on the power method in max algebra abstract the eigenvalue problem for an irreducible nonnegative matrix a a ij in the max algebra system is a otimes x lambda x where a otimes x i mathop max j a ij x j and lambda turns out to be the maximum circuit geometric mean mu a a power method algorithm is given to compute mu a and eigenvector x the algorithm is developed by using results on the convergence of max powers of a which are proved using nonnegative matrix theory in contrast to an algorithm developed in 4 this new method works for any irreducible nonnegative a and calculates eigenvectors in a simpler and more efficient way some asymptotic formulas relating mu a the spectral radius and norms are also given

hopf bifurcation and turing instability in spatial homogeneous and inhomogeneous predator prey models abstract this paper is concerned with two species spatial homogeneous and inhomogeneous predator prey models with beddington deangelis functional response for the spatial homogeneous model the asymptotic behavior of the interior equilibrium and the existence of hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered furthermore the direction of hopf bifurcation and the stability of bifurcated periodic solutions are investigated for the model with no flux boundary conditions turing instability of the interior equilibrium solution is studied in particular turing instability region regarding the parameters is established finally to verify our theoretical results some numerical simulations are also included

global well posedness and stability analysis of prey predator model with indirect prey taxis abstract this paper deals with a prey predator model with indirect prey taxis which means chemical of prey causes the directional movement of the predator we prove the global existence and uniform boundedness of solutions to the model for general functional responses in any spatial dimensions moreover through linear stability analysis it turns out that prey taxis is an essential factor in generating pattern formations this result differs in that the destabilizing effect of taxis does not occur in the direct prey taxis case in addition we show the global stability of the semi trivial steady state and coexistence steady state for some specific functional responses we give numerical examples to support the analytic results

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

exploiting sparsity in adaptive filters this paper studies a class of algorithms called natural gradient ng algorithms the least mean square lms algorithm is derived within the ng framework and a family of lms variants that exploit sparsity is derived this procedure is repeated for other algorithm families such as the constant modulus algorithm cma and decision directed dd lms mean squared error analysis stability analysis and convergence analysis of the family of sparse lms algorithms are provided and it is shown that if the system is sparse then the new algorithms will converge faster for a given total asymptotic mse simulations are provided to confirm the analysis in addition bayesian priors matching the statistics of a database of real channels are given and algorithms are derived that exploit these priors simulations using measured channels are used to show a realistic application of these algorithms

turing instability for a ratio dependent predator prey model with diffusion abstract ratio dependent predator prey models have been increasingly favored by field ecologists where predator prey interactions have to be taken into account the process of predation search in this paper we study the conditions of the existence and stability properties of the equilibrium solutions in a reaction diffusion model in which predator mortality is neither a constant nor an unbounded function but it is increasing with the predator abundance we show that analytically at a certain critical value a diffusion driven turing type instability occurs i e the stationary solution stays stable with respect to the kinetic system the system without diffusion we also show that the stationary solution becomes unstable with respect to the system with diffusion and that turing bifurcation takes place a spatially non homogenous non constant solution structure or pattern arises a numerical scheme that preserve the positivity of the numerical solutions and the boundedness of prey solution will be presented numerical examples are also included

pattern formation in a two component reaction diffusion system with delayed processes on a network abstract reaction diffusion systems with time delay defined on complex networks have been studied in the framework of the emergence of turing instabilities the use of the lambert w function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters the time delay the network topology and the diffusion coefficients depending on these parameters the analysis predicts whether the system will evolve towards a stationary turing pattern or rather to a wave pattern associated to a hopf bifurcation the possible outcomes of the linear analysis overcome the respective limitations of the single species case with delay and that of the classical activator inhibitor variant without delay numerical results gained from the mimura murray model support the theoretical approach

hopf bifurcation and turing instability of 2 d lengyel epstein system with reaction diffusion terms in this paper we study 2 d lengyel epstein l e reaction diffusion system with homogeneous neumann boundary condition by employing stability theory hopf bifurcation theorem and turing s theory we present some sufficient conditions ensuring the equilibrium point of system to be stable and derive conditions on the parameters so that spatial homogenous hopf bifurcation and turing instability occur these conditions obtained have important leading significance in applications of l e system finally we show some numerical examples to verity the theoretical analysis

steady states of a predator prey model with prey taxis abstract we study the steady states of a predator prey model with prey taxis incorporating holling type ii functional response under the homogeneous neumann boundary condition the stability of equilibrium points and the existence of non constant steady states are investigated we obtain that the prey tactic sensitivity coefficient delays the stability of the unique positive constant solution but for other equilibrium points stability the prey tactic sensitivity coefficient does not influence on it furthermore we derive some sufficient conditions relative to the prey tactic sensitivity coefficient which confines the existence of steady states and find that even if the interaction coefficient is sufficiently large there also exist non constant positive steady states under some conditions

global dynamics of a reaction diffusion waterborne pathogen model with general incidence rate abstract in this paper we first propose a reaction diffusion waterborne pathogen model with general incidence rate incorporating both direct and indirect transmission pathways then using the basic reproduction number we investigate the global dynamical behaviors of the continuous model by nonstandard finite difference scheme we derive a discrete counterpart of the continuous model then the global properties of the discretized model are investigated finally we conclude the paper by an example and numerical simulations

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

turing pattern of the oregonator model abstract the oregonator model is the mathematical dynamics which describes the field koros noyes mechanics of the famous belousov zhabotinskii reaction in this work we establish some fundamental analytic properties of this dynamics and its corresponding steady state under various conditions on the parameters and the size of the reactor we examine the existence and non existence of non constant steady states in particular for some properly chosen parameter ranges we prove the occurrence of the turing pattern generated by this oregonator model our results exhibit interesting and very different roles of the diffusion rates and the reactor in the formation of the turing pattern our mathematical analysis mainly relies on a priori estimates and the topological degree argument

an efficient algorithm for critical circuits and finite eigenvectors in the max plus algebra abstract we consider the eigenvalue problem in the max plus algebra for matrices in r n n but with eigenvectors in r n the problem is relaxed to a linear optimization lo problem of which the dual problem is solved by finding a maximal average weight circuit in the graph of the matrix the floyd warshall procedure is used to find such a circuit this procedure also provides an efficient algorithm for finding an eigenvector

multiple stability switches and hopf bifurcations induced by the delay in a lengyel epstein chemical reaction system abstract this paper examines the dynamical analysis of the lengyel epstein system with a discrete delay in detail under the assumption that the unique positive equilibrium of the model is locally asymptotically stable in the absence of the delay the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail it is found that under suitable conditions on the other parameters the delay doesn t affect the stability of the equilibrium namely the equilibrium is absolutely stable while under the other conditions on the other parameters the equilibrium will become ultimately unstable after passing through multiple stability switches and hopf bifurcations at some certain critical values of delay particularly by means of the normal form method and the center manifold reduction for retarded functional differential equations the explicit formulae determining the direction of hopf bifurcations and the stability of the bifurcating periodic solutions are obtained to verify our theoretical conclusions some numerical simulations for specific examples are also included at the end of this article

periodic solution of a prey predator model with nonlinear state feedback control assume that when the number of pests reaches the certain threshold pest management strategy will be taken to control pests based on this assumption in this paper we propose a pest management model with nonlinear state feedback control we then analyze the dynamic behavior of the model more precisely we first investigate the singularity of the model by using method of qualitative analysis secondly the existence of periodic solution of the model is studied by using successor functions and poincare bendixson theorem and then it is followed by the study of the stability of periodic solution finally an example with numerical simulations is given to illustrate our conclusions

complexity reduction of the nlms algorithm via selective coefficient update this article proposes an algorithm for partial update of the coefficients of the normalized least mean square nlms finite impulse response fir adaptive filter it is shown that while the proposed algorithm reduces the complexity of the adaptive filter it maintains the closest performance to the full update nlms filter for a given number of updates analysis of the mse convergence and steady state performance for independent and identically distributed i i d signals is provided for the extreme case of one update iteration

a simplex like method to compute the eigenvalue of an irreducible max system abstract in this paper we present an alternative method to compute the eigenvalue of an irreducible max system the method resembles the well known simplex method in linear programming in the sense that the eigenvalue and a corresponding eigenvector are obtained by going along the boundary of a polygon like set while increasing the number of equalities in some max algebraic eigenvalue eigenvector expression until only equalities are left over the latter is unlike the normal linear programming approach where going along the boundary of a polygon like set a linear functional is optimized

partial update lms algorithms partial updating of lms filter coefficients is an effective method for reducing computational load and power consumption in adaptive filter implementations this paper presents an analysis of convergence of the class of sequential partial update lms algorithms s lms under various assumptions and shows that divergence can be prevented by scheduling coefficient updates at random which we call the stochastic partial update lms algorithm spu lms specifically under the standard independence assumptions for wide sense stationary signals the s lms algorithm converges in the mean if the step size parameter spl mu is in the convergent range of ordinary lms relaxing the independence assumption it is shown that s lms and lms algorithms have the same sufficient conditions for exponential stability however there exist nonstationary signals for which the existing algorithms s lms included are unstable and do not converge for any value of spl mu on the other hand under broad conditions the spu lms algorithm remains stable for nonstationary signals expressions for convergence rate and steady state mean square error of spu lms are derived the theoretical results of this paper are validated and compared by simulation through numerical examples

spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law in this paper a reaction diffusion bimolecular model with autocatalysis and saturation law is investigated firstly we provide some conditions for the stability turing instability of the constant positive solution then we mainly consider hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system our results suggest the existence of spatially non homogeneous periodic orbit and non constant positive steady state solutions which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model numerical examples are presented to support our theoretical analysis furthermore non existence of non constant steady state is investigated in terms of parameters

on the asymptotic stability of the time fractional lengyel epstein system abstract this paper concerns a time fractional version of the conventional lengyel epstein cima reaction model we define the invariant regions of the system and establish sufficient conditions for the unique equilibrium s local and global asymptotic stability numerical results are presented to illustrate the effect of the fractional order on system dynamics

spatial patterns of a fractional type cross diffusion holling tanner model abstract in this paper a fractional type cross diffusion holling tanner model under homogeneous neumann boundary condition is considered using the maximum principle and the harnack inequality we obtain a priori estimates of the positive stationary solution nonexistence and existence of nonconstant positive stationary solution are obtained which reveal that fractional type cross diffusion can create spatial patterns but the random diffusion fails

steady states of a leslie gower model with diffusion and advection abstract this paper focuses on a stationary leslie gower model with diffusion and advection firstly some existence conditions of nonconstant positive solutions are obtained by means of the leray schauder degree theory as diffusion and advection of one of the species both tend to infinity we obtain a limiting system which is a semi linear elliptic equation with nonlocal constraint in the simplified 1d case the global bifurcation structure of nonconstant solutions of the limiting system is classified

turing hopf bifurcations in a predator prey model with herd behavior quadratic mortality and prey taxis abstract in this paper we propose a predator prey model with herd behavior and prey taxis then we analyze the stability and bifurcation of the positive equilibrium of the model subject to the homogeneous neumann boundary condition by using an abstract bifurcation theory and taking prey tactic sensitivity coefficient as the bifurcation parameter we obtain a branch of stable nonconstant solutions bifurcating from the positive equilibrium our results show that prey taxis can yield the occurrence of spatial patterns

the power algorithm in max algebra abstract it is proved that under certain conditions an algorithm resembling the power algorithm in conventional linear algebra can be used to find the eigenvector and eigenvalue of a max algebra system if the conditions are not satisfied the eigenvector can be found using an extension of the power algorithm

limit sets and the poincare bendixson theorem in impulsive semidynamical systems we consider semidynamical systems with impulse effects at variable times and we discuss some properties of the limit sets of orbits of these systems such as invariancy compactness and connectedness as a consequence we obtain a version of the poincare bendixson theorem for impulsive semidynamical systems

turing instability and pattern induced by cross diffusion in a predator prey system with allee effect to have spatial patterns form small allee effect admits relatively large range of predator mortality rate found cross diffusion is the key mechanism of spatial pattern formation obtained amplitude equations suggest both supercritical and subcritical bifurcation may occur weak allee effect admits supercritical bifurcation while strong one allows subcritical bifurcation in this paper we first propose a mathematical model for a spatial predator prey system with allee effect and then by using the proposed model we investigate the turing instability and the phenomena of pattern formation we show how cross diffusion destabilizes the spatially uniform steady state the method of multiple time scales is employed to derive the amplitude equations which is the cubic stuart landau equation in the supercritical case and the quintic in the subcritical case based on the amplitude equations we obtain the asymptotic solutions of the model close to the onset of instability

hopf bifurcation and steady state bifurcation for an autocatalysis reaction diffusion model abstract this paper is concerned with an autocatalysis model subject to no flux boundary conditions the existence of hopf bifurcation are firstly obtained then by the center manifold theory and the normal form method the direction of hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established on the other hand the steady state bifurcations from simple and double eigenvalues are intensively studied the techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues finally some numerical simulations are shown to verify the analytical results

efficient numerical methods for spatially extended population and epidemic models with time delay abstract reaction diffusion models with time delay have been widely applied in population biology as well as epidemiology this type of models can possibly exhibit complex dynamical behaviors such as traveling wave self organized spatial pattern or chaos numerical methods play an essential role in the study of these dynamical behaviors this paper concerns the finite element approximation for reaction diffusion models with time delay two fully discrete schemes and corresponding a priori error estimates are derived generally the research on evolution of population and epidemic needs to survey long time dynamical behaviors of these models so that it is important to improve the speed of numerical simulation to this end interpolation technique is used in our schemes to avoid numerical integration of reaction term an outstanding advantage of using interpolation of reaction term is that it improves the operation speed greatly meanwhile does not reduce convergence order applications are given to some model problems arising from population biology and epidemiology from these simulations some interesting phenomena can be found and we try to explain them in biological significance

qualitative analysis of a ratio dependent predator prey system with diffusion ratio dependent predator prey models are favoured by many animal ecologists recently as they better describe predator prey interactions where predation involves a searching process when densities of prey and predator are spatially homogeneous the so called michaelis menten ratio dependent predator prey system which is an ordinary differential system has been studied by many authors the present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous neumann boundary condition its main purpose is to study qualitative properties of solutions to this reaction diffusion partial differential system in particular we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary differential equation dynamics non constant positive steady states exist for the partial differential equation model this demonstrates that stationary patterns arise as a result of diffusion

exponential stability and extended dissipativity criteria for generalized discrete time neural networks with additive time varying delays abstract this paper is concerned with exponential stability and extended dissipativity criteria for generalized discrete time neural networks gdnns with additive time varying delays the generalized dissipativity analysis combines a few previous results into a framework such as l 2 l performance h performance passivity performance strictly q s r dissipative and strictly q s r dissipative the definition of exponential stability for gdnns is given with a new and more appropriate expression a novel augmented lyapunov krasovskii functional lkf which involves more information about the additive time varying delays is constructed by introducing more zero equalities and using a new double summation inequality together with finsler s lemma an improved delay dependent exponential stability and extended dissipativity criterion are derived in terms of convex combination technique cct finally numerical examples are given to illustrate the usefulness and advantages of the proposed methods

simple and superlattice turing patterns in reaction diffusion systems bifurcation bistability and parameter collapse abstract we use equivariant bifurcation theory to investigate pattern selection at the onset of a turing instability in a general two component reaction diffusion system the analysis is restricted to patterns that periodically tile the plane in either a square or hexagonal fashion both simple periodic patterns stripes squares hexagons and rhombs and superlattice patterns are considered the latter correspond to patterns that have structure on two disparate length scales the short length scale is dictated by the critical wave number from linear theory while the periodicity of the pattern is on a larger scale analytic expressions for the coefficients of the leading nonlinear terms in the bifurcation equations are computed from the general reaction diffusion system using perturbation theory we show that no matter how complicated the reaction kinetics might be the nonlinear reaction terms enter the analysis through just four parameters moreover for hexagonal problems all patterns bifurcate unstably unless a particular degeneracy condition is satisfied and at this degeneracy we find that the number of effective system parameters drops to two allowing a complete characterization of the possible bifurcation results at this degeneracy for example we find that rhombs squares and superlattice patterns always bifurcate unstably we apply these general results to some specific model equations including the lengyel epstein cima model to investigate the relative stability of patterns as a function of system parameters and to numerically test the analytical predictions

limiting structure of steady states to the lotka volterra competition model with large diffusion and advection abstract this paper is concerned with the neumann problem of a stationary lotka volterra competition model with diffusion and advection first we obtain some sufficient conditions of the existence of nonconstant solutions by the leray schauder degree theory next we derive a limiting system as diffusion and advection of one of the species tend to infinity the limiting system can be reduced to a semilinear elliptic equation with nonlocal constraint in the simplified 1d case the global bifurcation structure of nonconstant solutions of the limiting system can be classified depending on the coefficients for example this structure involves a global bifurcation curve which connects two different singularly perturbed states boundary layer solutions and internal layer solutions our proof employs a levelset analysis for the associate integral mapping

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

a diffusive one prey and two competing predator system with a ratio dependent functional response ii stationary pattern formation abstract this paper is a continuation of ko and ahn 2013 1 which investigates the stability at all non negative equilibria and long time behavior of solutions for a ratio dependent reaction diffusion system incorporating one prey and two competing predator species under homogeneous neumann boundary conditions we examine the nonexistence and the appearance of stationary patterns in the time independent system in achieving these we deal with the system only when the competition state between two competing predators is weak strong in particular the results explain the phenomenon of a stationary pattern being induced by the introduction of a new predator species in the one prey and one predator system with no stationary pattern

spatiotemporal dynamics in the single population model with memory based diffusion and nonlocal effect abstract to incorporate spatial memory and nonlocal effect of animal movements we propose and investigate the spatiotemporal dynamics of the single population model with memory based diffusion and nonlocal reaction we first study the stability of a positive equilibrium and the steady state bifurcation induced by diffusion and nonlocality we then investigate the impact of the averaged memory period on stability and bifurcation and show that the combination of the averaged memory period and the diffusion can lead to the occurrence of turing hopf and double hopf bifurcations the paper originally derives the normal form theory for turing hopf bifurcation in the general reaction diffusion equation with memory based diffusion and nonlocal reaction this novel algorithm can be widely used to classify the spatiotemporal dynamics near the turing hopf bifurcation point finally we apply the obtained results to a model proposed by britton and numerically illustrate the spatiotemporal patterns induced by hopf turing hopf and double hopf bifurcations stable spatially homogeneous nonhomogeneous periodic solutions homogeneous nonhomogeneous steady states and the transition from one of these solutions to another are provided in this paper we additionally acquire the coexistence of two stable spatially nonhomogeneous steady states or two spatially nonhomogeneous periodic solutions near the turing hopf bifurcation point

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

turing patterns of a strongly coupled predator prey system with diffusion effects abstract the main purpose of this work is to investigate the effects of cross diffusion in a strongly coupled predator prey system by a linear stability analysis we find the conditions which allow a homogeneous steady state stable for the kinetics to become unstable through a turing mechanism in particular it is shown that turing instability of the reaction diffusion system can disappear due to the presence of the cross diffusion which implies that the cross diffusion induced stability can be regarded as the cross stability of the corresponding reaction diffusion system furthermore we consider the existence and non existence results concerning non constant positive steady states patterns of the system we demonstrate that cross diffusion can create non constant positive steady state solutions these results exhibit interesting and very different roles of the cross diffusion in the formation and the disappearance of the turing instability

global stability of the steady states of an sis epidemic reaction diffusion model abstract in this work we investigate the sis epidemic reaction diffusion model under heterogeneous environment studied by 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 a 21 1 2008 1 20 in the two cases 1 the diffusion rate d s of the susceptible individuals is equal to the diffusion rate d i of the infected individuals 2 x r x for any fixed constant r 0 where x and x respectively represent the rates of disease transmission and disease recovery we completely determine the global stability of the disease free equilibrium and the unique endemic equilibrium if it exists our results partially answer the conjecture proposed by allen et al

hopf bifurcation analysis of a reaction diffusion sel kov system abstract a reaction diffusion system known as the sel kov model subject to the homogeneous neumann boundary condition is investigated where detailed hopf bifurcation analysis is performed we not only show the existence of the spatially homogeneous non homogeneous periodic solutions of the system but also derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution

spatio temporal dynamics near the steady state of a planktonic system abstract the study of spatio temporal behaviour of ecological systems is fundamentally important as it can provide deep understanding of species interaction and predict the effects of environmental changes in this paper we first propose a spatial model with prey taxis for planktonic systems in which we also consider the herb behaviour in prey and effect of the hyperbolic mortality rate applying the homogeneous neumann boundary condition to the model and using prey tactic sensitivity coefficient as bifurcation parameter we then detailedly analyse the stability and bifurcation of the steady state of the system firstly we carry out a study of the equilibrium bifurcation showing the occurrence of fold bifurcation hopf bifurcation and the bt bifurcation then by using an abstract bifurcation theory and taking prey tactic sensitivity coefficient as the bifurcation parameter we investigate the turing hopf bifurcation obtaining a branch of stable non constant solutions bifurcating from the positive equilibrium and our results show that prey taxis can yield the occurrence of spatio temporal patterns finally numerical simulations are carried out to illustrate our theoretical results showing the existence of a periodic solution when the prey tactic sensitivity coefficient is away from the critical value

hopf bifurcation analysis in the 1 d lengyel epstein reaction diffusion model abstract the lengyel epstein model with diffusion and homogeneous neumann boundary condition is considered in this paper we give the existence of multiple spatially non homogeneous periodic solutions though all the parameters of the system are spatially homogeneous

pattern formation in the three dimensional reaction diffusion systems existing group theoretic analysis of pattern formation in three dimensions t k callahan e knobloch symmetry breaking bifurcations on cubic lattices nonlinearity 10 1997 1179 1216 is used to make specific predictions about the formation of three dimensional patterns in two models of the turing instability the brusselator model and the lengyel epstein model spatially periodic patterns having the periodicity of the simple cubic sc face centered cubic fcc or body centered cubic bcc lattices are considered an efficient center manifold reduction is described and used to identify parameter regimes permitting stable lamellae sc fcc double diamond hexagonal prism bcc and bcci states both models possess a special wavenumber k at which the normal form coefficients take on fixed model independent ratios and both are described by identical bifurcation diagrams this property is generic for two species chemical reaction diffusion models with a single activator and inhibitor 1999 elsevier science b v all rights reserved

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

cross diffusion induced turing instability for a competition model with saturation effect abstract competitive behavior like predation widely exists in the world which has influences on the dynamics of ecosystems though a good knowledge for the patten formation and selection of predator prey models with diffusion is known in the literature little is known for that of a diffused competition model as a result a competition model with saturation effect and self and cross diffusion is proposed and analyzed in this paper firstly by making use of nullcline analysis we derive the condition of existence and stability of positive equilibrium and prove global stability then using the linear stability analysis the turing bifurcation critical value and the condition of the occurrence of turing pattern are obtained when control parameter is selected finally we deduce the amplitude equations around the turing bifurcation point by using the standard multiple scale analysis meanwhile a series of numerical simulations are given to expand our theoretical analysis this work shows that cross diffusion plays a key role in the formation of spatial patterns for competitive model with saturation effect

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

spatiotemporal dynamics in a diffusive ratio dependent predator prey model near a hopf turing bifurcation point abstract spatiotemporal dynamics in a ratio dependent predator prey model with diffusion is studied by analytical methods normal forms associated with codimension two hopf turing bifurcation are derived which can be used to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the hopf turing bifurcation point in the vicinity of this degenerate point a wealth of complex spatiotemporal dynamics are observed our theoretical results are confirmed by numerical simulations

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

investigating the interactions between allee effect and harvesting behaviour of a single species model an evolutionary dynamics approach abstract in this study we propose an ecological model which is based on the smith growth model and subject to multiplicative allee effect we then investigate the equilibrium bifurcation induced by the harvesting behaviour and the allee effect and pitch fork bifurcation and saddle node bifurcation may occur assuming that the carrying capacity and harvesting rate obey a gaussian distribution we then consider the evolutionary dynamics by using continuous strategy evolutionary game theory our study shows that strong allee effect and low harvesting rate can induce bistability it also suggests that evolution prevents extinction even if population has strong allee effect numerical analysis shows that the evolutionary model can admit up to nine equilibrium points including up to four evolutionary stable strategy equilibria

leaky lms algorithm mse analysis for gaussian data despite the widespread usage of the leaky lms algorithm there has been no detailed study of its performance this paper presents an analytical treatment of the mean square error mse performance for the leaky lms adaptive algorithm for gaussian input data the common independence assumption regarding w n and x n is also used exact expressions that completely characterize the second moment of the coefficient vector and algorithm steady state excess mse are developed rigorous conditions for mse convergence are also established analytical results are compared with simulation and are shown to agree well

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

positivity and boundedness preserving schemes for space time fractional predator prey reaction diffusion model the semi implicit schemes for the nonlinear predator prey reaction diffusion model with the space time fractional derivatives are discussed where the space fractional derivative is discretized by the fractional centered difference and wsgd scheme the stability and convergence of the semi implicit schemes are analyzed in the l norm we theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model and show that the numerical solutions preserve the positivity and boundedness the presented numerical examples confirm the theoretical results and convergence orders

permanence and periodicity of a delayed ratio dependent predator prey model with holling type functional response and stage structure a periodic and delayed ratio dependent predator prey system with holling type iii functional response and stage structure for both prey and predator is investigated it is assumed that immature predator and mature individuals of each species are divided by a fixed age and immature predator do not have the ability to attack prey sufficient conditions are derived for the permanence and existence of positive periodic solution of the model numerical simulations are presented to illustrate the feasibility of our main results

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

evolutionary dynamics of single species model with allee effect abstract by considering a continuous strategy evolutionary game theory egt a single species population subject to additive allee effect is investigated we assume that the carrying capacity and predator evasion ability following a gaussian distribution this assumption shows that evolution prevents extinction even if population has strong allee effect the ratio of competition and anti predation phenotypes has an important effect on determining global dynamics of the egt model the numerical simulations present the model can have nine equilibria and the model has multiple evolutionarily stable strategy equilibria ess

patterns formations in a diffusive ratio dependent predator prey model of interacting populations abstract the present investigation deals with the analysis of the spatial pattern formation of a diffusive predator prey system with ratio dependent functional response involving the influence of intra species competition among predators within two dimensional space the appropriate condition of turing instability around the interior equilibrium point of the present model has been determined the emergence of complex patterns in the diffusive predator prey model is illustrated through numerical simulations these results are based on the existence of bifurcations of higher codimension such as turing hopf turing saddle node turing transcritical bifurcation and the codimension 3 turing takens bogdanov bifurcation the paper concludes with discussions of our results in ecology

delay induced turing instability in reaction diffusion equations time delays have been commonly used in modeling biological systems and can significantly change the dynamics of these systems quite a few works have been focused on analyzing the effect of small delays on the pattern formation of biological systems in this paper we investigate the effect of any delay on the formation of turing patterns of reaction diffusion equations first for a delay system in a general form we propose a technique calculating the critical value of the time delay above which a turing instability occurs then we apply the technique to a predator prey model and study the pattern formation of the model due to the delay for the model in question we find that when the time delay is small it has a uniform steady state or irregular patterns which are not of turing type however in the presence of a large delay we find spiral patterns of turing type for such a model we also find that the critical delay is a decreasing function of the ratio of carrying capacity to half saturation of the prey density