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Paradox Of Enrichment
The paradox of enrichment is a term from population ecology coined by Michael Rosenzweig in 1971. He described an effect in six predator–prey models where increasing the food available to the prey caused the predator's population to destabilize. A common example is that if the food supply of a prey such as a rabbit is overabundant, its population will grow unbounded and cause the predator population (such as a lynx) to grow unsustainably large. That may result in a crash in the population of the predators and possibly lead to local eradication or even species extinction. The term 'paradox' has been used since then to describe this effect in slightly conflicting ways. The original sense was one of irony; by attempting to increase the carrying capacity in an ecosystem, one could fatally imbalance it. Since then, some authors have used the word to describe the difference between modelled and real predator–prey interactions. Rosenzweig used ordinary differential equation models ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Ecology
Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment, such as birth and death rates, and by immigration and emigration. The discipline is important in conservation biology, especially in the development of population viability analysis which makes it possible to predict the long-term probability of a species persisting in a given patch of habitat. Although population ecology is a subfield of biology, it provides interesting problems for mathematicians and statisticians who work in population dynamics. History In the 1940s ecology was divided into autecology—the study of individual species in relation to the environment—and synecology—the study of groups of species in relation to the environment. The term autecology (from Ancient Greek: αὐτο, ''aúto'', "self"; οίκος, ''oíkos'', "household"; and λόγος, ''lógos'', "knowledge"), refers to roughly the same fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hartman–Grobman Theorem
In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to Philip Hartman and David M. Grobman. The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyse its behaviour around equilibria. Main theorem Consider a system evolving in time with state u(t)\in\mathbb R^n that satisfies the differential equation du/ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Association For The Advancement Of Science
The American Association for the Advancement of Science (AAAS) is an American international non-profit organization with the stated goals of promoting cooperation among scientists, defending scientific freedom, encouraging scientific responsibility, and supporting scientific education and science outreach for the betterment of all humanity. It is the world's largest general scientific society, with over 120,000 members, and is the publisher of the well-known scientific journal ''Science''. History Creation The American Association for the Advancement of Science was created on September 20, 1848, at the Academy of Natural Sciences in Philadelphia, Pennsylvania. It was a reformation of the Association of American Geologists and Naturalists. The society chose William Charles Redfield as their first president because he had proposed the most comprehensive plans for the organization. According to the first constitution which was agreed to at the September 20 meeting, the goal of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Science (journal)
''Science'', also widely referred to as ''Science Magazine'', is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals. It was first published in 1880, is currently circulated weekly and has a subscriber base of around 130,000. Because institutional subscriptions and online access serve a larger audience, its estimated readership is over 400,000 people. ''Science'' is based in Washington, D.C., United States, with a second office in Cambridge, UK. Contents The major focus of the journal is publishing important original scientific research and research reviews, but ''Science'' also publishes science-related news, opinions on science policy and other matters of interest to scientists and others who are concerned with the wide implications of science and technology. Unlike most scientific journals, which focus on a specific field, ''Science'' and its rival ''Nature (journal), Nature'' c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paradox Of The Pesticides
The paradox of the pesticides is a paradox that states that applying pesticide to a pest may end up increasing the abundance of the pest if the pesticide upsets natural predator–prey dynamics in the ecosystem. Lotka–Volterra equation To describe the paradox of the pesticides mathematically, the Lotka–Volterra equation, a set of first-order, nonlinear, differential equations, which are frequently used to describe predator–prey interactions, can be modified to account for the additions of pesticides into the predator–prey interactions. Without pesticides The variables represent the following: : \begin H & = \text \\ P & = \text \\ c & = \text \\ r & = \text \\ a & = \text \\ m & = \text \\ \end The following two equations are the original Lotka–Volterra equation, which describe the rate of change of each respective population as a function of the population of the other organism: : \begin \frac & = rH - cHP \\ \frac & = acHP - mP \\ \end By setting each equat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braess's Paradox
Braess's paradox is the observation that adding one or more roads to a road network can slow down overall traffic flow through it. The paradox was discovered by the German mathematician Dietrich Braess in 1968. The paradox may have analogies in electrical power grids and biological systems. It has been suggested that, in theory, the improvement of a malfunctioning network could be accomplished by removing certain parts of it. The paradox has been used to explain instances of improved traffic flow when existing major roads are closed. Discovery and definition Dietrich Braess, a mathematician at Ruhr University, Germany, noticed the flow in a road network could be impeded by adding a new road, when he was working on traffic modelling. His idea was that if each driver is making the optimal self-interested decision as to which route is quickest, a shortcut could be chosen too often for drivers to have the shortest travel times possible. More formally, the idea behind Braess's di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arditi–Ginzburg Equations
The Arditi–Ginzburg equations describes ratio dependent predator–prey dynamics. Where ''N'' is the population of a prey species and ''P'' that of a predator, the population dynamics are described by the following two equations: \begin \frac & = f(N)\,N-gP \\ pt\frac & = e \,gP-uP \end Here ''f''(''N'') captures any change in the prey population not due to predator activity including inherent birth and death rates. The per capita effect of predators on the prey population (the harvest rate) is modeled by a function ''g'' which is a function of the ratio ''N''/''P'' of prey to predators. Predators receive a reproductive payoff, ''e,'' for consuming prey, and die at rate ''u''. Making predation pressure a function of the ratio of prey to predators contrasts with the prey dependent Lotka–Volterra equations, where the effect of predators on the prey population is simply a function of the magnitude of the prey population ''g''(''N''). Because the number of prey harvested by each pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Bifurcation
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis. Under reasonably generic assumptions about the dynamical system, a small-amplitude limit cycle branches from the fixed point. A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf. Overview Supercritical and subcritical Hopf bifurcations The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical. The normal form of a Hopf bifurcation is: ::\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Manifold Theorem
In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1. Stable manifold theorem Let :f: U \subset \mathbb^n \to \mathbb^n be a smooth map with hyperbolic fixed point at p. We denote by W^(p) the stable set and by W^(p) the unstable set of p. The theorem states that * W^(p) is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of f at p. * W^(p) is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of f at p. According ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Community Matrix
In mathematical biology, the community matrix is the linearization of the Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the Lyapunov stability, stability of the equilibrium point. The Lotka–Volterra predator–prey model is : \begin \dfrac &=& x(\alpha - \beta y) \\ \dfrac &=& - y(\gamma - \delta x), \end where ''x''(''t'') denotes the number of prey, ''y''(''t'') the number of predators, and ''α'', ''β'', ''γ'' and ''δ'' are constants. By the Hartman–Grobman theorem the non-linear system is Topological conjugacy, topologically equivalent to a linearization of the system about an equilibrium point (''x''*, ''y''*), which has the form : \begin \frac \\ \frac \end = \mathbf \begin u \\ v \end, where ''u'' = ''x'' − ''x''* and ''v'' = ''y'' − ''y''*. In mathematical biology, the Jacobian matrix \mathbf evaluated at the equilibrium point (''x''*, ''y''*) is called the community matrix. By the stable manifold theorem, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
