characterization of log del pezzo pairs via anticanonical models we give a characterization of log del pezzo pairs in terms of their anticanonical models as a first application we show that every non rational del pezzo surface with log canonical singularities is of picard number one and contains a unique singularity which is simple elliptic as a second application we verify a conjecture of smith and schwede in dimension two which says that every surface of globally f regular type is of fano type

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

convergence of path measures arising from a mean field or polaron type interaction we discuss the limiting path measures of markov processes with either a mean field or a polaron type interaction of the paths in the polaron type situation the strength is decaying at large distances on the time axis and so the interaction is of short range in time in contrast in the mean field model the interaction is weak but of long range in time donsker and varadhan proved that for the partition functions there is a transition from the polaron type to the mean field interaction when passing to a limit by letting the strength tend to zero while increasing the range the discussion of the path measures is more subtle we treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples

optimal harvesting problems for age dependent interacting species with diffusion we investigate the optimal harvesting problems of a predator prey system for two age dependent populations with diffusion the existence and uniqueness of solutions for the system are proven using the fixed point theorem the existence of optimal control and maximum principles for problems are obtained

a predator prey system with stage structure for predator and nonlocal delay abstract this paper deals with the behavior of solutions to the reaction diffusion system under homogeneous neumann boundary condition which describes a prey predator model with nonlocal delay sufficient conditions for the global stability of each equilibrium are derived by the lyapunov functional and the results show that the introduction of stage structure into predator positively affects the coexistence of prey and predator numerical simulations are performed to illustrate the results

global convergence of a reaction diffusion predator prey model with stage structure and nonlocal delays in this paper a lotka volterra type reaction diffusion predator prey model with stage structure for the prey and nonlocal delays due to gestation of the predator is investigated in the case of a general domain sufficient conditions are obtained for the global convergence of positive solutions of the proposed problem by using the energy function method numerical simulations are carried out to illustrate the main results

a reaction diffusion predator prey model with stage structure and nonlocal delay abstract in this paper a reaction diffusion predator prey model with nonlocal delay due to gestation of the predator and stage structure for the prey is investigated in the case of a general domain sufficient conditions are derived for the global convergence of positive solutions to the proposed problem by using the energy function method numerical simulations are carried out to illustrate the main results

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

complex dynamic behavior of a rumor propagation model with spatial temporal diffusion terms a spatial temporal rumor propagation model with discrete and non local delays is regarded stability and hopf bifurcation are proved in a strict mathematical way sensitivity analysis is applied to search active parameters hopf bifurcation causes an oscillation phenomenon in this model numerical simulations provide a new insight into rumor spreading in online social networks rumor propagation as a typical form of social communication in online social networks has had a significant negative impact on a harmonious and stable society with the rapid development of mobile communication equipments traditional rumor propagation models which depend on ordinary differential equations ode may not be suitable for describing rumor propagation in an online social network in this paper based on reaction diffusion equations we propose a novel epidemic like model with both discrete and nonlocal delays for investigating the spatial temporal dynamics of rumor propagation by analyzing the corresponding characteristic equations of this model the local stability conditions of a boundary equilibrium point and a positive equilibrium point are established by applying the linear approximation method of nonlinear systems sufficient conditions are derived for the existence of hopf bifurcation at the above two kinds of equilibrium points moreover a sensitivity analysis method based on the density of spreading users is proposed and then in theoretical and experimental aspect we identify some sensitive parameters in the process of rumor propagation finally numerical simulations are performed to illustrate the theoretical results

the optimal harvesting problems of a stage structured population in this paper we consider a stage structured population model with two life stages immature and mature the optimal harvesting policy for the considered population is made we obtain the optimal harvesting effort the maximum sustainable yield the maximum sustainable economic rent and the corresponding population density

optimal control of harvesting for single population the aim of this paper is to study the optimal harvesting policies for single population model that is to say we assume that the objective of the harvester is to find a harvesting plan making the harvesting yield maximum or maintaining the constant yield and making the terminal population level maximum when the harvesting has happened in a rigid time interval

persistence and extinction in general non autonomous logistic model with delays and stochastic perturbation this is a continuation of the paper liu and wang 2012 14 firstly we consider a general non autonomous logistic model with delays and stochastic perturbation then sufficient conditions for extinction are established as well as nonpersistence in the mean weak persistence and stochastic permanence the threshold between weak persistence and extinction is obtained finally numerical simulations are introduced to support the theoretical analysis results

optimal harvesting policy for general stochastic logistic population model abstract we consider some optimal harvesting policies for a general stochastic logistic population model for two management objectives that are maximum sustainable yield and the maximum retained profits the optimal harvesting policies are obtained meanwhile the optimal harvest effort the maximum of expectation of sustainable yield or retained profits and the corresponding variance are given

correlation structure of intermittency in the parabolic anderson model consider the cauchy problem u x t t u x t x d t 0 with initial condition u x 0 1 and with the anderson hamiltonian here is the discrete laplacian 0 is a diffusion constant and x x d is an i i d random field taking values in gartner and molchanov 1990 have shown that if the law of 0 is nondegenerate then the solution u is asymptotically intermittent in the present paper we study the structure of the intermittent peaks for the special case where the law of 0 is in the vicinity of the double exponential prob 0 s exp e s s here 0 is a parameter that can be thought of as measuring the degree of disorder in the field our main result is that for fixed x y d and t the correlation coefficient of u x t and u y t converges to w 2 l2 z d w x z w y z in this expression while w d is given by w v d with v the unique centered ground state i e the solution in l2 with minimal l 2 norm of the 1 dimensional nonlinear equation v 2 v log v 0 the uniqueness of the ground state is actually proved only for large but is conjectured to hold for any 0 empty it turns out that if the right tail of the law of 0 is thicker or thinner than the double exponential then the correlation coefficient of u x t and u y t converges to x y resp the constant function 1 thus the double exponential family is the critical class exhibiting a nondegenerate correlation structure

the effect of stage structure on the permanence of a predator prey system with time delay abstract a delayed predator prey system with stage structure for the prey and time delay due to the gestation of the predator is investigated by using an iteration technique and comparison arguments respectively sufficient conditions are derived for the global stability of the positive equilibrium and the two boundary equilibria of the proposed model numerical simulations are presented to illustrate the main results

parabolic problems for the anderson model the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

spatio temporal delays in a nutrient plankton model on a finite domain linear stability and bifurcations the issue of how to incorporate time delays into a mathematical model in which individuals are moving around requires careful consideration any time delay term must also involve a weighted spatial averaging to account for movement of individuals during the time delay period most of the current literature on this subject is on reaction diffusion equations and concentrates on the simplest case when the spatial domain is infinite in this paper we consider what changes arise when the domain is finite spatial averaging kernels are computed explicitly for the case of a finite one dimensional domain to illustrate the ideas we concentrate on a diffusive nutrient plankton model the model is analysed in terms of the local stability of the steady states and bifurcations the results of some numerical simulations are also presented

large deviations estimates for self intersection local times for simple random walk in mathbb z 3 we obtain large deviations estimates for the self intersection local times for a simple random walk in dimension 3 also we show that the main contribution to making the self intersection large in a time period of length n comes from sites visited less than some power of log n this is opposite to the situation in dimensions larger or equal to 5 finally we present an application of our estimates to moderate deviations for random walk in random sceneries

instability in diffusive ecological models with nonlocal delay effects abstract we propose a reaction diffusion system with nonlocal delays to model the growth of plankton communities feeding on a limiting nutrient supplied at a constant rate two delays are incorporated into the model one describes the delayed nutrient recycling process and the other models the delayed growth response of the plankton it is assumed that both delays are nonlocal in the sense that there are delayed not only in time but also in space local and bifurcation analyses are carried out it has been shown that turing type spatial patterns occur when the diffusion coefficients vary and temporal or spatial temporal patterns occur when the delay involved in the growth response changes

statistical mechanics of the 2 dimensional focusing nonlinear schrodinger equation we study a natural construction of an invariant measure for the 2 dimensional periodic focusing nonlinear schrodinger equation with the critical cubic nonlinearity we find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased but that the natural construction successful for the 1 dimensional case and for the 2 dimensional defocusing case cannot produce an invariant measure our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of berlin and kac

global asymptotic stability of lotka volterra 3 species reaction diffusion systems with time delays abstract this paper is concerned with three 3 species time delayed lotka volterra reaction diffusion systems and their corresponding ordinary differential systems without diffusion the time delays may be discrete or continuous and the boundary conditions for the reaction diffusion systems are of neumann type the goal of the paper is to obtain some simple and easily verifiable conditions for the existence and global asymptotic stability of a positive steady state solution for each of the three model problems these conditions involve only the reaction rate constants and are independent of the diffusion effect and time delays the result of global asymptotic stability implies that each of the three model systems coexists is permanent and the trivial and all semitrivial solutions are unstable our approach to the problem is based on the method of upper and lower solutions for a more general reaction diffusion system which gives a common framework for the 3 species model problems some global stability results for the 2 species competition and prey predator reaction diffusion systems are included in the discussion

dynamics of nonlinear parabolic systems with time delays abstract the dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system the same conclusions are obtained for a coupled system of parabolic ordinary equations with time delays applications are given to three model problems arising from ecology and nuclear engineering these model problems possess multiple steady state solutions and sufficient conditions are given to ensure the stability and instability of these solutions

globally f regular and log fano varieties abstract we prove that every globally f regular variety is log fano in other words if a prime characteristic variety x is globally f regular then it admits an effective q divisor such that k x is ample and x has controlled kawamata log terminal in fact globally f regular singularities a weak form of this result can be viewed as a prime characteristic analog of de fernex and hacon s new point of view on kawamata log terminal singularities in the non q gorenstein case we also prove a converse statement in characteristic zero every log fano variety has globally f regular type our techniques apply also to f split varieties which we show to satisfy a log calabi yau condition we also prove a kawamata viehweg vanishing theorem for globally f regular pairs

potentially non klt locus and its applications we introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor the potentially non klt locus is a subset of x which is birationally transformed precisely into the non klt locus on a k x minimal model of x we prove basic properties of potentially non klt locus in comparison with those of classical non klt locus as applications we give a new characterization of varieties of fano type and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor

mean field interaction of brownian occupation measures ii a rigorous construction of the pekar process we consider mean field interactions corresponding to gibbs measures on interacting brownian paths in three dimensions the interaction is self attractive and is given by a singular coulomb potential the logarithmic asymptotics of the partition function for this model were identified in the 1980s by donsker and varadhan in terms of the pekar variational formula which coincides with the behavior of the partition function of the polaron problem under strong coupling based on this in 1986 spohn made a heuristic observation that the strong coupling behavior of the polaron path measure on certain time scales should resemble a process named as the pekar process whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean field measures under interest whose rigorous analysis remained open the present paper is devoted to a precise analysis of these mean field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the pekar variational problem this leads to a rigorous construction of the aforementioned pekar process and hence is a contribution to the understanding of the mean field approximation of the polaron problem on the level of path measures the method of our proof is based on the compact large deviation theory developed by mukherjee and varadhan in 2016 its extension to the uniform strong metric for the singular coulomb interaction was carried out by konig and mukherjee in 2015 as well as an idea inspired by a partial path exchange argument appearing in 1997 in work by bolthausen and schmock 2016 wiley periodicals inc

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

optimal harvesting policy for stochastic logistic population model abstract in this paper we consider some optimal harvesting policies for single population models in which the harvest effort and the intrinsic growth rate are disturbed by environment noises we choose the maximum sustainable yield and the maximum retained profits as two management objectives and obtain the optimal harvesting policies respectively for the two objectives we give the optimal harvest effort that maximizes the sustainable yield or retained profits the maximum of expectation of sustainable yield or retained profits and the corresponding variance their explicit expressions are determined by the coefficients of equation and the disturbance intensity

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

on self attracting d dimensional random walks let x t t geq 0 be a symmetric nearest neighbor random walk on mathbb z d with exponential holding times of expectation 1 d starting at the origin for a potential v mathbb z d to 0 infty with finite and nonempty support define transformed path measures by d hat mathbb p t equiv exp t 1 int 0 t int 0 t v x s x t ds dt d mathbb p z t for t 0 where z t is the normalizing constant if d 1 or if the self attraction is sufficiently strong then x t infty has an exponential moment under mathbb p t which is uniformly bounded for t 0 and t in 0 t we also prove that x t t geq 0 under suitable subsequences of hat mathbb p t t 0 behaves for large t asymptotically like a mixture of space inhomogeneous ergodic random walks for special cases like a sufficiently strong dirac type interaction we even prove convergence of the transformed path measures and the law of x t as well as of the law of the empirical measure l t under hat mathbb p t t 0

localization of a two dimensional random walk with an attractive path interaction we consider an ordinary symmetric continuous time random walk on the two dimensional lattice mathbb z 2 the distribution of the walk is transformed by a density which discounts exponentially the number of points visited up to time t this introduces a self attracting interaction of the paths we study the asymptotic behavior for t rightarrow infty it turns out that the displacement is asymptotically of order t 1 4 the main technique for proving the result is a refined analysis of large deviation probabilities a partial discussion is given also for higher dimensions

optimal harvesting policy of logistic population model in a randomly fluctuating environment abstract in this paper we investigate the optimal harvest policy of a stochastic logistic population model the value of this study lies in two aspects mathematically we establish a stochastic threshold theorem to govern whether the population persists or not in the case of population persistence we prove the existence uniqueness and global asymptotic stability of the invariant density of the fokker planck equation associated with the sde model and we further show the relation between these two models in addition we give the unique optimal harvesting effort and the corresponding maximum of expectation of sustainable yield which gives us the profile of the optimal harvesting policy of the sde model ecologically we find that big harvesting effort or big intensity of noise will lead the population to extinct risk almost surely in addition we find that under a fixed randomness strategy and proper harvesting the maximum sustainable yield increases systematically as the harvesting effort increases but overexploitation will reduce the level of maximize sustainable yield and eventually make the whole population extinct with probability one hence in order to obtain the optimal harvesting policy we must decrease the harvesting effort and the intensity of noise furthermore we find that our parameter perturbation method in this paper is more beneficial to the exploitation of renewable resources than the classical one given by beddington and may 1977 the results show that different perturbation method can exhibit different stochastic population dynamics

existence of fast positive wavefronts for a non local delayed reaction diffusion equation abstract we establish the existence of a continuous family of fast positive wavefronts u t x x c t 0 for the non local delayed reaction diffusion equation u t t x u x x t x u t x r k x w g u t h w d w here 0 and 0 are fixed points of g c 2 r r and the non negative k is such that r k w e w d w is finite for every real we also prove that the fast wavefronts are non monotone if g h e h 1 1

a predator prey system with stage structure for predator abstract this paper studies the asymptotoc behavior of a predator prey model with stage structure it is found that an orbitally asymptotically stable periodic orbit exists in that model when time delay due to gestation of predator and time delay from crowding effect of prey are incorporated we establish the condition for the permanence of populations and sufficient conditions under which positive equilibrium of the model is globally stable

spatial structures and periodic travelling waves in an integro differential reaction diffusion population model an integro differential reaction diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase it is claimed that this is inherently more realistic than the usual kind of reaction diffusion model for mobile populations three kinds of bifurcation from the uniform steady state solution are considered i to steady spatially periodic structures ii to periodic standing wave solutions and iii to periodic travelling wave solutions these correspond to aggregation and motion of populations

a holling s type ii prey predator model with stage structure and nonlocal delay abstract in this paper a prey predator model with reaction diffusion is investigated under homogenous neumann boundary condition by taking food ingestion and species moving into account model is further coupled with holling s type ii function response and nonlocal delay sufficient conditions for the global stability of three equilibria i e positive semi trivial and trivial steady states are mainly derived by lyapunov functional respectively results show that intra specific competition benefits the coexistence of prey and predator numerical simulations are performed to illustrate the analytical results

uniruledness of stable base loci of adjoint linear systems via mori theory we explain how to deduce from recent results in the minimal model program a general uniruledness theorem for base loci of adjoint divisors as a special case we recover previous results by takayama

phase transition in reinforced random walk and rwre on trees a random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future using exchangeability theory the process is shown to be equivalent to a random walk in a random environment rwre that is to say a mixture of markov chains criteria are given to determine whether a rwre is transient or recurrent these criteria apply to show that the reinforced random walk can vary from transient to recurrent depending on the value of an adjustable parameter measuring the strength of the feedback the value of the parameter at the phase transition is calculated