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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] |
Mathematical Biology
Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development and behavior of the systems, as opposed to experimental biology which deals with the conduction of experiments to prove and validate the scientific theories. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. Theoretical biology focuses more on the development of theoretical principles for biology while mathematical biology focuses on the use of mathematical tools to study biological systems, even though the two terms are sometimes interchanged. Mathematical biology aims at the mathematical representation and modeling of biological processes, using techniques and tools of applied mathematics. It can be useful in both theoretical and prac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearization
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization of a function Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that f(x) is differentiable on , b/math> (or , a/math>) and that a is close to b. In short, linearization approximates the output of a function near x = a. For example, \sqrt = 2. However, what would be a good appro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilibrium Point
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The point \tilde\in \mathbb^n is an equilibrium point for the differential equation :\frac = \mathbf(t,\mathbf) if \mathbf(t,\tilde)=\mathbf for all t. Similarly, the point \tilde\in \mathbb^n is an equilibrium point (or fixed point) for the difference equation :\mathbf_ = \mathbf(k,\mathbf_k) if \mathbf(k,\tilde)= \tilde for k=0,1,2,\ldots. Equilibria can be classified by looking at the signs of the eigenvalues of the linearization of the equations about the equilibria. That is to say, by evaluating the Jacobian matrix at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances), by finding th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x_e stay near x_e forever, then x_e is Lyapunov stable. More strongly, if x_e is Lyapunov stable and all solutions that start out near x_e converge to x_e, then x_e is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs ... [...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] |
Topological Conjugacy
In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially. To illustrate this directly: suppose that f and g are iterated functions, and there exists a homeomorphism h such that :g = h^ \circ f \circ h, so that f and g are topologically conjugate. Then one must have :g^n = h^ \circ f^n \circ h, and so the iterated systems are topologically conjugate as well. Here, \circ denotes function composition. Definition f\colon X \to X, g\colon Y \to Y, and h\colon Y \to X are continuous functions on topological spaces, X and Y. f being topologically semiconjugate to g means, by definition, that h is a surjection such t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdots ... [...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] |
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] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
