chemotaxis with logistic source very weak global solutions and their boundedness properties abstract we consider the chemotaxis system u t u u v g u x t 0 0 v v u x t 0 in a smooth bounded domain r n where 0 and g generalizes the logistic function g u a u b u with 1 a 0 and b 0 a concept of very weak solutions is introduced and global existence of such solutions for any nonnegative initial data u 0 l 1 is proved under the assumption that 2 1 n moreover boundedness properties of the constructed solutions are studied inter alia it is shown that if b is sufficiently large and u 0 l has small norm in l for some n 2 then the solution is globally bounded finally in the case that additionally n 2 holds a bounded set in l can be found which eventually attracts very weak solutions emanating from arbitrary l 1 initial data the paper closes with numerical experiments that illustrate some of the theoretically established results

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

spatio temporal oscillations in the keller segel system with logistic growth abstract the keller segel system with the logistic growth term is discussed from the spatio temporal oscillation point of view this system exhibits two different types of spatio temporal oscillations in certain distinct parameter regimes in this paper we study the difference between the two types of spatio temporal oscillations in particular the characteristic properties of the behaviors become clear in a limiting system when a certain parameter value tends to zero moreover we demonstrate that the onset of one of the spatio temporal oscillatory patterns is an infinite dimensional relaxation oscillation that consists of slow and fast dynamics

how far does small chemotactic interaction perturb the fisher kpp dynamics abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the fully parabolic chemotaxis growth system u e t u e e u e v e u e 1 u e x t 0 v e t v e v e u e x t 0 with positive small parameter e 0 in a bounded convex domain r n n 1 with smooth boundary the solutions converge to the solution u to the fisher kpp equation as e 0 it is shown that for all 0 and any suitably regular nonnegative initial data u i n i t v i n i t there are some constants e 0 0 and c 0 such that sup t 0 u e t u t l c e for all e 0 e 0

progressive scattered data filtering given a finite point set z rd the covering radius of a nonempty subset x z is the minimum distance r x z such that every point in z is at a distance of at most r x z from some point in x this paper concerns the construction of a sequence of subsets of decreasing sizes such that their covering radii are small to this end a method for progressive data reduction referred to as scattered data filtering is proposed the resulting scheme is a composition of greedy thinning a recursive point removal strategy and exchange a postprocessing local optimization procedure the paper proves adaptive a priori lower bounds on the minimal covering radii which allows us to control for any current subset the deviation of its covering radius from the optimal value at run time important computational aspects of greedy thinning and exchange are discussed the good performance of the proposed filtering scheme is finally shown by numerical examples

a positivity preserving finite element method for chemotaxis problems in 3d we present an implicit finite element method for a class of chemotaxis models in three spatial dimensions the proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density to enforce the discrete maximum principle the standard galerkin discretization is constrained using a local extremum diminishing flux limiter to demonstrate the efficiency and robustness of this approach we solve blow up problems in a 3d chemostat domain to give a flavor of more complex and realistic chemotactic applications we investigate the pattern dynamics and aggregating behavior of the bacteria escherichia coli and salmonella typhimurium the obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature

a best possible heuristic for the k center problem in this paper we present a 2 approximation algorithm for the k center problem with triangle inequality this result is best possible since for any 2 the existence of approximation algorithm would imply that p np it should be noted that no approximation algorithm for any constant has been reported to date linear programming duality theory provides interesting insight to the problem and enables us to derive in o e log e time a solution with value no more than twice the k center optimal value a by product of the analysis is an o e algorithm that identifies a dominating set in g2 the square of a graph g the size of which is no larger than the size of the minimum dominating set in the graph g the key combinatorial object used is called a strong stable set and we prove the np completeness of the corresponding decision problem

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth recently wang and hillen chaos 17 3 2007 037108 037108 introduced a modified keller segel model with logistic growth that exhibits complex spatiotemporal dynamics of spikes these dynamics are driven by merging of spikes on one hand and spike insertion on the other in this paper we analyse the basic mechanisms that initiate and sustain these events we identify two distinguished regimes in the first regime a single interior spike drifts towards a boundary this instability is responsible for spike merging the same regime further exhibits spike insertion we identify a fold point bifurcation which is precursor to spike insertion event in the second regime we show that it is possible to stabilize a single interior spike and we compute analytically a critical threshold which is responsible for spike stabilization in particular our calculation characterizes a stable spike in the keller segel model with logistic growth this is in contrast to the classical keller segel model where the interior spike is known to be unstable

on a class of keller segel chemotaxis systems with cross diffusion abstract in this paper we study a class of keller segel chemotaxis systems with cross diffusion by using the entropy dissipation method and assuming mainly the chemotactic sensitivity separates the cell density and the chemical signal we first establish the existence of global weak solutions with the effects of cross diffusion included in 3 d then we show there is a critical cross diffusion rate c such that no patterns may be expected for c while patterns are formed for c and their stability is also derived in particular in 1 d patterns are always formed whenever c and the chemotactic coefficient is larger than an expressible bifurcation value and there is another critical cross diffusion rate c c such that cells with cross diffusion rate c c are stable while for cells with c to be stable their degradation rate must be less than a threshold value hence in some sense cross diffusion is harmful to enable pattern formation while it is helpful to stabilize the cells once patterns are formed finally we show that the cross diffusion plays a role in regularizing the cell aggregation phenomenon for large chemotactic coefficient our results provide global dynamics and insights on how the biological parameters especially the cross diffusion affect pattern formations

adaptive thinning for bivariate scattered data this paper studies adaptive thinning strategies for approximating a large set of scattered data by piecewise linear functions over triangulated subsets our strategies depend on both the locations of the data points in the plane and the values of the sampled function at these points adaptive thinning all our thinning strategies remove data points one by one so as to minimize an estimate of the error that results by the removal of a point from the current set of points this estimate is termed anticipated error the thinning process generates subsets of most significant points such that the piecewise linear interpolants over the delaunay triangulations of these subsets approximate progressively the function values sampled at the original scattered points and such that the approximation errors are small relative to the number of points in the subsets we design various methods for computing the anticipated error at reasonable cost and compare and test the performance of the methods it is proved that for data sampled from a convex function with the strategy of convex triangulation the actual error is minimized by minimizing the best performing measure of anticipated error it is also shown that for data sampled from certain quadratic polynomials adaptive thinning is equivalent to thinning which depends only on the locations of the data points nonadaptive thinning based on our numerical tests and comparisons two practical adaptive thinning algorithms are proposed for thinning large data sets one which is more accurate and another which is faster

boundedness and global existence in the higher dimensional parabolic parabolic chemotaxis system with without growth source abstract in this paper we are concerned with a general class of quasilinear parabolic parabolic chemotaxis systems with without growth source under homogeneous neumann boundary conditions in a smooth bounded domain r n with n 2 it is recently known that blowup is possible even in the presence of superlinear growth restrictions here we derive new and interesting characterizations on the growth versus the boundedness we show that the hard task of proving the l boundedness of the cell density can be reduced to proving its l r boundedness in other words we show that the l r boundedness of the cell density can successfully guarantee its l boundedness and hence its global boundedness where r n or n 2 depending on whether the growth restriction is essentially linear including no growth or superlinear hence a blowup solution also blows up in l p norm for any suitably large p more detailed information on how the growth source affects the boundedness of the solution is derived these results reveal deep understandings of blowup mechanism for chemotaxis models then we use these criteria to establish uniform boundedness and hence global existence of the underlying models logistic source in 2 d cubic source as initially proposed by mimura and tsujikawa in 3 d n 1 st source in n d with n 4 as a consequence in a chemotaxis growth model blowup is impossible if the growth effect is suitably strong finally we underline that our results remove the commonly assumed convexity on the domain

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

on a parabolic elliptic chemotactic system with non constant chemotactic sensitivity abstract we study a parabolic elliptic chemotactic system describing the evolution of a population s density u and a chemoattractant s concentration v the system considers a non constant chemotactic sensitivity given by n u for n 0 and a source term of logistic type u 1 u the existence of global bounded classical solutions is proved for any 0 n 0 and 0 by using a comparison argument we analyze the stability of the constant steady state u 1 v 1 for a range of parameters for n 1 and n 2 any positive and bounded solution converges to the steady state for n 1 the steady state is locally asymptotically stable and for n the steady state is globally asymptotically stable

higher dimensional localized patterns in excitable media abstract the excitable reaction diffusion equation model of the form e u t e 2 2 u u v v t 2 v u v is considered when u is assumed to be of mckean s piecewise linear type the interfacial approach can be applied to the stability of various localized patterns in higher dimensional spaces it is shown that a band shaped localized pattern is destabilized into a zig zag mode or a varicose mode and that a disk shaped localized pattern is destabilized into static modes and when is small into an oscillatory mode like a breather motion numerical simulations are performed to confirm such destabilizations for more general nonlinear functions u

dimension estimate of the global attractor for a semi discretized chemotaxis growth system by conservative upwind finite element scheme abstract in this paper we continue systematic study of the dimension estimate of the global attractors for the chemotaxis growth system and its finite element approximations utilizing a conservative upwind finite element scheme we managed significantly to improve dimension estimates with respect to the chemotactic parameter

singular limit of a reaction diffusion equation with a spatially inhomogeneous reaction term we study reaction diffusion equations with a spatially inhomogeneous reaction term if the coefficient of these reaction term is much larger than the diffusion coefficient a sharp interface appears between two different phases we show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations in particular we show that the same rich spatial patterns observed for a chemotaxis growth model can be realized by a system without a drift term

spatial pattern formation in a chemotaxis diffusion growth model abstract mimura and one of the authors 1996 proposed a mathematical model for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis for this model tello and winkler 2007 22 obtained infinitely many local branches of nonconstant stationary solutions bifurcating from a positive constant solution while kurata et al 2008 numerically showed several spatio temporal patterns in a rectangle motivated by their work we consider some qualitative behaviors of stationary solutions from global and local bifurcation viewpoints in the present paper first we study the asymptotic behavior of stationary solutions as the chemotactic intensity grows to infinity next we construct local bifurcation branches of stripe and hexagonal stationary solutions in the special case when the habitat domain is a rectangle for this case the directions of the branches near the bifurcation points are also obtained finally we exhibit several numerical results for the stationary and oscillating patterns

on the convergence rate of a reaction diffusion equation with chemotaxis to a viscosity solution abstract the singular limit of an advection reaction diffusion equation arising from a chemotaxis growth model equation is investigated for a very specific initial data the convergence to the viscosity solution of a limit free boundary problem is known henry et al 1999 15 in this work we not only allow rather general initial data but also construct refined barriers in some special cases these barriers provide an o e ln e estimate of the location and the thickness measured in space time of the transition layers

efficient accurate and flexible finite element solvers for chemotaxis problems in the framework of finite element discretizations we introduce a fully nonlinear newton like method and a linearized second order approach in time applied to certain partial differential equations for chemotactic processes incorporating two entities a chemical agent and the reacting population of certain biological organisms species we investigate the benefit of a corresponding monolithic approach and the decoupled variant in particular we analyze accuracy efficiency and stability of different methods and their dependences on certain parameters in order to identify a well suited finite element solver for chemotaxis problems

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

local factorization of differential forms on frechet schwartz spaces abstract the reduction of the problem on a frechet nuclear space to the study of the operator on a hilbert space produces a global solutionuwhen the second memberwfactors globally through this hilbert space easy counterexamples show that this global factorization is not in general possible and hence we can expect only local factorization ofw we show that c p 1differential forms on a frechet schwartz space factor locally through a normed space as c pdifferential forms the counterexamples deal with arbitrary c differential formsw while our results holds for certain c second members in particular for all holomorphicw more precisely we construct using local solutions of u w due to raboin and colombeau and perrot a c solutionu e c whenwis a holomorphic 0 1 form on a frechet nuclear spacee

the problem for a 0 2 form in a d f n space abstract we show that every holomorphic 0 2 form w in a d f n space e can be written as t6u w where u is a c 0 1 form of uniform bounded type

spatial pattern formation in the keller segel model with a logistic source this paper deals with a neumann boundary value problem in a d dimensional box t d 0 p d d 1 2 3 for the chemotaxis diffusion growth model u t d u u gu v ru 1 u k v t d v 2v au bv which describes the movement of cells in response to the presence of a chemical signal substance it is proved that given any general perturbation of magnitude d its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes over a time period of the order ln1 d each initial perturbation certainly can behave drastically differently from another which gives rise to the richness of patterns our results provide a mathematical characterization for the early stage pattern formation in the keller segel model

asymptotic behavior of solutions to two dimensional chemotaxis system with logistic source and singular sensitivity abstract in this paper we study the asymptotic behavior of solutions to the parabolic elliptic chemotaxis system with singular sensitivity and logistic source u t u u v v r u u 2 0 v v u subject to homogeneous neumann boundary conditions in a bounded domain r 2 with smooth boundary where 0 and r r it is proved that for any nonnegative initial date u 0 c such that u 0 1 16 2 2 with the constant relying on the solution u t v t converges asymptotically to the constant equilibrium r r in the l norm as t if r 2 1 1 2 2 16

an afc stabilized implicit finite element method for partial differential equations on evolving in time surfaces in this article we present a new implicit numerical scheme for reaction diffusion advection equations on an evolving in time hypersurface t the partial differential equations are solved on a stationary quadrilateral resp hexahedral mesh the zero level set of the time dependent indicator function t implicitly describes the position of t the dominating convective like terms which are due to the presence of chemotaxis transport of the cell density and surface evolution may lead to the non positiveness of a given numerical scheme and in such a way cause appearance of negative values and give rise of nonphysical oscillations in the numerical solution the proposed finite element method is constructed to avoid this problem implicit treatment of corresponding discrete terms in combination with the algebraic flux correction afc techniques make it possible to obtain a sufficiently accurate solution for reaction diffusion advection pdes on evolving surfaces

global existence of solutions to a parabolic parabolic system for chemotaxis with weak degradation abstract we study the global existence of solutions to a parabolic parabolic system for chemotaxis with a logistic source in a two dimensional domain where the degradation order of the logistic source is weaker than quadratic we introduce nonlinear production of a chemoattractant and show the global existence of solutions under certain relations between the degradation and production orders

convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source abstract we study the global attractors to the chemotaxis system with logistic source u t u u v a u b u 2 v t v v u in r subject to the homogeneous neumann boundary conditions where smooth bounded domain r n with b 0 a r and 0 1 for the parabolic elliptic case with 0 and n 3 we obtain that the positive constant equilibrium a b a b is a global attractor if a 0 and b max n 2 n a 4 under the assumption n 3 it is proved that for either the parabolic elliptic case with 0 a 0 b max 3 a 4 or the parabolic parabolic case with 1 a 0 b a 4 large enough the system admits the positive constant equilibrium a b a b as a global attractor while the trivial equilibrium 0 0 is a global attractor if a 0 and b 0 it is pointed out that here the convergence rates are established for all of them the results of the paper mainly rely on parabolic regularity theory and lyapunov functionals carefully constructed

numerical simulations for the chemotaxis models on surfaces via a novel characteristic finite element method abstract in this paper the chemotaxis partial differential equations models which describe the movement by one community in reaction to one chemical or biological signal are given and solved numerically on surfaces the models and the numerical method concerned are in terms of a generic form of chemotaxis models we develop a new semi implicit finite element scheme based on the gradient and laplacian recoveries to linearize the equations meanwhile we modify the proposed semi implicit scheme as a characteristic form and present a novel strategy for the discretization of characteristic derivative on surface the lumped mass modification is employed for positivity preservation and the related analysis results are provided we investigate the accuracy and convergence of the proposed method by numerical tests and the simulations of blowing up solution pattern formulations and aggregations of bacteria demonstrate the applicability of the proposed methods

convergence to a propagating front in a degenerate fisher kpp equation with advection abstract we consider a fisher kpp equation with density dependent diffusion and advection arising from a chemotaxis growth model we study its behavior as a small parameter related to the thickness of a diffuse interface tends to zero we analyze for small times the emergence of transition layers induced by a balance between reaction and drift effects then we investigate the propagation of the layers convergence to a free boundary limit problem is proved and a sharp estimate of the thickness of the layers is provided