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{{citation style|date=February 2014}}
[[File:Saddle point.png|thumb|right|200px|A saddle point on the graph of z=x<sup>2</sup>−y<sup>2</sup> (in red)]]
The '''unified neutral theory of biodiversity and biogeography''' (here '''"Unified Theory"''' or '''"UNTB"''') is a hypothesis and the title of a monograph by [[ecology|ecologist]] [[Stephen Hubbell]].<ref name=Hubbell2001>{{Cite web
[[File:Saddle pt.jpg|thumb|150px|right|Saddle point between two hills (the intersection of the figure-eight <math>z</math>-contour)]]
| author = Hubbell, SP
In [[mathematics]], a '''saddle point''' is a point in the [[domain (mathematics)|domain]] of a [[function (mathematics)|function]] that is a [[stationary point]] but not a [[local extremum]]. The name derives from the fact that the prototypical example in two dimensions is a [[surface]] that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a [[saddle]] or a [[mountain pass]]. In terms of [[contour line]]s, a saddle point in two dimensions gives rise to a contour that appears to intersect itself.
| year = 2001
| title = The Unified Neutral Theory of Biodiversity and Biogeography (MPB-32)
| url = https://pup.princeton.edu/chapters/s7105.html
}}</ref> The hypothesis aims to explain the diversity and relative abundance of species in ecological communities, although like other neutral theories of ecology, Hubbell's hypothesis assumes that the differences between members of an ecological community of [[trophic level|trophically]] similar species are "neutral," or irrelevant to their success.  This implies that biodiversity arises at random, as each species follows a [[random walk]].<ref name="McGill2003">{{cite doi|10.1038/nature01583}}</ref> The hypothesis has sparked controversy,<ref>Science Daily (2014). [http://www.sciencedaily.com/releases/2014/05/140527101346.htm New biodiversity study throws out controversial scientific theory. May 27]</ref><ref>Connolly S.R., MacNeil M.A., Caley M.J., Knowlton N., Cripps E., Hisano M., Thibaut L.M., Bhattacharya B.D., Benedetti-Cecchi L., Brainard R.E., Brandt A., Bulleri F., Ellingsen K.E., K (2014). Commonness and rarity in the marine biosphere.  Proceedings of the National Academy of Sciences. [http://www.pnas.org/content/early/2014/05/21/1406664111.abstract Abstract], [http://www.pnas.org/content/early/2014/05/21/1406664111.full.pdf+html paper].</ref> and some authors consider it a more complex version of other null models that fit the data better.<ref name="Nee2003"/>


''Neutrality'' means that at a given [[trophic level]] in a [[food web]], species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis.<ref name=Hubbell2005>{{cite doi|10.1666/0094-8373(2005)031[0122:TNTOBA]2.0.CO;2}}</ref>  This can be considered a [[null hypothesis]] to [[ecological niche|niche theory]].  Hubbell built on earlier neutral concepts, including [[Robert MacArthur|MacArthur]] & [[E.O. Wilson|Wilson]]'s theory of [[island biogeography]]<ref name="Hubbell2001"/> and [[Steven Jay Gould|Gould]]'s concepts of symmetry and null models.<ref name=Hubbell2005/>
== Mathematical Discussion ==


An ''ecological community'' is a group of trophically similar, [[sympatric]] [[species]] that actually or potentially compete in a local area for the same or similar resources.<ref name=Hubbell2001/>  Under the Unified Theory, complex [[ecological]] interactions are permitted among individuals of an ecological community (such as competition and cooperation), provided that all individuals obey the same rules. Asymmetric phenomena such as [[parasite|parasitism]] and [[predator|predation]] are ruled out by the terms of reference; but cooperative strategies such as swarming, and negative interaction such as competing for limited food or light are allowed (so long as all individuals behave in the same way).
A simple criterion for checking if a given stationary point of a real-valued function ''F''(''x'',''y'') of two real variables is a saddle point is to compute the function's [[Hessian matrix]] at that point: if the Hessian is [[Positive-definite matrix#Indefinite|indefinite]], then that point is a saddle point. For example, the Hessian matrix of the function <math>z=x^2-y^2</math> at the stationary point <math>(0, 0)</math> is the matrix
 
: <math>\begin{bmatrix}
The Unified Theory also makes predictions that have profound implications for the management of [[biodiversity]], especially the management of rare species.{{Citation needed|date=November 2007}}
2 & 0\\
 
0 & -2 \\
The theory predicts the existence of a fundamental biodiversity constant, conventionally written ''θ'', that appears to govern species richness on a wide variety of spatial and temporal scales.
\end{bmatrix}
 
==Saturation==
 
Although not strictly necessary for a neutral theory, many [[stochastic]] models of biodiversity assume a fixed, finite community size.  There are unavoidable physical constraints on the total number of individuals that can be packed into a given space (although space ''per se'' isn't necessarily a resource, it is often a useful surrogate variable for a limiting resource that is distributed over the landscape; examples would include [[sunlight]] or hosts, in the case of parasites). 
 
If a wide range of species is considered (say, [[giant sequoia]] trees and [[duckweed]], two species that have very different saturation densities), then the assumption of constant community size might not be very good, because density would be higher if the smaller species were monodominant.
 
However, because the Unified Theory refers only to communities of trophically similar, competing species, it is unlikely that population density will vary too widely from one place to another. 
 
Hubbell considers the fact that population densities are constant and interprets it as a general principle: ''large landscapes are always biotically saturated with individuals''.  Hubbell thus treats communities as being of a fixed number of individuals, usually denoted by ''J''.
 
Exceptions to the saturation principle include disturbed ecosystems such as the [[Serengeti]], where saplings are trampled by [[elephant]]s and [[Blue wildebeest]]s; or [[garden]]s, where certain species are systematically removed.
 
===Species abundances===
 
When abundance data on natural populations are collected, two observations are almost universal:
 
*  The most common species accounts for a substantial fraction of the individuals sampled;
*  A substantial fraction of the species sampled are very rare.  Indeed, a substantial fraction of the species sampled are singletons, that is, species which are sufficiently rare for only a single individual to have been sampled.
 
Such observations typically generate a large number of questions.  Why are the rare species rare? Why is the most abundant species so much more abundant than the median species abundance?
 
A non neutral explanation for the rarity of rare species might suggest that rarity is a result of poor adaptation to local conditions.  The UNTB implies that such considerations may be neglected from the perspective of population biology (because the explanation cited implies that the rare species behaves differently from the abundant species).
 
Species composition in any community will change randomly with time.  However, any  particular abundance structure will have an associated probability.  The UNTB  predicts that the probability of a community of ''J'' individuals composed of ''S''  distinct species with abundances <math>n_1</math> for species 1, <math>n_2</math> for  species 2, and so on up to <math>n_S</math> for species ''S'' is given by
 
:<math>
\Pr(n_1,n_2,\ldots,n_S| \theta, J)=
\frac{J!\theta^S}
{
  1^{\phi_1}2^{\phi_2}\cdots J^{\phi_J}
  \phi_1!\phi_2!\cdots\phi_J!
  \Pi_{k=1}^J(\theta+k-1)
}
</math>
 
where <math>\theta=2J\nu</math> is the fundamental biodiversity number (<math>\nu</math> is the speciation rate), and <math>\phi_i</math> is the number of species that have ''i'' individuals in the sample.
 
This equation shows that the UNTB implies a nontrivial dominance-diversity equilibrium between speciation and extinction.
 
As an example, consider a community with 10 individuals and three species "a", "b", and "c" with abundances 3, 6 and 1 respectively.  Then the formula above would allow us to assess the [[likelihood function|likelihood]] of different values of ''θ''.  There are thus ''S''&nbsp;=&nbsp;3 species and <math>\phi_1=\phi_3=\phi_6=1</math>, all other <math>\phi</math>'s being zero.  The formula would give
 
:<math>
\Pr(3,6,1| \theta,10)=
\frac{10!\theta^3}{
1^1\cdot 3^1\cdot 6^1
\cdot
1!1!1!
\cdot
\theta(\theta+1)(\theta+2)\cdots(\theta+9)}
</math>
 
which could be maximized to yield an estimate for ''θ'' (in practice, [[numerical methods]] are used).  The [[maximum likelihood]] estimate for ''θ'' is about 1.1478.
 
We could have labelled the species another way and counted the abundances being 1,3,6 instead (or 3,1,6, etc. etc.).  Logic tells us that the probability of observing a pattern of abundances will be the same observing any [[permutation]] of those abundances.  Here we would have
 
: <math>\Pr(3;3,6,1)=\Pr(3;1,3,6)=\Pr(3;3,1,6)</math>
 
and so on.
 
To account for this, it is helpful to consider only ranked abundances (that is, to sort the abundances before inserting into the formula).  A ranked dominance-diversity configuration is usually written as <math>\Pr(S;r_1,r_2,\ldots,r_s,0,\ldots,0)</math> where <math>r_i</math> is the abundance of the ''i''th most abundant species: <math>r_1</math> is the abundance of the most abundant, <math>r_2</math> the abundance of the second most abundant species, and so on.  For convenience, the expression is usually "padded" with enough zeros to ensure that there are ''J'' species (the zeros indicating that the extra species have zero abundance).
 
It is now possible to determine the [[Expected value|expected]] abundance of the ''i''th most abundant species:
 
:<math>
E(r_i)=\sum_{k=1}^C
r_i(k)\cdot \Pr(S;r_1,r_2,\ldots,r_s,0,\ldots,0)
</math>
 
where ''C'' is the total number of configurations, <math>r_i(k)</math> is the abundance of the ''i''th ranked species in the ''k''th configuration,  and <math>Pr(\ldots)</math> is the dominance-diversity probability.  This formula is difficult to manipulate mathematically, but relatively simple to simulate computationally.
 
The model discussed so far is a model of a regional community, which Hubbell calls the [[metacommunity]]. Hubbell also acknowledged that on a local scale, dispersal plays an important role. For example, seeds are more likely to come from nearby parents than from distant parents. Hubbell introduced the parameter m, which denotes the probability of immigration in the local community from the metacommunity. If m = 1, dispersal is unlimited; the local community is just a random sample from the metacommunity and the formulas above apply. If m < 1, however, dispersal is limited and the local community is a dispersal-limited sample from the metacommunity for which different formulas apply.
 
In a paper in [[Nature (journal)|Nature]] in 2003 [http://www.nature.com/nature/journal/v424/n6952/abs/nature01883.html], it is shown that <math>\langle \phi_n \rangle </math>, the expected number of species with abundance n, may be calculated by
 
:<math>
\theta\frac{J!}{n!(J-n)!}
\frac{\Gamma(\gamma)}{\Gamma(J+\gamma)}
\int_{y=0}^\gamma
\frac{\Gamma(n+y)}{\Gamma(1+y)}
\frac{\Gamma(J-n+\gamma-y)}{\Gamma(\gamma-y)}
\exp(-y\theta/\gamma)\,dy
</math>
 
where ''θ'' is the fundamental biodiversity number, ''J'' the community size,  <math>\Gamma</math> is the [[gamma function]], and <math>\gamma=(J-1)m/(1-m)</math>. This formula is however an approximation.  The correct formula is derived in a series of papers, reviewed and synthesized by Etienne & Alonso 2005 [http://www.blackwell-synergy.com/doi/abs/10.1111/j.1461-0248.2005.00817.x]:
 
:<math>
\frac{\theta }{(I)_{J}} {J \choose n}
\int_{0}^{1}(Ix)_{n}(I(1-x))_{J-n}\frac{(1-x)^{\theta -1}}{x}\,dx 
</math>
 
where <math>I = (J-1)*m/(1-m)</math> is a parameter that measures dispersal limitation.
 
<math>\langle \phi_n \rangle</math> is zero for ''n''&nbsp;>&nbsp;''J'', as there cannot be more species than individuals.
 
This formula is important because it allows a quick evaluation of the Unified Theory. It is not suitable for testing the theory. For this purpose, the approptiate likelihood function should be used. For the metacommunity this was given above. For the local community with dispersal limitation it is given by:
 
:<math>
\Pr(n_1,n_2,\ldots,n_S| \theta, m, J)=
\frac{J!}{\prod_{i=1}^{S}n_{i}
\prod_{j=1}^{J}\Phi_{j}!}\frac{\theta ^{S}}{(I)_{J}}
\sum_{A=S}^{J}K(\overrightarrow{D},A)\frac{I^{A}}{(\theta) _{A}}
</math>
 
Here, the <math>K(\overrightarrow{D},A)</math> for <math>A=S,...,J</math> are coefficients fully determined by the data, being defined as
 
:<math>
K(\overrightarrow{D},A):=\sum_{\{a_{1},...,a_{S}|\sum_{i=1}^{S}a_{i}=A\}}
\prod_{i=1}^{S}\frac{\overline{s}\left( n_{i},a_{i}\right) \overline{s}
\left( a_{i},1\right) }{\overline{s}\left( n_{i},1\right) }
</math>
 
This seemingly complicated formula involves [[Stirling number]]s and [[Pochhammer symbol]]s, but can be easily calculated. See [http://www.blackwellpublishing.com/products/journals/suppmat/ELE/ELE717/ELE717sm.htm]
 
An example of a species abundance curve can be found here: [http://www.sciam.com/article.cfm?articleID=000656C3-8F00-1CDA-B4A8809EC588EEDF].
 
==Stochastic modelling of species abundances under the UNTB==
 
UNTB distinguishes between a dispersal-limited local community of size <math>J</math> and a so-called metacommunity from which species can (re)immigrate and which acts as a [[heat bath]] to the local community. The distribution of species in the metacommunity is given by a dynamic equilibrium of speciation and extinction. Both community dynamics are modelled by appropriate [[urn problem|urn processes]], where each individual is represented by a ball with a color corresponding to its species. With a certain rate <math>r</math> randomly chosen individuals reproduce, i.e. add another ball of their own color to the urn. Since one basic assumption is saturation, this reproduction has to happen at the cost of another random individual from the urn which is removed. At a different rate <math>\mu</math> single individuals in the metacommunity are replaced by mutants of an entirely new species. Hubbell calls this simplified model for [[speciation]] a [[point mutation]], using the terminology of the [[Neutral theory of molecular evolution]]. The urn scheme for the metacommunity of <math>J_M</math> individuals is the following.
 
At each time step take one of the two possible actions :
 
# With probability <math>(1-\nu)</math> draw an individual at random and replace another random individual from the urn with a copy of the first one.
# With probability <math>\nu</math> draw an individual and replace it with an individual of a new species.
 
Note that the size <math>J_M</math> of the metacommunity does not change. Note also that this is a [[point process]] in time. The length of the time steps is distributed exponentially. For simplicity one can, however, assume that each time step is as long as the mean time between two changes which can be derived from the reproduction and mutation rates <math>r</math> and <math>\mu</math>. The probability <math>\nu</math> is given as <math>\nu=\mu/(r+\mu)</math>.
 
The species abundance distribution for this urn process is given by [[Ewens's sampling formula]] which was originally derived in 1972 for the distribution of [[allele]] under neutral mutations. The expected number <math>S_M(n)</math> of species in the metacommunity having exactly <math>n</math> individuals is [Vallade, Houchmandzadeh 2003 [http://prola.aps.org/abstract/PRE/v68/i6/e061902]
 
:<math>
S_M(n) = \frac{\theta}{n}\frac{\Gamma(J_M+1)\Gamma(J_M+\theta-n)}{\Gamma(J_M+1-n)\Gamma(J_M+\theta)}
</math>
 
where <math>\theta=(J_M-1)\nu/(1-\nu)\approx J_M\nu</math> is called the fundamental biodiversity number. For large
metacommunities and <math>n\ll J_M</math> one recovers the [[Logarithmic distribution|Fisher Log-Series]] as species distribution.
 
:<math>
S_M(n) \approx \frac{\theta}{n} \left( \frac{J_M}{J_M+\theta} \right)^n
</math>
 
The urn scheme for the local community of fixed size <math>J</math> is very similar to the one for the metacommunity.
 
At each time step take one of the two actions :
 
# With probability <math>(1-m)</math> draw an individual at random and replace another random individual from the urn with a copy of the first one.
# With probability <math>m</math> replace a random individual with an immigrant drawn from the metacommunity.
 
The metacommunity is changing on a much larger timescale and is assumed to be fixed during the evolution of the local community. The resulting distribution of species in the local community and expected values depend on four parameters, <math>J</math>, <math>J_M</math>, <math>\theta</math> and <math>m</math> (or <math>I</math>) and are derived in [Etienne, Alonso 2005 [http://www.blackwell-synergy.com/doi/full/10.1111/j.1461-0248.2005.00817.x], including several simplifying limit cases like the one presented in the previous section (there called <math>\langle\phi_n\rangle</math>). The parameter <math>m</math> is a dispersal parameter. If <math>m=1</math> then the local community is just a sample from the metacommunity. For <math>m=0</math> the local community is completely isolated from the metacommunity and all species will go extinct except one. This case has been analyzed by Hubbell himself [Hubbell 2001]. The case <math>0<m<1</math> is characterized by a [[unimodal]] species distribution in a [[Preston Diagram]] and often fitted by a [[log-normal distribution]]. This is understood as an intermediate state between domination of the most common species and a sampling from the metacommunity, where singleton species are most abundant. UNTB thus predicts that in dispersal limited communities rare species become even rarer. The log-normal distribution describes the maximum and the abundance of common species very well but underestimates the number of very rare species considerably which becomes only apparent for very large sample sizes [Hubbell 2001].
 
==Species-area relationships==
 
The Unified Theory unifies ''biodiversity'', as measured by species-abundance curves, with ''biogeography'', as measured by species-area curves. Species-area relationships show the rate at which species diversity increases with area.  The topic is of great interest to conservation biologists in the design of reserves, as it is often desired to harbour as many species as possible.
 
The most commonly encountered relationship is the power law given by
 
:<math>
S=cA^z</math>
 
where ''S'' is the number of species found, ''A'' is the area sampled, and ''c'' and ''z'' are constants.  This relationship, with different constants, has been found to fit a wide range of empirical data.
 
From the perspective of Unified Theory, it is convenient to consider ''S'' as a function of total community size  ''J''.  Then <math>S=kJ^z</math> for some constant ''k'', and if this relationship were exactly true, the species area line would be straight on log scales.  It is typically found that the curve is not straight, but the slope changes from being steep at small areas, shallower at intermediate areas, and steep at the largest areas.
 
The formula for species composition <!--derived above ('''not done this bit yet''')--> may be used to calculate the expected number of species present in a community under the assumptions of the Unified Theory.  In symbols
 
:<math>
E\left\{S|\theta,J\right\}=
\frac{\theta}{\theta  }+
\frac{\theta}{\theta+1}+
\frac{\theta}{\theta+2}+
\cdots +
\frac{\theta}{\theta+J-1}
</math>
</math>
which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point <math>(0, 0)</math> is a saddle point for the function <math>z=x^4-y^4,</math> but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite.


where θ is the fundamental biodiversity number.  This formula specifies the expected number of species sampled in a community of size ''J''. The last term, <math>\theta/(\theta+J-1)</math>, is the expected number of ''new'' species encountered when adding one new individual to the community.  This is an increasing function of θ and a decreasing function of ''J'', as expected.
In the most general terms, a '''saddle point''' for a [[smooth function]] (whose [[graph of a function|graph]] is a [[curve]], [[surface]] or [[hypersurface]]) is a stationary point such that the curve/surface/etc. in the [[neighborhood (mathematics)|neighborhood]] of that point is not entirely on any side of the [[tangent space]] at that point.
[[File:x cubed plot.svg|thumb|150px|The plot of ''y''&nbsp;=&nbsp;''x''<sup>3</sup> with a saddle point at 0]]


By making the substitution <math>J=\rho A</math> (see section on saturation above), then the expected number of species becomes <math>\Sigma\theta/(\theta+\rho A-1)</math>.
In one dimension, a saddle point is a [[Point (geometry)|point]] which is both a [[stationary point]] and a [[Inflection point|point of inflection]]. Since it is a point of inflection, it is not a [[local extremum]].
 
The formula above may be approximated to an [[integral]] giving
 
:<math>
S(\theta)=
1+\theta\ln\left(1+\frac{J-1}{\theta}\right).
</math>
 
This formulation is predicated on a random placement of individuals.
 
===Example===
 
Consider the following (synthetic) dataset, of 27 individuals:
 
a, a,a, a,a, a,a, a,a, a,b, b,b, b,c, c,c, c,d, d,d, d,e, f,g, h,i
 
There are thus 27 individuals of 9 species ("a" to "i") in the sample. Tabulating this would give:
 
  a  b  c  d  e  f  g  h  i
10  4  4  4  1  1  1  1  1
 
indicating that species "a" is the most abundant with 10 individuals and species "e" to "i" are singletons.  Tabulating the table gives:
 
species abundance    1    2    3    4    5    6    7    8    9    10
number of species    5    0    0    3    0    0    0    0    0    1
 
On the second row, the 5 in the first column means that five species, species "e" through "i", have abundance one.  The following two zeros in columns 2 and 3 mean that zero species have abundance 2 or 3.  The 3 in column 4 means that three species, species "b", "c", and "d", have abundance four.  The final 1 in column 10 means that one species, species "a", has abundance 10.
 
This type of dataset is typical in biodiversity studies.  Observe how more than half the biodiversity (as measured by species count) is due to singletons. 
 
For real datasets, the species abundances are binned into logarithmic categories, usually using base 2, which gives bins of  abundance 0-1, abundance 1-2, abundance 2-4, abundance 4-8, etc.  Such abundance classes are called ''octaves''; early developers of this concept included [[Frank W. Preston|F. W. Preston]] and histograms showing number of species as a function of abundance octave are known as [[Preston diagram]]s.
 
These bins are not mutually exclusive: a species with abundance 4, for example, could be considered as lying in the 2-4 abundance class or the 4-8 abundance class.  Species with an abundance of an exact power of 2 (i.e.  2,4,8,16, etc.) are conventionally considered as having 50% membership in the lower abundance class 50% membership in the upper class.  Such species are thus considered to be evenly split between the two adjacent classes (apart from singletons which are classified into the rarest category).  Thus in the example above, the Preston abundances would be
 
abundance class 1    1-2  2-4  4-8  8-16
species        5    0    1.5  1.5  1
 
The three species of abundance four thus appear, 1.5 in abundance class 2-4, and 1.5 in 4-8.
 
The above method of analysis cannot account for species that are unsampled: that is, species sufficiently rare to have been recorded zero times.  Preston diagrams are thus truncated at zero abundance. [[Frank W. Preston|Preston]] called this the ''veil line'' and noted that the cutoff point would move as more individuals are sampled.
 
==Dynamics under neutral hypothesis==
 
All biodiversity patterns previously described are related to time-independent quantities. However, for biodiversity evolution and species preservation, it is crucial to compare the dynamics of ecosystems with models (Leigh, 2007). An easily accessible index of the underlying evolution is the so-called species turnover distribution (STD), defined as the probability P(r,t) that the population of any species has varied by a fraction r after a given time t.
 
A neutral model that can analytically predict both the relative species abundance (RSA) at steady-state and the STD at time t has been presented in [http://www.nature.com/nature/journal/v444/n7121/abs/nature05320.html]. Within this framework the population of any species is represented by a continuous (random) variable x, whose evolution is governed by the following Langevin equation:
:<math>
\dot{x}=b-x/\tau+\sqrt{Dx}\xi(t)
</math>
where b is the immigration rate from a large regional community, <math>-x/\tau</math> represents competition for finite resources and D is related to demographic stochasticity; <math>\xi(t)</math> is a Gaussian white noise. The model can also be derived as a continuous approximation of a master equation, where birth and death rates are independent of species, and predicts that at steady-state the RSA is simply a gamma distribution.
 
From the exact time-dependent solution of the previous equation, one can exactly calculate the STD at time t under stationary conditions:
:<math>
P(r,t)=A\frac{\lambda+1}{\lambda}\frac{(e^{t/\tau})^{b/2D}}{1-e^{-t/\tau}}\left(\frac{\sinh(\frac{t}{2\tau})}{\lambda}\right)^{\frac{b}{D}+1}\left(\frac{4\lambda^2}{(\lambda+1)^2e^{t/\tau}-4\lambda}\right)^{\frac{b}{D}+\frac{1}{2}}.
</math>
This formula provides good fits of data collected in the Barro Colorado tropical forest from 1990 to 2000. From the best fit one can estimate <math>\tau</math> ~ 3500 years with a broad uncertainty due to the relative short time interval of the sample. This parameter can be interpreted as the relaxation time of the system, i.e. the time the system needs to recover from a perturbation of species distribution. In the same framework, the estimated mean species lifetime is very close to the fitted temporal scale <math>\tau</math>. This suggests that the neutral assumption could correspond to a scenario in which species originate and become extinct on the same timescales of fluctuations of the whole ecosystem.


== Testing the theory ==
== Other uses ==


The theory has provoked much controversy as it "abandons" the role of ecology when modelling ecosystems.<ref name="Ricklefs2006">{{cite doi|10.1890/0012-9658(2006)87[1424:TUNTOB]2.0.CO;2}}</ref> The theory has been criticized as it requires an equilibrium, yet climatic and geographical conditions are thought to change too frequently for this to be attained.<ref name="Ricklefs2006" />
In [[dynamical systems]], a ''saddle point'' is a [[periodic point]] whose [[stable manifold|stable]] and [[unstable manifold]]s have a [[dimension]] that is not zero. If the dynamic is given by a [[differentiable map]] ''f'' then a point is hyperbolic if and only if the differential of ''&fnof;'' <sup>''n''</sup> (where ''n'' is the period of the point) has no eigenvalue on the (complex) [[unit circle]] when computed at the point.
Tests on bird and tree abundance data demonstrate that the theory is usually a poorer match to the data than alternative null hypotheses that use fewer parameters (a log-normal model with two tunable parameters, compared to the neutral theory's three<ref name="Nee2003">{{cite doi|10.1016/S0169-5347(03)00196-4}}</ref>), and are thus more parsimonious.<ref name="McGill2003" />  The theory also fails to describe coral reef communities<ref name="Dornelas2006">{{cite doi|10.1038/nature04534}}</ref> and is a poor fit to data in intertidal communities.<ref name="Timothy Wootton2005">{{cite doi|10.1038/nature03211}}</ref> It also fails to explain why families of tropical trees have statistically highly correlated numbers of species in phylogenetically unrelated and geographically distant forest plots in Central and South America, Africa, and South East Asia <ref name="Ricklefs2012">{{cite DOI: 10.1126/science.1215182}}</ref>


== Application ==
In a two-player [[zero-sum (game theory)|zero sum]] game defined on a continuous space, the [[Nash equilibrium|equilibrium]] point is a saddle point.


While the theory has been heralded as a valuable tool for palaeontologists,<ref name=Hubbell2005/> little work has so far been done to test the theory against the fossil record.<ref name="Bonuso2007">{{cite doi|10.2110/palo.2007.S05}}</ref>
A saddle point is an element of the matrix which is both the largest element in its column and the smallest element in its row.


==See also==
For a second-order linear autonomous systems, a [[critical point (mathematics)|critical point]] is a saddle point if the [[Characteristic equation (calculus)|characteristic equation]] has one positive and one negative real eigenvalue.<ref>{{harvnb|von Petersdorff|2006}}</ref>
* [[Biodiversity Action Plan]]
* [[Warren Ewens]]
* [[Ewens's sampling formula]]
* [[Metabolic theory of ecology]]
* [[Neutral theory of molecular evolution]]


==External links==
== See also ==
* [http://www.sciam.com/article.cfm?articleID=000D401A-8D9D-1CDA-B4A8809EC588EEDF Scientific American Interview with Steve Hubbell]
* [[Saddle-point method]] is an extension of [[Laplace's method]] for approximating integrals
* [http://finzi.psych.upenn.edu/R/library/untb/html/untb.package.html R package for implementing UNTB]
* [[Extremum]]
* [http://news.cell.com/discussions/trends-in-ecology-and-evolution/ecological-neutral-theory-useful-model-or-statement-of-ignorance "Ecological neutral theory: useful model or statement of ignorance?" in Cell Press Discussions]
* [[First derivative test]]
* [[Second derivative test]]
* [[Higher-order derivative test]]
* [[Saddle surface]]
* [[Hyperbolic equilibrium point]]
* [[Sion's minimax theorem]]
* [[Mountain pass]]
* [[Max–min inequality]]


==References==
== Notes ==
{{reflist|33em}}
<references/>


* {{cite journal|last=Azaele|first=S.|author2=Pigolotti, S. |author3=Banavar, J. R. |author4=Maritan, A. |title=Dynamical evolution of ecosystems|journal=Nature|volume=444|pages=926–928|doi=10.1038/nature05320| year=2006|pmid=17167485|issue=7121|bibcode = 2006Natur.444..926A }}
== References ==
* {{cite journal|last=Etienne|first=R.S. |author2=Alonso, D. |title=A dispersal-limited sampling theory for species and alleles|journal=Ecology Letters|volume=8|pages=1147–1156|year=2005|doi=10.1111/j.1461-0248.2005.00817.x|url=http://www.ourbiosphere.org/~dalonso/SelectPubl/ecoLet05_EtienneAlonso.pdf|issue=11|pmid=21352438}} ({{cite journal|title=Errata |journal=Ecology Letters|volume=9|pages=500|doi=10.1111/j.1461-0248.2006.00917.x|year=2006|issue=4}})
*{{cite journal|last=Gilbert|first=B |author2=Lechowicz MJ |year=2004|title=Neutrality, niches, and dispersal in a temperate forest understory|journal=PNAS|volume=101|pages=7651–7656|doi=10.1073/pnas.0400814101|pmid=15128948|issue=20|pmc=419661|bibcode = 2004PNAS..101.7651G }}
* {{cite book |author=[[Stephen P. Hubbell|Hubbell, S.P.]] |year=2001 |title=The Unified Neutral Theory of Biodiversity and Biogeography|publisher=[[Princeton University Press]]| isbn=0-691-02128-7}}
* {{cite journal|author=Leigh E.G. (Jr) | year=2007| title=Neutral theory: a historical perspective| journal=[[Journal of Evolutionary Biology]] |volume=20 |pages=2075–2091| doi=10.1111/j.1420-9101.2007.01410.x|pmid=17956380|issue=6|url=http://si-pddr.si.edu/dspace/bitstream/10088/6669/1/Leigh0305neutral_theory_a_historic.pdf}}
* {{cite journal|last=Preston|first=F. W.|title=The Canonical Distribution of Commonness and Rarity: Part I|journal=Ecology|volume=43|issue=2|pages=185–215|year=1962|doi=10.2307/1931976|publisher=Ecology, Vol. 43, No. 2|jstor=1931976}}
* {{cite journal |last=Pueyo |first=S. |author2=He, F. |author3=Zillio, T.  |year=2007 |title= The maximum entropy formalism and the idiosyncratic theory of biodiversity |journal=Ecology Letters |volume=10 |pages=1017–1028 |url=http://www.blackwell-synergy.com/doi/pdf/10.1111/j.1461-0248.2007.01096.x |doi=10.1111/j.1461-0248.2007.01096.x |pmid=17692099 |issue=11 |pmc=2121135}}
* {{cite journal|last=Vallade|first=M.|author2=Houchmandzadeh, B. |title=Analytical solution of a neutral model of biodiversity|journal=Phys. Rev. E|volume=68|issue=061902|year=2003|doi=10.1103/PhysRevE.68.061902|pages=061902|bibcode = 2003PhRvE..68f1902V }}
* {{cite journal|last=Volkov|first=I.|author2=Banavar, J. R. |author3=Hubbell, S. P. |author4=Maritan, A. |title=Neutral theory and relative species abundance in ecology|journal=Nature|volume=424|pages=1017–1028|doi=10.1038/nature01883| year=2003|pmid=12944964|issue=6952|url=http://sirismm.si.edu/ctfs/volkov_et_al_2003_nature.pdf|arxiv = q-bio/0504018 |bibcode = 2003Natur.424.1035V }}


{{modelling ecosystems|expanded=other}}
* {{citation |last1=Gray |first1=Lawrence F.|last2=Flanigan|first2=Francis J.|last3=Kazdan|first3=Jerry L.|last4=Frank|first4=David H|last5=Fristedt|first5=Bert |title=Calculus two: linear and nonlinear functions |publisher=Springer-Verlag |location=Berlin |year=1990 |pages= page 375|isbn=0-387-97388-5 |oclc= |doi=}}
* {{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | last2=Cohn-Vossen | first2=Stephan | author2-link=Stephan Cohn-Vossen | title=Geometry and the Imagination | publisher=Chelsea | location=New York | edition=2nd | isbn=978-0-8284-1087-8 | year=1952 }}
* {{citation|first=Tobias|last=von Petersdorff|url=http://www2.math.umd.edu/~petersd/246/stab.html|chapter=Critical Points of Autonomous Systems|year=2006|title=Differential Equations for Scientists and Engineers (Math 246 lecture notes)}}
* {{citation |last=Widder|first=D. V. |title=Advanced calculus |publisher=Dover Publications |location=New York |year=1989 |pages=page 128 |isbn=0-486-66103-2 |oclc= |doi=}}
* {{citation |last=Agarwal|first=A. |title=Study on the Nash Equilibrium (Lecture Notes)|url=http://www.cse.iitd.ernet.in/~rahul/cs905/lecture3/nash1.html#SECTION00041000000000000000}}


[[Category:Biodiversity]]
{{DEFAULTSORT:Saddle Point}}
[[Category:Ecological theories]]
[[Category:Differential geometry of surfaces]]
[[Category:Multivariable calculus]]
[[Category:Stability theory]]
[[Category:Analytic geometry]]

Revision as of 13:24, 12 August 2014

A saddle point on the graph of z=x2−y2 (in red)
Saddle point between two hills (the intersection of the figure-eight -contour)

In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a saddle or a mountain pass. In terms of contour lines, a saddle point in two dimensions gives rise to a contour that appears to intersect itself.

Mathematical Discussion

A simple criterion for checking if a given stationary point of a real-valued function F(x,y) of two real variables is a saddle point is to compute the function's Hessian matrix at that point: if the Hessian is indefinite, then that point is a saddle point. For example, the Hessian matrix of the function at the stationary point is the matrix

which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point is a saddle point for the function but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite.

In the most general terms, a saddle point for a smooth function (whose graph is a curve, surface or hypersurface) is a stationary point such that the curve/surface/etc. in the neighborhood of that point is not entirely on any side of the tangent space at that point.

The plot of y = x3 with a saddle point at 0

In one dimension, a saddle point is a point which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum.

Other uses

In dynamical systems, a saddle point is a periodic point whose stable and unstable manifolds have a dimension that is not zero. If the dynamic is given by a differentiable map f then a point is hyperbolic if and only if the differential of ƒ n (where n is the period of the point) has no eigenvalue on the (complex) unit circle when computed at the point.

In a two-player zero sum game defined on a continuous space, the equilibrium point is a saddle point.

A saddle point is an element of the matrix which is both the largest element in its column and the smallest element in its row.

For a second-order linear autonomous systems, a critical point is a saddle point if the characteristic equation has one positive and one negative real eigenvalue.[1]

See also

Notes

  1. ↑ Template:Harvnb

References

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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