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

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

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

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

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

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

closed characteristics on non compact hypersurfaces in r 2n viterbo demonstrated that any 2n 1 dimensional compact hypersur face m r 2n of contact type has at least one closed characteristic this result proved the weinstein conjecture for the standard symplectic space r 2n various extensions of this theorem have been obtained since all for compact hypersurfaces in this paper we consider non compact hypersurfaces m r 2n coming from mechanical hamiltonians and prove an analogue of viterbo s result the main result provides a strong connection between the top half homology groups hi m i n 2n 1 and the existence of closed characteristics in the non compact case including the compact case

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

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

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

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

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

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

many closed symplectic manifolds have infinite hofer zehnder capacity we exhibit many examples of closed symplectic manifolds on which there is an autonomous hamiltonian whose associated flow has no nonconstant periodic orbits the only previous explicit example in the literature was the torus t2n n 2 with an irrational symplectic structure the underlying smooth manifolds of our examples include for instance the k3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it infinitely many minimal four manifolds having any given finitely presented group as their fundamental group and simply connected minimal four manifolds realizing all but finitely many points in the first quadrant of the geography plane below the line corresponding to signature 3 the examples are constructed by performing symplectic sums along suitable tori and then perturbing the symplectic form in such a way that hypersurfaces near the neck in the symplectic sum have no closed characteristics we conjecture that any closed symplectic four manifold with b 1 admits symplectic forms with a similar property

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

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

a c 2 smooth counterexample to the hamiltonian seifert conjecture in r 4 we construct a proper c 2 smooth function on r 4 such that its hamiltonian flow has no periodic orbits on at least one regular level set this result can be viewed as a c 2 smooth counterexample to the hamiltonian seifert conjecture in dimension four

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

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

convexity conditions of kantorovich function and related semi infinite linear matrix inequalities the kantorovich function x tax x ta 1x where a is a positive definite matrix is not convex in general from a matrix or convex analysis point of view it is interesting to address the question when is this function convex in this paper we prove that the 2 dimensional kantorovich function is convex if and only if the condition number of its matrix is less than or equal to 3 22 thus the convexity of the function with two variables can be completely characterized by the condition number the upper bound 3 22 is turned out to be a necessary condition for the convexity of the kantorovich function in any finite dimensional spaces we also point out that when the condition number of the matrix which can be any dimensional is less than or equal to 5 26 the kantorovich function is convex furthermore we prove that this general sufficient convexity condition can be improved to 2 3 in 3 dimensional space our analysis shows that the convexity of the function is closely related to some modern optimization topics such as the semi infinite linear matrix inequality or robust positive semi definiteness of symmetric matrices in fact our main result for 3 dimensional cases has been proved by finding an explicit solution range to some semi infinite linear matrix inequalities

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

existence tests for solutions of nonlinear equations using borsuk s theorem we show how interval arithmetic can be used in connection with borsuk s theorem to computationally prove the existence of a solution of a system of nonlinear equations it turns out that this new test which can be checked computationally in several different ways is more general than an existing test based on miranda s theorem in the sense that it is successful for a larger set of situations a numerical example is included

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

generalized counterexamples to the seifert conjecture using the theory of plugs and the self insertion construction due to the second author we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise linear fashion so that all minimal sets have codimension 1 in particular the 3 sphere s 3 has a real analytic dynamical system such that all limit sets are 2 dimensional we also prove that a 1 dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise linear fashion so that there are no closed leaves but all minimal sets are 1 dimensional these theorems provide new counterexamples to the seifert conjecture which asserts that every dynamical system on s 3 with no singular points has a periodic trajectory

a hybrid approach for efficient robust design of dynamic systems we propose a novel approach for the parametrically robust design of dynamic systems the approach can be applied to system models with parameters that are uncertain in the sense that values for these parameters are not known precisely but only within certain bounds the novel approach is guaranteed to find an optimal steady state that is stable for each parameter combination within these bounds our approach combines the use of a standard solver for constrained optimization problems with the rigorous solution of nonlinear systems the constraints for the optimization problems are based on the concept of parameter space normal vectors that measure the distance of a tentative optimum to the nearest known critical point i e a point where stability may be lost such normal vectors are derived using methods from nonlinear dynamics after the optimization the rigorous solver is used to provide a guarantee that no critical points exist in the vicinity of the optimum or to detect such points in the latter case the optimization is resumed taking the newly found critical points into account this optimize and verify procedure is repeated until the rigorous nonlinear solver can guarantee that the vicinity of the optimum is free from critical points and therefore the optimum is parametrically robust in contrast to existing design methodologies our approach can be automated and does not rely on the experience of the designing engineer a simple model of a fermenter is used to illustrate the concepts and the order of activities arising in a typical design process

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

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

taubes s proof of the weinstein conjecture in dimension three does every smooth vector field on a closed three manifold for example the three sphere have a closed orbit no according to counterexamples by k kuperberg and others on the other hand there is a special class of vector fields called reeb vector fields which are associated to contact forms the three dimensional case of the weinstein conjecture asserts that every reeb vector field on a closed oriented three manifold has a closed orbit this conjecture was recently proved by taubes using seiberg witten theory we give an introduction to the weinstein conjecture the main ideas in taubes s proof and the bigger picture into which it fits taubes s proof of the weinstein conjecture is the culmination of a large body of work both by taubes and by others in an attempt to make this story accessible to nonspecialists much of the present article is devoted to background and context and taubes s proof itself is only partially explained hopefully this article will help prepare the reader to learn the full story from taubes s paper 62 more exposition of this subject which was invaluable in the preparation of this article can be found in the online video archive from the june 2008 msri hot topics workshop 44 and in the article by auroux 5 below in 1 3 we introduce the statement of the weinstein conjecture and discuss some examples in 4 6 we discuss a natural strategy for approaching the weinstein conjecture which proves it in many but not all cases and provides background for taubes s work in 7 we give an overview of the big picture surrounding taubes s proof of the weinstein conjecture readers who already have some familiarity with the weinstein conjecture may wish to start here in 8 9 we recall necessary material from seiberg witten theory in 10 we give an outline of taubes s proof and in 11 we explain some more details of it to conclude in 12 we discuss some further results and open problems related to the weinstein conjecture 1 statement of the weinstein conjecture the weinstein conjecture asserts that certain vector fields must have closed orbits before stating the conjecture at the end of this section we first outline its origins this is discussion is only semi historical because only a sample of the relevant works will be cited and not always in chronological order 1 1 closed orbits of vector fields let y be a closed manifold in this article all manifolds and all objects defined on them are smooth unless otherwise stated 2000 mathematics subject classification 57r17 57r57 53d40 partially supported by nsf grant dms 0806037

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

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

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

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

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

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