Spectral theory

From Wikipedia, the free encyclopedia

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

In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

Mathematical background

The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous. Hilbert himself was surprised by the unexpected application of this theory, noting that "I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis' without any presentiment that it would later find application to the actual spectrum of physics."

There have been three main ways to formulate spectral theory, each of which find use in different domains. After Hilbert's initial formulation, the later development of abstract Hilbert spaces and the spectral theory of single normal operators on them were well suited to the requirements of physics, exemplified by the work of von Neumann. The further theory built on this to address Banach algebras in general. This development leads to the Gelfand representation, which covers the commutative case, and further into non-commutative harmonic analysis.

The difference can be seen in making the connection with Fourier analysis. The Fourier transform on the real line is in one sense the spectral theory of differentiation as a differential operator. But for that to cover the phenomena one has already to deal with generalized eigenfunctions (for example, by means of a rigged Hilbert space). On the other hand it is simple to construct a group algebra, the spectrum of which captures the Fourier transform's basic properties, and this is carried out by means of Pontryagin duality.

One can also study the spectral properties of operators on Banach spaces. For example, compact operators on Banach spaces have many spectral properties similar to that of matrices.

Physical background

The background in the physics of vibrations has been explained in this way:

Spectral theory is connected with the investigation of localized vibrations of a variety of different objects, from atoms and molecules in chemistry to obstacles in acoustic waveguides. These vibrations have frequencies, and the issue is to decide when such localized vibrations occur, and how to go about computing the frequencies. This is a very complicated problem since every object has not only a fundamental tone but also a complicated series of overtones, which vary radically from one body to another.

Such physical ideas have nothing to do with the mathematical theory on a technical level, but there are examples of indirect involvement (see for example Mark Kac's question Can you hear the shape of a drum?). Hilbert's adoption of the term "spectrum" has been attributed to an 1897 paper of Wilhelm Wirtinger on Hill differential equation (by Jean Dieudonné), and it was taken up by his students during the first decade of the twentieth century, among them Erhard Schmidt and Hermann Weyl. The conceptual basis for Hilbert space was developed from Hilbert's ideas by Erhard Schmidt and Frigyes Riesz. It was almost twenty years later, when quantum mechanics was formulated in terms of the Schrödinger equation, that the connection was made to atomic spectra; a connection with the mathematical physics of vibration had been suspected before, as remarked by Henri Poincaré, but rejected for simple quantitative reasons, absent an explanation of the Balmer series. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous, rather than being an object of Hilbert's spectral theory.

A definition of spectrum

Main article: Spectrum (functional analysis)

Consider a bounded linear transformation T defined everywhere over a general Banach space. We form the transformation:

$${\displaystyle R_{\zeta }={\left(\zeta I-T\right)}^{-1}.}$$

Here I is the identity operator and ζ is a complex number. The inverse of an operator T, that is T⁻¹, is defined by:

$${\displaystyle T{T}^{-1}=T^{-1}T=I.}$$

If the inverse exists, T is called regular. If it does not exist, T is called singular.

With these definitions, the resolvent set of T is the set of all complex numbers ζ such that R_ζ exists and is bounded. This set often is denoted as ρ(T). The spectrum of T is the set of all complex numbers ζ such that R_ζ fails to exist or is unbounded. Often the spectrum of T is denoted by σ(T). The function R_ζ for all ζ in ρ(T) (that is, wherever R_ζ exists as a bounded operator) is called the resolvent of T. The spectrum of T is therefore the complement of the resolvent set of T in the complex plane. Every eigenvalue of T belongs to σ(T), but σ(T) may contain non-eigenvalues.

This definition applies to a Banach space, but of course other types of space exist as well; for example, topological vector spaces include Banach spaces, but can be more general. On the other hand, Banach spaces include Hilbert spaces, and it is these spaces that find the greatest application and the richest theoretical results. With suitable restrictions, much can be said about the structure of the spectra of transformations in a Hilbert space. In particular, for self-adjoint operators, the spectrum lies on the real line and (in general) is a spectral combination of a point spectrum of discrete eigenvalues and a continuous spectrum.

Spectral theory briefly

Main article: Spectral theorem

See also: Eigenvalue, eigenvector and eigenspace

In functional analysis and linear algebra the spectral theorem establishes conditions under which an operator can be expressed in simple form as a sum of simpler operators. As a full rigorous presentation is not appropriate for this article, we take an approach that avoids much of the rigor and satisfaction of a formal treatment with the aim of being more comprehensible to a non-specialist.

This topic is easiest to describe by introducing the bra–ket notation of Dirac for operators. As an example, a very particular linear operator L might be written as a dyadic product:

${\displaystyle L=|k_1⟩⟨b_1|,}$

in terms of the "bra" ⟨b₁| and the "ket" |k₁⟩. A function f is described by a ket as |f ⟩. The function f(x) defined on the coordinates ${\displaystyle (x_1,x_2,x_3,\dots )}$ is denoted as

${\displaystyle f(x)=⟨x,f⟩}$

and the magnitude of f by

${\displaystyle \Vert f{\Vert }^2=⟨f,f⟩=\int ⟨f,x⟩⟨x,f⟩dx=\int f^{\ast }(x)f(x)dx}$

where the notation (*) denotes a complex conjugate. This inner product choice defines a very specific inner product space, restricting the generality of the arguments that follow.

The effect of L upon a function f is then described as:

${\displaystyle L|f⟩=|k_1⟩⟨b_1|f⟩}$

expressing the result that the effect of L on f is to produce a new function ${\displaystyle |k_1⟩}$ multiplied by the inner product represented by ${\displaystyle ⟨b_1|f⟩}$.

A more general linear operator L might be expressed as:

${\displaystyle L={\lambda }_1|e_1⟩⟨f_1|+{\lambda }_2|e_2⟩⟨f_2|+{\lambda }_3|e_3⟩⟨f_3|+\dots ,}$

where the ${\displaystyle \{{\lambda }_i\}}$ are scalars and the ${\displaystyle \{|e_i⟩\}}$ are a basis and the ${\displaystyle \{⟨f_i|\}}$ a reciprocal basis for the space. The relation between the basis and the reciprocal basis is described, in part, by:

${\displaystyle ⟨f_i|e_j⟩={\delta }_{ij}}$

If such a formalism applies, the ${\displaystyle \{{\lambda }_i\}}$ are eigenvalues of L and the functions ${\displaystyle \{|e_i⟩\}}$ are eigenfunctions of L. The eigenvalues are in the spectrum of L.

Some natural questions are: under what circumstances does this formalism work, and for what operators L are expansions in series of other operators like this possible? Can any function f be expressed in terms of the eigenfunctions (are they a Schauder basis) and under what circumstances does a point spectrum or a continuous spectrum arise? How do the formalisms for infinite-dimensional spaces and finite-dimensional spaces differ, or do they differ? Can these ideas be extended to a broader class of spaces? Answering such questions is the realm of spectral theory and requires considerable background in functional analysis and matrix algebra.

Resolution of the identity

See also: Borel functional calculus § Resolution of the identity

This section continues in the rough and ready manner of the above section using the bra–ket notation, and glossing over the many important details of a rigorous treatment. A rigorous mathematical treatment may be found in various references. In particular, the dimension n of the space will be finite.

Using the bra–ket notation of the above section, the identity operator may be written as:

${\displaystyle I=\sum _{i=1}^n|e_i⟩⟨f_i|}$

where it is supposed as above that ${\displaystyle \{|e_i⟩\}}$ are a basis and the ${\displaystyle \{⟨f_i|\}}$ a reciprocal basis for the space satisfying the relation:

${\displaystyle ⟨f_i|e_j⟩={\delta }_{ij}.}$

This expression of the identity operation is called a representation or a resolution of the identity. This formal representation satisfies the basic property of the identity:

${\displaystyle I^k=I}$

valid for every positive integer k.

Applying the resolution of the identity to any function in the space ${\displaystyle |\psi ⟩}$, one obtains:

${\displaystyle I|\psi ⟩=|\psi ⟩=\sum _{i=1}^n|e_i⟩⟨f_i|\psi ⟩=\sum _{i=1}^n{c}_i|e_i⟩}$

which is the generalized Fourier expansion of ψ in terms of the basis functions { e_i }. Here ${\displaystyle c_i=⟨f_i|\psi ⟩}$.

Given some operator equation of the form:

${\displaystyle O|\psi ⟩=|h⟩}$

with h in the space, this equation can be solved in the above basis through the formal manipulations:

${\displaystyle O|\psi ⟩=\sum _{i=1}^n{c}_i\left(O|e_i⟩\right)=\sum _{i=1}^n|e_i⟩⟨f_i|h⟩,}$

${\displaystyle ⟨f_j|O|\psi ⟩=\sum _{i=1}^n{c}_i⟨f_j|O|e_i⟩=\sum _{i=1}^n⟨f_j|e_i⟩⟨f_i|h⟩=⟨f_j|h⟩,\mathrm{\forall }j}$

which converts the operator equation to a matrix equation determining the unknown coefficients c_j in terms of the generalized Fourier coefficients ${\displaystyle ⟨f_j|h⟩}$ of h and the matrix elements ${\displaystyle O_{ji}=⟨f_j|O|e_i⟩}$ of the operator O.

The role of spectral theory arises in establishing the nature and existence of the basis and the reciprocal basis. In particular, the basis might consist of the eigenfunctions of some linear operator L:

${\displaystyle L|e_i⟩={\lambda }_i|e_i⟩;}$

with the { λ_i } the eigenvalues of L from the spectrum of L. Then the resolution of the identity above provides the dyad expansion of L:

${\displaystyle LI=L=\sum _{i=1}^{n}L|e_i⟩⟨f_i|=\sum _{i=1}^n{\lambda }_i|e_i⟩⟨f_i|.}$

Resolvent operator

Main article: Resolvent formalism

See also: Green's function and Dirac delta function

Using spectral theory, the resolvent operator R:

${\displaystyle R=(\lambda I-L{)}^{-1},}$

can be evaluated in terms of the eigenfunctions and eigenvalues of L, and the Green's function corresponding to L can be found.

Applying R to some arbitrary function in the space, say ${\displaystyle \phi }$,

${\displaystyle R|\phi ⟩=(\lambda I-L{)}^{-1}|\phi ⟩=\sum _{i=1}^n\frac{1}{\lambda -{\lambda }_i}|e_i⟩⟨f_i|\phi ⟩.}$

This function has poles in the complex λ-plane at each eigenvalue of L. Thus, using the calculus of residues:

${\displaystyle \frac{1}{2\pi i}{\oint }_{C}R|\phi ⟩d\lambda =-\sum _{i=1}^n|e_i⟩⟨f_i|\phi ⟩=-|\phi ⟩,}$

where the line integral is over a contour C that includes all the eigenvalues of L.

Suppose our functions are defined over some coordinates {x_j}, that is:

${\displaystyle ⟨x,\phi ⟩=\phi (x_1,x_2,...).}$

Introducing the notation

${\displaystyle ⟨x,y⟩=\delta (x-y),}$

where δ(x − y) = δ(x₁ − y₁, x₂ − y₂, x₃ − y₃, ...) is the Dirac delta function, we can write

${\displaystyle ⟨x,\phi ⟩=\int ⟨x,y⟩⟨y,\phi ⟩dy.}$

Then:

${\displaystyle \begin{array}{rl}⟨x,\frac{1}{2\pi i}{\oint }_C\frac{\phi }{\lambda I-L}d\lambda ⟩& =\frac{1}{2\pi i}{\oint }_{C}d\lambda ⟨x,\frac{\phi }{\lambda I-L}⟩\\ & =\frac{1}{2\pi i}{\oint }_{C}d\lambda \int dy⟨x,\frac{y}{\lambda I-L}⟩⟨y,\phi ⟩\end{array}}$

The function G(x, y; λ) defined by:

${\displaystyle \begin{array}{rl}G(x,y;\lambda )& =⟨x,\frac{y}{\lambda I-L}⟩\\ & =\sum _{i=1}^n\sum _{j=1}^n⟨x,e_i⟩⟨f_i,\frac{e_j}{\lambda I-L}⟩⟨f_j,y⟩\\ & =\sum _{i=1}^n\frac{⟨x,e_i⟩⟨f_i,y⟩}{\lambda -{\lambda }_i}\\ & =\sum _{i=1}^n\frac{e_i(x)f_i^{\ast }(y)}{\lambda -{\lambda }_i},\end{array}}$

is called the Green's function for operator L, and satisfies:

${\displaystyle \frac{1}{2\pi i}{\oint }_{C}G(x,y;\lambda )d\lambda =-\sum _{i=1}^n⟨x,e_i⟩⟨f_i,y⟩=-⟨x,y⟩=-\delta (x-y).}$

Operator equations

See also: Spectral theory of ordinary differential equations and Integral equation

Consider the operator equation:

${\displaystyle (O-\lambda I)|\psi ⟩=|h⟩;}$

in terms of coordinates:

${\displaystyle \int ⟨x,(O-\lambda I)y⟩⟨y,\psi ⟩dy=h(x).}$

A particular case is λ = 0.

The Green's function of the previous section is:

${\displaystyle ⟨y,G(\lambda )z⟩=⟨y,(O-\lambda I{)}^{-1}z⟩=G(y,z;\lambda ),}$

and satisfies:

${\displaystyle \int ⟨x,(O-\lambda I)y⟩⟨y,G(\lambda )z⟩dy=\int ⟨x,(O-\lambda I)y⟩⟨y,(O-\lambda I{)}^{-1}z⟩dy=⟨x,z⟩=\delta (x-z).}$

Using this Green's function property:

${\displaystyle \int ⟨x,(O-\lambda I)y⟩G(y,z;\lambda )dy=\delta (x-z).}$

Then, multiplying both sides of this equation by h(z) and integrating:

${\displaystyle \int dzh(z)\int dy⟨x,(O-\lambda I)y⟩G(y,z;\lambda )=\int dy⟨x,(O-\lambda I)y⟩\int dzh(z)G(y,z;\lambda )=h(x),}$

which suggests the solution is:

${\displaystyle \psi (x)=\int h(z)G(x,z;\lambda )dz.}$

That is, the function ψ(x) satisfying the operator equation is found if we can find the spectrum of O, and construct G, for example by using:

${\displaystyle G(x,z;\lambda )=\sum _{i=1}^n\frac{e_i(x)f_i^{\ast }(z)}{\lambda -{\lambda }_i}.}$

There are many other ways to find G, of course. See the articles on Green's functions and on Fredholm integral equations. It must be kept in mind that the above mathematics is purely formal, and a rigorous treatment involves some pretty sophisticated mathematics, including a good background knowledge of functional analysis, Hilbert spaces, distributions and so forth. Consult these articles and the references for more detail.

Spectral theorem and Rayleigh quotient

Optimization problems may be the most useful examples about the combinatorial significance of the eigenvalues and eigenvectors in symmetric matrices, especially for the Rayleigh quotient with respect to a matrix M.

Theorem Let M be a symmetric matrix and let x be the non-zero vector that maximizes the Rayleigh quotient with respect to M. Then, x is an eigenvector of M with eigenvalue equal to the Rayleigh quotient. Moreover, this eigenvalue is the largest eigenvalue of M.

Proof Assume the spectral theorem. Let the eigenvalues of M be ${\displaystyle {\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_n}$. Since the ${\displaystyle \{v_i\}}$ form an orthonormal basis, any vector x can be expressed in this basis as

${\displaystyle x=\sum _i{v}_i^{T}x{v}_i}$

The way to prove this formula is pretty easy. Namely,

${\displaystyle \begin{array}{rl}{v}_j^T\sum _i{v}_i^{T}x{v}_i=& \sum _i{v}_i^{T}x{v}_j^T{v}_i\\ =& (v_j^{T}x)v_j^T{v}_j\\ =& v_j^{T}x\end{array}}$

evaluate the Rayleigh quotient with respect to x:

${\displaystyle \begin{array}{rl}{x}^{T}Mx=& {\left(\sum _i(v_i^{T}x)v_i\right)}^{T}M\left(\sum _j(v_j^{T}x)v_j\right)\\ =& \left(\sum _i(v_i^{T}x)v_i^T\right)\left(\sum _j(v_j^{T}x)v_j{\lambda }_j\right)\\ =& \sum _{i,j}(v_i^{T}x)v_i^T(v_j^{T}x)v_j{\lambda }_j\\ =& \sum _j(v_j^{T}x)(v_j^{T}x){\lambda }_j\\ =& \sum _j(v_j^{T}x{)}^2{\lambda }_j\le {\lambda }_n\sum _j(v_j^{T}x{)}^2\\ =& {\lambda }_n{x}^{T}x,\end{array}}$

where we used Parseval's identity in the last line. Finally we obtain that

${\displaystyle \frac{x^{T}Mx}{x^{T}x}\le {\lambda }_n}$

so the Rayleigh quotient is always less than ${\displaystyle {\lambda }_n}$.

Peripatric speciation

From Wikipedia, the free encyclopedia

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

 

Figure 1: Peripatric speciation

 

Figure 2: Centrifugal speciation

Diagrams representing the process of peripatric and centrifugal speciation. In peripatry, a small population becomes isolated on the periphery of the central population evolving reproductive isolation (blue) due to reduced gene flow. In centrifugal speciation, an original population (green) range expands and contracts, leaving an isolated fragment population behind. The central population (changed to blue) evolves reproductive isolation in contrast to peripatry.

Peripatric speciation is a mode of speciation in which a new species is formed from an isolated peripheral population. Since peripatric speciation resembles allopatric speciation, in that populations are isolated and prevented from exchanging genes, it can often be difficult to distinguish between them. Nevertheless, the primary characteristic of peripatric speciation proposes that one of the populations is much smaller than the other. The terms peripatric and peripatry are often used in biogeography, referring to organisms whose ranges are closely adjacent but do not overlap, being separated where these organisms do not occur—for example on an oceanic island compared to the mainland. Such organisms are usually closely related (e.g. sister species); their distribution being the result of peripatric speciation.

The concept of peripatric speciation was first outlined by the evolutionary biologist Ernst Mayr in 1954. Since then, other alternative models have been developed such as centrifugal speciation, that posits that a species' population experiences periods of geographic range expansion followed by shrinking periods, leaving behind small isolated populations on the periphery of the main population. Other models have involved the effects of sexual selection on limited population sizes. Other related models of peripherally isolated populations based on chromosomal rearrangements have been developed such as budding speciation and quantum speciation.

The existence of peripatric speciation is supported by observational evidence and laboratory experiments. Scientists observing the patterns of a species biogeographic distribution and its phylogenetic relationships are able to reconstruct the historical process by which they diverged. Further, oceanic islands are often the subject of peripatric speciation research due to their isolated habitats—with the Hawaiian Islands widely represented in much of the scientific literature.

History

Main article: History of speciation

Peripatric speciation was originally proposed by Ernst Mayr in 1954, and fully theoretically modeled in 1982. It is related to the founder effect, where small living populations may undergo selection bottlenecks. The founder effect is based on models that suggest peripatric speciation can occur by the interaction of selection and genetic drift, which may play a significant role. Mayr first conceived of the idea by his observations of kingfisher populations in New Guinea and its surrounding islands. Tanysiptera galatea was largely uniform in morphology on the mainland, but the populations on the surrounding islands differed significantly—referring to this pattern as "peripatric". This same pattern was observed by many of Mayr's contemporaries at the time such as by E. B. Ford's studies of Maniola jurtina. Around the same time, the botanist Verne Grant developed a model of quantum speciation very similar to Mayr's model in the context of plants.

In what has been called Mayr's genetic revolutions, he postulated that genetic drift played the primary role that resulted in this pattern. Seeing that a species cohesion is maintained by conservative forces such as epistasis and the slow pace of the spread of favorable alleles in a large population (based heavily on J. B. S. Haldane's calculations), he reasoned that speciation could only take place in which a population bottleneck occurred. A small, isolated, founder population could be established on an island for example. Containing less genetic variation from the main population, shifts in allele frequencies may occur from different selection pressures. This to further changes in the network of linked loci, driving a cascade of genetic change, or a "genetic revolution"—a large-scale reorganization of the entire genome of the peripheral population. Mayr did recognize that the chances of success were incredibly low and that extinction was likely; though noting that some examples of successful founder populations existed at the time.

Shortly after Mayr, William Louis Brown, Jr. proposed an alternative model of peripatric speciation in 1957 called centrifugal speciation. In 1976 and 1980, the Kaneshiro model of peripatric speciation was developed by Kenneth Y. Kaneshiro which focused on sexual selection as a driver for speciation during population bottlenecks.

Models

Peripatric

Peripatric speciation models are identical to models of vicariance (allopatric speciation). Requiring both geographic separation and time, speciation can result as a predictable byproduct. Peripatry can be distinguished from allopatric speciation by three key features:

• The size of the isolated population
• Strong selection caused by the dispersal and colonization of novel environments,
• The effects of genetic drift on small populations.

The size of a population is important because individuals colonizing a new habitat likely contain only a small sample of the genetic variation of the original population. This promotes divergence due to strong selective pressures, leading to the rapid fixation of an allele within the descendant population. This gives rise to the potential for genetic incompatibilities to evolve. These incompatibilities cause reproductive isolation, giving rise to—sometimes rapid—speciation events. Furthermore, two important predictions are invoked, namely that geological or climatic changes cause populations to become locally fragmented (or regionally when considering allopatric speciation), and that an isolated population's reproductive traits evolve enough as to prevent interbreeding upon potential secondary contact.

The peripatric model results in, what have been called, progenitor-derivative species pairs, whereby the derivative species (the peripherally isolated population)—geographically and genetically isolated from the progenitor species—diverges. A specific phylogenetic signature results from this mode of speciation: the geographically widespread progenitor species becomes paraphyletic (thereby becoming a paraspecies), with respect to the derivative species (the peripheral isolate). The concept of a paraspecies is therefore a logical consequence of the evolutionary species concept, by which one species gives rise to a daughter species. It is thought that the character traits of the peripherally isolated species become apomorphic, while the central population remains pleisomorphic.

Modern cladistic methods have developed definitions that have incidentally removed derivative species by defining clades in a way that assumes that when a speciation event occurs, the original species no longer exists, while two new species arise; this is not the case in peripatric speciation. Mayr warned against this, as it causes a species to lose their classification status. Loren H. Rieseberg and Luc Brouillet recognized the same dilemma in plant classification.

Quantum and budding speciation

The botanist Verne Grant proposed the term quantum speciation that combined the ideas of J. T. Gulick (his observation of the variation of species in semi-isolation), Sewall Wright (his models of genetic drift), Mayr (both his peripatric and genetic revolution models), and George Gaylord Simpson (his development of the idea of quantum evolution). Quantum speciation is a rapid process with large genotypic or phenotypic effects, whereby a new, cross-fertilizing plant species buds off from a larger population as a semi-isolated peripheral population. Interbreeding and genetic drift takes place due to the reduced population size, driving changes to the genome that would most likely result in extinction (due to low adaptive value). In rare instances, chromosomal traits with adaptive value may arise, resulting in the origin of a new, derivative species. Evidence for the occurrence of this type of speciation has been found in several plant species pairs: Layia discoidea and L. glandulosa, Clarkia lingulata and C. biloba, and Stephanomeria malheurensis and S. exigua ssp. coronaria.

A closely related model of peripatric speciation is called budding speciation—largely applied in the context of plant speciation. The budding process, where a new species originates at the margins of an ancestral range, is thought to be common in plants—especially in progenitor-derivative species pairs.

Centrifugal speciation

William Louis Brown, Jr. proposed an alternative model of peripatric speciation in 1957 called centrifugal speciation. This model contrasts with peripatric speciation by virtue of the origin of the genetic novelty that leads to reproductive isolation. A population of a species experiences periods of geographic range expansion followed by periods of contraction. During the contraction phase, fragments of the population become isolated as small refugial populations on the periphery of the central population. Because of the large size and potentially greater genetic variation within the central population, mutations arise more readily. These mutations are left in the isolated peripheral populations, promoting reproductive isolation. Consequently, Brown suggested that during another expansion phase, the central population would overwhelm the peripheral populations, hindering speciation. However, if the species finds a specialized ecological niche, the two may coexist. The phylogenetic signature of this model is that the central population becomes derived, while the peripheral isolates stay pleisomorphic—the reverse of the general model. In contrast to centrifugal speciation, peripatric speciation has sometimes been referred to as centripetal speciation (see figures 1 and 2 for a contrast). Centrifugal speciation has been largely ignored in the scientific literature, often dominated by the traditional model of peripatric speciation. Despite this, Brown cited a wealth of evidence to support his model, of which has not yet been refuted.

Peromyscus polionotus and P. melanotis (the peripherally isolated species from the central population of P. maniculatus) arose via the centrifugal speciation model. Centrifugal speciation may have taken place in tree kangaroos, South American frogs (Ceratophrys), shrews (Crocidura), and primates (Presbytis melalophos). John C. Briggs associates centrifugal speciation with centers of origin, contending that the centrifugal model is better supported by the data, citing species patterns from the proposed 'center of origin' within the Indo-West Pacific

Kaneshiro model

In the Kaneshiro model, a sample of a larger population results in an isolated population with less males containing attractive traits. Over time, choosy females are selected against as the population increases. Sexual selection drives new traits to arise (green), reproductively isolating the new population from the old one (blue).

When a sexual species experiences a population bottleneck—that is, when the genetic variation is reduced due to small population size—mating discrimination among females may be altered by the decrease in courtship behaviors of males. Sexual selection pressures may become weakened by this in an isolated peripheral population, and as a by-product of the altered mating recognition system, secondary sexual traits may appear. Eventually, a growth in population size paired with novel female mate preferences will give rise to reproductive isolation from the main population-thereby completing the peripatric speciation process. Support for this model comes from experiments and observation of species that exhibit asymmetric mating patterns such as the Hawaiian Drosophila species or the Hawaiian cricket Laupala. However, this model has not been entirely supported by experiments, and therefore, it may not represent a plausible process of peripatric speciation that takes place in nature.

Evidence

See also: Allopatric speciation § Observational evidence

Observational evidence and laboratory experiments support the occurrence of peripatric speciation. Islands and archipelagos are often the subject of speciation studies in that they represent isolated populations of organisms. Island species provide direct evidence of speciation occurring peripatrically in such that, "the presence of endemic species on oceanic islands whose closest relatives inhabit a nearby continent" must have originated by a colonization event. Comparative phylogeography of oceanic archipelagos shows consistent patterns of sequential colonization and speciation along island chains, most notably on the Azores islands, Canary Islands, Society Islands, Marquesas Islands, Galápagos Islands, Austral Islands, and the Hawaiian Islands—all of which express geological patterns of spatial isolation and, in some cases, linear arrangement. Peripatric speciation also occurs on continents, as isolation of small populations can occur through various geographic and dispersion events. Laboratory studies have been conducted where populations of Drosophila, for example, are separated from one another and evolve in reproductive isolation.

Hawaiian archipelago

Colonization events of species from the genus Cyanea (green) and species from the genus Drosophila (blue) on the Hawaiian island chain. Islands age from left to right, (Kauai being the oldest and Hawaii being the youngest). Speciation arises peripatrically as they spatiotemporally colonize new islands along the chain. Lighter blue and green indicate colonization in the reverse direction from young-to-old.

 

A map of the Hawaiian archipelago showing the colonization routes of Theridion grallator superimposed. Purple lines indicate colonization occurring in conjunction with island age where light purple indicates backwards colonization. T. grallator is not present on Kauai or Niihau so colonization may have occurred from there, or the nearest continent.

 

The sequential colonization and speciation of the ‘Elepaio subspecies along the Hawaiian island chain.

Drosophila species on the Hawaiian archipelago have helped researchers understand speciation processes in great detail. It is well established that Drosophila has undergone an adaptive radiation into hundreds of endemic species on the Hawaiian island chain; originating from a single common ancestor (supported from molecular analysis). Studies consistently find that colonization of each island occurred from older to younger islands, and in Drosophila, speciating peripatrically at least fifty percent of the time. In conjunction with Drosophila, Hawaiian lobeliads (Cyanea) have also undergone an adaptive radiation, with upwards of twenty-seven percent of extant species arising after new island colonization—exemplifying peripatric speciation—once again, occurring in the old-to-young island direction.

Other endemic species in Hawaii also provide evidence of peripatric speciation such as the endemic flightless crickets (Laupala). It has been estimated that, "17 species out of 36 well-studied cases of [Laupala] speciation were peripatric". Plant species in genera's such as Dubautia, Wilkesia, and Argyroxiphium have also radiated along the archipelago. Other animals besides insects show this same pattern such as the Hawaiian amber snail (Succinea caduca), and ‘Elepaio flycatchers.

Tetragnatha spiders have also speciated peripatrically on the Hawaiian islands, Numerous arthropods have been documented existing in patterns consistent with the geologic evolution of the island chain, in such that, phylogenetic reconstructions find younger species inhabiting the geologically younger islands and older species inhabiting the older islands (or in some cases, ancestors date back to when islands currently below sea level were exposed). Spiders such as those from the genus Orsonwelles exhibit patterns compatible with the old-to-young geology. Other endemic genera such as Argyrodes have been shown to have speciated along the island chain. Pagiopalus, Pedinopistha, and part of the family Thomisidae have adaptively radiated along the island chain, as well as the wolf spider family, Lycosidae.

A host of other Hawaiian endemic arthropod species and genera have had their speciation and phylogeographical patterns studied: the Drosophila grimshawi species complex, damselflies (Megalagrion xanthomelas and Megalagrion pacificum), Doryonychus raptor, Littorophiloscia hawaiiensis, Anax strenuus, Nesogonia blackburni, Theridion grallator, Vanessa tameamea, Hyalopeplus pellucidus, Coleotichus blackburniae, Labula, Hawaiioscia, Banza (in the family Tettigoniidae), Caconemobius, Eupethicea, Ptycta, Megalagrion, Prognathogryllus, Nesosydne, Cephalops, Trupanea, and the tribe Platynini—all suggesting repeated radiations among the islands.

Other islands

Phylogenetic studies of a species of crab spider (Misumenops rapaensis) in the genus Thomisidae located on the Austral Islands have established the, "sequential colonization of [the] lineage down the Austral archipelago toward younger islands". M. rapaensis has been traditionally thought of as a single species; whereas this particular study found distinct genetic differences corresponding to the sequential age of the islands. The figwart plant species Scrophularia lowei is thought to have arisen through a peripatric speciation event, with the more widespread mainland species, Scrophularia arguta dispersing to the Macaronesian islands. Other members of the same genus have also arisen by single colonization events between the islands.

Species patterns on continents

The southern chestnut-tailed antbird, Sciaphylax hemimelaena

 

Satellite image of the Noel Kempff Mercado National Park (outlined in green) in Bolivia, South America. The white arrow indicates the location of the isolated forest fragment.

The occurrence of peripatry on continents is more difficult to detect due to the possibility of vicariant explanations being equally likely. However, studies concerning the Californian plant species Clarkia biloba and C. lingulata strongly suggest a peripatric origin. In addition, a great deal of research has been conducted on several species of land snails involving chirality that suggests peripatry (with some authors noting other possible interpretations).

The chestnut-tailed antbird (Sciaphylax hemimelaena) is located within the Noel Kempff Mercado National Park (Serrania de Huanchaca) in Bolivia. Within this region exists a forest fragment estimated to have been isolated for 1000–3000 years. The population of S. hemimelaena antbirds that reside in the isolated patch express significant song divergence; thought to be an "early step" in the process of peripatric speciation. Further, peripheral isolation "may partly explain the dramatic diversification of suboscines in Amazonia".

The montane spiny throated reed frog species complex (genus: Hyperolius) originated through occurrences of peripatric speciation events. Lucinda P. Lawson maintains that the species' geographic ranges within the Eastern Afromontane Biodiversity Hotspot support a peripatric model that is driving speciation; suggesting that this mode of speciation may play a significant role in "highly fragmented ecosystems".

In a study of the phylogeny and biogeography of the land snail genus Monacha, the species M. ciscaucasica is thought to have speciated peripatrically from a population of M. roseni. In addition, M. claussi consists of a small population located on the peripheral of the much larger range of M. subcarthusiana suggesting that it also arose by peripatric speciation.

Foliage and cones of Picea mariana

 

Foliage and cones of Picea rubens

Red spruce (Picea rubens) has arisen from an isolated population of black spruce (Picea mariana). During the Pleistocene, a population of black spruce became geographically isolated, likely due to glaciation. The geographic range of the black spruce is much larger than the red spruce. The red spruce has significantly lower genetic diversity in both its DNA and its mitochondrial DNA than the black spruce. Furthermore, the genetic variation of the red spruce has no unique mitochondrial haplotypes, only subsets of those in the black spruce; suggesting that the red spruce speciated peripatrically from the black spruce population. It is thought that the entire genus Picea in North America has diversified by the process of peripatric speciation, as numerous pairs of closely related species in the genus have smaller southern population ranges; and those with overlapping ranges often exhibit weak reproductive isolation.

Using a phylogeographic approach paired with ecological niche models (i.e. prediction and identification of expansion and contraction species ranges into suitable habitats based on current ecological niches, correlated with fossil and molecular data), researchers found that the prairie dog species Cynomys mexicanus speciated peripatrically from Cynomys ludovicianus approximately 230,000 years ago. North American glacial cycles promoted range expansion and contraction of the prairie dogs, leading to the isolation of a relic population in a refugium located in the present day Coahuila, Mexico. This distribution and paleobiogeographic pattern correlates with other species expressing similar biographic range patterns such as with the Sorex cinereus complex.

Laboratory experiments

Species	Replicates	Year
Drosophila adiastola	1	1979
Drosophila silvestris	1	1980
Drosophila pseudoobscura	8	1985
Drosophila simulans	8	1985
Musca domestica	6	1991
Drosophila pseudoobscura	42	1993
Drosophila melanogaster	50	1998
Drosophila melanogaster	19; 19	1999
Drosophila grimshawi	1	N/A

 

See also: Laboratory experiments of speciation

Peripatric speciation has been researched in both laboratory studies and nature. Jerry Coyne and H. Allen Orr in Speciation suggest that most laboratory studies of allopatric speciation are also examples of peripatric speciation due to their small population sizes and the inevitable divergent selection that they undergo. Much of the laboratory research concerning peripatry is inextricably linked to founder effect research. Coyne and Orr conclude that selection's role in speciation is well established, whereas genetic drift's role is unsupported by experimental and field data—suggesting that founder-effect speciation does not occur. Nevertheless, a great deal of research has been conducted on the matter, and one study conducted involving bottleneck populations of Drosophila pseudoobscura found evidence of isolation after a single bottleneck.

The table is a non-exhaustive table of laboratory experiments focused explicitly on peripatric speciation. Most of the studies also conducted experiments on vicariant speciation as well. The "replicates" column signifies the number of lines used in the experiment—that is, how many independent populations were used (not the population size or the number of generations performed).