modified curvature motion for image smoothing and enhancement we formulate a general modified mean curvature based equation for image smoothing and enhancement the key idea is to consider the image as a graph in some r sup 72 and apply a mean curvature type motion to the graph we consider some special cases relevant to grey scale and color images

individual based model with global competition interaction fluctuation effects in pattern formation we present some numerical results obtained from a simple individual based model that describes clustering of organisms caused by competition our aim is to show that even when a deterministic description developed for continuum models predicts no pattern formation an individual based model displays well defined patterns as a consequence of fluctuation effects caused by the discrete nature of the interacting agents

computation of homoclinic solutions to periodic orbits in a reduced water wave problem abstract this paper concerns homoclinic solutions to periodic orbits in a fourth order hamiltonian system arising from a reduction of the classical water wave problem in the presence of surface tension these solutions correspond to travelling solitary waves which converge to non decaying ripples at infinity an analytical result of amick and toland 1992 showing the existence of such homoclinic orbits to small amplitude periodic orbits in a singular limit is extended numerically also a related result by amick and mcleod 1991 showing the non existence of homoclinic solutions to zero is motivated geometrically a general boundary value method is constructed for continuation of homoclinic orbits to periodic orbits in hamiltonian and reversible systems numerical results are presented using the path following software auto showing that the amick toland solutions persist well away from the singular limit and for large amplitude periodic orbits special account is taken of the phase shift between the two periodic solutions in the asymptotic limits furthermore new multi modal homoclinic solutions to periodic orbits are shown to exist under a transversality hypothesis which is verified a posteriori by explicit computation continuation of these new solutions reveals limit points with respect to the singular parameter

partially implicit schemes for the numerical solutions of some non linear differential equations as numerical methods are one of the main tools used to discover the non linear behaviour characteristics of continuous dynamical systems a study on the use of numerical schemes which delineate the true physics from numerical artefact is desirable as a step in that direction partially implicit rational schemes based on first order methods and a second order method are derived and analysed from the point of view of dynamical systems for a scalar autonomous ordinary differential equation the second order method is developed by taking a linear combination of first order methods though implicit in nature the methods are applied explicitly the approach adopted is extended to obtain numerical schemes for the solution of a class of reaction diffusion partial differential equations the methods are tested on differential equations with cubic and quadratic reaction terms it is found that for certain sets of initial conditions the first order partially implicit schemes converge monotonically to the correct fixed points for all time steps unlike the euler method and the second order method in other words for the same steady state solution the associated domain of attraction of the discrete problem and its continuous counterpart are exactly the same however for other initial conditions it is found that the domains of attraction of the second order method mimic more closely the domains of attraction of the continuous differential equations the comparative study of the various explicit runge kutta methods of orders 2 6 shows that it is the discretized form with its specific algebraic form recursive relation which are the source of spurious steady state solutions and numerical chaos from these methods it is found however that the use in general of fifth order runge kutta methods is seen to be preferred to other runge kutta methods as they have an extended stability range in particular the fifth order lawson method which has a far superior range of stability than the other runge kutta methods

young measure solutions of a class of forward backward diffusion equations abstract this paper is devoted to young measure solutions of a class of forward backward diffusion equations inspired by the idea from a recent work of demoulini we first discuss the regular case by introducing the young measure solutions and prove the existence for such solutions and then approximate the extreme case by the approach of regularization and establish the existence of young measure solutions in the class of functions with bounded variation

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

cyclical interactions with alliance specific heterogeneous invasion rates we study a six species lotka volterra type system on different two dimensional lattices when each species has two superior and two inferior partners the invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator prey pair whereby cyclic symmetries within the two three species defensive alliances are conserved monte carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance specific invasion rates this behavior is qualitatively reproduced by a four point mean field approximation the study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences

bifurcation studies in reaction diffusion ii and estimated the critical lengths of the domain at which bifurcation occurs in the cases b 0 a and 1 also we were able to characterise some of the non constant steady state solutions which appear beyond the points of instability by a steady state solution of the form u x

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

a general framework for low level vision we introduce a new geometrical framework based on which natural flows for image scale space and enhancement are presented we consider intensity images as surfaces in the x i space the image is thereby a two dimensional 2 d surface in three dimensional 3 d space for gray level images and 2 d surfaces in five dimensions for color images the new formulation unifies many classical schemes and algorithms via a simple scaling of the intensity contrast and results in new and efficient schemes extensions to multidimensional signals become natural and lead to powerful denoising and scale space algorithms

implicit spectral methods for wave propagation problems abstract the numerical solution of a non linear wave equation can be obtained by using spectral methods to resolve the unknown in space and the standard crank nicolson differencing scheme to advance the solution in time we have analyzed iterative techniques for solving the non linear equations that arise from such implicit time stepping schemes for the k dv and the k p equations we derived predictor corrector method that retain the full accuracy of the implicit method with minimal stability restrictions on the size of the time step some numerical examples show the propagation of interacting solitons

asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics we consider a spatially aggregating population model which provides the homogenizing process due to density dependent diffusion and the dehomogenizing one due to a certain long range transport the result asserts that by a balance between two processes an initial distribution of populations forms itself into a traveling solitary wave pattern for large time which exhibits phenomenologically a kind of aggregation of a species

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

interior gradient estimates for mean curvature equations the interior gradient estimate for the prescribed mean curvature equation has been extensively studied see 9 and the references therein for high order mean curvature equations it has also been obtained in 11 18 in most articles such estimates were obtained by carrying out analysis on the graphs of solutions and so the arguments depend on the invariance of the equations under rigid motion from the view point of partial differential equations such estimate should hold for equations with similar structural conditions in 13 6 the gradient estimate was obtained for certain fully nonlinear elliptic equations under various conditions different proofs for the gradient estimate for mean curvature equations have been given in 1 3 in this note we show that for curvature and hessian equations the interior gradient estimate can be obtained very easily our proof which is based on suitable choice of auxiliary functions is elementary and avoids geometric computations on the graph of solutions the technique in this note has actually been widely used in literature see e g 9

a comparison of basis functions for the pseudo spectral method for a model reaction diffusion problem abstract different types of basis functions are considered for the pseudospectral method applied to the equation u t u u in 1 and 2 space dimensions for large n where n is the number of basis functions the choice of cheybyshev polynomials gives much greater accuracy than the eigenfunctions of the laplacian however for small n 7 the eigenfunction expansion can be more accurate even with weakly nonperiodic boundary conditions

weakly nonlinear stability analyses of prototype reaction diffusion model equations comparisons are made between the results obtained by the application of various methods of weakly nonlinear stability analysis to several prototype reaction diffusion equations which among them reproduce most of the salient features representative of such investigations of model systems for dissipative phenomena emphasis is placed upon the subtleties involved in scaling these equations and deducing asymptotically valid solutions phenomenological interpretations of the bifurcation behavior of the model equations are offered based on the pattern formational aspects of convective instabilities in hydrodynamics morphological instabilities in solidification and diffusive instabilities in ecology

non parametric mean curvature evolution with boundary conditions soit un domaine borne de r n n 2 de frontiere lisse et soit a l operateur de surface minimale donne par a d i a i p a i p i 1 p 2 1 2 on considere des surfaces dont la vitesse en direction de leur normale unite est egale a la courbure moyenne on etudie l equation d evolution qui en resulte u 1 du 2 1 2 au 0 dans 0 t

on a perturbation of the benjamin ono equation abstract we prove using the fourier restriction norm method that the initial value problem associated to a perturbation of the benjamin ono equation u t u u x h u x x h u x u x x 0 where x r t 0 0 and h denotes the usual hilbert transform is locally well posed in the sobolev space h s r for any s 1 2 and globally well posed in h s r when s 0 for s 1 we also prove some ill posedness issues

gradient estimates for solutions of parabolic equations and systems 1 statement of the problem let 52 c r n 2 be a bounded domain with smooth boundary r 2 in this note we study scalar quasilinear parabolic differential equations u div y g ivul vu f u 0 1 u a u f u o 2 and systems u da u f u 0 3 on q x 0 co with initial conditions u 0 u0 resp u 0 u0 and homogeneous dirichlet boundary conditions u i x o co o for 1 and 2 resp 4 u1r x 0 00 0 for 3 5

dissipative toda rayleigh lattice and its oscillatory modes a detailed theoretical and experimental analysis of the possible oscillatory regimes of the dissipative toda rayleigh lattice system is provided it is shown that the system has n 1 oscillatory modes with different space time scales and two rotatory modes using its analog electronic circuit implementation we also show with a simple and robust method how switching between modes occurs

self similar entropy solutions of a singular diffusion equation in arbitrary spatial dimension abstract in this paper we consider a singular diffusion equation arising in phase transition and investigate its self similar entropy solutions with jump hypersurfaces the existence nonexistence and uniqueness theorems of such solutions are established we also discuss some properties of this kind of solutions including monotonicity and asymptotic behavior

multi hump stationary waves for a korteweg de vries equation with nonlocal perturbations abstract periodic and solitary wave solutions are investigated numerically for a perturbed korteweg de vries equation with unstable and dissipation terms in hilbert transform u t uu x u xxx h u x h u xxx 0 a family of solitary wave solutions s 1 s 2 s n whose members are distinguished by the number of humps is numerically identified the tails of these waves decay as o 1 x 2 when x irrespective of the magnitude of it is also found that for a given there exist families of periodic wave solutions p 1 p 2 p n which originate from one near sinusoidal wave and end up in the infinite periodicity to the corresponding solitary waves the numerical results are consistent with the theoretical estimates based on the conservation properties

dynamical approach study of spurious steady state numerical solutions of nonlinear differential equations i the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics abstract the goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady state numerical solutions when using the time dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms this interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as the dynamics of numerics and the numerics of dynamics at the present time this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications this is the first of a series of research papers under the same topic our hope is to reach researchers in the fields of computational fluid dynamics cfd and in particular hypersonic and combustion related cfd by simple examples in which the exact solutions of the governing equations are known the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results one striking phenomenon is that with the same initial data the continuum and its discretized counterpart can asymptotically approach different stable solutions this behavior is especially important for employing a time dependent approach to the steady state since the initial data are usually not known and a freestream condition or an intelligent guess for the initial conditions is often used with the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart it is not easy to delineate the true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum part i concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in cfd and to serve as an introduction for parts 11 and iii of the same series of research papers

similar entropy solutions of a singular diffusion equation in this paper we study the similar entropy solutions of the singular diffusion equation u t x j u x with j s s 1 s 2 these kinds of solutions have nonvertical jump lines we establish the existence and uniqueness and also discuss some properties of these kinds of solutions

instability in diffusive ecological models with nonlocal delay effects abstract we propose a reaction diffusion system with nonlocal delays to model the growth of plankton communities feeding on a limiting nutrient supplied at a constant rate two delays are incorporated into the model one describes the delayed nutrient recycling process and the other models the delayed growth response of the plankton it is assumed that both delays are nonlocal in the sense that there are delayed not only in time but also in space local and bifurcation analyses are carried out it has been shown that turing type spatial patterns occur when the diffusion coefficients vary and temporal or spatial temporal patterns occur when the delay involved in the growth response changes

a turbulence model with stochastic soliton motion a dissipative benjamin ono equation is used to study fluid and plasma turbulence the system is studied by an exact nonlinear mode truncation method in which a finite number of poles are used to present the solution the justification of the pole expansion approach is discussed with the proof of a completeness theorem the stability and spectrum analysis show that asymptotic behavior of the system is completely represented by a finite number of nonlinear modes the behavior of those nonlinear modes resembles solitons and exhibits a wide range of bifurcation phenomena and routes to turbulence

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

turing instability in competition models with delay i linear theory turing instability in two component predator prey and reaction diffusion models including diffusion and volterra type distributed delays in the interspecies interaction terms is considered for general functional forms of the prey birthrate predator deathrate reaction terms and delays modeled by the weak generic kernel a exp au or the strong generic kernel au exp au the structure of the diffusively unstable space is shown to be completely altered by the inclusion of delays the necessary and sufficient conditions for turing instability are derived using the weak generic kernel and are found to be significantly different from the classical conditions with no delay the structure of the turing space where steady states may be diffusionally driven unstable initiating spatial pattern is delineated for several specific models and compared to the corresponding regimes in the absence of delay an alternative bifurcation theoretic derivation of the boundary of the turing unstable domain

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

pattern generation in space and aspect a survey is presented of theories for the generation and maintenance of spatial pattern in reaction diffusion equations and their generalizations applications are selected from the biological sciences and physical chemistry special emphasis is placed on nonlocal interaction as manifested by the inclusion of terms involving higher derivatives or integrals it is stressed that traditional ideas of spatial pattern generation can usefully be extended to the study of pattern in general descriptive aspect variables particularly in understanding ecological diversity and heterogeneity

probing a subcritical instability with an amplitude expansion an exploration of how far one can get abstract we explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite wavelength instability we do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers this is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic laminar state we apply the method to two model equations a subcritical generalization of the swift hohenberg equation and a novel extension of the kuramoto sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate from infinity and that can nevertheless be probed perturbatively we explore the reliability and robustness of such an expansion with a particular focus on the use of these methods for determining the existence and approximate properties of finite amplitude stationary solutions such methods obviously are to be used with caution the expansions are often only asymptotic approximations and if they converge their radius of convergence may be small nevertheless expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

a numerical study of spontaneous bifurcation abstract in the study of nonlinear boundary value problems it has been observed that bifurcations may take place which are not local to the origin in function space in such problems the analysis of the bifurcation and the computation of solutions is complicated by the nonlocal nature of the bifurcation and the multiplicity of solutions in this work a model problem exhibiting this phenomena which has been of recent interest is numerically studied the method of alternative problems is employed to provide a complete numerical description of the bifurcation occurring in the model problem

emergent singular solutions of nonlocal density magnetization equations in one dimension we investigate the emergence of singular solutions in a nonlocal model for a magnetic system we study a modified gilbert type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the nonlocal effects we pass to a coupled density magnetization model and perform a linear stability analysis noting the effect of the length scales of nonlocality on the system s stability properties we carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data the singular solutions represent a collection of interacting particles clumpons by restricting ourselves to the two clumpon case we are reduced to a two dimensional dynamical system that is readily analyzed and thus we classify the different clumpon interactions possible

chaos free numerical solutions of reaction diffusion equations two numerical methods which do not bring contrived chaos into the solution are proposed for the solution of the riccati logistic equation though implicit in nature with the resulting improvements in stability the methods are applied explicitly when extended to the numerical solution of fisher s equation in which the quadratic polynomial representing the derivative in the riccati equation appears as the reaction term the solution is found by solving a linear system of algebraic equations at each time step as opposed to solving a nonlinear system which frequently happens when solving nonlinear partial differential equations the approaches adopted are extended to an ordinary differential equation in which the derivative is expressed as a cubic polynomial in the dependent variable the solution of this initial value problem is not available in closed form for finite values of the independent variable t under the conditions stated numerical solutions are seen to converge to the correct steady state solution a nonlinear partial differential equation which governs the conduction of electrical impulses along a nerve axon and which has the aforementioned cubic polynomial as its reaction term is solved by applying the numerical methods developed for solving the ordinary differential equation the solution to this nonlinear reaction diffusion equation is determined by solving a linear algebraic system at each time step

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

infinite time blow up of solutions to a nonlinear parabolic problem abstract we investigate the blow up of solutions of nonuniformly parabolic equations it will be shown that the gradient of the solution tends to infinity as time goes to infinity even though thel norm of the solution may remain uniformly bounded this phenomenon is in contrast with a result for semilinear heat equations where it follows from regularity theory that the solution converges to an equilibrium if it is uniformly bounded in thel norm

threshold phenomena and solitary traveling waves in a class of reaction diffusion systems in this paper we consider a class of two species reaction diffusion systems in which one species may vary on a faster time scale than the other the system is assumed to have a unique stable steady state and conditions on the reaction terms ensure that one of the nullclines of the diffusionless system has a kink in it we exploit the existence of the two time scales in order to analyze the system asymptotically and show that it exhibits threshold behavior in the spatially uniform case and a solitary traveling wave in the spatially nonuniform case we consider examples of physical systems which exhibit these phenomena and show that each may be modeled by a reaction diffusion system satisfying the conditions of the paper

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

existence and regularity of extremal solutions for a mean curvature equation we study a class of mean curvature equations mu h up where m denotes the mean curvature operator and for p 1 we show that there exists an extremal parameter such that this equation admits a minimal weak solutions for all 0 while no weak solutions exists for weak solutions will be defined as critical points of a suitable functional in the radially symmetric case we then show that minimal weak solutions are classical solutions for all 0 and that another branch of classical solutions exists in a neighborhood of

temporal evolutions and stationary waves for dissipative benjamin ono equation abstract initial value problems as well as stationary solitary and periodic waves are investigated for dissipative benjamin ono dbo equation multi hump stationary waves and their structures are identified numerically and the stability regions of stationary periodic waves are also examined numerically these results elucidate a close relation between irregular behaviours in the initial value problem and the multiplicity of stationary waves

stationary travelling wave solutions of an unstable kdv burgers equation abstract both solitary and periodic stationary travelling wave solutions are investigated numerically for an unstable korteweg de vries burgers equation u t uu x u xxx u u xx 0 0 a family of stationary solitary wave solutions whose members are distinguished by the number of humps is found for a given corresponding to each solitary wave thus found a family of stationary periodic waves with the same number of humps exists under periodic condition and ends up in the infinite periodicity to the corresponding solitary wave the numerical results are consistent with the theoretical estimates based on the conservation properties

traveling waves in a nonlocal dispersal population model with age structure abstract this paper is concerned with the traveling waves in a single species population model which is derived by considering the nonlocal dispersal and age structure if the birth function is monotone then the existence of traveling wavefront is reduced to the existence of a pair of super and subsolutions without the requirement of smoothness it is proved that the traveling wavefront is strictly increasing and unique up to a translation the asymptotic behavior of traveling wavefronts is also obtained if the birth function is not monotone the existence of traveling wave solution is affirmed by introducing two auxiliary nonlocal dispersal equations with quasi monotonicity

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

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

a picture of the global bifurcation diagram in ecological interacting and diffusing systems abstract this paper concerns global phenomena of pattern formation in stationary reaction diffusion equations possessing turing s diffusion induced instability which appear typically in mathematical biology global bifurcation diagrams with respect to two diffusion parameters are presented by integrating two complementary approaches analytical and numerical simple and double bifurcation analysis using group theoretic methods for the compact lie group d x results from singular perturbations when one diffusion constant is sufficiently small and a global existence theorem on primary bifurcated branches when the other diffusion is sufficiently large are the main analytical results a new numerical method is used to trace all bifurcating branches we examine interrelations between those local and semi global results under the light of global pictures obtained by numerical studies a variety of interesting and new diagrams near double eigenvalues are observed coexistence of multiple stable stationary states and global extension and deformation of local double structures are one of the main conclusions

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

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

some new observations on the classical logistic equation with heredity abstract several new and significant observations are presented pertaining to the classical problem of single population growth with hereditary influences in its conventional form the resulting equation with heredity is mathematically represented by a nonlinear volterra integro differential equation in this paper we propose a new differential formulation where the dependent variable is now defined in terms of the integral of the unknown population this formulation allows us to develop novel analyses leading to enlightening results some particular findings include the development and analysis of an integrated phase plane the elucidation of the exact value for the extremum of the population and several other important functional relations at that corresponding time the development of two analytic expressions for determining the time at which the population peaks the determination of the upper asymptote for the cumulative population and the development of an accurate early time solution as obtained from a riccati equation additionally we illustrate that an analytical solution based on taylor series expansions can be developed with the aid of mathematica tm a pure numerical solution is offered for comparison with the analytic solution

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

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

pattern formation for a volume filling chemotaxis model with logistic growth abstract this paper is devoted to investigating the pattern formation of a volume filling chemotaxis model with logistic cell growth we first apply the local stability analysis to establish sufficient conditions of destabilization for the uniform steady state solution then the weakly nonlinear analysis with multi scales is used to deal with the emerging process of patterns near the bifurcation point for the single unstable mode case we derive the stuart landau equations describing the evolution of the amplitude and thus the asymptotic expressions of patterns are obtained in both supercritical case and subcritical case while for the case of multiple unstable modes we also derive coupled amplitude equations to study the competitive behavior between two unstable modes through the phase plane analysis in particular we find that the initial data play a dominant role in the competition all the theoretical and numerical results are in excellently qualitative agreement and better quantitative agreement than that in 10 moreover in the subcritical case we confirm the existence of stationary patterns with larger amplitudes when the bifurcation parameter is less than the first bifurcation point which gives an positive answer to the open problem proposed in 9

laminarizing effects of dispersion in an active dissipative nonlinear medium abstract the kuramoto sivashinsky equation has become a popular prototype for systems which exhibit spatial temporal chaos we show here that a linear dispersion term h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures as is consistent with prior numerical and experimental observations the study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation it is shown that by 1 1 the infinite families of stationary periodic waves of the kuramoto sivashinsky equation each ending in a solitary wave have been annihilated successively such that only a lone family of periodic waves consisting of one hump kdv pulses for 3 7 remains as the only periodic stationary wave attractors of the system these periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial temporal chaos in an extended domain with significant dispersion

bifurcation diagrams of population models with nonlinear diffusion we develop analytical and numerical tools for the equilibrium solutions of a class of reaction diffusion models with nonlinear diffusion rates such equations arise from population biology and material sciences we obtain global bifurcation diagrams for various nonlinear diffusion functions and several growth rate functions

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