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Integrodifference Equation
In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: : n_(x) = \int_ k(x, y)\, f(n_t(y))\, dy, where \\, is a sequence in the function space and \Omega\, is the domain of those functions. In most applications, for any y\in\Omega\,, k(x,y)\, is a probability density function on \Omega\,. Note that in the definition above, n_t can be vector valued, in which case each element of \ has a scalar valued integrodifference equation associated with it. Integrodifference equations are widely used in mathematical biology, especially theoretical ecology, to model the dispersal and growth of populations. In this case, n_t(x) is the population size or density at location x at time t, f(n_t(x)) describes the local population growth at location x and k(x,y), is the probability of moving from point y to point x, often referred to as the dispersal kernel. Integrodifference equations are most commonly used to describe univoltine Vol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function ''space''. In linear algebra Let be a vector space over a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define \begin (f+g)(x) &= f(x)+g(x) \\ (c\cdot f)(x) &= c\cdot f(x) \end When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if is also a vector space over , th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed ... [...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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Ecology
Theoretical ecology is the scientific discipline devoted to the study of ecological systems using theoretical methods such as simple conceptual models, mathematical models, computational simulations, and advanced data analysis. Effective models improve understanding of the natural world by revealing how the dynamics of species populations are often based on fundamental biological conditions and processes. Further, the field aims to unify a diverse range of empirical observations by assuming that common, mechanistic processes generate observable phenomena across species and ecological environments. Based on biologically realistic assumptions, theoretical ecologists are able to uncover novel, non-intuitive insights about natural processes. Theoretical results are often verified by empirical and observational studies, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world. The field is broad and includes foundations in appli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biological Dispersal
Biological dispersal refers to both the movement of individuals (animals, plants, fungi, bacteria, etc.) from their birth site to their breeding site ('natal dispersal'), as well as the movement from one breeding site to another ('breeding dispersal'). Dispersal is also used to describe the movement of propagules such as seeds and spores. Technically, dispersal is defined as any movement that has the potential to lead to gene flow. The act of dispersal involves three phases: departure, transfer, settlement and there are different fitness costs and benefits associated with each of these phases. Through simply moving from one habitat patch to another, the dispersal of an individual has consequences not only for individual fitness, but also for population dynamics, population genetics, and species distribution. Understanding dispersal and the consequences both for evolutionary strategies at a species level, and for processes at an ecosystem level, requires understanding on the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voltinism
Voltinism is a term used in biology to indicate the number of broods or generations of an organism in a year. The term is most often applied to insects, and is particularly in use in sericulture, where silkworm varieties vary in their voltinism. * Univoltine (monovoltine) – (adjective) referring to organisms having one brood or generation per year * Bivoltine (divoltine) – (adjective) referring to organisms having two broods or generations per year *Trivoltine – (adjective) referring to organisms having three broods or generations per year * Multivoltine (polyvoltine) – (adjective) referring to organisms having more than two broods or generations per year * Semivoltine – There are two meanings: :* (''biology'') Less than univoltine; having a brood or generation less often than once per year :* or (adjective) referring to organisms whose generation time is more than one year. Examples The speckled wood butterfly is univoltine in the northern part of its range, e.g. nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Population Dynamics
Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 220 years,Malthus, Thomas Robert. An Essay on the Principle of Population: Library of Economics although over the last century the scope of mathematical biology has greatly expanded. The beginning of population dynamics is widely regarded as the work of Malthus, formulated as the Malthusian growth model. According to Malthus, assuming that the conditions (the environment) remain constant (''ceteris paribus''), a population will grow (or decline) exponentially. This principle provided the basis for the subsequent predictive theories, such as the demographic studies such as the work of Benjamin Gompertz and Pierre François Verhulst in the early 19th century, who refined and adjusted the Malthusian demographic mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metapopulation
A metapopulation consists of a group of spatially separated populations of the same species which interact at some level. The term metapopulation was coined by Richard Levins in 1969 to describe a model of population dynamics of insect pests in agricultural fields, but the idea has been most broadly applied to species in naturally or artificially fragmented habitats. In Levins' own words, it consists of "a population of populations". A metapopulation is generally considered to consist of several distinct populations together with areas of suitable habitat which are currently unoccupied. In classical metapopulation theory, each population cycles in relative independence of the other populations and eventually goes extinct as a consequence of demographic stochasticity (fluctuations in population size due to random demographic events); the smaller the population, the more chances of inbreeding depression and prone to extinction. Although individual populations have finite life-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical And Theoretical 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 pra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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