When optimal foragers meet in a game theoretical conflict: A model of kleptoparasitism

Kleptoparasitism can be considered as a game theoretical problem and a foraging tactic at the same time, so the aim of this paper is to combine the basic ideas of two research lines: evolutionary game theory and optimal foraging theory. To unify these theories, firstly, we take into account the fact that kleptoparasitism between foragers has two consequences: the interaction takes time and affects the net energy intake of both contestants. This phenomenon is modeled by a matrix game under time constraints. Secondly, we also give freedom to each forager to avoid interactions, since in optimal foraging theory foragers can ignore each food type (we have two prey types: either a prey item in possession of another predator or a free prey individual is discovered). The main question of the present paper is whether the zero-one rule of optimal foraging theory (always or never select a prey type) is valid or not, in the case where foragers interact with each other? In our foraging game we consider predators who engage in contests (contestants) and those who never do (avoiders), and in general those who play a mixture of the two strategies. Here the classical zero-one rule does not hold. Firstly, the pure avoider phenotype is never an ESS. Secondly, the pure contestant can be a strict ESS, but we show this is not necessarily so. Thirdly, we give an example when there is mixed ESS

Entropy and Hausdorff Dimension in Random Growing Trees

We investigate the limiting behavior of random tree growth in preferential attachment models. The tree stems from a root, and we add vertices to the system one-by-one at random, according to a rule which depends on the degree distribution of the already existing tree. The so-called weight function, in terms of which the rule of attachment is formulated, is such that each vertex in the tree can have at most K children. We define the concept of a certain random measure mu on the leaves of the limiting tree, which captures a global property of the tree growth in a natural way. We prove that the Hausdorff and the packing dimension of this limiting measure is equal and constant with probability one. Moreover, the local dimension of mu equals the Hausdorff dimension at mu-almost every point. We give an explicit formula for the dimension, given the rule of attachment

Essential correlatedness and almost independence

This note characterizes all pairs of random variables (X1, X2 for which there exist no Borel measurable injections f1, f 2 such that f1(X1) and f2(X2) are uncorrelated.Essential correlatedness almost independence

When is predator’s opportunism remunerative?

When an opportunistic predator is looking for a given type of prey and encounters another one from different species, it tries to utilize this random opportunity. We characterize the optimal levels of this opportunism in the framework of stochastic models for the two prey-one predator case. We consider the spatial dispersal of preys and the optimal diet choice of predator as well. We show that when both preys have no handling time, the total opportunism provides maximal gain of energy for the predator. When handling times differ with prey, we find a conditional optimal behavior: for small density of both prey species the predator prefers the more valuable one and is entirely opportunistic. However, when the density of the more valuable prey is higher than that of the other species, then the predator prefers the first one and intentionally neglects the other. Furthermore, when the density of the less valuable prey is high and that of the other one is small, then predator will look for the less valuable prey and is therefore totally opportunistic. We demonstrate that prey preference is remunerative whenever the advantage of a proper prey preference is larger than the average cost of missed prey preference. We also propose a dynamics which explicitly contains two sides of shared predation: apparent mutualism and apparent competition, and we give conditions when the rare prey goes extinct

An extremal property of rectangular distributions

It is proved that the correlation coefficient between the elements of the order statistics is maximal for a rectangularly (uniformly) distributed population.rectangular distribution uniform distribution order statistics correlation coefficient maximal correlation Jacobi polynomials

Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

We study a novel class of affine invariant and consistent tests for normality in any dimension. The tests are based on a characterization of the standard $d$-variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schr\"odinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood-of-model validation procedure.Comment: 29 pages, 1 figure, 7 table