feedback mediated control of spiral waves abstract a theoretical approach is developed which allows one to describe the drift of a spiral wave mediated by a feedback signal taken as an integral of a system variable over a certain domain of an excitable medium the underlying reaction diffusion equations are reduced to an autonomous system of ordinary differential equations for the coordinates of the core center of the spiral wave a new method to determine the drift velocity field of the spiral core is proposed based on a superposition principle this approach is applied to analyze the drift velocity field for a feedback signal generated from a one point detector and from integration domains of different size and shape including global feedback it is shown that a variation of the shape of the integration domain can induce bifurcations that change the attractor structure of the drift velocity field these theoretical predictions are in good quantitative agreement with numerical simulations performed for the oregonator model and the complex ginzburg landau equation the results are also confirmed by experiments with the light sensitive belousov zhabotinsky reaction

on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

the instability of spiky steady states for a competing species model with cross diffusion this paper is concerned with the stability instability of a class of positive spiky steady states for a quasi linear cross diffusion system describing two species competition by detailed spectral analysis it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross diffusion system are non linearly unstable

a criterion for stability of matrices a necessary and sufficient condition for the stability ofn nmatrices with real entries is proved applications to asymptotic stability of equilibria for vector fields are considered the results offer an alternative to the well known routh hurwitz conditions

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

global solutions to a class of multi species reaction diffusion systems with cross diffusions arising in population dynamics in this paper an n species strongly coupled cooperating diffusive system is considered in a bounded smooth domain subject to homogeneous neumann boundary conditions employing the method of energy estimates we obtain some conditions on the diffusion matrix and inter specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution the globally asymptotical stability of the constant positive steady state is also discussed as a consequence all the results hold true for multi species lotka volterra type competition model and prey predator model

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

effect of a protection zone in the diffusive leslie predator prey model abstract in this paper we consider the diffusive leslie predator prey model with large intrinsic predator growth rate and investigate the change of behavior of the model when a simple protection zone 0 for the prey is introduced as in earlier work y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 y du x liang a diffusive competition model with a protection zone j differential equations 244 2008 61 86 we show the existence of a critical patch size of the protection zone determined by the first dirichlet eigenvalue of the laplacian over 0 and the intrinsic growth rate of the prey so that there is fundamental change of the dynamical behavior of the model only when 0 is above the critical patch size however our research here reveals significant difference of the model s behavior from the predator prey model studied in y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 with the same kind of protection zone we show that the asymptotic profile of the population distribution of the leslie model is governed by a standard boundary blow up problem and classical or degenerate logistic equations

on a predator prey system with cross diffusion representing the tendency of predators in the presence of prey species abstract in this paper a predator prey system with cross diffusion representing the tendency of predators to avoid the group defense by a large number of prey or diffuse in the direction of higher concentration of the prey species under homogeneous dirichlet boundary condition is considered using the method of upper and lower solutions developed by pao c v pao strongly coupled elliptic systems and applications to lotka volterra models with cross diffusion nonlinear anal 60 2005 1197 1217 sufficient conditions for the existence of positive solutions are provided when the induced cross diffusion coefficient is sufficiently small furthermore the investigation of non existence of positive solutions is also presented

stationary patterns caused by cross diffusion for a three species prey predator model we study a system arising from a three component prey predator model with prey dependent and ratio dependent functional responses where cross diffusion is included in such a way that predator chases the prey and the prey runs away from the predator we prove that the cross diffusion can generate the stationary patterns nonconstant positive steady states

robust feed back control of travelling waves in a class of reaction diffusion distributed biological systems abstract reaction diffusion rd mechanisms can describe many biological phenomena such as neuron firing in the brain the heartbeat cellular organization activities or even biological disorders such as fibrillation the fitzhugh nagumo fhn model is a particular case of rd systems it is able to capture the key features of many biological processes and since it is relatively simple it has been widely employed during recent years some examples of its predictive capabilities include the representation of the normal behavior of some physiological phenomena related to a travelling plane wave as well as biological disorders associated with spiral or irregular fronts the objective of this work is to design a control law that is able to stabilize complex behaviors travelling plane wave in biological systems using the fhn model as a case study since in biological systems there usually exists a lack of detailed information on the system structure our control law will be designed to be robust i e it must be able to reach the predefined reference regardless the presence of structural uncertainties to this purpose we will extend some classical results on the finite dimensional robust control theory to rd systems by means of order reduction techniques in particular the proper orthogonal decomposition method

green s function evaluation for three dimensional exponentially graded elasticity the numerical implementation of the green s function for an isotropic exponentially graded three dimensional elastic solid is reported the formulas for the nonsingular lq grading term rq in this green s function originally deduced by martin et al emph proc r soc lond a 458 1931 1947 2000 are quite complicated and a small error in one of the formulas is corrected the evaluation of the fundamental solution is tested by employing indirect boundary integral formulation using a galerkin approximation to solve several problems having analytic solutions the numerical results indicate that the green s function formulas and their evaluation are correct

how many correlation lengths for multifractals abstract it is shown here in the context of percolation that one can have multifractal behavior and at the same time a single correlation length this length sets the scale below which non trivial scaling behavior occurs and it is controlled by a few relevant operators as in ordinary critical phenomena all other correlation lengths may be obtained from simple metric factors the quantities which lead to an infinite set of exponents are described by a second renormalization group which is slaved to the first one which determines the correlation length the results are relevant also to the problem of noise in percolating systems

non constant positive steady states of a predator prey system with non monotonic functional response and diffusion this paper deals with non constant positive steady state solutions of a predator prey system with non monotonic functional response also called holling type iv interaction terms and diffusion under the homogeneous neumann boundary condition we first establish positive upper and lower bounds for such solutions and then study their non existence global existence and bifurcation

second order and stability analysis for optimal sparse control of the fitzhugh nagumo equation optimal sparse control problems are considered for the fitzhugh nagumo system including the so called schlogl model the nondifferentiable objective functional of tracking type includes a quadratic tikhonov regularization term and the l 1 norm of the control that accounts for the sparsity though the objective functional is not differentiable a theory of second order sufficient optimality conditions is established for tikhonov regularization parameter nu 0 and also for the case nu 0 in this context also local minima are discussed that are strong in the sense of the calculus of variations the second order conditions are used as the main assumption for proving the stability of locally optimal solutions with respect to nu to 0 and with respect to perturbations of the desired state functions the theory is confirmed by numerical examples that are resolved with high precision to confirm that the optimal solution obeys the system of necessary optimality conditions

traveling waves for a class of cross diffusion systems with small parameters abstract this paper is concerned with the existence of the travelling wave solutions for some cross diffusion systems with small parameters by using singular perturbation method we prove the existence of the travelling wave solutions with a speed c which extend the results of 22 and 23

wavy fronts and speed bifurcation in excitable systems with cross diffusion a bifurcation of excitation fronts induced by cross diffusion in two component bistable reaction diffusion systems of activator inhibitor type is discovered this bifurcation is similar to the nonequilibrium ising bloch bifurcation a different type of fronts whose spatial profiles are characterized by oscillating tails are associated with this bifurcation these fronts are described using exact analytical solutions of piecewise linear reaction diffusion equations

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

on the persistence of a nonautonomous n species lotka volterra cooperative system in this paper a special nonautonomous n species lotka volterra cooperative system is studied sufficient conditions for the uniform strong persistence uniform weak average persistence and uniform strong average persistence of population are obtained

synchronization of diffusively coupled limit cycle oscillators we develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems we examine two classes of systems reaction diffusion pdes with neumann boundary conditions and compartmental odes where compartments are interconnected through diffusion terms with adjacent compartments in both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle we provide two time scale averaging methods for certifying stability of spatially homogeneous time periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion we highlight cases where diffusion stabilizes or destabilizes such trajectories

a sufficient condition for additive d stability and application to reaction diffusion models abstract the matrix a is said to be additively d stable if a d remains hurwitz for all non negative diagonal matrices d in reaction diffusion models additive d stability of the matrix describing the reaction dynamics guarantees the stability of the homogeneous steady state thus ruling out the possibility of diffusion driven instabilities we present a new criterion for additive d stability using the concept of compound matrices we first give conditions under which the second additive compound matrix has non negative off diagonal entries we then use this metzler property of the compound matrix to prove additive d stability with the help of an additional determinant condition this result is then applied to investigate the stability of cyclic reaction networks in the presence of diffusion finally a reaction network structure that fails to achieve additive d stability is exhibited

correlations in multifractals we present an analysis of the implicit assumptions that lie at the basis of the scaling relations for fractal measures proposed by cates et al 1 furthermore the scaling relations which have been shown to hold for trivial multifractals like the generalized cantor set 2 are tested numerically for critical wave functions of an incommensurate system

analytic solutions for monotonic and oscillating fronts in a reaction diffusion system under external fields abstract a piecewise linear reaction diffusion system including an external field is considered analytic solutions are obtained for the propagating front the front velocity the perturbed front and the growth rate of perturbations it is found that when the ratio of the time scales null cline parameter and the external field are small enough the moving front has an oscillating behaviour similar to that of a front in an oscillatory medium a comparison with an oscillatory front invading an unstable state at small velocity is made and it is argued that there are crucial differences between the two cases in particular we find that for fronts oscillating about a stable state both concentrations must oscillate in two variable models whereas fronts oscillating about an unstable state can approach it with one of the two variables being monotonic

two point statistics of multifractal measures abstract the relationship between the function of a multifractal and the spatial correlations of its singularity strengths is examined an expression to compute the probability of observing two different singularities and within a distance r is derived for measures arising from isotropic random multiplicative processes the correlation of s is shown to decay logarithmically with distance possible applications to turbulence models are discussed and the results are illustrated for a binomial measure where the scaling of two point correlation functions is shown to exhibit a phase transition

on the dynamics of predator prey models with the beddington deangelis functional response abstract the dynamics of predator prey models with the beddington deangelis functional response are analyzed primarily from the viewpoint of permanence uniform persistence the beddington deangelis functional response is similar to the holling type 2 functional response but contains an extra term describing mutual interference by predators both spatially homogeneous models based on ordinary differential equations and reaction diffusion models are considered criteria for permanence and for predator extinction are derived for systems without diffusion or with no flux boundary conditions criteria are derived for the existence of a globally stable coexistence equilibrium or alternatively for the existence of periodic orbits

green s function for a two dimensional exponentially graded elastic medium the freespace green function for a twodimensional exponentially graded elastic medium is derived the shear modulus is assumed to be an exponential function of the cartesian coordinates x y i e

global stability for reaction diffusion equations in biology the aim of this work is to study the global stability for some diffusion equations in biology by constructing lyapunov functionals these lyapunov functionals are obtained from those for ordinary differential equations several examples from virology and epidemiology are given to illustrate our method

positive steady states for a prey predator model with some nonlinear diffusion terms abstract this paper discusses a prey predator system with strongly coupled nonlinear diffusion terms we give a sufficient condition for the existence of positive steady state solutions our proof is based on the bifurcation theory some a priori estimates for steady state solutions will play an important role in the proof

stationary patterns of strongly coupled prey predator models abstract we study some elliptic systems arising from 3 component predator prey models where cross diffusions are included in such a way that predator chases the prey and the prey runs away from the predator we establish the existence and non existence of non constant positive solutions our results show that the cross diffusions can create the stationary patterns

on stability of two degenerate reaction diffusion systems abstract in this paper we construct a partially degenerate reaction diffusion equation subject to the neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable in this way we solve an open problem proposed by casten and holland 1977 4 moreover we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross diffusion effects which cannot be determined by a simple spectral analysis based on the well developed semigroup theory

nonlinear stability in reaction diffusion systems via optimal lyapunov functions we define optimal lyapunov functions to study nonlinear stability of constant solutions to reaction diffusion systems a computable and finite radius of attraction for the initial data is obtained applications are given to the well known brusselator model and a three species model for the spatial spread of rabies among foxes

diffusion driven instability in reaction diffusion systems abstract for a stable matrix a with real entries sufficient and necessary conditions for a d to be stable for all non negative diagonal matrices d are obtained implications of these conditions for the stability and instability of constant steady state solutions to reaction diffusion systems are discussed and an example is given to show applications

a strongly coupled predator prey system with non monotonic functional response abstract the predator prey system with non monotonic functional response is an interesting field of theoretical study in this paper we consider a strongly coupled partial differential equation model with a non monotonic functional response a holling type iv function in a bounded domain with no flux boundary condition we prove a number of existence and non existence results concerning non constant steady states patterns of the underlying system in particular we demonstrate that cross diffusion can create patterns when the corresponding model without cross diffusion fails

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

slow switching in globally coupled oscillators robustness and occurrence through delayed coupling the phenomenon of slow switching in populations of globally coupled oscillators is discussed this characteristic collective dynamics which was first discovered in a particular class of the phase oscillator model is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population we argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom we also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators although a heteroclinic loop might seem to arise only exceptionally we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony we argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle node bifurcations slow switching arises when a system with a heteroclinic loop is weakly perturbed this will be demonstrated with a vector model by applying weak noises other types of weak symmetry breaking perturbations can also cause slow switching

coexistence of three species in a strongly coupled elliptic system abstract a strongly coupled elliptic system which describes three interacting species with homogeneous dirichlet boundary conditions is considered it is shown that there is no coexistence state if diffusion rates are strong or if the intrinsic growth rates are slow making use of the schauder fixed point theory we derive some sufficient conditions to have a semi coexistence or a coexistence state for the strongly coupled elliptic problem moreover our results reveal that this problem possesses at least one coexistence state if cross diffusions and intra specific competitions are weak

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

solitary waves in excitable systems with cross diffusion we consider a fitzhughnagumo system of equations where the traditional diffusion terms are replaced with linear cross diffusion of components this system describes solitary waves that have unusual

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

a boundary element formulation for the heterogeneous functionally graded viscoelastic structures structural analysis of the heterogeneous viscoelastic materials is among the most important subjects in the engineering design stages several attempts have been performed so far for employing the integral equation approach to heterogeneous viscoelastic structures in the present research employing two viscoelastic constitutive equations and a weighted residual technique a simple but effective boundary element formulation is implemented for heterogeneous kelvin voigt and boltzmann viscoelastic solid models the boltzmann model provides proper behavior predictions for realistic structures manifest viscoelastic natures present formulation needs only fundamental solution of the heterogeneous elastostatic problem with materials whose properties are prescribed as explicit functions of time heterogeneous exponentially graded materials and quasistatic time variations are considered in the present formulation present approach does not require using the relaxation functions and mathematical transformations and is capable of solving quasistatic viscoelastic problems with arbitrary load time dependencies and boundary conditions some numerical examples are provided to verify the proposed boundary element approach

on global existence of solutions to a cross diffusion system abstract we consider a strongly coupled nonlinear parabolic system which arises in population dynamics in n dimensional domains n 1 we prove the global existence of classical solutions to the system for n 10

existence of stationary solutions with transition layers for a class of cross diffusion systems this paper is concerned with the existence of stationary solutions for some cross di usion systems with small parameters using a singular perturbation method we prove the existence of stationary solutions with boundary and interior layers which extends the results of fife and ito to the cross di usion cases

shaping wave patterns in reaction diffusion systems we present a method to control the two dimensional shape of traveling wave solutions to reaction diffusion systems such as interfaces and excitation pulses control signals that realize a pregiven wave shape are determined analytically from nonlinear evolution equation for isoconcentration lines as the perturbed nonlinear phase diffusion equation or the perturbed linear eikonal equation while the control enforces a desired wave shape perpendicular to the local propagation direction the wave profile along the propagation direction itself remains almost unaffected provided that the one dimensional wave profile of all state variables and its propagation velocity can be measured experimentally and the diffusion coefficients of the reacting species are given the new approach can be applied even if the underlying nonlinear reaction kinetics are unknown

some remarks on matrix stability with application to turing instability abstract an example of a 4 4 matrix is given that provides a counterexample to a result on turing diffusion driven instability and also answers negatively a conjecture on strong stability such instability is shown to arise from nonreal eigenvalues the relevance and the connection of our example to classes of matrix stability known in the literature are discussed

positive steady state solutions of a competing reaction diffusion system with large cross diffusion coefficients abstract this paper deals with a coupled system of reaction diffusion equations modeling the competition of two species with cross diffusion we discuss the existence of positive steady state solutions in relation to the intrinsic birth rates a 1 a 2 of species and the cross diffusion coefficients 12 21 our results show that positive steady state solutions exist if a 1 and a 2 lie in certain range or if 12 and 21 are sufficiently large

propagation phenomena in a bistable reaction diffusion system consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical biochemical systems these equations are of the fitzhugh nagumo type and contain three parameters for certain ranges of the parameters the system exhibits two stable singular points a singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves a complete rigorous description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity numerical results of solutions to these equations are also presented these calculations illustrate how waves are generated from initial data how they interact during collisions and how they are influenced by local disturbances and boundary conditions in one example a righltward moving

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

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 asymptotic stability of a ratio dependent predator prey system with diffusion this paper is concerned with a predator prey system with diffusion the global asymptotic stability of the unique positive constant equilibrium is obtained under certain conditions the method used here is the upper and lower solutions combined with the monotone iteration and constructing suitable lyapunov functions

turing pattern formation in a three species model with generalist predator and cross diffusion abstract in a natural ecosystem specialist predators feed almost exclusively on one species of prey but generalist predators feed on many types of species consequently their dynamics is not coupled to the dynamics of a specific prey population however the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species this is described by cross diffusion in a generalist predator prey model first the positive equilibrium solution is globally asymptotically stable for the ode system and for the reaction diffusion system without cross diffusion respectively hence it does not belong to the classical turing instability scheme but it becomes linearly unstable only when cross diffusion also plays a role this implies that cross diffusion can lead to the occurrence and disappearance of the instability our results exhibit some interesting combining effects of cross diffusion predations and intra species interactions 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

stationary patterns created by cross diffusion for the competitor competitor mutualist model abstract in this paper we consider a competitor competitor mutualist model with cross diffusion we prove some existence and non existence results concerning non constant positive steady states patterns in particular we demonstrate that the cross diffusion can create patterns when the corresponding model without cross diffusion fails

global asymptotic stability of a diffusive predator prey model with ratio dependent functional response abstract this paper is concerned with a diffusive leslie gower predator prey system with ratio dependent holling type iii functional response under homogeneous neumann boundary conditions the uniform persistence of the solutions semiflows the existence of global attractors local and global asymptotic stability of the positive constant steady state of the reaction diffusion model are discussed by using comparison principle the linearization method and the lyapunov functional method respectively the global asymptotic stability of the positive constant steady state shows that the prey and predator will be spatially homogeneously distributed as time converges to infinities

front propagation and segregation in a reaction diffusion model with cross diffusion abstract a study of front propagation and segregation in a system of reaction diffusion equations with cross diffusion is presented the reaction models predator prey dynamics involving two fields the diffusive part is nonlinear in the sense that the diffusion coefficient instead of being a constant as in the well studied case depends on one of the fields a key element of the model is a cross diffusion term according to which the flux of one of the fields is driven by gradients of the other field the original motivation of the model was the study of the turbulence shear flow interaction in plasmas the model also bears some similarities with models used in the study of spatial segregation of interacting biological species the system has three nontrivial fixed points and a study of traveling fronts solutions joining these states is presented depending on the stability properties of the fixed points the fronts are uniform or have spatial structure in the latter case a cross diffusion driven pattern forming k 0 instability leads to segregation in the wake of the front the segregated state consists of layered structures a ginzburg landau amplitude equation is used to describe the dynamics near marginal stability

bifurcation analysis to the lugiato lefever equation in one space dimension we study the stability and bifurcation of steady states for a certain kind of damped driven nonlinear schrodinger equation with cubic nonlinearity and a detuning term in one space dimension mathematically in a rigorous sense it is known by numerical simulations that the system shows lots of coexisting spatially localized structures as a result of subcritical bifurcation since the equation does not have a variational structure unlike the conservative case we cannot apply a variational method for capturing the ground state hence we analyze the equation from a viewpoint of bifurcation theory in the case of a finite interval we prove the fold bifurcation of nontrivial stationary solutions around the codimension two bifurcation point of the trivial equilibrium by exact computation of a fifth order expansion on a center manifold reduction in addition we analyze the steady state mode interaction and prove the bifurcation of mixed mode solutions which will be a germ of localized structures on a finite interval finally we study the corresponding problem on the entire real line by use of spatial dynamics we obtain a small dissipative soliton bifurcated adequately from the trivial equilibrium

analysis of a population model with strong cross diffusion in unbounded domains we study a parabolic population model in the full space and prove the global in time existence of a weak solution this model consists of two strongly coupled diffusion equations describing the population densities of two competing species the system features intrinsic growth inter and intra specific competition of the species as well as diffusion cross diffusion and self diffusion and drift terms related to varying environment quality the cross diffusion terms can be large making the system non parabolic for large initial data for the first time solutions in unbounded domains are studied the method of our proof is a combination of a time semi discretization a transformation of the system guaranteeing the positivity of the solution a special entropy symmetrizing the system and compactness arguments 2000 mathematics subject classification 35k55 primary 92d25 35d05 secondary

some results on multifractal correlations abstract we have performed a numerical and analytical study of multifractal correlations in a two scale random cantor set at large distances for not very large we find qualitative agreement between canonical simulation results also counting results and the formula recently proposed by meneveau and chhabra following work of cates and deutsch for large a new correlation scaling function which is linear in log r is obtained due to a transition of the moment scaling function from cates and deutsch behavior to a differing behavior for some part of its domain this linear is consistent with numerical results

segregation and pursuit waves in activator inhibitor systems we investigate the effects of cross diffusion on propagating waves in an activator inhibitor system the model consists of a piecewise linear approximation of fitzhugh nagumo kinetics and a cross diffusion term for either the activator or the inhibitor we obtain exact analytic solutions for traveling fronts and solitary pulses and discuss the corresponding speed diagrams a detailed comparison with the corresponding rinzel keller model for the usually studied case of self diffusion is performed

linear stability of delayed reactiondiffusion systems a common feature of pattern formation in both space and time is the destabilization of a stable equilibrium solution of an ordinary differential equation by adding diffusion or delay or both here we study linear stability of general reactiondiffusion systems with off diagonal time delays we show that a delay stable system cannot be destabilized by diffusion and that a diffusion stable system is also stable with respect to delay if the diffusion is sufficiently fast a system with direct negative feedback which is strongly stable with respect to diffusion can be destabilized by off diagonal delay

existence uniqueness and stability of positive periodic solution for a nonlinear prey competition model with delays a nonlinear periodic predator prey model with m preys and n m predators and delays is proposed in this paper which can be seen as the modification of the traditional lotka volterra prey competition model sufficient conditions which guarantee the existence of a unique globally attractive positive periodic solution of the system are obtained

exhibiting cross diffusion induced patterns for reaction diffusion systems on evolving domains and surfaces the aim of this manuscript is to present for the first time the application of the finite element method for solving reaction diffusion systems with cross diffusion on continuously evolving domains and surfaces furthermore we present pattern formation generated by the reaction diffusion systemwith cross diffusion on evolving domains and surfaces a two component reaction diffusion system with linear cross diffusion in both u and v is presented the finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles for surfaces the vertices of the triangulation lie on the continuous surface a finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface to demonstrate the role of cross diffusion to the theory of pattern formation we compute patterns with model kinetic parameter values that belong only to the cross diffusion parameter space these do not belong to the standard parameter space for classical reaction diffusion systems numerical results exhibited show the robustness flexibility versatility and generality of our methodology the methodology can deal with complicated evolution laws of the domain and surface and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface