differentiability of t functionals of location and scatter the paper aims at finding widely and smoothly defined nonparametric location and scatter functionals as a convenient vehicle maximum likelihood estimation of the location vector and scatter matrix e of an elliptically symmetric t distribution on r d with degrees of freedom v i extends to an m functional defined on all probability distributions p in a weakly open weakly dense domain u here u consists of p putting not too much mass in hyperplanes of dimension d as shown for empirical measures by kent and tyler ann statist 19 1991 2102 2119 it will be seen here that e is analytic on u for the bounded lipschitz norm or for d 1 for the sup norm on distribution functions for k 1 2 and other norms depending on k and more directly adapted to t functionals one has continuous differentiability of order k allowing the delta method to be applied to for any p in u which can be arbitrarily heavy tailed these results imply asymptotic normality of the corresponding m estimators n n in dimension d 1 only the t v functional a extends to be defined and weakly continuous at all p

uniform central limit theorems for kernel density estimators let mathbb p n ast k h n x n 1 h n d sum i 1 n k left x x i h n right be the classical kernel density estimator based on a kernel k and n independent random vectors x i each distributed according to an absolutely continuous law mathbb p on mathbb r d it is shown that the processes f longmapsto sqrt n int fd mathbb p n ast k h n mathbb p f in mathcal f converge in law in the banach space ell infty mathcal f for many interesting classes mathcal f of functions or sets some mathbb p donsker some just mathbb p pregaussian the conditions allow for the classical bandwidths h n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error thus showing that subject to some natural conditions kernel density estimators are plug in estimators in the sense of bickel and ritov ann statist 31 1033 1053 2003 some new results on the uniform central limit theorem for smoothed empirical processes needed in the proofs are also included

hopf bifurcation and turing instability in spatial homogeneous and inhomogeneous predator prey models abstract this paper is concerned with two species spatial homogeneous and inhomogeneous predator prey models with beddington deangelis functional response for the spatial homogeneous model the asymptotic behavior of the interior equilibrium and the existence of hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered furthermore the direction of hopf bifurcation and the stability of bifurcated periodic solutions are investigated for the model with no flux boundary conditions turing instability of the interior equilibrium solution is studied in particular turing instability region regarding the parameters is established finally to verify our theoretical results some numerical simulations are also included

spatiotemporal dynamics and spatial pattern in a diffusive intraguild predation model with delay effect abstract based on biological meaning a kind of diffusive intraguild predation igp resource ig prey and ig predator model with delay effect is investigated in this paper the model has holling type i functional response between resource ig prey and resource ig predator holling type ii functional response between ig prey and ig predator we first give sufficient conditions on the stability of possible nonnegative constant steady state solutions for the proposed model which give us a complete picture of the global dynamics then we investigate hopf bifurcation near the unique positive constant steady state solution by taking delay as bifurcation parameter and derive the hopf bifurcation threshold it is shown that the delay can induce three types of bistability node node bistability node cycle bistability and cycle cycle bistability periodic oscillations and irregular oscillations triggering spatiotemporal chaos in the diffusive igp model numerical simulations are performed to illustrate our theoretical results and suggest that delay can even trigger the emergence of self organised spatiotemporal patterns which evolve from spiral patterns to irregular spatial patterns via spatiotemporal hopf bifurcation in addition the impact of diffusion on the model s dynamics under certain time delay are also explored

cyclical interactions with alliance specific heterogeneous invasion rates we study a six species lotka volterra type system on different two dimensional lattices when each species has two superior and two inferior partners the invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator prey pair whereby cyclic symmetries within the two three species defensive alliances are conserved monte carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance specific invasion rates this behavior is qualitatively reproduced by a four point mean field approximation the study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences

multiple focus and hopf bifurcations in a predator prey system with nonmonotonic functional response in this paper we develop a criterion to calculate the multiplicity of a multiple focus for general predator prey systems as applications of this criterion we calculate the largest multiplicity of a multiple focus in a predator prey system with nonmonotonic functional response p x frac x ax 2 bx 1 studied by zhu campbell and wolkowicz siam j appl math 63 2002 pp 636 682 and prove that the degenerate hopf bifurcation is of codimension two furthermore we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles the inner one unstable and outer one stable numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus are also given in support of the criterion

class number bounds and catalan s equation we improve a criterion of inkeri and show that if there is a solution to catalan s equation 1 x p y q 1 with p and q prime numbers greater than 3 and both congruent to 3 mod 4 then p and q form a double wieferich pair further we refine a result of schwarz to obtain similar criteria when only one of the exponents is congruent to 3 mod 4 indeed in light of the results proved here it is reasonable to suppose that if q 3 mod 4 then p and q form a double wieferich pair

new donsker classes several classes of functions are shown to be donsker by an argument based on partitioning the sample space one example is the class of all nondecreasing functions f r r such that 0 f f for a given function f with f 2 dp p

quantitative concentration inequalities for empirical measures on non compact spaces we establish quantitative concentration estimates for the empirical measure of many independent variables in transportation distances as an application we provide some error bounds for particle simulations in a model mean field problem the tools include coupling arguments as well as regularity and moment estimates for solutions of certain diffusive partial differential equations

donsker type theorems for nonparametric maximum likelihood estimators let mathcal p be a nonparametric probability model consisting of smooth probability densities and let hat p n be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law mathbb p with hat mathbb p n denoting the measure induced by the density hat p n define the stochastic process hat nu n f longmapsto sqrt n int fd hat mathbb p n mathbb p where f ranges over some function class mathcal f we give a general condition for donsker classes mathcal f implying that the stochastic process hat nu n is asymptotically equivalent to the empirical process in the space ell infty mathcal f of bounded functions on mathcal f this implies in particular that hat nu n converges in law in ell infty mathcal f to a mean zero gaussian process we verify the general condition for a large family of donsker classes mathcal f we give a number of applications convergence of the probability measure hat mathbb p n to mathbb p at rate sqrt n in certain metrics metrizing the topology of weak star convergence a unified treatment of convergence rates of the mle in a continuous scale of sobolev norms sqrt n efficient estimation of nonlinear functionals defined on mathcal p limit theorems at rate sqrt n for the maximum likelihood estimator of the convolution product mathbb p ast p

hopf bifurcations in a reaction diffusion population model with delay effect abstract a reaction diffusion population model with a general time delayed growth rate per capita is considered the growth rate per capita can be logistic or weak allee effect type from a careful analysis of the characteristic equation the stability of the positive steady state solution and the existence of forward hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem where the time delay is used as the bifurcation parameter the general results are applied to a food limited population model with diffusion and delay effects as well as a weak allee effect population model

stability switch and hopf bifurcation for a diffusive prey predator system with delay the increasing time delay usually destabilizes any dynamical system in this paper we give an example that in some special cases the opposite effect can be experienced if the time delay is sufficiently great we investigate the effect of both the parameter in the time delay kernel and diffusion coefficient on the stability of the positive steady state for a diffusive prey predator system with delay we obtain the condition of the occurrence of the stability switches of the positive steady state

empirical processes indexed by lipschitz functions on obtient des conditions necessaires et suffisantes sur p pour que les boules unites de bl r et lip r soient des classes de p donsker fonctionnelles

stability hopf bifurcations and spatial patterns in a delayed diffusive predator prey model with herd behavior in this paper we consider a delayed diffusive predator prey model with herd behavior firstly by choosing the appropriate bifurcation parameter the stability of the positive equilibria and the existence of hopf bifurcations induced by diffusion and delay respectively are investigated by analyzing the corresponding characteristic equation then applying the normal form theory and the center manifold argument for partial functional differential equations the formula determining the properties of the hopf bifurcation are obtained furthermore the instability of the hopf bifurcation leads to the emergence of spatial patterns finally some numerical simulations are also carried out to illustrate and expand the theoretical results

the central limit theorem for empirical processes under local conditions the case of radon infinitely divisible limits without gaussian component on obtient des resultats de convergence faible pour des processus empiriques indices par des classes f de fonctions dans le cas de limites de radon purement de poisson infiniment divisibles

complex dynamic behavior in a viral model with delayed immune response the rich dynamics of a viral infection model is studied under the assumption that the immune response is retarded it is shown that if the basic reproductive ratio of the virus is less than one the infection free equilibrium is globally asymptotically stable analytical and numerical results show that if the basic reproductive ratio of the virus is greater than one the combined effect of the strength of the lytic component the time delay of the immune response and the birth rate of susceptible host cells is to create a rich dynamics which includes the occurrence of stable periodic solutions and chaotic dynamical behavior the route from periodic oscillations to chaos is investigated these results can be used to explain irregular real time series data on the immune state of patients

diffusion effect and stability analysis of a predator prey system described by a delayed reaction diffusion equations in this paper we consider a delayed reaction diffusion equations which describes a two species predator prey system with diffusion terms and stage structure by using the linearization method and the method of upper and lower solutions we study the local and global stability of the constant equilibria respectively the results show that the free diffusion of the delayed reaction diffusion equations has no effect on the populations when the diffusion is too slow otherwise the free diffusion has a certain influence on the populations however the influence can be eliminated by improving the parameters to satisfy some suitable conditions

stability and hopf bifurcation for a delayed prey predator system with diffusion effects abstract this paper is concerned with a delayed lotka volterra prey predator system with diffusion effects and neumann boundary conditions the main purpose is to investigate the stability of spatially homogeneous positive equilibrium and give the explicit formulae determining the direction and stability of hopf bifurcation by linearizing the system at positive equilibrium and analyzing the associated characteristic equation the stability of positive equilibrium and the existence of hopf bifurcation are demonstrated by means of the normal form theory and the center manifold reduction for partial functional differential equations pfdes the direction and stability of periodic solutions occurring through hopf bifurcation are determined finally in order to verify our theoretical results some numerical simulations are also included

bifurcation and spatiotemporal patterns in a homogeneous diffusive predator prey system abstract a diffusive predator prey system with holling type ii predator functional response subject to neumann boundary conditions is considered hopf and steady state bifurcation analysis are carried out in details in particular we show the existence of multiple spatially non homogeneous periodic orbits while the system parameters are all spatially homogeneous our results and global bifurcation theory also suggest the existence of loops of spatially non homogeneous periodic orbits and steady state solutions these results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation

stability and hopf bifurcation analysis on a simplified bam neural network with delays a delay differential equation modelling a bidirectional associative memory bam neural network with three neurons is investigated its dynamics are studied in terms of local analysis and hopf bifurcation analysis by analyzing the associated characteristic equation its linear stability is investigated and hopf bifurcations are demonstrated the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold reduction numerical simulation results are given to support the theoretical predictions

effect of cross diffusion on the stationary problem of a predator prey system with a protection zone abstract this paper is concerned with the steady state problem of a predator prey cross diffusion system with herd effect allee effect and a protection zone some sufficient conditions for the existence of positive steady state solutions are given our proof is based on the local and global bifurcation theory and some a priori estimates some limiting behavior of positive steady states with respect to the allee effect constant or the diffusion coefficient d are discussed

hopf bifurcation and global stability of a diffusive gause type predator prey models abstract this paper mainly provides hopf bifurcation formulas for a general gause type predator prey system with diffusion and neumann boundary condition by using the center manifold theory and normal form method where the spectral and stability analysis around an equilibrium is addressed and our results can be applied to the case without diffusion as an application of these results we give a complete and rigorous analysis of the global dynamics of a diffusive predator prey model with herd behavior especially the hopf bifurcation and its direction and the stability of the bifurcating periodic solutions

hopf bifurcation and steady state bifurcation for an autocatalysis reaction diffusion model abstract this paper is concerned with an autocatalysis model subject to no flux boundary conditions the existence of hopf bifurcation are firstly obtained then by the center manifold theory and the normal form method the direction of hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established on the other hand the steady state bifurcations from simple and double eigenvalues are intensively studied the techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues finally some numerical simulations are shown to verify the analytical results

on hopf bifurcation and control for a delay systems abstract hopf bifurcation of a delay differential equation modeling two enterprise interaction mechanism is investigated for the demands of computer simulation applications the discrete time two enterprise interaction model is proposed by using the forward euler scheme by discussing the characteristic equation of linearized part the local stability criterion of the positive fixed point is presented time lag is selected as the bifurcation parameter then it is demonstrated that variation of time lag may result in hopf bifurcation furthermore derivations for the directions and stability of hopf bifurcation are also given for the purpose of controlling hopf bifurcation a time delayed feedback control scheme is proposed finally we simulate the main results

a diffusive toxin producing phytoplankton model with maturation delay and three dimensional patch the dynamics of a diffusive toxin producing phytoplankton model with maturation delay and three dimensional patch subject to neumann boundary condition is investigated in this paper the global stability of boundary equilibrium is obtained the local stability of the coexistent equilibrium and the existence of hopf bifurcation are investigated the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived

efficient maximum likelihood estimation in semiparametric mixture models we consider maximum likelihood estimation in several examples of semiparametric mixture models including the exponential frailty model and the errors in variables model the observations consist of a sample of size n from the mixture density int p theta x z d eta z the mixing distribution is completely unknown we show that the first component hat theta n of the joint maximum likelihood estimator hat theta n hat eta n is asymptotically normal and asymptotically efficient in the semiparametric sense

spatiotemporal dynamics in a diffusive ratio dependent predator prey model near a hopf turing bifurcation point abstract spatiotemporal dynamics in a ratio dependent predator prey model with diffusion is studied by analytical methods normal forms associated with codimension two hopf turing bifurcation are derived which can be used to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the hopf turing bifurcation point in the vicinity of this degenerate point a wealth of complex spatiotemporal dynamics are observed our theoretical results are confirmed by numerical simulations

bifurcation branch of stationary solutions in a general predator prey system with prey taxis abstract in this paper a general reaction diffusive predator prey system with prey taxis subject to the homogeneous neumann boundary condition is considered firstly we investigate the local stability of the unique positive equilibrium by analyzing the characteristic equation and study a priori estimates of positive solutions by the iterative technique and then choosing the prey tactic sensitivity coefficient as bifurcation parameter we proved that a branch of nonconstant solutions can bifurcate from the unique positive equilibrium when the prey tactic sensitivity is repulsive moreover we find the stable bifurcating solutions near the bifurcation point by the spectrum theory under some suitable conditions our results show that prey taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns

phase diagrams for three strategy evolutionary prisoner s dilemma games on regular graphs evolutionary prisoner s dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one the players follow one of the three strategies tit for tat unconditional cooperation and defection the simplified payoff matrix is characterized by two parameters the temptation b to choose defection and the cost c of inspection reducing the income of tit for tat the strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise using monte carlo simulations and the extended versions of pair approximation we have evaluated the b c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence absorbing and oscillatory states including continuous and discontinuous phase transitions by reasonable costs the tit for tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner s dilemma game irrespective of the connectivity structure we also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance

a complete analysis of the global dynamics of a diffusive predator and toxic prey model considering many species can release toxic substances to protect themselves against predators a diffusive predator and toxic prey system with spatial heterogeneity in predator and prey populations has been investigated for this system we give a complete and rigorous analysis of the global dynamics with the boundedness globally asymptotical stability transcritical bifurcation hopf bifurcation and its direction and the stability of the bifurcating periodic solutions meanwhile we consider the effects of toxins produced by the prey on the dynamic behavior the consequence of the global stability of the coexistence equilibrium is that the toxin s intrinsic characteristic will not change the stability of the system irreversibly our results show that the toxin produced by the prey phytoplankton may be used as a bio control agent for the harmful algal bloom problems