turing patterns in the lengyel epstein system for the cima reaction the first experimental evidence of the turing pattern was observed by de kepper and her associates 1990 on the cima reaction in an open unstirred gel reactor almost 40 years after turing s prediction lengyel and epstein characterized this famous experiment using a system of reaction diffusion equations in this paper we report some fundamental analytic properties of the lengyel epstein system our result also indicates that if either of the initial concentrations of the reactants the size of the reactor or the effective diffusion rate are not large enough then the system does not admit nonconstant steady states a priori estimates are fundamental to our approach for this nonexistence result the degree theory was combined with the a priori estimates to derive existence of nonconstant steady states

stability analysis of impulsive stochastic nonlinear systems this paper studies stochastic input to state stability and stochastic global stability for impulsive stochastic nonlinear systems using fixed dwell time condition and lyapunov based approach sufficient conditions are established for the stability properties two cases are studied 1 the case that the continuous dynamics is stable and 2 the case that the impulsive effects are stable furthermore the relations among different dwell time conditions are studied finally two examples are used to illustrate the developed theory

stability and hopf bifurcation of the stationary solutions to an epidemic model with cross diffusion in the previous paper cai and wang 2015 we investigated the stationary solutions of a cross diffusion epidemic model with vertical transmission in a spatially heterogeneous environment with neumann boundary condition and proved that the set of positive stationary solutions forms a bounded bifurcation branch which is monotone s or fish hook shaped with respect to the bifurcation parameter in the present paper we give some criteria on the stability of solutions on we prove that the stability of the solutions changes only at every turning point of while in a different case that a k and x are sufficiently large original stable positive stationary solutions at certain point may lose their stability and hopf bifurcation can occur these results are very different from those of the spatially homogeneous or without cross diffusion cases

noise to state stability for a class of random systems with state dependent switching this note is intended to investigate noise to state stability for random nonlinear systems with state dependent switching under some mild and easily verified conditions the existence of global solution to random switched systems can be proved based on a reasonable requirement for the random disturbance the criteria on noise to state stability of random switched systems are presented by the aid of single lyapunov function technique the reasonability of the obtained results is illustrated by using a mechanical model in random vibration environment

spatiotemporal patterns in a reaction diffusion model with the degn harrison reaction scheme abstract spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades in this work we are concerned with a reaction diffusion model with the degn harrison reaction scheme which accounts for the qualitative feature of the respiratory process in a klebsiella aerogenes bacterial culture we study the global stability of the constant steady state existence and nonexistence of nonconstant steady states as well as the hopf and steady state bifurcations in particular our results show the existence of turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous these results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns

input output to state stability and state norm estimators for switched nonlinear systems in this paper the concepts of input output to state stability ioss and state norm estimators are considered for switched nonlinear systems under average dwell time switching signals we show that when the average dwell time is large enough a switched system is ioss if all of its constituent subsystems are ioss moreover under the same conditions a non switched state norm estimator exists for the switched system furthermore if some of the constituent subsystems are not ioss we show that still ioss can be established for the switched system if the activation time of the non ioss subsystems is not too big again under the same conditions a state norm estimator exists for the switched system however in this case the state norm estimator is a switched system itself consisting of two subsystems we show that this state norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for

leader following consensus of a class of stochastic delayed multi agent systems with partial mixed impulses in this paper the exponential leader following consensus problem is investigated for a class of nonlinear stochastic networked multi agent systems with partial mixed impulses and unknown time varying but bounded delays the main feature of partial mixed impulses is that time varying impulses are not only composed of synchronizing and desynchronizing impulses simultaneously but they are also injected into a fraction of nodes in multi agent systems three kinds of partial mixed impulses are proposed fixed partial mixed impulses periodic partial mixed impulses and try once discard like partial mixed impulses by means of the lyapunov function theory and the comparison principle conditions are derived for ensuring global exponential leader following consensus under the presented three kinds of partial mixed impulses simulations of leader following consensus of robotic systems are provided to validate the effectiveness of the proposed results and to show the advantages of the proposed partial mixed impulses

the modularity of k3 surfaces with non symplectic group actions we consider complex k3 surfaces with a non symplectic group acting trivially on the algebraic cycles vorontsov and kond classified those k3 surfaces with transcendental lattice of minimal rank the purpose of this note is to study the galois representations associated to these k3 surfaces the rank of the transcendental lattices is even and varies from 2 to 20 excluding 8 and 14 we show that these k3 surfaces are dominated by fermat surfaces and hence they are all of cm type we will establish the modularity of the galois representations associated to them also we discuss mirror symmetry for these k3 surfaces in the sense of dolgachev and show that a mirror k3 surface exists with one exception

dynamical behavior of an epidemic model with a nonlinear incidence rate abstract in this paper we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action by carrying out global qualitative and bifurcation analyses it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region when such a region exists it is shown that the model undergoes a bogdanov takens bifurcation i e it exhibits a saddle node bifurcation hopf bifurcations and a homoclinic bifurcation existence of none one or two limit cycles is also discussed

stability of nontrivial solution of delay differential equations with state dependent impulses this paper considers the stability problem of nontrivial solutions of delay differential equations with state dependent impulses by using the concepts of reduced system and quasi stability the stability problem of a nontrivial solution of the impulsive delay differential equation with state dependent impulses is converted to that of the trivial solution of the impulsive delay differential equation with fixed impulses several stability criteria are then established utilizing the known results for systems with fixed impulses

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

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

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

a stochastic sirs epidemic model with logistic growth and general nonlinear incidence rate abstract in this paper we consider a stochastic sirs epidemic model with logistic growth and general nonlinear incidence rate we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic lyapunov function which provides us a good description of persistence we find that the hypothetical conditions on the nonlinear function are relative weak and valid for many forms of incidence rate

input to state stability of impulsive and switching hybrid systems with time delay this paper investigates input to state stability iss and integral input to state stability iiss of impulsive and switching hybrid systems with time delay using the method of multiple lyapunov krasovskii functionals it is shown that even if all the subsystems governing the continuous dynamics in the absence of impulses are not iss iiss impulses can successfully stabilize the system in the iss iiss sense provided that there are no overly long intervals between impulses i e the impulsive and switching signal satisfies a dwell time upper bound condition moreover these impulsive iss iiss stabilization results can be applied to systems with arbitrarily large time delays conversely in the case when all the subsystems governing the continuous dynamics are iss iiss in the absence of impulses the iss iiss properties can be retained if the impulses and switching do not occur too frequently i e the impulsive and switching signal satisfies a dwell time lower bound condition several illustrative examples are presented with their numerical simulations to demonstrate the main results

stabilization of switched linear systems with quantized output and switching delays this paper addresses the problem of designing time varying quantizers for the stabilization of switched linear systems with quantized output and switching delays the detection delays of switches are assumed to be time varying but bounded and the dwell time of the switching signal is assumed to be larger than the maximum delay given a switching controller we analyze reachable sets of the closed loop state by using a common lyapunov function and then construct a quantizer that guarantees asymptotic stability a sufficient condition for the existence of such a quantizer is characterized by the maximum switching delay and the dwell time

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

asymptotic profiles of the positive steady state for an sis epidemic reaction diffusion model part i abstract to capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease allen et al in l j s allen b m bolker y lou a l nevai asymptotic profiles of the steady states for an sis epidemic reaction diffusion model discrete contin dyn syst ser a 21 1 2008 1 20 proposed a spatial sis susceptible infected susceptible reaction diffusion model and studied the existence uniqueness and particularly the asymptotic behavior of the endemic equilibrium as the diffusion rate of the susceptible individuals goes to zero in the case where a so called low risk subhabitat is created in this work we shall provide further understanding of the impacts of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease which leads us to determine the asymptotic behaviors of the endemic equilibrium when the diffusion rate of either the susceptible or infected population approaches to infinity or zero in the remaining cases consequently our results reveal that in order to eliminate the infected population at least in low risk area it is necessary that one will have to create a low risk subhabitat and reduce at least one of the diffusion rates to zero in this case our results also show that different strategies of controlling the diffusion rates of individuals may lead to very different spatial distributions of the population moreover once the spatial environment is modified to include a low risk subhabitat the optimal strategy of eradicating the epidemic disease is to restrict the diffusion rate of the susceptible individuals rather than that of the infected ones

input to state stability of impulsive stochastic delayed systems under linear assumptions in this paper the input to state stability iss integral iss iiss and stochastic iss siss are investigated for impulsive stochastic delayed systems by means of the lyapunov krasovskii function and the average impulsive interval approach the conditions for iss type properties are derived under linear assumptions respectively for destabilizing and stabilizing impulses it is shown that if the continuous stochastic delayed system is iss and the impulsive effects are destabilizing then the hybrid system is iss with respect to a lower bound of the average impulsive interval moreover it is unveiled that if the continuous stochastic delayed system is not iss the impulsive effects can successfully stabilize the system for a given upper bound of the average impulsive interval an improved comparison principle is developed for impulsive stochastic delayed systems which facilitates the derivations of our results for iss iiss siss an example of networked control systems is provided to illustrate the effectiveness of the proposed results

a stochastic sirs epidemic model with infectious force under intervention strategies abstract in this paper we extend a classical sirs epidemic model with the infectious forces under intervention strategies from a deterministic framework to a stochastic differential equation sde one through introducing random fluctuations the value of our study lies in two aspects mathematically by using the markov semigroups theory we prove that the reproduction number r 0 s can be used to govern the stochastic dynamics of sde model if r 0 s 1 under mild extra conditions the sde system has a disease free absorbing set which means the extinction of disease with probability one if r 0 s 1 under mild extra conditions it has an endemic stationary distribution which leads to the stochastical persistence of the disease epidemiologically we find that random fluctuations can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

turing patterns in a reaction diffusion model with the degn harrison reaction scheme abstract in this paper we consider a reaction diffusion model with degn harrison reaction scheme some fundamental analytic properties of nonconstant positive solutions are first investigated we next study the stability of constant steady state solution to both ode and pde models our result also indicates that if either the size of the reactor or the effective diffusion rate is large enough then the system does not admit nonconstant positive solutions finally we establish the global structure of steady state bifurcations from simple eigenvalues by bifurcation theory and the local structure of the steady state bifurcations from double eigenvalues by the techniques of space decomposition and implicit function theorem

stability analysis of impulsive switched time delay systems with state dependent impulses this paper studies the stability for impulsive switched time delay systems with state dependent impulses since the impulses and the switches are not necessarily synchronous we start from a stability analysis of impulsive switched time delay systems with time dependent impulses sufficient conditions are derived to guarantee the stability property which extends the previous results for the synchronous switch and impulse case for the state dependent impulse case using the b equivalent method impulsive switched time delay systems with state dependent impulses are transformed into impulsive switched time delay systems with time dependent impulses the equivalence between the original system and the transformed system is established and stability conditions are obtained for impulsive switched time delay systems with state dependent impulses finally a numerical example is given to demonstrate the obtained results

lyapunov conditions for input to state stability of impulsive systems this paper introduces appropriate concepts of input to state stability iss and integral iss for impulsive systems i e dynamical systems that evolve according to ordinary differential equations most of the time but occasionally exhibit discontinuities or impulses we provide a set of lyapunov based sufficient conditions for establishing these iss properties when the continuous dynamics are iss but the discrete dynamics that govern the impulses are not the impulses should not occur too frequently which is formalized in terms of an average dwell time adt condition conversely when the impulse dynamics are iss but the continuous dynamics are not there must not be overly long intervals between impulses which is formalized in terms of a novel reverse adt condition we also investigate the cases where i both the continuous and discrete dynamics are iss and ii one of these is iss and the other only marginally stable for the zero input while sharing a common lyapunov function in the former case we obtain a stronger notion of iss for which a necessary and sufficient lyapunov characterization is available the use of the tools developed herein is illustrated through examples from a micro electro mechanical system mems oscillator and a problem of remote estimation over a communication network

positive steady states in an epidemic model with nonlinear incidence rate abstract in this paper we investigate a diffusive epidemic model with nonlinear incidence rate under homogeneous neumann boundary condition the value of this study lies in two aspects mathematically we show the stability of the constant positive steady state solution the existence and nonexistence the local and global structure of nonconstant positive steady states and epidemiologically we find that the system exhibits turing patterns controlled by diffusion