularization China

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