dynamics in a diffusive predator prey system with ratio dependent predator influence abstract a diffusive predator prey system with predator functional response subject to neumann boundary conditions is considered existence nonexistence and boundedness of positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation hopf and steady state bifurcation analyses are carried out in detail these results provide theoretical evidences to the complex spatio temporal dynamics found by numerical simulation

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

mesa type patterns in the one dimensional brusselator and their stability the brusselator is a generic reaction diffusion model for a tri molecular chemical reaction we consider the case when the input and output reactions are slow in this limit we show the existence of k periodic spatially bi stable structures mesas and study their stability using singular perturbation techniques we find a threshold for the stability of k mesas this threshold occurs in the regime where the exponentially small tails of the localized structures start to interact by comparing our results with turing analysis we show that in the generic case a turing instability is followed by a slow coarsening process whereby logarithmically many mesas are annihilated before the system reaches a steady equilibrium state we also study a breather type instability of a mesa which occurs due to a hopf bifurcation full numerical simulations are shown to confirm the analytical results in 1952 turing proposed that the formation of spatial patterns during morphogenesis could be explained in terms of the instability of a homogeneous steady state solution of a reaction diffusion network describing the evolution of a set of morphogens 31 turing himself illustrated his ideas on two chemical models turing s original work is primarily concerned with the stability analysis of the homogeneous steady state solution of the rate equations for the interacting morphogens 22 the main point of biological interest however is whether stable spatial structures may be generated beyond the instability i e whether the rate equations admit stable and positive inhomogeneous solutions exhibiting the most characteristic features of morphogenetic patterns this point has been taken up seriously in the early seventies more systematic numerical studies of turing s model have been performed showing irregular spatial structures 4 this led to serious reservations about the relevance of turing s theory in developmental biology particularly its ability to generate regular patterns but these criticisms originate from a somewhat unfortunate choice of turing s example and they do not touch the essential points of turing s theory more specifically the obvious requirement that the rate equations must admit positive and bounded solutions is not satisfied in turing s example 11

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

oscillation behavior of second order nonlinear neutral differential equations with deviating arguments oscillation criteria are established for second order nonlinear neutral differential equations with deviating arguments of the form r t z t a 1z t f t x s t 0 t t 0 where a 0 and z t x t p t x t t our results improve and extend some known results in the literature some illustrating examples are also provided to show the importance of our results

oscillation criteria for nonlinear differential equations with damping oscillation criteria are established for the nonlinear second order damped differential equation rx px qf x 0 which are different from most known results

bifurcation analysis of an enzyme catalyzed reaction diffusion system abstract in this paper we concern about the dynamics of a diffusive enzyme catalyzed system arising from glycolysis describing a biochemical reaction in which a substrate is converted into a product with positive feedback and into a branched sink the temporal and spatiotemporal dynamics of the system under homogeneous neumann boundary conditions are studied preliminary analysis on the local asymptotic stability and hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented for the reaction diffusion model firstly the parameter regions for the stability or instability of the unique constant steady state are discussed finally bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied numerical simulations are presented to verify and illustrate the theoretical results

interval oscillation of a general class of second order nonlinear impulsive differential equations with nonlinear damping in this paper a general class of second order nonlinear impulsive differential equations with damping is studied several interval oscillation criteria which generalize and extend some known ones are established moreover examples are given to illustrate the effectiveness and non emptiness of our results

oscillation criteria for half linear elliptic inequalities with p x laplacians via riccati method abstract riccati inequality for half linear elliptic inequalities with p x laplacians is established and oscillation criteria are derived by using the riccati inequality our method is to reduce oscillation problems for half linear elliptic inequalities with p x laplacians to one dimensional riccati inequality with variable exponents generalizations to more general elliptic inequalities with p x laplacians are also discussed

spectral properties of the canonical solution operator in this paper we consider a hilbert space h 0 1 of 0 1 forms and a space l2 of square integrable functions with respect to a measure on a rotation invariant open set in cn we give necessary and sufficient conditions in terms of the moments of the measure for the canonical solution operator of the equation to be bounded compact and in the schatten p class from h 0 1 into l2 examples of h 0 1 can be chosen to be the space of 0 1 forms with coefficients in one of the classical hilbert spaces of holomorphic functions such as the weighted bergman space the hardy space the hardy sobolev space or the mobius invariant space

integral inequalities and second order linear oscillation on the interval 0 a u 0 below by a solution we mean a non trivial solution a solution is oscillatory if it has an infinite number of zeros numerous oscillation criteria are known and diversified approaches have proved fruitful among these is the use of the associated riccati equation although the theory of differential and integral inequalities has been used by some investigators see e g hartman 4 71 the authors believe that it has not been fully exploited the non oscillation of 1 1 is equivalent to the existence of a solution of the associated riccati equation on a half line thus one technique of proving oscillation is to show that any solution of the riccati equation diverges to i co at a finite point the theory of differential and integral inequalities comes in handy in the estimation of the growth of such a solution below we describe some known results that are relevant to results established in this paper although these are stated here for the case i co in the rest of the paper we consider both cases a 0 and a 00 it is well known that if q is large in the mean then 1 1 is oscillatory a typical example of such a criterion is due to fite leighton and wintner if

oscillations of second order impulses nonlinear fde with forcing term a kind of second order impulses nonlinear fde with forcing term is studied in this paper by means of kartsatos technique we reduce it to a second order nonlinear impulsive homogeneous equation several criteria on the oscillations of solutions are given we find some suitable impulse functions such that all the solutions of the equation are oscillatory under the impulse control at last we give some examples to demonstrate our results

interval oscillation criteria for second order neutral nonlinear differential equations some oscillation criteria for the second order neutral nonlinear differential equation y t p t y s t email protected i 1 nq i t f i y t i t 0 t t 0are established new oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem

interval criteria for oscillation of second order linear ordinary differential equations abstract new oscillation criteria are established for the equation py qy 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem and sharper than some previous results and can be applied to extreme cases such as t 0 q t dt

instabilities and patterns in coupled reaction diffusion layers we study instabilities and pattern formation in reaction diffusion layers that are diffusively coupled for two layer systems of identical two component reactions we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem there are eight possible primary bifurcation scenarios including a turing turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling for systems of n component layers and nonidentical layers the linear problem s block form allows approximate decomposition into lower dimensional linear problems if the coupling is sufficiently weak as an example we apply these results to a two layer brusselator system the competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system selecting a sqrt 2 1 length scale ratio produces an unusual steady square pattern

weighted means and oscillation conditions for second order matrix differential equations on considere le systeme d equations differentielles d ordre 2 p t y t q t y t 0 pour t 0 ou p et q sont des matrices symetriques n n reelles sur 0 et y un vecteur colonne on suppose que p t est definie positive pour tout t tel que p t 1 on donne des conditions d oscillations

spatiotemporal dynamics induced by delay and diffusion in a predator prey model with mutual interference among the predator abstract in this work we study a reaction diffusion predator prey model with mutual interference among the predators while searching for food we prove that the model exhibits bistability which indicates that there are no patterns for our model when time delay is incorporated into the model multiple stability switches phenomenon of positive constant steady state emerged by taking delay as a bifurcation parameter the hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay the algorithm for determining the direction and the stability of the bifurcating periodic solutions is also derived the delay diffusion driven turing instability of the positive constant steady state is investigated our results show that delay and diffusion can create periodic oscillatory patterns of spatially homogeneous and inhomogeneous and turing patterns

interval oscillation criteria for second order linear differential equations with impulsive effects abstract this paper discusses how the oscillation of solutions changes by adding an impulsive term to a second order linear differential equation having only a restoring force term but no damping term due to the effect of this impulsive term the moving speed of a mass point of the equation of motion described by the second order linear differential equation changes discontinuously oscillation theorems are given for this impulsive differential equation as is well known when the restoring force term is small all nontrivial solutions of the equation with no impulsive effects do not oscillate even when the restoring force term is small if the action of the impulsive term compensates the oscillation criteria obtained are satisfied and all nontrivial solutions oscillate several examples are given to confirm this fact and some figures are shown to depict the solution curves of these examples as shown by the obtained oscillation theorems there are cases that the impulsive effect promotes the oscillation of solutions but on the contrary it may suppress the oscillation of solutions it is shown by a simulation that there is a case where the impulsive effect suppress the oscillation of solutions finally the obtained results are compared with the previous ones

new oscillation criteria for a class of second order nonlinear forced differential equations abstract by using classical variational principle and averaging technique new oscillation criteria are established for second order nonlinear forced differential equations r t x t x t q t f x t e t t t 0 where the functions r q e c t 0 r and f c r r improve and extend some recent results

interval oscillation criteria for second order nonlinear differential equations with damping new oscillation criteria are established for the nonlinear damped differential equation ry py qf y 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t0 rather than on the whole half line our results are sharper than some previous results and extend and improve the recent interval criteria of kong and huang and can be applied to extreme cases such as t0 q s ds in particular several examples that dwell upon the sharp conditions of our results are also included

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

qualitative analysis of a predator prey model with holling type ii functional response incorporating a prey refuge abstract we study a predator prey model with holling type ii functional response incorporating a prey refuge under homogeneous neumann boundary condition we show the existence and non existence of non constant positive steady state solutions depending on the constant m 0 1 which provides a condition for protecting 1 m u of prey u from predation moreover we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions

note on wong s paper using the method of interval integral average we establish some new oscillation criteria which improve the oscillation criteria due to james s w wong

interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations abstract we present new interval oscillation criteria related to integral averaging technique for certain classes of second order nonlinear differential equations that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are sharper than some previous results and handle the cases which are not covered by known criteria in particular several examples that dwell upon the sharp conditions of our results are also included

global stability of prey taxis systems abstract in this paper we prove the global boundedness and stability of the predator prey system with prey taxis in a two dimensional bounded domain with neumann boundary conditions by deriving an entropy like equality and a boundedness criterion we show that the intrinsic interaction between predators and preys is sufficient to prevent the population overcrowding even the prey taxis is included and strong furthermore by constructing appropriate lyapunov functionals we show that prey only steady state is globally asymptotically stable if the predation is weak and the co existence steady state is globally asymptotically stable under some conditions like the prey taxis is weak or the prey diffuses fast if the predation is strong the convergence rates of solutions to the steady states are derived in the paper

interval oscillation criteria for a forced second order nonlinear differential equations with damping abstract in this paper we are concerned with the oscillations in a class of forced second order differential equations with nonlinear damping terms by using classical variational principle and averaging technique several new interval oscillation criteria for the equations are established which improve and extend some known results example is also given to illustrate the results

oscillation criteria for a forced mixed type emden fowler equation with impulses some oscillation criteria for a forced mixed type emden fowler equation with impulses are given when the impulses are dropped our results extend those of sun and meng y g sun f w meng interval criteria for oscillation of second order differential equations with mixed nonlinearities appl math comput 15 2008 375 381 sun and wong y g sun j s w wong oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities j math anal appl 334 2007 549 560 for second order forced ordinary differential equation with mixed nonlinearities nasr a h nasr sufficient conditions for the oscillation of forced superlinear second order differential equations with oscillatory potential proc am math soc 126 1998 123 125 yang q yang interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential appl math comput 135 2003 49 64 for forced superlinear emden fowler equation kong q kong interval criteria for oscillation of second order linear differential equations j math anal appl 229 1999 483 492 for unforced second order linear differential equations and wong j s w wong oscillation criteria for a forced second order linear differential equation j math anal appl 231 1999 235 240 for forced second order linear differential equation

forced oscillations for second order delay differential equations with impulses this paper is devoted to the study of the oscillations for forced second order delay differential equations with impulses the results gained here are based on the information of the forcing term on a sequence of subintervals of t 0 which develops some well known results for the equations without impulses and the equations without delay

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

dynamics and pattern formation in a diffusive predator prey system abstract this paper is devoted to study a general diffusive predator prey system with strong allee effect in the prey a priori estimate and nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation the hopf bifurcation and steady state bifurcation from the positive constant solution branch are investigated these results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulations

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

on global bifurcation for quasilinear elliptic systems on bounded domains general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between sobolev spaces it is shown that the linearized mapping is a fredholm operator of index zero this and the abstract global bifurcation theorem of jacobo pejsachowicz patrick j rabier degree theory for c1 fredholm mappings of index 0 j anal math 76 1998 289 319 allow us to carry out bifurcation analysis directly on these elliptic systems at the abstract level we establish a unilateral global bifurcation result that is needed when studying positive solutions finally we supply two examples of cross diffusion population model and chemotaxis model to demonstrate how the theory can be applied

oscillation criteria for second order linear differential equations abstract some oscillation criteria are given for the second order linear differential equation r t u t c t u t 0 where r t c 1 t 0 0 and c t c t 0 r these results improve oscillation criteria of wintner hartman kamenev yan and philos by using a generalized riccati transformation t a t r t u t u t f t where f c 1 is a given function on t 0 and a t exp 2 t f s ds

interval oscillation of a general class of second order nonlinear differential equations with nonlinear damping abstract the paper is concerned with the oscillation of a class of general type second order differential equations with nonlinear damping terms several new interval oscillation criteria are established for such a class of differential equations under quite general assumptions examples are also given to illustrate the results in particular it is shown that under some very mild conditions on k 1 k 2 and f the equation k 1 x x cos tk 2 x x x sin tf x 0 is oscillatory which extends the result obtained by kwong and wong j math anal appl 288 2003 15

complex dynamics of a diffusive predator prey model with strong allee effect and threshold harvesting abstract in this paper complex dynamics of a diffusive predator prey model is investigated where the prey is subject to strong allee effect and threshold harvesting the existence and stability of nonnegative constant steady state solutions are discussed the existence and nonexistence of nonconstant positive steady state solutions are analyzed to identify the ranges of parameters of pattern formation spatially homogeneous and nonhomogeneous hopf bifurcation and discontinuous hopf bifurcation are proved these results show that the introduction of strong allee effect and threshold harvesting increases the system spatiotemporal complexity finally numerical simulations are presented to validate the theoretical results

oscillation criteria for second order forced elliptic differential equations with mixed nonlinearities in the case of oscillatory potentials we establish some oscillation criteria for the second order forced elliptic differential equation with mixed nonlinearities div a x y c x y e x when n 1 our results extend those of kong q kong interval criteria for oscillation of second order linear ordinary differential equations j math anal appl 229 1999 258 270 for certain second order linear ordinary differential equations sun and meng y g sun f w meng interval criteria for oscillation of second order differential equations with mixed nonlinearities appl math comput 2007 doi 10 10 1016 j amc 2007 08 042 for certain second order differential equations with mixed nonlinearities yang q yang interval oscillation criteria for a second order nonlinear ordinary differential equations with oscillatory potential appl math comput 135 2003 49 64 for a forced second order superlinear ordinary differential equation when n 2 our theorems are more general and improve the oscillation criteria of zhuang r k zhuang annual oscillation criteria for second order nonlinear elliptic differential equations j comput anal math 2007 doi 10 1016 j cam 2007 06 031 for a forced second order undamped elliptic differential equation

interval oscillation of second order half linear functional differential equations by using an inequality due to hardy littlewood and polya and averaging functions new interval oscillation criteria are established for the half linear functional differential equationr t y t a 1y t q t y t t a 1y t t 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results extend and improve some previous oscillation criteria and handle the cases which are not covered by known results and implies that the delay t t t t does not effect the oscillation where t 0 is a constant in particular several examples that dwell upon the sharp conditions of our results are also included

dynamics and pattern formation in a diffusive predator prey system with strong allee effect in prey abstract the dynamics of a reaction diffusion predator prey system with strong allee effect in the prey population is considered nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied these results show that the impact of the allee effect essentially increases the system spatiotemporal complexity

monotonicity oscillations and stability of a solution to a nonlinear equation modelling the capillary rise abstract in this paper we analyse a singular second order nonlinear ode which models the capillary rise of a fluid inside a tubular column we prove global existence uniqueness and find several approximations along with the asymptotic behaviour of the solution moreover we are able to find a critical value of the nondimensional parameter for which the solution exhibits a transition in its behaviour from being monotone to oscillatory this is an analytical rigorous proof of the experimentally and numerically confirmed phenomenon

kamenev type and interval oscillation criteria for second order linear differential equations on a measure chain abstract we establish kamenev type criteria and interval criteria for oscillation of the second order scalar differential equation p t x t q t x t 0 on a measure chain our results cover those for differential equations and provide new oscillation criteria for difference equations several examples are given to show the significance of the results

global bifurcation analysis and pattern formation in homogeneous diffusive predator prey systems abstract the dynamics of a general diffusive predator prey system is considered existence and nonexistence of non constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non constant steady state solutions are studied

new oscillation criteria of second order nonlinear differential equations by employing a class of new functions f f t s l and a generalized riccati technique some new oscillation and interval oscillation criteria are established for the second order nonlinear differential equation r t y t q t y t y t 0 the obtained interval oscillation criteria can be applied to equations with forcing term two examples are also included to show the significance of our results

integral averaging technique for the interval oscillation criteria of certain second order nonlinear differential equations abstract we present new interval oscillation criteria related to integral averaging technique for certain classes of second order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line they generalize and improve some known results examples are also given to illustrate the importance of our results

interval criteria for oscillation of certain matrix differential systems by employing a generalized riccati technique and an averaging technique interval oscillation criteria are established for the second order matrix differential system of the form p t y q t y 0 that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t0 rather than on the whole half line our results are presented in the form of a high degree of generality handle the cases which are not covered by known criteria and can be applied to extreme cases such as limt 1 t0tq s ds in particular several interesting examples that dwell upon the importance of our results are included

bifurcation branch of stationary solutions in a general predator prey system with prey taxis abstract in this paper a general reaction diffusive predator prey system with prey taxis subject to the homogeneous neumann boundary condition is considered firstly we investigate the local stability of the unique positive equilibrium by analyzing the characteristic equation and study a priori estimates of positive solutions by the iterative technique and then choosing the prey tactic sensitivity coefficient as bifurcation parameter we proved that a branch of nonconstant solutions can bifurcate from the unique positive equilibrium when the prey tactic sensitivity is repulsive moreover we find the stable bifurcating solutions near the bifurcation point by the spectrum theory under some suitable conditions our results show that prey taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns

non constant positive steady states of the sel kov model abstract this paper deals with the reaction diffusion system known as the sel kov model with the homogeneous neumann boundary condition this model has been applied to various problems in chemistry and biology we first give a priori estimates positive upper and lower bounds of positive steady states and then study the non existence bifurcation and global existence of non constant positive steady states as the parameters and are varied