Contenu de sensagent
Dictionnaire et traducteur pour mobile
Nouveau : sensagent est maintenant disponible sur votre mobile
dictionnaire et traducteur pour sites web
Une fenêtre (pop-into) d'information (contenu principal de Sensagent) est invoquée un double-clic sur n'importe quel mot de votre page web. LA fenêtre fournit des explications et des traductions contextuelles, c'est-à-dire sans obliger votre visiteur à quitter votre page web !
Avec la boîte de recherches Sensagent, les visiteurs de votre site peuvent également accéder à une information de référence pertinente parmi plus de 5 millions de pages web indexées sur Sensagent.com. Vous pouvez Choisir la taille qui convient le mieux à votre site et adapter la charte graphique.
Solution commerce électronique
Augmenter le contenu de votre site
Ajouter de nouveaux contenus Add à votre site depuis Sensagent par XML.
Parcourir les produits et les annonces
Obtenir des informations en XML pour filtrer le meilleur contenu.
Indexer des images et définir des méta-données
Fixer la signification de chaque méta-donnée (multilingue).
Renseignements suite à un email de description de votre projet.
Jeux de lettres
Lettris est un jeu de lettres gravitationnelles proche de Tetris. Chaque lettre qui apparaît descend ; il faut placer les lettres de telle manière que des mots se forment (gauche, droit, haut et bas) et que de la place soit libérée.
Il s'agit en 3 minutes de trouver le plus grand nombre de mots possibles de trois lettres et plus dans une grille de 16 lettres. Il est aussi possible de jouer avec la grille de 25 cases. Les lettres doivent être adjacentes et les mots les plus longs sont les meilleurs. Participer au concours et enregistrer votre nom dans la liste de meilleurs joueurs ! Jouer
Dictionnaire de la langue française
La plupart des définitions du français sont proposées par SenseGates et comportent un approfondissement avec Littré et plusieurs auteurs techniques spécialisés.
Le dictionnaire des synonymes est surtout dérivé du dictionnaire intégral (TID).
L'encyclopédie française bénéficie de la licence Wikipedia (GNU).
Les jeux de lettres anagramme, mot-croisé, joker, Lettris et Boggle sont proposés par Memodata.
Le service web Alexandria est motorisé par Memodata pour faciliter les recherches sur Ebay. La SensagentBox est offerte par sensAgent.
Changer la langue cible pour obtenir des traductions.
Astuce: parcourir les champs sémantiques du dictionnaire analogique en plusieurs langues pour mieux apprendre avec sensagent.
The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. They evolve in time according to the pair of equations:
The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator-prey interactions, competition, disease, and mutualism.
The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka “in the theory of autocatalytic chemical reactions” in 1910. This was effectively the logistic equation, which was originally derived by Pierre François Verhulst. In 1920 Lotka extended, via Kolmogorov (see above), the model to "organic systems" using a plant species and a herbivorous animal species as an example  and in 1925 he utilised the equations to analyse predator-prey interactions in his book on biomathematics  arriving at the equations that we know today. Vito Volterra, who made a statistical analysis of fish catches in the Adriatic  independently investigated the equations in 1926.
C.S. Holling extended this model yet again, in two 1959 papers, in which he proposed the idea of functional response. Both the Lotka-Volterra model and Holling's extensions have been used to model the moose and wolf populations in Isle Royale National Park, which with over 50 published papers is one of the best studied predator-prey relationships.
The Lotka–Volterra equations have a long history of use in economic theory; their initial application is commonly credited to Richard Goodwin in 1965 or 1967. In economics, links are between many if not all industries; a proposed way to model the dynamics of various industries has been by introducing trophic functions between various sectors, and ignoring smaller sectors by considering the interactions of only two industrial sectors.
The Lotka-Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:
When multiplied out, the prey equation becomes:
The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero then there can be no predation.
With these two terms the equation above can be interpreted as: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.
The predator equation becomes:
In this equation, δxy represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). γy represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey.
Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.
The equations have periodic solutions and do not have a simple expression in terms of the usual trigonometric functions. However, a linearization of the equations yields a solution similar to simple harmonic motion with the population of predators following that of prey by 90°.
Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 80 baboons and 40 cheetahs, one can plot the progression of the two species over time. The choice of time interval is arbitrary.
One can also plot solutions without representing time, but with one axis representing the number of prey and the other axis representing the number of predators. The solutions are closed curves, and there is a quantity V which is conserved on each curve:
These graphs clearly illustrate a serious problem with this as a biological model: in each cycle, the baboon population is reduced to extremely low numbers yet recovers (while the cheetah population remains sizeable at the lowest baboon density). With chance fluctuations, discrete numbers of individuals, and the family structure and lifecycle of baboons, the baboons actually go extinct and by consequence the cheetahs as well. This modelling problem has been called the "atto-fox problem", an atto-fox being an imaginary 10−18 of a fox, in relation to rabies modelling in the UK.
In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low the prey population will increase again. These dynamics continue in a cycle of growth and decline.
Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0.
When solved for x and y the above system of equations yields
Hence, there are two equilibria.
The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters, α, β, γ, and δ.
The Jacobian matrix of the predator-prey model is
When evaluated at the steady state of (0, 0) the Jacobian matrix J becomes
The eigenvalues of this matrix are
In the model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point.
The stability of this fixed point is of importance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, we find that the extinction of both species is difficult in the model. (In fact, this can only occur if the prey are artificially completely eradicated, causing the predators to die of starvation. If the predators are eradicated, the prey population grows without bound in this simple model).
Evaluating J at the second fixed point we get
The eigenvalues of this matrix are
and the level curves, where K = const, are closed trajectories surrounding the fixed point. Consequently, the levels of the predator and prey populations cycle, and oscillate around this fixed point.
The largest value of the constant K can be obtained by solving the optimization problem
The maximal value of K is attained at the stationary point and it is given by
where e is Euler's Number.