parasite population delay model of malaria type with stochastic perturbation and environmental criterion for limitation of disease we present a stochastic delay model of an infectious disease malaria transmitted by a vectors mosquitoes after an incubation time a criterion for limitation of disease is found

dynamical behavior of a one predator and two independent preys system with stochastic perturbations abstract a one predator and two independent preys system with stochastic perturbations is considered first we show that there is a positive solution of this system then we investigate the long time behavior of this system sufficient criteria for extinction and persistence in time average are established moreover it is also shown that this system has a stationary distribution and it is ergodicity finally examples and simulations are carried on to verify these results

stochastic periodic solutions of stochastic differential equations driven by levy process abstract in this paper a class of periodic stochastic differential equations driven by levy processes sdeslp is studied firstly we give the existence uniqueness theorem for a large class of sdeslp then using the properties of the periodic markov processes sufficient conditions for the existence and uniqueness of periodic solution to the periodic sdeslp are given finally an example is introduced to illustrate the results developed

analysis of a hybrid switching svir epidemic model with vaccination and levy noise abstract this paper is concerned with the dynamical behavior of a hybrid switching svir epidemic model with vaccination and levy noise besides a standard geometric brownian motion another two driving processes are considered a stationary poisson point process and a continuous time finite state markov chain to begin with we obtain sufficient conditions for persistence in the mean of the disease then we establish sufficient conditions for extinction of the disease furthermore in the case of persistence we also obtain sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic lyapunov function with regime switching

boundedness and convergence of some self attracting diffusions we consider some self attracting diffusions and study the behaviour of their paths when time tends to infinity we prove that in case the interaction is nonlocal the paths are bounded a s even if they don t converge otherwise we generalize the convergence result of m cranston and y le jan

stationary distribution of a stochastic delayed sveir epidemic model with vaccination and saturation incidence abstract in this paper we develop and analyze a stochastic delayed sveir epidemic model with vaccination and saturation incidence by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model the existence of a stationary distribution implies stochastic weak stability numerical simulations are introduced to illustrate the analytical result

nonconvergence to unstable points in urn models and stochastic approximations a particle in mathbf r d moves in discrete time the size of the n th step is of order 1 n and when the particle is at a position mathbf v the expectation of the next step is in the direction mathbf f mathbf v for some fixed vector function mathbf f of class c 2 it is well known that the only possible points mathbf p where mathbf v n may converge are those satisfying mathbf f mathbf p mathbf 0 this paper proves that convergence to some of these points is in fact impossible as long as the noise the difference between each step and its expectation is sufficiently omnidirectional the points where convergence is impossible are the unstable critical points for the autonomous flow d dt mathbf v t mathbf f v t this generalizes several known results that say convergence is impossible at a repelling node of the flow

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

self interacting diffusions iii symmetric interactions let m be a compact riemannian manifold a self interacting diffusion on m is a stochastic process solution to dx t dw t x t 1 t t 0 v xs x t ds dt where w t is a brownian vector field on m and v x y v x y a smooth function let t 1 t t 0 xs ds denote the normalized occupation measure of x t we prove that when v is symmetric t converges almost surely to the critical set of a certain nonlinear free energy functional j furthermore j has generically finitely many critical points and t converges almost surely toward a local minimum of j each local minimum has a positive probability to be selected

a class of self interacting processes with applications to games and reinforced random walks this paper studies a class of non markovian and nonhomogeneous stochastic processes on a finite state space relying on a recent paper by benaim hofbauer and sorin siam j control optim 44 2005 pp 328 348 it is shown that under certain assumptions the asymptotic behavior of occupation measures can be described in terms of a certain set valued deterministic dynamical system this provides a unified approach to simulated annealing type processes and permits the study of new models of vertex reinforced random walks and new models of learning in games such as markovian fictitious play

asymptotic behaviors of a cell to cell hiv 1 infection model perturbed by white noise abstract in this paper we analyze a mathematical model of cell to cell hiv 1 infection to c d 4 t cells perturbed by stochastic perturbations first of all we investigate that there exists a unique global positive solution of the system for any positive initial value then by using lyapunov analysis methods we study the asymptotic property of this solution moreover we discuss whether there is a stationary distribution for this system and if it owns the ergodic property numerical simulations are presented to illustrate the theoretical results

stationary distribution of stochastic population systems under regime switching abstract this paper is concerned with n species model of facultative mutualism in random environments the environment variability in this study is characterized with both white noise and color noise modeled by markovian switching we established new sufficient conditions that ensuring that the system model is positive recurrent we also showed the existence of a unique ergodic stationary distribution the presented results are demonstrated by numerical simulations

a periodic epidemic model in a patchy environment an epidemic model in a patchy environment with periodic coefficients is investigated in this paper by employing the persistence theory we establish a threshold between the extinction and the uniform persistence of the disease further we obtain the conditions under which the positive periodic solution is globally asymptotically stable at last we present two examples and numerical simulations

a climate based malaria transmission model with structured vector population in this paper we present a malaria transmission model with periodic birth rate and age structure for the vector population we first introduce the basic reproduction ratio for this model and then show that there exists at least one positive periodic state and that the disease persists when mathcal r 0 1 it is also shown that the disease will die out if mathcal r 0 1 provided that the invasion intensity is not strong we further use these analytic results to study the malaria transmission cases in kwazulu natal province south africa some sensitivity analysis of mathcal r 0 is performed and in particular the potential impact of climate change on seasonal transmission and populations at risk of the disease is analyzed

properties of solutions of stochastic differential equations with continuous state dependent switching abstract this work is concerned with several properties of solutions of stochastic differential equations arising from hybrid switching diffusions the word hybrid highlights the coexistence of continuous dynamics and discrete events the underlying process has two components one component describes the continuous dynamics whereas the other is a switching process representing discrete events one of the main features is the switching component depending on the continuous dynamics in this paper weak continuity is proved first then continuous and smooth dependence on initial data are demonstrated in addition it is shown that certain functions of the solutions verify a system of kolmogorov s backward differential equations moreover rates of convergence of numerical approximation algorithms are dealt with

self interacting diffusions a simulated annealing version we study asymptotic properties of processes x living in a riemannian compact manifold m solution of the stochastic differential equation sde dx t dw t x t beta t nabla v mu t x t dt with w a brownian vector field t alog t 1 mu t frac 1 t int 0 t delta x s ds and v mu t x frac 1 t int 0 t v x x s ds v being a smooth function we show that the asymptotic behavior of t can be described by a non autonomous differential equation this class of processes extends simulated annealing processes for which v x y is only a function of x in particular we study the case m mathbb s n the n dimensional sphere and v x y cos d x y where d x y is the distance on mathbb s n which corresponds to a process attracted by its past trajectory in this case it is proved that t converges almost surely towards a dirac measure

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

self repelling diffusions on a riemannian manifold let m be a compact connected oriented riemannian manifold the purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space m times mathbb r n which is obtained via a natural change of variable from a self repelling diffusion taking the form begin aligned dx t sigma db t x t int 0 t nabla v x s x t dsdt qquad x 0 x end aligned where b t is a brownian vector field on m sigma 0 and v x y v x y is a diagonal mercer kernel we prove that the induced semi group enjoys the strong feller property and has a unique invariant probability mu given as the product of the normalized riemannian measure on m and a gaussian measure on mathbb r n we then prove an exponential decay to this invariant probability in l 2 mu and in total variation

ergodic property of a lotka volterra predator prey model with white noise higher order perturbation under regime switching abstract in this paper we investigate a classical lotka volterra predator prey model with telephone noise and a higher order perturbation of white noise the existence of a unique positive solution is discussed and sufficient conditions for the existence of an ergodic stationary distribution is established some simulation figures are presented to illustrate the analytical findings

stochastic persistence and stationary distribution in a holling tanner type prey predator model abstract in this paper we study a stochastic predator prey model with beddington deangelis type functional response and logistic growth for predators the deterministic model is already well studied and we recall some important results here we construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations for the stochastic model we show that the system admits unique positive global solution starting from the positive initial value then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes finally we show that the system has a stationary distribution under certain parametric restrictions numerical simulations are carried out to substantiate the analytical results

mosquito stage structured malaria models and their global dynamics we formulate mosquito stage structured continuous time compartmental malaria models which include four distinct metamorphic stages of mosquitoes we derive a formula for the reproductive number of infection and investigate the existence of endemic equilibria we determine conditions under which the models undergo either forward or backward bifurcation we carry out rigorous mathematical analysis on the model dynamics by proving the global stability of the infection free equilibrium while a forward bifurcation occurs and the global stability of the endemic equilibrium as the reproductive number is greater than one in certain cases our study can be also applied to other diseases such as chikungunya fever

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

stock trading an optimal selling rule trading in stock markets consists of three major steps select a stock purchase a number of shares and eventually sell them to make a profit the timing to buy and sell is extremely crucial a selling rule can be specified by two preselected levels a target price and a stop loss limit this paper is concerned with an optimal selling rule based on the model characterized by a number of geometric brownian motions coupled by a finite state markov chain such a policy can be obtained by solving a set of two point boundary value differential equations moreover the corresponding expected target period and probability of making money and that of losing money are derived analytic solutions are obtained in one and two dimensional cases finally a numerical example is considered to demonstrate the effectiveness of our method

dynamics in a periodic competitive model with stage structure in this paper we consider a time delayed periodic system which describes the competition among mature populations by appealing to theories of monotone dynamical systems periodic semiflows and uniform persistence we analyze the evolutionary behavior of the system and establish sufficient conditions for competitive coexistence and exclusion

global dynamic analysis of a model for vector borne diseases on bipartite networks abstract in this paper an sis si model on bipartite networks for vector borne disease is developed the basic reproduction number r 0 is identified and analyzed the global dynamics are completely determined by the reproduction number r 0 it is shown that if r 0 1 the disease free equilibrium is globally asymptotically stable if r 0 1 there is a unique endemic equilibrium which is globally asymptotically stable finally numerical simulations are performed to validate the theoretical results and reveal the influence of network structure on basic reproduction number the transmission scale and propagation speed

self avoiding random walk a brownian motion model with local time drift a natural model for a self avoiding brownian motion inr d when specialised and simplified tod 1 becomes the stochastic differential equation x t b t int limits 0 t g x s l s x s ds where l t x t 0 x r is the local time process ofx thoughx is not markovian an analogue of the ray knight theorem holds for l x x r which allows one to prove in many cases of interest that mathop lim limits t to infty x t t exists almost surely and to identify the limit

dynamics of a stochastic siqr epidemic model with standard incidence abstract in this paper a stochastic siqr epidemic model with quarantine and standard incidence is investigated we first consider the existence and uniqueness of the global positive solution to the stochastic system further we obtain conclusions that the large noise can make the disease die out exponentially and the stationary distribution will exist under certain conditions

the global dynamics for a stochastic sis epidemic model with isolation abstract in this paper we investigate the dynamical behavior for a stochastic sis epidemic model with isolation which is as an important strategy for the elimination of infectious diseases it is assumed that the stochastic effects manifest themselves mainly as fluctuation in the transmission coefficient the death rate and the proportional coefficient of the isolation of infective it is shown that the extinction and persistence in the mean of the model are determined by a threshold value r 0 s that is if r 0 s 1 then disease dies out with probability one and if r 0 s 1 then the disease is stochastic persistent in the means with probability one furthermore the existence of a unique stationary distribution is discussed and the sufficient conditions are established by using the lyapunov function method finally some numerical examples are carried out to confirm the analytical results

on the creation growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme the equations which govern a simple pooled chemical reaction scheme are analysed in detail in terms of a nondimensional parameter mu which represents the amount of the pooled chemical originally present it is shown that there is one finite equilibrium point with a hopf bifurcation occurring at mu 1 the phase plane at infinity is then examined and it is shown that there are equilibrium points at infinity at the positive ends of both axes the nature of which are discussed this together with a knowledge of the global phase portraits for mu ll 1 and mu gg 1 enables the global phase portrait to be constructed for all positive mu from this it emerges that the stable limit cycle created at mu 1 by a hopf bifurcation is destroyed at mu 0 mu 0 1 by an infinite period bifurcation due to the formation of a heteroclinic orbit by the separatrices from the equilibrium points at infinity the form of this heteroclinic orbit is then discussed in detail and in particular i

a mathematical analysis of the effects of control strategies on the transmission dynamics of malaria we formulate a deterministic model with two latent periods in the non constant host and vector populations in order to theoretically assess the potential impact of personal protection treatment and possible vaccination strategies on the transmission dynamics of malaria the thresholds and equilibria for the model are determined the model is analysed qualitatively to determine criteria for control of a malaria epidemic and is used to compute the threshold vaccination and treatment rates necessary for community wide control of malaria in addition to having a disease free equilibrium which is locally asymptotically stable when the basic reproductive number is less than unity the model exhibits the phenomenon of backward bifurcation where a stable disease free equilibrium coexists with a stable endemic equilibrium for a certain range of associated reproductive number less than one from the analysis we deduce that personal protection has a positive impact on disease control but to eradicate the disease in the absence of any other control measures the efficacy and compliance should be very high our results show that vaccination and personal protection can suppress the transmission rates of the parasite from human to vector and vice versa if the treated populations are infectious then certain conditions should be satisfied for treatment to reduce the spread of malaria in a community among the interesting dynamical behaviours of the model numerical simulations show a backward bifurcation which gives a challenge to the designing of effective control measures

a periodic age structured epidemic model with a wide class of incidence rates abstract in this paper a time delayed epidemic model is formulated to describe the dynamics of seasonal diseases with age structure by the method of the spectral radius of an integral operator we define the basic reproduction number r 0 of the model it is shown that the disease is uniformly persistent and there exists at least one positive periodic state when r 0 1 while the disease will die out if r 0 1 the presented case study not only confirms the theoretical results but also demonstrates that the epidemic peak is very sensitive to the maturation period and the magnitude of seasonality which is different from the dynamics of the model without considering age heterogeneities these findings contribute to better understanding the epidemiological properties of the disease with age structure

threshold behavior in a stochastic siqr epidemic model with standard incidence and regime switching abstract this paper is concerned with the dynamical behavior of a stochastic siqr epidemic model with standard incidence which is disturbed by both white and telegraph noises firstly we obtain sufficient conditions for persistence in the mean of the disease then we establish sufficient conditions for extinction of the disease in addition in the case of persistence we get sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic lyapunov function with regime switching meanwhile the threshold between persistence in the mean and extinction of the stochastic system is also obtained finally some numerical simulations are introduced to demonstrate the analytical results

some vector borne diseases with structured host populations extinction and spatial spread we derive from a structured population model a system of delay differential equations describing the interaction of five subpopulations namely susceptible and infected adult and juvenile reservoirs and infected adult vectors for a vector borne disease with particular reference to west nile virus and we also incorporate spatial movements by considering the analogue reaction diffusion systems with nonlocal delayed terms specific conditions for the disease eradication and sharp conditions for the local stability of the disease free equilibrium are obtained using comparison techniques coupled with the spectral theory of monotone linear semi flows a formal calculation of the minimal wave speed for the traveling waves is given and compared with field observation data

global stability for a delayed hiv 1 infection model with nonlinear incidence of infection abstract in this paper a delayed hiv 1 infection model with nonlinear incidence of infection is reinvestigated it is shown that if the reproduction number r 1 then the system is permanent and the infective equilibrium of the system is globally asymptotically stable thus the global dynamics of the system is completely determined by the reproduction number r the results obtained enrich and improve the corresponding results given by wang et al x wang y tao x song a delayed hiv 1 infection model with beddington deangelis functional response nonlinear dynamics 62 2010 67 72 the conclusions we established also verify the numerical simulation results on the global asymptotic stability of the infective equilibrium in the paper d li w ma asymptotic properties of an hiv 1 infection model with time delay j math anal appl 335 2007 683 691

the effect of diffusion on the time varying logistic population growth abstract in this paper we consider the effect of diffusion on the species that live in changing patches environment different from the former studies 1 4 we pay attention to the more important situation in conservation biology that species live in a weak patches environment in the sense that species in some of the isolated patches will be extinct without the contribution from other patches we obtain an interesting result the identical species can persist for some diffusion rates and can also vanish for another set of restriction on diffusion rates

survival and stationary distribution of a sir epidemic model with stochastic perturbations abstract in this paper the dynamics of a sir epidemic model is investigated first we show that the system admits a unique positive global solution starting from the positive initial value then when r 0 1 we show that the stochastic model has a stationary distribution under certain parametric restrictions in particular we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence when r 0 1 a result on fluctuation of the solution around the disease free equilibrium of deterministic system is established under suitable conditions finally numerical simulations are carried out to illustrate the theoretical results

stationary distributions for diffusions with inert drift consider reflecting brownian motion in a bounded domain in mathbb r d that acquires drift in proportion to the amount of local time spent on the boundary of the domain we show that the stationary distribution for the joint law of the position of the reflecting brownian motion and the value of the drift vector has a product form moreover the first component is uniformly distributed on the domain and the second component has a gaussian distribution we also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential

asymptotic behavior of brownian polymers we consider a system that models the shape of a growing polymer our basic problem concerns the asymptotic behavior ofx t the location of the end of the polymer at timet we obtain bounds onx t in the physically uninteresting case thatd 1 and the interaction functionf x 0 if in addition f x behaves for largex likecx with 1 we obtain a strong law that gives the exact growth rate

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

the application of singularity theory to isothermal autocatalytic open systems the elementary scheme a mb m 1 b in a well stirred open system isothermal c s t r is determined via singularity theory methods these methods allow all of the possible patterns for the dependence of the stationary state extent of conversion x on the residence time da i e the x da bifurcation diagrams to be identified and located the cubic rate law has a cusp singularity separating diagrams with a unique dependence of x on da from those which display a simple s shaped hysterisis loop with multiple stationary states this behaviour is qualitatively similar to that shown by a simple exothermic reaction in an adiabatic c s t r i e the two systems are contact equivalent if the autocatalyst is not infinitely stable but instead undergoes a simple decay b c rate k2b a wider range of bifurcation diagrams is possible with isolas or mushroom patterns these arise as the system is unfolded from its winged cusp singularity by varying parameters such as the catalyst lifetime and inflow concentration it is shown that these are also the only patterns possible for a generalized order of autocatalysis i e for a rate proportional to abm with m taking any value greater than one integral or non integral the ranges of the above parameters over which t he different responses are found are also given analytically for the general m these parameters cannot however give a complete unfolding so certain additional bifurcation diagrams that are found for the exothermic reaction in a nonisothermal non adiabatic reactor which also has a winged cusp singularity are not found in the autocatalytic system

the strong law of large numbers for a brownian polymer we prove the strong law of large numbers for a continuous time version of reinforced random walk this extends previous results of durrett and rogers

on strong feller recurrence and weak stabilization of regime switching diffusions focusing on two component markov processes having continuous dynamics and discrete events this work develops asymptotic properties of regime switching diffusions first strong feller property is established next classifying the underlying processes as recurrence positive recurrence or null recurrence or transience sufficient conditions for such processes are obtained in addition to providing general criteria for recurrent and transient switching diffusions for processes that are linearizable with respect to the continuous component easily verifiable conditions are established also provided are conditions for null recurrent processes furthermore conditions for controlled switching diffusions in which the feedback controls ensure weak stability or ergodicity are given finally simple examples are used for demonstration

self interacting diffusions this paper is concerned with a general class of self interacting diffusions x t t 0 living on a compact riemannian manifold m these are solutions to stochastic differential equations of the form dx t brownian increments drift term depending on x t and t the normalized occupation measure of the process it is proved that the asymptotic behavior of t can be precisely related to the asymptotic behavior of a deterministic dynamical semi flow t t 0 defined on the space of the borel probability measures on m in particular the limit sets of t are proved to be almost surely attractor free sets for these results are applied to several examples of self attracting repelling diffusions on the n sphere for instance in the case of self attracting diffusions our results apply to prove that t can either converge toward the normalized riemannian measure or to a gaussian measure depending on the value of a parameter measuring the strength of the attraction

threshold of a regime switching sirs epidemic model with a ratio dependent incidence rate abstract in this paper we study the dynamical behavior of a regime switching sirs epidemic model with a ratio dependent incidence rate we propose a stochastic reproduction number r 0 s which can be regarded as a threshold to use in identifying the stochastic extinction and persistence if r 0 s 1 the disease dies out exponentially with probability one while if r 0 s 1 there exists a unique ergodic stationary distribution of the positive solutions to the system which implies the stochastic persistence of the infectious disease

analysis of degenerate hopf bifurcations for a nonlinear model of calcium metabolism results of the in viva studies related to calcium metabolism have been generally analysed within the framework of linear mathematical models e g classical compartmental analysis 11 this approach assumes the maintenance of the extracellular calcium concentration at a nearly constant value under physiological conditions plasma calcium homeostasis and the adequacy of a sum of exponential terms the analytical solution of a first order linear differential system to fit kinetic data such as the time course of tracer concentration nonlinear models which exhibit a self oscillatory behaviour have been proposed recently a model of the hormonal control of calcium homeostasis in chicken includes the consideration of a growth function and takes into account complex interrelationships among the calciumregulating hormones and between these hormones and specific metabolic functions 2 another model constructed in our laboratory is a self oscillating model for calcium metabolism in the rat it includes some nonlinear processes involved in bone calcium metabolism and adequately reproduces the kinetic behaviour of radiocalcium in plasma including circadian variations 3 in contrast to linear systems for which the simple dynamic behaviour patterns are predictable and their analytical solutions are commonly applied in calcium metabolism field it is wellestablished that nonlinear systems have potentially numerous and complex spatio temporal expressions excitability periodic oscillations chaos etc therefore the relevance of the proposed nonlinear models and their significance with regard to the control of calcium metabolism require a concomitant knowledge of the set of the dynamic patterns that in theory such nonlinear structures are able to produce the variety of the dynamic behaviours exhibited by the self oscillating model we have established in rat or rather its nonlinear sub structure 4 51 motivated this theoretical work our model of calcium metabolism has been constructed in the general context of compartmental formalism fig 1 briefly this model was based on the explicit consideration the nonlinear properties of the phase transition and nucleation mechanisms associated with bone calcification and their formulation through a single second order autocatalytic process 5 in addition to its ability to generate periodic temporal variations fitting to experimental data fig 2 the model correctly approximates the growth curve of calcium mass as a function of age and proposes a dynamic organization for bone calcium metabolism which distinguishes

permanence of a regime switching malaria transmission model abstract malaria transmission is a complex system and lots of malaria transmission models have been showed by determined differential equations considering that variability of the environment may have an important impact on its transmission we introduce the regime switching to a malaria transmission model with susceptible infectious recovered susceptible sirs structure for human population and susceptible infectious si structure for mosquito population first we show the existence of positive solutions for the model then the dynamics of the model is carried out by constructing a suitable lyapunov function we show that there exists a stationary distribution and so the threshold for the disease to prevail is given at last an example with two states is given to illustrate the results

stability of an seir epidemic model with independent stochastic perturbations abstract for an epidemic model of the type mentioned we prove a theorem on almost sure exponential stability of the disease free equilibrium for small values of the diffusion parameter we describe the stability of the disease free equilibrium point in terms of an appropriate analogue r of the basic reproduction number r 0 of the deterministic special case whenever 0 then r r 0 for small values of the stability theorem guarantees almost sure exponential stability whenever r 1 we also discuss the effect of increasing

stationary distribution of stochastic np ecological model under regime switching abstract marine ecosystem is inevitably disturbed by random factors this article discusses a nutrient phytoplankton np model with tpp which is disturbed by both white and telegraph noises firstly we get sufficient conditions for the existence of ergodic stationary distribution by constructing a suitable stochastic lyapunov function with regime switching and then we establish sufficient conditions for extinction of the phytoplankton finally the results of this paper are verified by numerical simulation

stability of markov semigroups and applications to parabolic systems abstract a new theorem for asymptotic stability of markov semigroups is proved this result is applied to semigroups generated by parabolic systems describing the evolution of densities of two state diffusion processes

critical result on the break even concentration in a single species stochastic chemostat model abstract in this paper the strictly critical parameter is identified as the break even concentration for the stochastic chemostat model s 0 suffices for extinction of the microorganism for any noise intensity 2 0 is proved firstly which completely solves the open problem proposed in 10 then by using the feller test we show that the microorganism will go extinct in probability if s 0 which has not been studied in the known literature when s 0 in addition to the previously discussed persistence in mean the stochastic persistence of the microorganism is studied by use of chebyshev s inequality besides the existence of the stationary distribution is proved for the considered model numerical simulations are introduced finally to support the obtained 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

mean field dispersal induced synchrony and stability in an epidemic model under patchy environment abstract in our highly connected modern world the dispersion of population is very important at the time when disease spread to evaluate its nature i e whether the disease will die out or grow therefore in this paper sir susceptible infected recovered model is framed to describe the dispersal of population for different patches communities which are interconnected using mean field diffusive coupling the effect of coupling on the dynamics of the epidemiology classes among different patches has been investigated the dynamics of the epidemiology classes is explored under the environment of symmetric coupling and asymmetric coupling it is shown that the synchronization and stability of disease free equilibrium or endemic equilibrium can be attained simultaneously through the mechanism of suppression of oscillations namely amplitude death ad and oscillation death od irrespective of coupling environment i e symmetric and asymmetric we have observed the tri stable state and its transition from different states through hopf bifurcation and transcritical bifurcation the basic reproduction number is also established for the stability of disease free equilibrium and endemic equilibrium through bifurcation analysis it has been found that the disease can become stable or it can die out through dispersal of population among patches

stationary distribution and periodic solutions for stochastic holling leslie predator prey systems abstract the stochastic autonomous and periodic predator prey systems with holling and leslie type functional response are investigated for the autonomous system we prove that there exists a unique stationary distribution which is ergodic by constructing a suitable lyapunov function under relatively small white noise the result shows that stationary distribution doesn t rely on the existence and the stability of the positive equilibrium in the undisturbed system furthermore for the corresponding non autonomous system we show that there exists a positive periodic markov process under relatively weaker condition finally numerical simulations illustrate our theoretical results

persistence and distribution of a stochastic susceptible infected removed epidemic model with varying population size abstract in this paper the dynamics of a stochastic susceptible infected removed model in a population with varying size is investigated we firstly show that the stochastic epidemic model has a unique global positive solution with any positive initial value then we verify that random perturbations lead to extinction when some conditions are being valid moreover we prove that the solution of the stochastic epidemic model is persistent in the mean by building up a suitable lyapunov function and using generalized ito s formula further the stochastic epidemic model admits a stationary distribution around the endemic equilibrium when parameters satisfy some sufficient conditions to end this contribution and to check the validity of the main results numerical simulations are separately carried out to illustrate these results

self attracting diffusions case of the constant interaction in this paper we study a self attracting diffusion in the case of a constant self attraction and for dimension larger than two we prove that this process converges almost surely

dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis abstract in this paper we propose new mathematical models with nonlinear incidence rate and double epidemic hypothesis then we dedicate to develop a method to obtain the threshold of the stochastic sis epidemic model to this end first we investigate the stability of the equilibria of the deterministic system and obtain the conditions for the extinction and the permanence of two epidemic diseases second we explore and obtain the threshold of a stochastic sis system for the extinction and the permanence in mean of two epidemic diseases the results show that a large stochastic disturbance can cause infectious diseases to go to extinction in other words the persistent infectious disease of a deterministic system can become extinct due to the white noise stochastic disturbance this implies that the stochastic disturbance is conducive to epidemic diseases control to illustrate the performance of the theoretical results we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients

almost sure stabilization for feedback controls of regime switching linear systems with a hidden markov chain this work is devoted to the almost sure stabilization of adaptive control systems that involve an unknown markov chain the control system displays continuous dynamics represented by differential equations and discrete events given by a hidden markov chain in the previous investigation on this class of problems averaging criteria were used which provides only the system behavior in some expectation sense a closer scrutiny of the system behavior necessarily requires the consideration of sample path properties different from previous work on stabilization of adaptive controlled systems with a hidden markov chain where average criteria were considered this work focuses on the almost sure stabilization or sample path stabilization of the underlying processes under simple conditions it is shown that as long as the feedback controls have linear growth in the continuous component the resulting process is regular moreover by appropriate choice of the lyapunov functions it is shown that the adaptive system is stabilizable almost surely as a by product it is also established that the controlled process is positive recurrent

dynamics of a stochastic sir epidemic model with saturated incidence in this paper the dynamics of a stochastic sir epidemic model with saturated incidence is investigated firstly we prove that the system has a unique global positive solution with any positive initial value then we verify that random effect may lead the disease to extinction under a simple condition thirdly we establish a sufficient condition for persistence in the mean of the disease moreover we show that there is a stationary distribution to the stochastic system under certain parametric restrictions finally some numerical simulations are carried out to confirm the analytical results

asymptotic behavior of a stochastic delayed seir epidemic model with nonlinear incidence abstract this paper is concerned with a stochastic delayed seir epidemic model with nonlinear incidence firstly we verify that there exists a unique global positive solution of the system then by constructing some suitable lyapunov functions we study the asymptotic behaviors of the disease free equilibrium and the endemic equilibrium respectively

a spatial seirs reaction diffusion model in heterogeneous environment abstract we propose a susceptible exposed infected recovered susceptible seirs reaction diffusion model where the disease transmission and recovery rates can be spatially heterogeneous the basic reproduction number r 0 is connected with the principal eigenvalue of a linear cooperative elliptic system threshold type results on the global dynamics in terms of r 0 are established the monotonicity of r 0 with respect to the diffusion rates of the exposed and infected individuals which does not hold in general is established in several cases finally the asymptotic profile of the endemic equilibrium is investigated when the diffusion rate of the susceptible individuals is small our results reveal the importance of the movement of the exposed and recovered individuals in disease dynamics as opposed to most of previous works which solely focused on the movement of the susceptible and infected individuals

the stationary distribution and ergodicity of a stochastic phytoplankton zooplankton model with toxin producing phytoplankton under regime switching abstract this paper addresses a stochastic phytoplankton zooplankton model with toxin producing phytoplankton under regime switching the stochastic model in this paper includes white and colored noises for the model we first prove the existence of the global positive solution of the model and then by using khasminskii s method and lyapunov function we derive sufficient conditions for the existence of a unique ergodic stationary distribution finally numerical simulations are carried out to support our results

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

a multipatch malaria model with logistic growth populations in this paper we propose a multipatch model to study the effects of population dispersal on the spatial spread of malaria between patches the basic reproduction number mathcal r 0 is derived and it is shown that the disease free equilibrium is locally asymptotically stable if mathcal r 0 1 bounds on the disease free equilibrium and mathcal r 0 are given a sufficient condition for the existence of an endemic equilibrium when mathcal r 0 1 is obtained for the two patch submodel the dependence of mathcal r 0 on the movement of exposed infectious and recovered humans between the two patches is investigated numerical simulations indicate that travel can help the disease to become endemic in both patches even though the disease dies out in each isolated patch however if travel rates are continuously increased the disease may die out again in both patches

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

analysis of a stochastic hybrid population model with allee effect abstract this research formulates and studies a stochastic population model with allee effect and markovian switching sufficient criteria for extinction stochastic permanence and the existence of a unique ergodic invariant measure of the model are established the result indicates that these long term behaviors of the model are intimately associated with the random perturbations

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

dynamical behaviors of a stochastic malaria model a case study for yunnan china abstract in this paper we investigate dynamical behaviors for a stochastic malaria model by introducing the effect of environmental white noise on transmission dynamics of malaria firstly the distance between stochastic solutions and the disease free equilibrium of the corresponding deterministic model is estimated in the time mean sense by constructing stochastic lyapunov functions particularly by the comparison theorem of stochastic differential equations and the law of large numbers sufficient conditions for the extinction of the disease are obtained in the special case next the existence of the unique ergodic stationary distribution is proved by constructing a suitable lyapunov function finally the model is used to simulate the human malaria data in yunnan province and predict the trend of malaria in yunnan

stationary distribution and extinction of the ds i a model disease with periodic parameter function and markovian switching this paper introduces the ds i a model with periodic parameter function and markovian switching first we will prove that the solution of the system is positive and global furthermore we draw a conclusion that there exists nontrivial positive periodic solution for the stochastic system and we establish sufficient conditions for extinction of system moreover we construct stochastic lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to ds i a model perturbed by white and telephone noises and we also establish sufficient conditions for extinction of system with regime switching finally we test our theory conclusion by simulations

the ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence abstract in this paper we include stochastic perturbations into sir and seir epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number r 0 the long time behavior of the two stochastic systems is studied mainly we utilize stochastic lyapunov functions to show under some conditions the solution has the ergodic property as r 0 1 while exponential stability as r 0 1 at last we make simulations to conform our analytical results

global analysis and numerical simulations of a novel stochastic eco epidemiological model with time delay abstract in this paper a new stochastic eco epidemiological model is proposed and investigated and the high dimensional model includes time delay and a general incidence rate which make the study more complex first the existence and uniqueness of the global positive solution with any positive initial value are verified second using stochastic analysis methods and some differential inequality techniques we explore the long time asymptotic properties of the stochastic delayed system furthermore we prove that the solution of the stochastic delayed system has a unique stationary distribution and it has ergodic property third the sufficient conditions of permanence in mean or extinction for the ecological populations are obtained finally we introduce a series of computer simulations to show the performance of the theoretical results

a reaction diffusion lyme disease model with seasonality this paper is devoted to the study of a reaction diffusion lyme disease model with seasonality in the case of a bounded habitat we obtain a threshold result on the global stability of either disease free or endemic periodic solution in the case of an unbounded habitat we establish the existence of the disease spreading speed and its coincidence with the minimal wave speed for time periodic traveling wave solutions we also estimate parameter values based on some published data and use them to study the lyme disease transmission in port dove ontario our numerical simulations are consistent with the obtained analytic results

a periodic vector bias malaria model with incubation period in this paper we propose a malaria model which takes into account the climate factors the extrinsic incubation period eip and the vector bias effect we first introduce the basic reproduction ratio r 0 and then prove that r 0 serves as a threshold parameter in determining the global dynamics of the model that is the disease free periodic solution is globally attractive if r 0 le1 and the system admits a unique positive periodic solution which is globally attractive if r 0 1 numerically we study the malaria transmission case in maputo province mozambique our numerical simulation results are consistent with the obtained analytic results in addition we observe that prolonging the length of the eip is helpful for the control of the disease

dynamical behavior of a stochastic svir epidemic model with vaccination abstract in this paper we investigate the dynamical behavior of svir models in random environments firstly we show that if r 0 s 1 the disease of stochastic autonomous svir model will die out exponentially if r 0 s 1 the disease will be prevail moreover this system admits a unique stationary distribution and it is ergodic when r 0 s 1 results show that environmental white noise is helpful for disease control secondly we give sufficient conditions for the existence of nontrivial periodic solutions to stochastic svir model with periodic parameters finally numerical simulations validate the analytical results

stochastic extinction and persistence of a parasite host epidemiological model abstract in this paper we investigate the stochastic extinction and persistence of a parasite host epidemiological model we show that the global dynamics of the stochastic model can be governed by the basic reproduction number r 0 s if r 0 s 1 under mild extra conditions the disease goes to extinction with probability one and the disease free dynamics occurs while r 0 s 1 under mild extra conditions the disease persists and endemic dynamics occurs almost surely the solutions of the stochastic model fluctuate around the steady state of the deterministic model and a unique stationary distribution can be found based on realistic parameters of daphnia microparasite system numerical simulations have been performed to verify extend our analytical results epidemiologically we find that 1 large environment fluctuations can suppress the outbreak of disease 2 the distributions are governed by r 0 s 3 the noise perturbations can be beneficial to control the spread of disease on average

analysis of an age structured model for tick populations subject to seasonal effects abstract we investigate an age structured hyperbolic equation model by allowing the birth and death functions to be density dependent and periodic in time with the consideration of seasonal effects by studying the integral form solution of this general hyperbolic equation obtained through the method of integration along characteristics we give a detailed proof of the uniqueness and existence of the solution in light of the contraction mapping theorem with additional biologically natural assumptions using the tick population growth as a motivating example we derive an age structured model with time dependent periodic maturation delays which is quite different from the existing population models with time independent maturation delays for this periodic differential system with seasonal delays the basic reproduction number r 0 is defined as the spectral radius of the next generation operator then we show the tick population tends to die out when r 0 1 while remains persistent if r 0 1 when there is no intra specific competition among immature individuals due to the sufficient availability of immature tick hosts the global stability of the positive periodic state for the whole model system of four delay differential equations can be obtained with the observation that a scalar subsystem for the adult stage size can be decoupled the challenge for the proof of such a global stability result can be overcome by introducing a new phase space based on which a periodic solution semiflow can be defined which is eventually strongly monotone and strictly subhomogeneous