Vito Volterra

How to use this course

Vito Volterra is remembered in several separate communities. Analysts know the Volterra operator and the integral equations with a variable upper limit. Functional analysts trace their subject to his “functions of lines”. Materials scientists use his construction of dislocations. Ecologists learn the Lotka–Volterra equations in their first course. Italians remember him as a senator, the first president of the National Research Council and one of twelve professors who would not take the Fascist oath.

This course connects these pieces. A single idea runs through most of them: a quantity can depend on the whole history or the whole shape of another function, and mathematics should treat such dependence as naturally as it treats dependence on a number.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs or derivations, and exercises with folded solutions. The order roughly follows his career: real analysis (1881), functionals (1887), integral equations (1896), heredity and elasticity (1905–1913), and population dynamics (1926–1931).

Timeline

Ancona and Florence, 1860–1878

Vito Volterra was born in Ancona on 3 May 1860, in the last months of papal rule over the city. His parents, Abramo Volterra and Angelica Almagià, belonged to the Jewish community of the Marche. Abramo, a cloth merchant, died in 1862, leaving the family poor. Angelica’s brother Alfonso Almagià took mother and son in. They lived for a time in Turin and then settled in Florence, where Vito spent his youth.

He read Adrien-Marie Legendre’s Geometry at eleven and was soon working through calculus on his own. His family wanted him to go into commerce. He attended the Dante Alighieri Technical School and the Galileo Galilei Technical Institute in Florence, where two teachers changed his course. The mathematician Cesare Arzelà recognised his ability. In 1877 the physicist Antonio Ròiti offered him a post at the Istituto di Studi Superiori in Florence, which gave the boy a small income and a reason to stay in science.

In the autumn of 1878, with the support of his teachers, Volterra entered the University of Pisa to study science.

Pisa: Betti and Dini, 1878–1883

In 1879 Volterra won a place at the Scuola Normale Superiore with the highest marks of his year. Pisa then had two of the best analysts in Italy. Ulisse Dini was building the rigorous theory of real functions in the style of Karl Weierstrass. Enrico Betti, who had been a close friend of Bernhard Riemann during Riemann’s stays in Pisa in the 1860s, taught mathematical physics: potential theory, elasticity and hydrodynamics.

Both influences show at once. In 1881, still a student, Volterra published notes on the strange functions that Dini’s school collected, including a function with a bounded derivative that the Riemann integral cannot recover (Lesson 04). His laurea thesis of 1882, under Betti, was on hydrodynamics; it contained results that George Stokes had obtained independently.

Betti made him his assistant in December 1882. A year later, at 23, Volterra won the competition for the chair of rational mechanics at Pisa. From the start he worked on two levels at once: careful analysis and physical problems. That pairing defines his career.

Pisa, Turin, Rome, 1883–1900

The Pisa years were his most original. In 1887 he published a series of notes in the Rendiconti of the Accademia dei Lincei on functions that depend on all the values of another function, which he called functions of lines (Lesson 05). In 1890 he extended the Hamilton–Jacobi method to other problems of mathematical physics, and between 1892 and 1894 he wrote on partial differential equations, especially the equation of cylindrical waves.

When Betti died in 1892, Volterra took over his chair of mathematical physics and became dean of the science faculty and editor of Il Nuovo Cimento. He soon moved to the chair of mechanics at Turin. There, in 1896, he published his notes on the inversion of definite integrals, which created the theory of integral equations with a variable limit (Lesson 06). In 1897 he founded the Italian Physical Society.

In November 1900 he succeeded Eugenio Beltrami in the chair of mathematical physics at Rome, which he held for 31 years. The same year he gave a plenary lecture at the International Congress of Mathematicians in Paris, on the analytic styles of Betti, Francesco Brioschi and Felice Casorati, and on 11 July he married his second cousin Virginia Almagià. Their son Edoardo became a leading historian of Roman law; their daughter Luisa married the marine biologist Umberto D’Ancona, a connection that would matter in 1925; their son Enrico became a professor of engineering in the United States.

A derivative that cannot be integrated

For the general reader. The fundamental theorem of calculus says that integrating a derivative gives back the original function. In 1881 the student Volterra built a function whose derivative exists everywhere and stays bounded, yet cannot be integrated in Riemann’s sense. Twenty years later Henri Lebesgue cited such examples to show that a better integral was needed.

The integrability criterion

Volterra worked before measure theory existed, but his construction is designed exactly to defeat this criterion. He needed a nowhere dense set of positive “content” and a derivative that oscillates near every point of it.

The construction

Start from \(f(x)=x^2\sin(1/x)\), \(f(0)=0\). It is differentiable everywhere, with \(f'(x)=2x\sin(1/x)-\cos(1/x)\) for \(x\neq0\) and \(f'(0)=0\). So \(f'\) exists but oscillates between values near \(\pm1\) in every neighbourhood of 0.

Let \(S\subset[0,1]\) be a Smith–Volterra–Cantor set: at stage \(n\) remove an open middle interval of length \(4^{-n}\) from each of the \(2^{n-1}\) remaining closed intervals. On each removed interval \((\alpha,\beta)\), place a copy of \(f\) grown from the left end, then frozen at a critical point \(x_0\) with \(x_0\le(\beta-\alpha)/2\) and \(f'(x_0)=0\), then reflected so it returns to 0 at the right end. Set the function to 0 on \(S\). The result \(V\) has these properties:

Why it works. Inside each removed interval, \(V\) is a smooth piece and \(V'\) is continuous. At a point \(s\in S\), \(|V(x)-V(s)|=|V(x)|\le(x-s)^2\) because each piece is bounded by the square of the distance to the nearest endpoint, so \(V'(s)=0\). But every \(s\in S\) is a limit of endpoints of removed intervals, and near each endpoint \(V'\) takes values close to \(1\) and \(-1\). So \(V'\) is discontinuous at every point of \(S\), a set of positive measure, and the criterion fails. Lebesgue’s integral repairs the theorem: if \(F'\) exists everywhere on \([a,b]\) and is bounded, then \(F'\) is Lebesgue integrable and \(\int_a^bF'=F(b)-F(a)\).

A second result from 1881

Proof in modern language. The continuity set of any function is a \(G_\delta\): it equals \(\bigcap_n U_n\) where \(U_n\) is the open set of points at which the oscillation of \(f\) is less than \(1/n\). If the continuity set were \(\mathbb Q\), then \(\mathbb Q\) would be a dense \(G_\delta\). The irrationals \(\bigcap_{q\in\mathbb Q}(\mathbb R\setminus\{q\})\) are also a dense \(G_\delta\). Their intersection, the empty set, would be a countable intersection of dense open sets, contradicting the Baire category theorem. Volterra’s argument predates Baire’s 1899 theorem and proves the needed special case directly.

Functions of lines

For the general reader. The area under a curve depends on the whole curve, not on a single number. So does the length of a path, the energy of a vibrating string, or the time a bead takes to slide down a wire. In 1887 Volterra proposed treating such quantities as functions in their own right, with derivatives and Taylor series. Jacques Hadamard later named them functionals.

The definition

Volterra considered a quantity \(F[y]\) that depends on all the values \(y(t)\), \(a\le t\le b\). He called it a function of lines because \(y\) is a curve. He required continuity: small changes in \(y\) give small changes in \(F\). His key definition is the derivative at a point \(t\).

Today this is the Fréchet derivative represented by a kernel, or the physicist’s functional derivative \(\delta F/\delta y(t)\). The extremals of the calculus of variations are the curves with \(F'[y;t]=0\) for all \(t\), and the Euler–Lagrange equations are what this condition becomes for integral functionals.

From finitely many variables to a continuum

Volterra’s heuristic was explicit. Divide \([a,b]\) into \(n\) pieces of length \(\Delta t\) with sample points \(t_i\). Then \(F[y]\) is approximately a function \(f(y_1,\dots,y_n)\) of the values \(y_i=y(t_i)\), and

So \(\partial f/\partial y_i\approx F'[y;t_i]\,\Delta t\): partial derivatives become a density in \(t\), and sums become integrals. This passage to the limit guided his work on integral equations as well (Lesson 06).

The Volterra series

The same heuristic, applied to Taylor series, gives the expansion now named after him:

with symmetric kernels \(k_n\). In the 1940s and 1950s Norbert Wiener used such series to describe nonlinear systems with memory, where \(y\) is an input signal and \(F\) the output at a given time. They remain a standard tool in nonlinear system identification.

Volterra integral equations

For the general reader. An integral equation asks for an unknown function that appears inside an integral. When the integral runs only over the past, from a fixed start up to the present moment, the equation describes a process whose current state depends on its history. Volterra showed in 1896 that such equations always have exactly one solution, and he showed how to compute it.

The second kind

Let \(K(x,t)\) be continuous on the triangle \(a\le t\le x\le b\) and \(f\) continuous on \([a,b]\). Consider

Define iterated kernels \(K_1=K\) and \(K_{n+1}(x,t)=\int_t^x K(x,s)\,K_n(s,t)\,ds\).

Proof. Let \(M=\max|K|\). By induction, \(|K_n(x,t)|\le M^n(x-t)^{n-1}/(n-1)!\): indeed

So \(\sum|\lambda|^{n-1}|K_n|\le M e^{|\lambda|M(b-a)}\), and the series converges uniformly for \(\lambda\) in any bounded set. Substituting shows that \(u\) solves the equation. For uniqueness, the difference \(w\) of two solutions satisfies \(w=\lambda\int_a^xKw\); iterating \(n\) times gives \(|w(x)|\le\|w\|_\infty(|\lambda|M(x-a))^n/n!\to0\). The factorial, which comes from integrating over a simplex of shrinking volume, is the whole point: it is the variable upper limit that makes the series converge for every \(\lambda\).

This is the decisive difference from the equations with fixed limits \(\int_a^b\) that Ivar Fredholm treated in 1900–1903. Those have eigenvalues, where the solution fails to exist or to be unique, and Fredholm’s alternative governs them. Volterra’s equations have none.

The first kind

Differentiate under the integral sign to obtain this. Niels Henrik Abel’s 1823 tautochrone problem leads to a first-kind equation with the singular kernel \((x-t)^{-1/2}\), which needs a separate treatment and was one of Volterra’s starting points.

The Volterra operator

On \(L^2[0,1]\) define \((Vu)(x)=\int_0^xu(t)\,dt\). It is compact. If \(Vu=\mu u\) with \(\mu\neq0\), then \(u\) is continuous, \(u(0)=0\) and \(u'=u/\mu\), so \(u\equiv0\). Hence \(V\) has no eigenvalues at all, and the spectrum of \(V\) is \(\{0\}\). Its norm is \(2/\pi\). It is the standard example of a quasinilpotent operator that is not nilpotent.

Composition and heredity

For the general reader. Bend a steel spring and release it, and it springs back. Bend a rubber band or a glass fibre, and part of the deformation lingers; the material remembers. Ludwig Boltzmann had proposed in 1874 that such memory adds up linearly over the past. From about 1909 Volterra built the mathematics of these hereditary phenomena on his integral equations and a new product of functions.

Composition

For kernels defined on \(x\le y\), Volterra defined the composition

It is associative and distributive, and it plays the role of matrix multiplication for continuous indices. The iterated kernels of Lesson 06 are powers \(K^{\ast n}\), and the resolvent is a geometric series in this algebra. Two kernels are permutable if \(F\ast G=G\ast F\). The most important permutable kernels are those of the closed cycle, \(F(x,y)=f(y-x)\); for them composition becomes the convolution

which is commutative. Volterra built an algebra of such functions, with powers, roots and analytic functions of them, a forerunner of operational calculus and of Jan Mikusiński’s later convolution quotients.

Hereditary elasticity

In a linear material with memory, the stress depends on the strain now and on the whole past strain history:

Inverting such a law to express strain through stress is a Volterra equation of the second kind. Combining it with Newton’s law gives integro-differential equations, which Volterra studied systematically and summarized in his 1913 Leçons sur les équations intégrales et les équations intégro-différentielles and in his 1928 plenary lecture at Bologna. His last paper, on energy in hereditary elastic phenomena, continued the theme.

Distortions of elastic bodies

For the general reader. Take a ring of rubber, cut it through, slide one face of the cut relative to the other, and glue it back. The ring is now stressed even though no external force acts on it. Volterra classified these self-stressed states in 1905–1907. Thirty years later, metallurgists found that the same objects, now called dislocations, explain why metals bend instead of shattering.

Compatibility and multivalued displacement

In linear elasticity a displacement field \(u\) gives a strain \(\varepsilon_{ij}=\tfrac12(\partial_iu_j+\partial_ju_i)\). Conversely, a strain field satisfying Adhémar Barré de Saint-Venant’s compatibility conditions comes from a displacement, but only locally. In a simply connected body the displacement is single-valued. In a multiply connected body, such as a hollow cylinder, integrating the strain around a loop can return a different displacement.

Volterra realized each type by a construction: cut along \(\Sigma\), move one face by the rigid motion, add or remove material as needed, and weld. For a hollow cylinder with the cut along a half-plane through the axis, the three translations and three rotations give six elementary distortions. Today the translational ones with \(b\) along the axis or perpendicular to it are the screw and edge dislocations; the rotational ones are disclinations. In 1934 Geoffrey Ingram Taylor, Egon Orowan and Michael Polanyi independently proposed crystal dislocations to explain plastic flow, and they used Volterra’s solutions for the elastic fields. The vector \(b\) is now called the Burgers vector.

The screw dislocation

In cylindrical coordinates \((r,\theta,z)\) around the \(z\)-axis, take

Going once around the axis increases \(\theta\) by \(2\pi\), so \(u_z\) jumps by \(b\). The only strains are \(\varepsilon_{\theta z}=\frac{1}{2r}\partial_\theta u_z=\frac{b}{4\pi r}\), and the stress is \(\sigma_{\theta z}=2\mu\varepsilon_{\theta z}=\frac{\mu b}{2\pi r}\), where \(\mu\) is the shear modulus. The stress is single-valued even though the displacement is not.

Predators and prey

For the general reader. During the First World War fishing in the upper Adriatic almost stopped. The biologist Umberto D’Ancona, Volterra’s son-in-law, found that the share of sharks, rays and other predatory fish in the catches landed in upper Adriatic ports had risen. Why should less fishing help predators more than prey? Volterra answered with two differential equations. Alfred Lotka had written the same system in 1920 and 1925, and Volterra acknowledged his priority.

The equations

Let \(N_1\) be the prey and \(N_2\) the predators. In Volterra’s notation,

with positive constants. Prey grow at rate \(\varepsilon_1\) without predators; predators die at rate \(\varepsilon_2\) without prey; encounters are proportional to \(N_1N_2\). The interior equilibrium is \(\big(q_1,q_2\big)=\big(\varepsilon_2/\gamma_2,\ \varepsilon_1/\gamma_1\big)\).

Proof. \(\frac{dH}{dt}=\big(\gamma_2-\frac{\varepsilon_2}{N_1}\big)N_1(\varepsilon_1-\gamma_1N_2)+\big(\gamma_1-\frac{\varepsilon_1}{N_2}\big)N_2(\gamma_2N_1-\varepsilon_2)\). The first term is \((\gamma_2N_1-\varepsilon_2)(\varepsilon_1-\gamma_1N_2)\) and the second is \((\gamma_1N_2-\varepsilon_1)(\gamma_2N_1-\varepsilon_2)\); they cancel. Convexity follows since each summand \(cN-d\ln N\) is convex. A solution on a closed level curve with no equilibrium on it must go round it periodically.

Proof. \(\frac{d}{dt}\ln N_1=\varepsilon_1-\gamma_1N_2\). Integrate over one period: the left side gives \(\ln N_1(T)-\ln N_1(0)=0\), so \(\gamma_1\int_0^TN_2=\varepsilon_1T\). The same argument with \(N_2\) gives the other average.

Fishing changes \(\varepsilon_1\) to \(\varepsilon_1-h\) and \(\varepsilon_2\) to \(\varepsilon_2+h\), so the averages become \((\varepsilon_2+h)/\gamma_2\) and \((\varepsilon_1-h)/\gamma_1\). Reducing fishing does the opposite, which is what D’Ancona saw. Volterra published the analysis in 1926 in the Memorie of the Lincei and in a short paper in Nature.

The struggle for life

For the general reader. After the predator–prey paper, Volterra built a general theory of interacting species. Two species that compete for one food cannot coexist: the weaker one dies out. Many species can form conservative systems, like frictionless mechanics, or dissipative ones that settle down. And populations can remember: births today depend on food eaten long ago. He gathered these ideas in his 1931 Leçons sur la théorie mathématique de la lutte pour la vie.

Competition for one resource

Suppose two species consume the same food, and the food available decreases as \(F(N_1,N_2)\) increases, where \(F\) is a positive increasing function of both populations. Volterra wrote

Proof. Eliminate \(F\): \(\gamma_2\frac{d}{dt}\ln N_1-\gamma_1\frac{d}{dt}\ln N_2=\gamma_2\varepsilon_1-\gamma_1\varepsilon_2=:c>0\). So \(N_1^{\gamma_2}/N_2^{\gamma_1}=Ae^{ct}\). Since \(N_1\) is bounded, \(N_2^{\gamma_1}\le N_1^{\gamma_2}A^{-1}e^{-ct}\to0\). The biologist Georgii Gause tested this prediction with protozoa in the 1930s, and it became known as the competitive exclusion principle.

Many species

For \(n\) species with \(\frac{dN_r}{dt}=N_r\big(\varepsilon_r+\beta_r^{-1}\sum_sa_{rs}N_s\big)\), where \(\beta_r>0\) are “equivalents” converting one species into another, Volterra called the system conservative when \(a_{rs}=-a_{sr}\) (what one species gains in an encounter, the other loses) and dissipative when the quadratic form \(\sum a_{rs}x_rx_s\) is negative definite. In the conservative case, with an interior equilibrium \(q\), the function

is constant along solutions; in the dissipative case it decreases, and solutions approach equilibrium. This is an early instance of a Lyapunov function in ecology.

Populations with memory

Volterra applied his hereditary mechanics to biology. A logistic population whose growth is depressed by its own past, for example through accumulated toxins, satisfies

Equations of this kind, studied in the Leçons and in later papers, are the ancestors of the delay and integro-differential models now used in ecology and epidemiology. In 1936 Andrey Kolmogorov generalized the two-species theory to arbitrary interaction functions in a paper addressed to Volterra’s theory.

The organizer of Italian science, 1900–1926

In Rome Volterra became the public face of Italian science. He was named a senator of the Kingdom in 1905. He helped revive the Italian Society for the Progress of Science and was its president in 1919. Abroad he was the best-known Italian mathematician of his generation: a plenary speaker at the International Congresses of 1900, 1908, 1920 and 1928, honorary doctor of Christiania (1902), Cambridge (1904) and Clark University (1909), and Foreign Member of the Royal Society from 1910. His Sorbonne lectures of 1912 became the Leçons sur les fonctions de lignes (1913), edited by his student Joseph Pérès. Paul Lévy and Maurice Fréchet developed functional analysis in Paris partly from his ideas.

In 1914–15 Volterra argued for Italy to abandon its alliance with Austria and Germany. When Italy entered the war in 1915 he enlisted, aged 55, as a lieutenant in the engineers and worked at the Central Institute of Aeronautics on firing guns from airships without igniting them. In 1917 he became head of the Office of Inventions, which linked the army, industry and the universities; it became the Office of Inventions and Research in 1918. He received the Italian War Cross, and in 1921 George V made him an honorary Knight Commander of the Order of the British Empire.

After the war he turned this experience into institutions. He presided over the Academy of the XL from 1918 to 1920, became vice-president of the Accademia dei Lincei in 1920 and its president in 1923, and was the first president of the National Research Council from its creation in 1923. Alongside these offices came the fish problem of 1925 and the population theory of Lessons 09 and 10, a new field begun at 65.

The oath, exile and legacy, 1926–1940

Volterra joined the senatorial opposition to Mussolini from 1922. In 1926, after the opposition parties were abolished, he left the presidency of the Lincei, and in 1927 the regime replaced him at the CNR with Guglielmo Marconi. From 1928 the police watched him as politically suspect. In 1931 the government required every university professor to swear loyalty to the Fascist regime. Of about 1,250 professors, twelve refused. Volterra was one of them, and he lost his chair at Rome. In 1934 he refused a second oath and was expelled from the Lincei.

He spent much of the following years abroad, lecturing in Paris, where he had many friends, and in Spain, Belgium, Switzerland, Romania and Czechoslovakia. He wrote Théorie générale des fonctionnelles (1936) with Joseph Pérès and Opérations infinitésimales linéaires (1938) with Bohuslav Hostinský. In 1936, on the initiative of Agostino Gemelli, he was appointed to the Pontifical Academy of Sciences, one of the few bodies that still welcomed him. The racial laws of 1938 removed him from the Istituto Lombardo and barred his sons from university posts. He died at his home in Rome on 11 October 1940 and was buried at Ariccia. The Pontifical Academy arranged his funeral. A postcard of his from the 1930s, often quoted as his epitaph, says that empires die but Euclid’s theorems stay young forever.

Legacy

• Functional analysis: functionals, the functional derivative and the Volterra series opened the path to Fréchet, Lévy and the analysis of infinite-dimensional spaces.
• Integral equations: the Volterra equation and operator are standard; renewal theory, viscoelasticity and the theory of delay equations all use them.
• Mechanics of materials: his distortions became dislocation theory, the basis of crystal plasticity.
• Mathematical biology: the Lotka–Volterra equations, competitive exclusion and hereditary population models founded theoretical ecology as a mathematical discipline.
• Institutions: the Italian Physical Society and the CNR. The Accademia dei Lincei published his collected works in five volumes (1954–1962). A lunar crater bears his name.

Connections

People linked to Vito Volterra across this collection. A name without a link has no page here yet. See Vito Volterra in the network of all 337 lives →

Teachers and advisors

• Enrico Betti 1882

Students

• Paul Lévy
• Cornelia Fabri
• Joseph Pérès

Influenced by

• Ludwig Boltzmann
• Cesare Arzelà
• Ulisse Dini

Influenced

• Andrey Kolmogorov 1936
• G. I. Taylor
• Ivar Fredholm
• Maurice Fréchet

Colleagues and collaborators

• Eugenio Beltrami
• Guglielmo Marconi
• Jacques Hadamard
• Tullio Levi-Civita 1919-1938
• Émile Picard
• Bohuslav Hostinský 1938

Rivals

• Alfred J. Lotka

Family

• Edoardo Volterra
• Umberto D'Ancona

Also mentioned in this life

• Niels Henrik Abel
• Georg Cantor
• Euclid
• Leonhard Euler
• Galileo Galilei
• Israel Gelfand
• William Rowan Hamilton
• David Hilbert
• Carl Gustav Jacob Jacobi
• Joseph-Louis Lagrange
• Pierre-Simon Laplace
• Henri Lebesgue
• Adrien-Marie Legendre
• Aleksandr Lyapunov
• Isaac Newton
• Bernhard Riemann
• George Gabriel Stokes
• Karl Weierstrass
• Norbert Wiener

Fields

• functional analysis 11 lives
• mathematical physics 45 lives
• elasticity 11 lives
• mathematical biology 3 lives

On CoursesHub

• Functional Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Vito Volterra, MacTutor History of Mathematics.
2. Vito Volterra, Wikipedia; Italian edition.
3. Volterra, Vito, Complete Dictionary of Scientific Biography.
4. J. R. Goodstein, The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician 1860–1940, AMS and LMS, 2007.
5. E. T. Whittaker, Vito Volterra 1860–1940, Obituary Notices of Fellows of the Royal Society 3 (1941).
6. G. Israel and A. Millán Gasca, The Biology of Numbers: The Correspondence of Vito Volterra on Mathematical Biology, Birkhäuser, 2002.

Mathematics

1. V. Volterra, Leçons sur les fonctions de lignes, ed. J. Pérès, Gauthier-Villars, 1913.
2. V. Volterra, Leçons sur les équations intégrales et les équations intégro-différentielles, Gauthier-Villars, 1913.
3. V. Volterra, Theory of Functionals and of Integral and Integro-Differential Equations, Blackie, 1930; Dover reprint, 1959.
4. V. Volterra, Sur l’équilibre des corps élastiques multiplement connexes, Annales scientifiques de l’École Normale Supérieure 24 (1907).
5. V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Memorie della R. Accademia dei Lincei (1926).
6. V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature 118 (1926).
7. V. Volterra, Leçons sur la théorie mathématique de la lutte pour la vie, Gauthier-Villars, 1931.
8. A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, 1925.
9. Volterra’s function and Lotka–Volterra equations, Wikipedia.
10. J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd ed., Wiley, 1982.