almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks stability of numerical solutions to stochastic delay differential equations have received an increasing attention but there has been so far little work on the stability analysis of numerical solutions to stochastic delay hopfield neural networks the aim of this paper is to study the almost sure exponential stability of numerical solutions to stochastic delay hopfield neural networks by using two approaches the euler method and the backward euler method under some reasonable conditions both the euler scheme and the backward euler scheme are proved to be almost sure exponential stability in particular the euler method and the backward euler method are mainly based on the semimartingale convergence theorem

dynamics of the deterministic and stochastic siqs epidemic model with non linear incidence the deterministic and stochastic siqs models with non linear incidence are introduced for deterministic model the basic reproductive rate r 0 is derived moreover if r 0 1 there exists a unique endemic equilibrium which is globally asymptotically stable for stochastic model sufficient condition for extinction of the disease that is regardless of the value of r 0 is presented in addition if the intensities of the white noises are sufficiently small and r 0 1 then there exists a unique stationary distribution to stochastic model numerical simulations are also carried out to confirm the analytical results

lasalle type theorem and its applications to infinite horizon optimal control of discrete time nonlinear stochastic systems based on discrete martingale theory the lasalle type theorem for general discrete time stochastic systems is obtained and the almost sure stability is in turn discussed as applications infinite horizon nonlinear optimal regulator is investigated and a dynamical programming equation called the hamilton jacobi bellman equation is also derived for discrete time nonlinear stochastic optimal control

multiplicity results for some elliptic systems near resonance with a nonprincipal eigenvalue abstract some multiplicity results of solutions are obtained for a class of elliptic systems which are near resonance with a nonprincipal eigenvalue by the classical saddle point theorem and a local saddle point theorem in critical point theory

persistence and extinction in stochastic delay logistic equation by incorporating ornstein uhlenbeck process abstract the persistence and extinction pe are interesting topics in mathematics this research analyzed pe of stochastic logistic equations sle by incorporating the ornstein uhlenbeck process slop and stochastic delay logistic equation sdle by incorporating the ornstein uhlenbeck process sldop firstly we proved that slop and sldop have positive solutions likewise for stochastic permanence sp weak persistence in the mean wpm non persistence in the mean npm and extinction the sufficient conditions are established for slop and sldop subsequently for numerical simulation we used 4 stage stochastic runge kutta srk4 to illustrate the effectiveness of the results

new criteria on persistence in mean and extinction for stochastic competitive lotka volterra systems with regime switching abstract this paper concerns persistence in mean and extinction for stochastic competitive lotka volterra systems with regime switching by using some novel stochastic analysis techniques sufficient criteria for the partial permanence and partial extinction are established some novel sufficient conditions on persistence in mean and extinction are also obtained nontrivial examples are provided to illustrate our results

new criterion of partial asymptotic stability in probability of stochastic differential equations autonomous system of stochastic differential equations dx t f x dt i 1 kg i x dw i t which has a zero solution x 0 is considered it is assumed that there exists a positive definite with respect to part of the variables function v x such that corresponding operator lv is nonpositive it is proved that if the set m 0 x lv 0 x v x 0 does not include entire semitrajectories of the system 2 1 almost surely then the zero solution is asymptotically stable in probability with respect to part of the variables

stability criteria of random nonlinear systems and their applications stochastic differential equations sdes are widely adopted to describe systems with stochastic disturbances while they are not necessarily the best models in some specific situations this paper considers the nonlinear systems described by random differential equations rdes the notions and the corresponding criteria of noise to state stability asymptotic gain and asymptotic stability are proposed in the m th moment or in probability several estimation methods of stochastic processes are presented to explain the reasonability of the assumptions used in theorems as applications of stability criteria some examples about stabilization regulation and tracking are considered respectively a theoretical framework on stability of rdes is finally constructed which is distinguished from the existing framework of sdes

a note on persistence and extinction of a randomized food limited logistic population model this paper addresses the issue of the asymptotic behavior for a non autonomous randomized food limited logistic population model several sufficient conditions are formulated and proved for p moment persistence and extinction of the population as well as in sense of almost sure results show that food limited assumption has an influence on the convergence rate of the solution to the equilibria for the deterministic and stochastic model some previously known results are improved numerical simulations are provided to support the results

extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls given a 1 periodic real potential q l1 r z we use 0 q to denote the smallest 1 periodic eigenvalue of the hill s equation x q t x 0 let b1 r be the ball centered at 0 of radius r in the l1 space l1 r z it is trivial that sup 0 q q b1 r r for all r 0 based on continuity of 0 q in q with the weak topology and continuous differentiability of 0 q in q with the l1 norm 1 we will apply scaling technique variational approach to extremal values in lp balls singular integrals and the limiting approach as p 1 to obtain i 0 q is bounded for q in any bounded set of l1 r z 1 and ii the minimal value l1 r inf 0 q q l1 r z q 1 r inf 0 q q l1 r z q 1 r is simply z0 1 r where z0 x 2 xtanh x 2 x 0 the extremal values of the smallest neumann eigenvalues for potentials in l1 balls are also found explicitly

remarks on stochastic permanence of population models abstract this paper examines the asymptotic behavior of the stochastic lotka volterra model under markovian switching we show that the stochastic lotka volterra model is stochastically permanent moreover we give another type of stochastic permanence and consider the relationship of two types of stochastic permanence under white noise perturbation

on p dunford integrable functions with values in banach spaces abstract let be a complete probability space x a banach space and 1 p in this paper we discuss several aspects of p dunford integrable functions f x special attention is paid to the compactness of the dunford operator of f we also study the p bochner integrability of the composition u f y where u is a p summing operator from x to another banach space y finally we also provide some tests of p dunford integrability by using w thick subsets of x

existence and uniqueness theorems for periodic markov process and applications to stochastic functional differential equations abstract this work is devoted to an m valued markov process with a periodic transition probability function where m is a polish space sufficient conditions for the existence and uniqueness of the periodic markov process are given these existence theorems are rather general and the corresponding results in 1 11 15 23 are improved and generalized the uniqueness theorems for the periodic markov process are original as applications we study the existence uniqueness and global attractivity of periodic solution for the periodic stochastic n species lotka voltrra competitive model with bounded delays and the periodic stochastic neural networks with infinite delay respectively numerical simulations are presented to illustrate the main results

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

almost sure permanence of stochastic single species models abstract a new definition of almost sure permanence of stochastic population systems is proposed in this paper we consider the stochastic logistic model d x t x t a t b t x t d t t x t d b t where b t is a standard brownian motion and is a positive constant under a simple assumption a interesting result is obtained as follows 0 lim inf t x t lim sup t x t a s the result is analogous to the result in the deterministic case and some numerical simulations are introduced to support our main results at the end

the improved lasalle type theorems for stochastic functional differential equations abstract the main aim of this paper is to improve some results obtained by mao x mao the lasalle type theorems for stochastic functional differential equations nonlinear stud 7 2000 307 328 our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above for example we need only the local lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions to guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation sfde under the weaker conditions imposed in this paper we establish a generalised existence and uniqueness theorem which covers a wider class of nonlinear sfdes as demonstrated by the examples discussed in this paper moreover from our improved results follow some new criteria on the stochastic asymptotic stability for sfdes

the failure of fatou s theorem on poisson integrals of pettis integrable functions in this paper we prove that for every infinite dimensional banach spacexand every 1 p there exists a strongly measurablex valuedp pettis integrable function on the unit circle t such that thex valued harmonic function defined as its poisson integral does not converge radially at any point of t not even in the weak topology we also show that this function does not admit a conjugate function an application to spaces of vector valued harmonic functions is given in the case thatxdoes not have finite cotype we can construct the function with the additional property of being analytic in the sense that its fourier coefficients of negative frequency are null in the general case we are able to give a countably additive vector measure analytic in the same sense

stability in distribution of neutral stochastic functional differential equations with markovian switching abstract stability in distribution of stochastic differential equations with markovian switching and stochastic differential delay equations with markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems there are several papers which study this stability for stochastic differential equations with markovian switching and stochastic differential delay equations with markovian switching technically in our paper we are concerned with the general neutral stochastic functional differential equations with markovian switching and we derive the sufficient conditions for stability in distribution at the end of our paper one example is established to illustrate the theory of our work

analysis of stochastic two prey one predator model with levy jumps abstract taking white noises and levy noises into account a two prey one predator model in random environments is proposed and investigated under some simple assumptions the critical value between persistence in the mean and extinction for each population is obtained then sufficient conditions for stability in distribution of the model are established finally some numerical examples are introduced to validate the analytical findings

stochastic mutualism model under regime switching with levy jumps abstract this paper is concerned with a stochastic mutualism model under regime switching with levy jumps to begin with the existence and uniqueness of the global positive solution is proved with any given positive initial value then the sufficient conditions for stochastic permanence are established the critical value between extinction and persistence in mean is also obtained in addition under some suitable conditions we proved that there is a unique stationary distribution for the system without levy jumps our method relies on the lyapunov function analysis and the fredholm alternative the results demonstrate that regime switching may contribute to the permanence but jump noise may suppress the permanence

the almost sure asymptotic stability and p th moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth this note is concerned with the asymptotic stability analysis for nonlinear stochastic differential systems sdss the systems coefficients are assumed to satisfy local lipschitz condition and polynomial growth condition by applying some novel techniques some easily verifiable conditions are obtained which ensure the almost sure asymptotic stability and pth moment asymptotic stability for such sdss we also provide the range of the order p a numerical example is provided to illustrate the effectiveness and the benefits of the proposed result

boundary values of functions in vector valued hardy spaces and geometry on banach spaces abstract the spaces of boundary values of vector valued functions in hardy spaces defined by either holomorphic functions on the disk or harmonic functions with maximal function in l p are characterized in terms of vector valued measures of bounded p variation we extend to the case p 1 a characterization of the radon nikodym property based on the existence of limits at the boundary for harmonic functions with maximal function in l 1 in the case 0 p 1 we find the umd property as the necessary and sufficient condition to make the spaces defined by maximal function and by conjugate poisson kernel coincide

p th moment noise to state stability of stochastic differential equations with persistent noise this paper studies the stability properties of stochastic differential equations subject to persistent noise including the case of additive noise which is noise that is present even at the equilibria of the underlying differential equation and does not decay with time the class of systems we consider exhibit disturbance attenuation outside a closed not necessarily bounded set we identify conditions based on the existence of lyapunov functions to establish the noise to state stability in probability and in pth moment of the system with respect to a closed set as part of our analysis we study the concept of two functions being proper with respect to each other formalized via a pair of inequalities with comparison functions we show that such inequalities define several equivalence relations for increasingly strong refinements on the comparison functions we also provide a complete characterization of the properties that a pair of functions must satisfy to belong to the same equivalence class th

unique stationary distribution and ergodicity of a stochastic logistic model with distributed delay abstract dynamics of a stochastic logistic model with distributed delay are considered we first transfer a scalar stochastic logistic model with strong kernel or weak kernel into an equivalent stochastic system through the linear chain technique then we obtain the sufficient and necessary conditions for extinction and persistence of the species with probability one moreover in the case of persistence we prove that there exists a unique stationary distribution by the markov semigroups theory the results show that the stronger white noise results in the extinction of the species and the weaker white noise guarantees the existence of a unique stationary distribution though for the deterministic model with strong kernel or weak kernel the average delay may induce the existence of a group of small amplitude periodic solutions

dissipativity theory for nonlinear stochastic dynamical systems in this paper we develop stochastic dissipativity theory for nonlinear dynamical systems using basic input output and state properties specifically a stochastic version of dissipativity using both an input output as well as a state dissipation inequality in expectation for controlled markov diffusion processes is presented the results are then used to derive extended kalman yakubovich popov conditions for characterizing necessary and sufficient conditions for stochastic dissipativity of stochastic dynamical systems using two times continuously differentiable storage functions in addition feedback interconnection stability in probability results for stochastic dynamical systems are developed thereby providing a generalization of the small gain and positivity theorems to stochastic systems

solutions to a gradient system with resonance at both zero and infinity in this paper we study the existence and multiplicity of nontrivial solutions for a gradient system with resonance at both zero and infinity via morse theory

on the dual space of hp x 1 p let x be a complex banach space and 1 p lp x is the lebesgue bochner space of x valued integrable functions on the circle and hp x its hardy type subspace f lp x tf n 0 n 0 after studying the analytic or riesz projection of x valued functions a representation of the dual space of hp x as a space of antianalytic functions on the disc is given which generalizes a result of bukhvalov examples are constructed for bad behaviour of the analytic projection and of functions in this dual space if x does not belong to the well known umd class of banach spaces

on a stochastic logistic equation with impulsive perturbations a stochastic logistic model with impulsive perturbations is proposed and investigated first we give a new definition of a solution of an impulsive stochastic differential equation isde which is more convenient for use than the existing one using this definition we show that our model has a global and positive solution and obtain its explicit expression then we establish the sufficient conditions for extinction non persistence in the mean weak persistence persistence in the mean and stochastic permanence of the solution the critical value between weak persistence and extinction is obtained in addition the limit of the average in time of the sample path of the solution is estimated by two constants afterwards the lower growth rate and the upper growth rate of the solution are estimated finally sufficient conditions for global attractivity are established

some contributions to stochastic asymptotic stability and boundedness via multiple lyapunov functions most of the existing results on stochastic stability use a single lyapunov function but we shall instead use multiple lyapunov functions in this paper to establish some sufficient criteria for locating the limit sets of solutions of stochastic differential equations from them follow many useful results on stochastic asymptotic stability and boundedness which enable us to construct the lyapunov functions much more easily in applications in particular the well known classical theorem on stochastic asymptotic stability is a special case of our more general results these show clearly the power of our new results

optimal harvesting of a stochastic mutualism model with levy jumps in this letter a stochastic mutualism model with levy jumps and harvesting is considered under some simple assumptions sufficient and necessary criteria for the existence of optimal harvesting policy are established the optimal harvesting effort and the maximum of sustainable yield are also obtained the effects of random noises on the optimal harvesting of the model are discussed and some numerical simulations are introduced to illustrate the main results

dynamics of a stochastic cell to cell hiv 1 model with distributed delay abstract in this paper we consider a stochastic cell to cell hiv 1 model with distributed delay firstly we show that there is a global positive solution of this model before exploring its long time behavior then sufficient conditions for extinction of the disease are established moreover we obtain sufficient conditions for the existence of an ergodic stationary distribution of the model by constructing a suitable stochastic lyapunov function the stationary distribution implies that the disease is persistent in the mean finally we provide some numerical examples to illustrate theoretical results

global stability and stochastic permanence of a non autonomous logistic equation with random perturbation abstract this paper discusses a randomized non autonomous logistic equation d n t n t a t b t n t d t t d b t where b t is a 1 dimensional standard brownian motion in d q jiang n z shi a note on non autonomous logistic equation with random perturbation j math anal appl 303 2005 164 172 the authors show that e 1 n t has a unique positive t periodic solution e 1 n p t provided a t b t and t are continuous t periodic functions a t 0 b t 0 and 0 t a s 2 s d s 0 we show that this equation is stochastically permanent and the solution n p t is globally attractive provided a t b t and t are continuous t periodic functions a t 0 b t 0 and min t 0 t a t max t 0 t 2 t by the way the similar results of a generalized non autonomous logistic equation with random perturbation are yielded

analysis of a stochastic autonomous mutualism model abstract an autonomous lotka volterra mutualism system with random perturbations is investigated under some simple conditions it is shown that there is a decreasing sequence k which has the property that if 1 1 then all the populations go to extinction i e lim t x i t 0 1 i n if k 1 k 1 then lim t x j t 0 j k 1 n whilst the remaining k populations are stable in the mean i e lim t t 1 0 t x i s d s a positive constant i 1 k if n 1 then all the species are stable in the mean sufficient conditions for stochastic permanence and global asymptotic stability are also established

asymptotic behavior of a stochastic non autonomous predator prey system with jumps in this paper a two species stochastic non autonomous food chain predator prey system with jumps is proposed and studied the asymptotic properties of the system are examined sufficient conditions for extinction non persistence in the mean weak persistence in the mean and weak persistence of the system are established results show that levy jumps are disadvantageous for the persistence of the populations furthermore we also show that the solution is stochastically ultimate bounded under certain conditions some examples together with numerical simulations are introduced to illustrate our main results

a recurrence principle for stochastic difference inclusions the invariance principle is extended to a recurrence principle and is developed for stochastic difference inclusions for these systems random solutions are not unique under appropriate lyapunov like conditions it is established for every random solution that almost every complete sample path converges to the largest weakly totally recurrent set contained in a level set of the lyapunov like function such a set is not larger and is sometimes smaller than the largest weakly invariant set contained in the level set the principle is useful for establishing robust uniform asymptotic stability in probability or robust uniform strong recurrence under weak lyapunov conditions for stochastic discrete time control systems that employ discontinuous feedback laws examples demonstrate the achieved results

stabilization of input disturbed stochastic port hamiltonian systems via passivity it is a universal phenomenon that the inputs of a real stochastic process are disturbed however this phenomenon is often ignored when stochastic port hamiltonian system sphs is used for modeling in which only the disturbance on the states is considered for this reason an extension of sphs named input disturbed sphs id sphs is proposed to describe the inputs disturbance as well as the states one the definition and some properties of id sphs including structure invariance and stochastic passivity are presented one by one to construct the basic framework of id sphs based on this framework a passivity based controller is further developed to stabilize id sphs at its steady state the control strategy is demonstrated in a thermodynamic heat conduction process with the results supporting strongly the proposed id sphs theory

sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching in this paper we prove that a stochastic logistic population under regime switching controlled by a markov chain is either stochastically permanent or extinctive and we obtain the sufficient and necessary conditions for stochastic permanence and extinction under some assumptions in the case of stochastic permanence we estimate the limit of the average in time of the sample path of the solution by two constants related to the stationary probability distribution of the markov chain and the parameters of the subsystems of the population model finally we illustrate our conclusions through two examples

a note on nonautonomous logistic equation with random perturbation abstract this paper discusses a randomized nonautonomous logistic equation d n t n t a t b t n t d t t d b t where b t is 1 dimensional standard brownian motion we show that e 1 n t has a unique positive t periodic solution e 1 n p t provided a t b t and t are continuous t periodic functions a t 0 b t 0 and 0 t a s 2 s d s 0

bifurcation for an elliptic system coupled in the linear part we study linearly coupled elliptic systems without any assumption on the signs of the coefficients of the linear part we derive first the leray schauder degree of the associated linear system then we obtain sufficient conditions on the nonlinear terms for having bifurcation to coexistence states our results include the sublinear case keys words linearly coupled elliptic systems bifurcation to positive solutions coexistence states leray schauder degree email jfleck cict fr rpardo sunma4 mat ucm es

multiplicity of solutions for resonant elliptic systems we establish the existence and multiplicity of solutions for some resonant elliptic systems the results are proved by applying minimax arguments and morse theory

stochastic periodic solution for a perturbed non autonomous predator prey model with generalized nonlinear harvesting and impulses abstract in this paper stochastic non autonomous predator prey models with and without impulses are investigated the effects of generalized nonlinear harvesting for prey and predator populations are considered for the stochastic system without impulses the existence and uniqueness of the positive solution is proven and sufficient conditions that guarantee the extinction and persistence of the population in the mean are achieved we show the existence of a nontrivial positive periodic solution by constructing appropriate lyapunov functions and using khasminskii s theory moreover the global attractiveness and stochastic persistence in probability of the stochastic model are discussed results show that the stronger noises and nonlinear harvesting component can significantly influence the dynamics of the system and lead to the extinction of the predator population additionally for the stochastic predator prey system with impulsive effect we prove that there exists a positive periodic solution numerical simulations are conducted to show the effectiveness and feasibility of the obtained results

existence results for some nonlinear elliptic systems abstract the paper is concerned with some nonlinear elliptic systems by using variational methods we obtain the existence of weak solutions for the semilinear elliptic system and the quasilinear elliptic system

qualitative analysis of a stochastic ratio dependent predator prey system a stochastic ratio dependent predator prey model is investigated in this paper by the comparison theorem of stochastic equations and ito s formula we obtain the global existence of a positive unique solution of the ratio dependent model besides a condition for species to be extinct is given and a persistent condition is established we also conclude that both the prey population and the ratio dependent function are stable in time average in the end numerical simulations are carried out to confirm our findings

almost sure stabilizability of controlled degenerate diffusions we develop a direct lyapunov method for the almost sure open loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes the infinitesimal decrease condition for a lyapunov function is a new form of hamilton jacobi bellman partial differential inequality of second order we give local and global versions of the first and second lyapunov theorems assuming the existence of a lower semicontinuous lyapunov function satisfying such an inequality in the viscosity sense an explicit formula for a stabilizing feedback is provided for affine systems with smooth lyapunov function several examples illustrate the theory

permanence and extinction for a single species system with jump diffusion abstract in this study we consider the effect of jump diffusion random environmental perturbations on the permanence and extinction of a single species dispersal periodic system in poor patchy environments with the possibility of species loss during their dispersion among patches first we prove that there is a unique global positive solution to the system with any initial positive value with probability 1 and we obtain the sufficient conditions that stochastically ensure the ultimate boundedness as well as the asymptotic polynomial growth of the population system next we establish the sufficient conditions for the almost sure permanence in the mean stochastic permanence and extinction of the system in particular by constructing an appropriate integrating factor we obtain the sufficient conditions for the almost sure weak permanence of the patch the conditions obtained for permanence generalize the sufficient conditions established previously on the system without random environmental perturbations finally we discuss the biological implications of the main results

analysis of autonomous lotka volterra competition systems with random perturbation abstract this paper discusses a randomized n species lotka volterra competition system we show that this system is stable in time average under certain conditions furthermore there is a stationary distribution of this system if extra conditions are satisfied also we give the extinction condition of this system finally numerical simulations are carried out to support our results

asymptotic properties of stochastic hybrid gilpin ayala system with jumps this paper focuses on studying the dynamics of the stochastic gilpin ayala model under regime switching with jumps the aim is to analyze what happens under the perturbations of regime switching and jumps some asymptotic properties are investigated and sufficient conditions for stochastic permanence extinction non persistence in the mean and weak persistence are provided the critical value among the extinction non persistence in the mean and weak persistence is obtained our results demonstrate that the dynamics of the model have close relations with the jumps and the stationary distribution of the markov chain

brief paper small gain control method for stochastic nonlinear systems with stochastic iiss inverse dynamics this paper investigates the small gain type conditions on stochastic iiss siiss systems and makes full use of these conditions in the design and analysis of the controller the contributions are as follows 1 a new proof of the stochastic lasalle theorem is provided 2 the small gain type conditions on siiss are developed and their relationship is discussed 3 based on the stochastic lasalle theorem and siiss small gain type conditions the adaptive controllers are designed to guarantee that all of the closed loop signals are bounded almost surely and the stochastic closed loop systems are globally asymptotically stable in probability

feedback control of quantum state reduction feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement we formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter we explore the use of stochastic lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability

existence and multiplicity results for some elliptic systems at resonance abstract the existence and multiplicity of weak solutions for some resonant elliptic systems are obtained by using ekeland s variational principle the mountain pass theorem and the saddle point theorem in critical point theory

a matrosov theorem for adversarial markov decision processes matrosov s relaxation of lyapunov conditions for uniform global asymptotic stability in time varying systems is extended to stochastic set valued discrete time systems nested matrosov functions are used to give conditions for stability that complement invariance principles for time invariant systems unlike invariance principles matrosov functions also can be applied to general time varying systems

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

a class of stochastic delayed sir epidemic models with generalized nonlinear incidence rate and temporary immunity abstract a class of sir epidemic model with generalized nonlinear incidence rate is presented in this paper temporary immunity and stochastic perturbation are also considered the existence and uniqueness of the global positive solution is achieved sufficient conditions guaranteeing the extinction and persistence of the epidemic disease are established moreover the threshold behavior is discussed and the threshold value r 0 is obtained we show that if r 0 1 the disease eventually becomes extinct with probability one whereas if r 0 1 then the system remains permanent in the mean

attraction stability and robustness for stochastic functional differential equations with infinite delay this paper establishes the stochastic lasalle theorem to locate limit sets for stochastic functional differential equations with infinite delay from which some criteria on attraction boundedness stability and robustness are obtained to illustrate the applications of our results clearly this paper considers a scalar stochastic integro differential equation with infinite delay as an example

analysis of a stochastic non autonomous food limited lotka volterra cooperative model in this paper a stochastic non autonomous food limited lotka volterra cooperative model is proposed and investigated sufficient conditions are formulated and verified for p moment persistence and extinction of the system results demonstrate that cooperative coefficients have important effects on the convergence rate of the solution to the equilibria of the stochastic model some numerical simulations are introduced to support the analytical results

lyapunov and converse lyapunov theorems for stochastic semistability abstract this paper develops lyapunov and converse lyapunov theorems for stochastic semistable nonlinear dynamical systems semistability is the property whereby the solutions of a stochastic dynamical system almost surely converge to not necessarily isolated lyapunov stable in probability equilibrium points determined by the system initial conditions specifically we provide necessary and sufficient lyapunov conditions for stochastic semistability and show that stochastic semistability implies the existence of a continuous lyapunov function whose infinitesimal generator decreases along the dynamical system trajectories and is such that the lyapunov function satisfies inequalities involving the average distance to the set of equilibria

asymptotic behaviors of stochastic periodic differential equation with markovian switching some asymptotic behaviors of hybrid stochastic periodic differential equations are investigated in this paper firstly by constructing the space of the periodic functions with probability measure values we prove that the solution processes converge to a stochastic process with periodic distribution under the conditions of the existence and uniqueness tightness of the transition probability and global attractivity of the solutions for stochastic periodic differential equations with markovian switching then we give the definition of stochastic periodic solution of stochastic periodic differential equations which can be regarded as the stochastic counterpart of the periodic solution for the deterministic systems for the stochastic periodic logistic equation we firstly give the expression of the unique explicit solution and other properties such as p moment boundedness global attractivity and asymptotic stability in distribution we then prove the existence uniqueness and stability of the unique stochastic periodic solution finally we simulate the sample trajectory of stochastic periodic logistic equation the results show a certain degree of periodicity in fact it is a simulation result for the mixture of periodicity and randomness

global stability of two group sir model with random perturbation abstract in this paper we discuss the two group sir model introduced by guo li and shuai h b guo m y li z shuai global stability of the endemic equilibrium of multigroup sir epidemic models can appl math q 14 2006 259 284 allowing random fluctuation around the endemic equilibrium we prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large in addition the stability condition is obtained by the construction of lyapunov function finally numerical simulations are presented to illustrate our mathematical findings

persistence and extinction in stochastic non autonomous logistic systems this paper studies two widely used stochastic non autonomous logistic models for the first system sufficient conditions for extinction non persistence in the mean weak persistence and stochastic permanence are established the critical number between weak persistence and extinction is obtained for the second system sufficient criteria for extinction non persistence in the mean weak persistence in the mean strong persistence in the mean and stochastic permanence are established the critical number between weak persistence in the mean and extinction is obtained it should be pointed out that this research is systematical and complete in fact the behaviors of the two models in every coefficient cases are cleared up by the results obtained in this paper

multiplicity of solutions for gradient systems with strong resonance at higher eigenvalues abstract in this paper we establish existence and multiplicity of solutions for an elliptic system which has strong resonance at higher eigenvalues we describe the resonance using an eigenvalue problem with indefinite weights in all results we use variational methods and the morse theory

a stochastic sirs epidemic model with nonlinear incidence rate a stochastic sirs model with ratio dependent incidence rate is developed the global dynamics of the deterministic model is shown the stochastic dynamics of the sde model is given the existence of a unique stationary distribution of the sde model is displayed in this paper we investigate the global dynamics of a general sirs epidemic model with a ratio dependent incidence rate and its corresponding stochastic differential equation version for the deterministic model we show that the basic reproduction number r0 determines whether there is an endemic outbreak or not if r0 1 the disease persists for the stochastic model we show that its related reproduction number r0s can determine whether there is a unique disease free stationary distribution or a unique endemic stationary distribution in addition we provide analytic results regarding the stochastic boundedness and permanence extinction one of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

a matrosov theorem for strong global recurrence a theorem on nested matrosov functions is presented for time varying stochastic set valued discrete time systems satisfying mild regularity conditions it establishes sufficient conditions for uniform strong global recurrence of an open bounded set in general matrosov functions are required to satisfy less rigid requirements than typical lyapunov functions that satisfy a strict decrease condition along trajectories

explicit conditions for asymptotic stability of stochastic lienard type equations with markovian switching this paper deals with the boundedness in probability and convergence for solutions of stochastic lienard type equations some explicit conditions to ensure that all solution is bounded in probability and convergent to some fixed sets are obtained we also show that white noise can affect the boundedness of corresponding deterministic lienard equations

recurrence principles and their application to stability theory for a class of stochastic hybrid systems in this paper we establish results similar in nature to the invariance principle for a class of stochastic hybrid systems modeled by set valued mappings in particular we characterize the set to which bounded random solutions converge in terms of a new concept called weak total recurrence a refinement of the result under the existence of a non increasing on average lyapunov like function is also established a comparison between weak total recurrence and the frequently studied invariance concepts are presented finally application of the main results to establish weak sufficient conditions for certifying stochastic stability properties like uniform global asymptotic stability in probability for compact sets and uniform global recurrence for open bounded sets are discussed

stochastic barbalat s lemma and its applications in the deterministic case a significant improvement on stability analysis of nonlinear systems is caused by introducing barbalat s lemma into control area after lyapunov s second method and lasalle s theorem were established this note considers the extension of barbalat s lemma to the stochastic case to this end the uniform continuity and the absolute integrability are firstly described in stochastic forms it is nevertheless a small generalization upon the existing references since our result can be used to adapted processes which are not necessarily ito diffusions when it is applied to ito diffusion processes many classical results on stochastic stability are covered as special cases

existence and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

global stability and stabilization of more general stochastic nonlinear systems abstract this paper considers the global stability and stabilization of more general stochastic nonlinear systems due to the absence of the conventional assumptions e g lipschitz condition the stochastic nonlinear systems under investigation may have more than one weak solution however the most associated results are only applicable to the stochastic systems having a unique strong solution and therefore it is meaningful to refine and extend the relevant concepts and methods to the more general case in this paper the concepts of stochastic stability in the more general sense are first introduced to cover the stochastic nonlinear systems having more than one weak solution then the generalized stochastic barbashin krasovskii theorem and lasalle theorem are established which present the criterions of stochastic stability for more general stochastic nonlinear systems as one of the main contributions in this paper we rigorously prove the generalized stochastic barbashin krasovskii theorem moreover based on the generalized theorems the output feedback and state feedback stabilization are accomplished respectively for two classes of high order stochastic nonlinear systems under rather weaker assumptions comparing to the existing literature

global stability analysis for stochastic coupled systems on networks this paper considers the global stability problem for some stochastic coupled systems on networks scsns we provide a systematic method for constructing a global lyapunov function for these scsns by using graph theory and the lyapunov second method consequently a new global stability principle which has a close relation to the topology property of the network is given as an application to the results we employ the principle to two well known coupled systems in physical and ecology and then some easy verified sufficient conditions which guarantee the global stability are obtained

a remark on stochastic logistic model with diffusion stochastic logistic system with diffusion is proposed and studied almost sufficient and necessary conditions for stability in time average and extinction are established some numerical examples are introduced to support the main results

population dynamical behavior of lotka volterra system under regime switching in this paper we investigate a lotka volterra system under regime switching dx t diag x 1 t x n t b r t a r t x t dt s r t db t where b t is a standard brownian motion the aim here is to find out what happens under regime switching we first obtain the sufficient conditions for the existence of global positive solutions stochastic permanence and extinction we find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the markov chain the limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients finally the main results are illustrated by several examples

a remark on a stochastic logistic model with levy jumps this note is concerned with a famous stochastic logistic equation with levy noises sufficient and necessary conditions for extinction and permanence are established the results reveal that the levy noise may change the properties of population dynamics significantly the results also reveal an important property of the levy noise it is unfavorable for the permanence of the population some numerical simulations are introduced to validate the analytical results

practical asymptotic stability of stochastic systems driven by levy processes and its application to control of tora systems two lyapunov type theorems are developed to investigate global practical asymptotic stability in probability for sdes driven by non vanishing levy processes which can be either affine or non a

existence uniqueness and global asymptotic stability of positive solutions of a predator prey system with holling ii functional response with random perturbation abstract this paper discusses a randomized two species predator prey system with holling ii functional response we show that the positive solution of the associated stochastic delay differential equation does not explode to infinity in a finite time in addition the existence uniqueness and global asymptotic stability of the positive solutions are studied

analysis of a general stochastic non autonomous logistic model with delays and levy jumps abstract in this paper a general stochastic non autonomous logistic model with delays and levy jumps is proposed and studied sufficient criteria for extinction non persistence in the mean and weak persistence of the model are established the threshold between weak persistence and extinction is also obtained finally numerical simulations are introduced to illustrate the theoretical analysis results our results illustrate that both persistence and extinction have close relationships with time dependent delay and levy noise

the asymptotic stability and exponential stability of nonlinear stochastic differential systems with markovian switching and with polynomial growth abstract this paper discusses the asymptotic stability and exponential stability of nonlinear stochastic differential systems with markovian switching sdswmss the systems coefficients are assumed to satisfy local lipschitz condition and polynomial growth condition by applying some novel techniques we propose some conditions under which such sdswmss are asymptotically stable and exponentially stable nontrivial examples are provided to illustrate our results