chemotactic cellular migration smooth and discontinuous travelling wave solutions a simple model of chemotactic cell migration gives rise to travelling wave solutions by varying the cellular growth rate and chemoattractant production rate travelling waves with both smooth and discontinuous fronts are found using phase plane analysis the phase plane exhibits a curve of singularities whose position relative to the equilibrium points in the phase plane determines the nature of the heteroclinic orbits where they exist smooth solutions have trajectories connecting the steady states lying to one side of the singular curve travelling shock waves arise by connecting trajectories passing through a special point in the singular curve and recrossing the singular curve by way of a discontinuity hyperbolic partial differential equation theory gives the necessary shock condition conditions on the parameter values determine when the solutions are smooth travelling waves versus discontinuous travelling wave solutions these conditions provide bounds on the travel ling wave speeds corresponding to bounds on the chemotactic velocity or bounds on cellular growth rate this analysis gives rise to the possibility of representing sharp fronts to waves of invading cells through a simple chemotactic term without introducing a nonlinear diffusion term this is more appropriate when cell populations are sufficiently dense

continuous time random walks on networks with vertex and time dependent forcing we have investigated the transport of particles moving as random walks on the vertices of a network subject to vertex and time dependent forcing we have derived the generalized master equations for this transport using continuous time random walks characterized by jump and waiting time densities as the underlying stochastic process the forcing is incorporated through a vertex and time dependent bias in the jump densities governing the random walking particles as a particular case we consider particle forcing proportional to the concentration of particles on adjacent vertices analogous to self chemotactic attraction in a spatial continuum our algebraic and numerical studies of this system reveal an interesting pair aggregation pattern formation in which the steady state is composed of a high concentration of particles on a small number of isolated pairs of adjacent vertices the steady states do not exhibit this pair aggregation if the transport is random on the vertices i e without forcing the manifestation of pair aggregation on a transport network may thus be a signature of self chemotactic like forcing

nonlinear stability of traveling waves to a hyperbolic parabolic system modeling chemotaxis we prove nonlinear stability of traveling waves of arbitrary amplitudes to a hyperbolic parabolic system modeling repulsive chemotaxis in contrast to the previous related results where various sm

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

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

on the fractional fisher information with applications to a hyperbolic parabolic system of chemotaxis we introduce new lower bounds for the fractional fisher information equipped with these bounds we study a hyperbolic parabolic model of chemotaxis and prove the global existence of solutions in certain dissipation regimes

global existence and large time behavior for a two dimensional chemotaxis shallow water system abstract this paper concerns the chemotaxis shallow water system in two dimensional whole spaces we first establish the global existence of strong solution under the condition that the initial data are close to the constant equilibrium state in h n r 2 n 3 framework then the optimal time decay rates of the global solution are obtained by the method of energy estimates if the initial data belong to h s r 2 0 s 1 additionally

singularity formation in chemotaxis a conjecture of nagai consider the initial boundary value problem for the system s ut uxx uvx x vt u av on an interval 0 1 for t 0 where a 0 with ux 0 t ux 1 t 0 suppose mu v0 are positive constants the corresponding spatially homogeneous global solution u t mu v t mu a v0 mu a exp at is stable in the sense that if mu v0 are positive constants the corresponding spatially homogeneous solution will be uniformly close to u cdot v cdot we consider in sequence space an approximate system s which is related to s in the following sense the chemotactic term uvx x is replaced by the inverse fourier transform of the finite part of the convolution integral for the fourier transform of uvx x here the finite part of the convolution on the line at a point x of two functions f g is defined as int 0 x f y g y x dy we prove the following if mu a then in every neighborhood of mu v0 there are spatially nonconstant initial data for which the solution of proble

numerical resolution of a reinforced random walk model arising in haptotaxis in this paper we study the numerical resolution of a reinforced random walk model arising in haptotaxis and the stabilization of solutions the model consists of a system of two differential equations one parabolic equation with a second order non linear term haptotaxis term coupled to an ode in a bounded two dimensional domain we assume radial symmetry of the solutions the scheme of resolution is based on the application of the characteristics method together with a finite element one we present some numerical simulations which illustrate some features of the numerical stabilization of solutions

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

mathematical models of tumour angiogenesis we first study a parabolic ode system modelling tumour growth proposed by othmer and stevens aggregation blowup and collapse the abc s of taxis in reinforced random walks siam j appl math 57 4 1997 1044 1081 according to levine and sleeman a system of reaction and diffusion equations arising in the theory of reinforced random walks siam j appl math 57 3 1997 683 730 we reduced it to a hyperbolic equation and showed the existence of collapse in a kubo t suzuki asymptotic behavior of the solution to a parabolic ode system modeling tumour growth differential integral equations 17 2004 721 736 we also deal with the system in case the reduced equation is elliptic and show the existence of collapse analogously next we apply the above result to another model proposed by anderson and chaplain arising from tumour angiogenesis and show the existence of collapse further we investigate a contact point between these two models and a common property to them

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

on a parabolic hyperbolic chemotaxis system with discontinuous data well posedness stability and regularity abstract the global dynamics and regularity of parabolic hyperbolic systems is an interesting topic in pdes due to the coupling of competing dissipation and hyperbolic effects this paper is concerned with the cauchy problem of a parabolic hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis it is shown that as time tends to infinity the cauchy problem with large amplitude discontinuous data admit global weak solutions which converge to a constant state resp a viscous shock wave if the asymptotic states of initial values at far field are equal resp unequal our results improve the previous results where initial value was required to be continuous and have small amplitude numerical simulations are performed to verify our analytical results illustrate the possible regularity of solutions and speculate the minimal regularity of initial data required to obtain the smooth classical solutions of the concerned parabolic hyperbolic system

many body theory of chemotactic cell cell interactions we consider an individual based stochastic model of cell movement mediated by chemical signaling fields this model is formulated using langevin dynamics which allows an analytic study using methods from statistical and many body physics in particular we construct a diagrammatic framework within which to study cell cell interactions in the mean field limit where statistical correlations between cells are neglected we recover the deterministic keller segel equations within exact perturbation theory in the chemotactic coupling epsilon statistical correlations are non negligible at large times and lead to a renormalization of the cell diffusion coefficient d r an effect that is absent at mean field level an alternative closure scheme based on the necklace approximation probes the strong coupling behavior of the system and predicts that d r is renormalized to zero at a critical value of epsilon indicating self localization of the cell stochastic simulations of the model give very satisfactory agreement with the perturbative result at higher values of the coupling simulations indicate that d r approximately epsilon 2 a result at odds with the necklace approximation we briefly discuss an extension of our model which incorporates the effects of short range interactions such as cell cell adhesion

simplified method for including spatial correlations in mean field approximations biological systems involving proliferation migration and death are observed across all scales for example they govern cellular processes such as wound healing as well as the population dynamics of groups of organisms in this paper we provide a simplified method for correcting mean field approximations of volume excluding birth death movement processes on a regular lattice an initially uniform distribution of agents on the lattice may give rise to spatial heterogeneity depending on the relative rates of proliferation migration and death many frameworks chosen to model these systems neglect spatial correlations which can lead to inaccurate predictions of their behavior for example the logistic model is frequently chosen which is the mean field approximation in this case this mean field description can be corrected by including a system of ordinary differential equations for pairwise correlations between lattice site occupancies at various lattice distances in this work we discuss difficulties with this method and provide a simplification in the form of a partial differential equation description for the evolution of pairwise spatial correlations over time we test our simplified model against the more complex corrected mean field model finding excellent agreement we show how our model successfully predicts system behavior in regions where the mean field approximation shows large discrepancies additionally we investigate regions of parameter space where migration is reduced relative to proliferation which has not been examined in detail before and find our method is successful at correcting the deviations observed in the mean field model in these parameter regimes

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

computational framework to capture the spatiotemporal density of cells with a cumulative environmental coupling abstract stochastic agent based models can account for millions of cells with spatiotemporal movement that can be a function of different factors however these simulations can be computationally expensive in this work we develop a novel computational framework to describe and simulate stochastic cellular processes that are coupled to the environment specifically through upscaling we derive a continuum governing equation that considers the cell density as a function of time space and a cumulative variable that is coupled to the environmental conditions for this new governing equation we consider the stability through an energy analysis as well as proving uniqueness and well posedness to solve the governing equations in free space we propose a numerical method using fundamental solutions as an application we study a cell moving in an infinite domain that contains a toxic chemical where a cumulative exposure above a critical value results in cell death we illustrate the validity of this new modeling framework and associated numerical methods by comparing the density of live cells to results from the corresponding agent based model

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel hilbert space in this paper approximate solutions to a class of fractional differential equations with delay are presented by using a semi analytical approach in hilbert function space further the uniqueness of the solution is proved in the space of real valued continuous functions as well as the existence of the solution is proved in hilbert function space we also prove convergence and perform an analysis error for the proposed approach sophisticated delay differential equations of fractional order are considered as test examples numerical results illustrate the efficiency of the proposed approach computationally

asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis abstract the asymptotic behavior of solutions to a singular chemotaxis system modeling the onset of tumor angiogenesis in two and three dimensional whole spaces is investigated in the paper by a cole hopf type transformation the singular chemotaxis is converted into a non singular hyperbolic system then we study the transformed system and establish the global existence asymptotic decay rates and diffusion convergence rate of solutions by the method of energy estimates the main novelty of our results is the finding of a hidden interactive dissipation structure in the system by which the energy dissipation is established

on the boundary conditions characterizing j selfadjoint extensions of j symmetric operators the study of boundary value problems involving linear differentiai equations with real valued coefficients is by now a well established area of analysis on the other hand much less is known about the solvability of such problems when the coefficients or boundary conditions are known to be complex examples of this latter type arise naturally for example in nuclear physics e g the so called optical mode1 for low energy scattering 1 p ijo electromagnetic field theory dielectric waveguides with heat loss c f 19 i or the propagation of radio waves through inhomogeneous media s and elsewhere in cases like these the relevant differential expressions are no longer formally symmetric and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available to facilitate the study of such problems glazman introduced in 4 the concept of a j symmetric operator in a complex hilbert space r let j be a given conjugation operator on 2 i e j is a conjugate linear involution with jx 3 u x for all x and y in 2 a closed densely defined linear operator t in 3 is said to be j symmetric if

boundary conditions for general ordinary differential operators and their adjoints a characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in the domains of these operators are described in terms of boundary conditions involving the solutions of mu wu and the adjoint equation the results include those of sun jiong 15 concerning self adjoint realisations of a symmetric m when the minimal operator has equal deficiency indices if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein another special case concerns the j self adjoint operators where j denotes complex conjugation and for this we recover the results of zai jiu shang in 16

point dynamics in a singular limit of the keller segel model 2 formation of the concentration regions this paper continues the analysis started in the first part of this article cf j j l velazquez siam j appl math 64 2004 pp 1198 1223 it was seen there using the method of matched asymptotics that a regularized version of the keller segel system admits for a suitable asymptotic limit solutions with some regions of high concentrations for the cell density this paper considers the relation between the phenomenon of blow up for the limit problem and the dynamics of the concentration regions described in j j l velazquez siam j appl math 64 2004 pp 1198 1223 in particular this paper analyzes the precise way in which the regularization introduced in the keller segel system stops the aggregation process and yields the formation of concentration regions

piecewise picard iteration method for solving nonlinear fractional differential equation with proportional delays abstract in this paper a numerical method for solving a class of nonlinear fractional differential equation with proportional delays is proposed in order to overcome the strongly nonlinear case we propose the piecewise picard iteration method ppim the convergence proof and error estimations of the picard and the ppim are obtained meanwhile a sufficient condition for the stability of the ppim is also given some numerical examples confirm the validity of the ppim it s worth noting that the ppim is quite effective for solving linear weakly nonlinear and some strongly nonlinear fractional differential equations with proportional delays

from a discrete model of chemotaxis with volume filling to a generalized patlak keller segel model we present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased rand

global solutions for a hyperbolic parabolic system of chemotaxis abstract we study a hyperbolic parabolic model of chemotaxis related to tumor angiogenesis in dimensions one and two we consider diffusions given by the fractional laplacian and in particular we prove the global existence of classical solutions in certain dissipation regimes

global generalized solutions to a keller segel system with nonlinear diffusion and singular sensitivity abstract we consider the chemotaxis system with nonlinear diffusion and singular sensitivity u t u m u v v x t 0 v t v u v x t 0 u v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x in a smooth bounded domain r n n 2 in this work it is shown that for all reasonably regular initial data u 0 0 and v 0 0 the corresponding neumann initial boundary value problem possesses a global generalized solution provided that m 1 n 2 2 n

existence and non existence of global solutions to initial boundary value problems for nonlinear evolution equations with strong dissipation abstract the main purpose in this paper is to investigate existence and non existence of global solutions of the initial dirichlet boundary value problem for evolution equations with strong dissipation many authors studied classes consisting of such types of equations for which initial boundary value problems possess global solutions for this purpose we consider a related problem and seek global solutions and blow up solutions of it depending on whether it belongs to such classes or not

boundary layer problem on a hyperbolic system arising from chemotaxis abstract this paper is concerned with the boundary layer problem for a hyperbolic system transformed via a cole hopf type transformation from a repulsive chemotaxis model with logarithmic sensitivity proposed in 23 34 modeling the biological movement of reinforced random walkers which deposit a non diffusible or slowly moving signal that modifies the local environment for succeeding passages by prescribing the dirichlet boundary conditions to the transformed hyperbolic system in an interval 0 1 we show that the system has the boundary layer solutions as the chemical diffusion coefficient e 0 and further use the formal asymptotic analysis to show that the boundary layer thickness is e 1 2 our work justifies the boundary layer phenomenon that was numerically found in the recent work 25 however we find that the original chemotaxis system does not possess boundary layer solutions when the results are reverted to the pre transformed system

perturbation of singular equilibria of hyperbolic two component systems a universal hydrodynamic limit we consider one dimensional locally finite interacting particle systems with two conservation laws which under the eulerian hydrodynamic limit lead to two by two systems of conservation laws t x u 0 t u x u 0

well posedness and ill posedness of a multidimensional chemotaxis system in the critical besov spaces abstract we study the cauchy problem for a multidimensional chemotaxis system in critical besov spaces b p 1 d p 2 r d b p 1 d p 1 r d d for 1 p 2 d we prove locally well posedness for large initial data and globally well posedness for small initial data of this system and more importantly we show the ill posedness in the sense that a norm inflation phenomenon occurs for p 2 d more precisely we construct a specific initial data which can be arbitrarily small in the besov spaces meanwhile the corresponding solution u can be arbitrarily large after an arbitrarily short time

initial boundary value problems for a system of hyperbolic balance laws arising from chemotaxis abstract this paper is devoted to the analytical study of initial boundary value problems for a system of hyperbolic balance laws derived from a repulsive chemotaxis model with logarithmic sensitivity in the first part of the paper we show that subject to the dirichlet boundary conditions classical solutions exist globally in time for large initial data asymptotically in time the solutions are shown to converge to their boundary data at an exponential rate as time goes to infinity numerical simulations are supplied to corroborate the analytical results the analytic approach developed herein can be utilized to handle a family of initial and initial boundary value problems of the model and related models with similar mathematical structure as a demonstration of the effectiveness of our approach in the second part of the paper we show that subject to the neumann dirichlet boundary conditions classical solutions exist and converge to constant equilibrium states for large initial data and for arbitrary values of the chemical diffusion coefficient this improves a previous result obtained in 30 where the smallness of the chemical diffusion coefficient was required

accurate discretization of advection diffusion equations we present an exact mathematical transformation which converts a wide class of advection diffusion equations into a form allowing simple and direct spatial discretization in all dimensions and thus the construction of accurate and more efficient numerical algorithms these discretized forms can also be viewed as master equations which provide an alternative mesoscopic interpretation of advection diffusion processes in terms of diffusion with spatially varying hopping rates

analysis and numerical methods for fractional differential equations with delay in this paper we consider fractional differential equations with delay we focus on linear equations we summarise existence and uniqueness theory based on the method of steps and we give a theorem on the propagation of derivative discontinuities we discuss the dependence of the solution on the parameters of the equation and conclude with a numerical treatment and examples based on the adaptation of a fractional backward difference method to the delay case

two dimensional cellular automata a largely phenomenological study of two dimensional cellular automata is reported qualitative classes of behavior similar to those in one dimensional cellular automata are found growth from simple seeds in two dimensiona cellular automata can produce patterns with complicated boundaries characterized by a variety of growth dimensions evolution from disordered states can give domains with boundaries that execute effectively continuous motions some global properties of cellular automata can be described by entropies and lyapunov exponents others are undecidable

fractional order anisotropic diffusion for image denoising this paper introduces a new class of fractional order anisotropic diffusion equations for noise removal these equations are euler lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function so the proposed equations can be seen as generalizations of second order and fourth order anisotropic diffusion equations we use the discrete fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain it is one important aspect of the algorithm that it considers the input image as a periodic image to overcome this problem we use a folded algorithm by extending the image symmetrically about its borders finally we list various numerical results on denoising real images experiments show that the proposed fractional order anisotropic diffusion equations yield good visual effects and better signal to noise ratio

on the efficacy of a control volume finite element method for the capture of patterns for a volume filling chemotaxis model in this paper a control volume finite element scheme for the capture of spatial patterns for a volume filling chemotaxis model is proposed and analyzed the diffusion term which generally involves an anisotropic and heterogeneous diffusion tensor is discretized by piecewise linear conforming triangular finite elements p1 fem the other terms are discretized by means of an upstream finite volume scheme on a dual mesh where the dual volumes are constructed around the vertices of each element of the original mesh the scheme ensures the validity of the discrete maximum principle under the assumption that the transmissibility coefficients are nonnegative the convergence analysis is based on the establishment of a priori estimates on the cell density these estimates lead to some compactness arguments in l 2 based on the use of the kolmogorov compactness theorem finally we show some numerical results to illustrate the effectiveness of the scheme to capture the pattern formation for the mathematical model

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

going from microscopic to macroscopic on nonuniform growing domains throughout development chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself these chemicals termed morphogens are often modeled using partial differential equations pdes the connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by baker yates and erban bull math biol 72 719 2010 in which the migration of individual particles was modeled as an on lattice position jump process we build on this work by incorporating a more physically reasonable description of domain growth instead of allowing underlying lattice elements to instantaneously double in size and divide we allow incremental element growth and splitting upon reaching a predefined threshold size such a description of domain growth necessitates a nonuniform partition of the domain we first demonstrate that an individual based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a pde model of the same phenomenon on a nongrowing domain providing the transition rates which we derive are chosen correctly and we partition the domain in the correct manner we extend this analysis to the case where the domain is allowed to change in size altering the transition rates as necessary through application of the master equation formalism we derive a pde for particle density on this growing domain and corroborate our findings with numerical simulations

oscillatory traveling wave solutions to an attractive chemotaxis system abstract this paper investigates oscillatory traveling wave solutions to an attractive chemotaxis system the convective part of this system changes its type when crossing a parabola in the phase space the oscillatory nature of the traveling wave comes from the fact that one far field state is in the elliptic region and another in the hyperbolic region such traveling wave solutions are shown to be linearly unstable detailed construction of some traveling wave solutions is presented

stability of solutions of chemotaxis equations in reinforced random walks in this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation the system arises in chemotaxis a process whereby living organisms respond to chemical substance by moving toward higher or lower concentrations of the chemical substance or by aggregating or dispersing we prove that stationary solutions of the system are asymptotically stable

superdiffusion of a random walk driven by an ergodic markov process with switching we propose a markov model with an ergodic two component switching mechanism that dynamically generates anomalous diffusion the first component plays the role of a hidden parameter the second one is the switching component generating the superdiffusion of a random walker and is itself non markovian the model is studied numerically using the monte carlo technique

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

asymptotic integration of linear differential equations abstract we establish a new result on the asymptotic integration of linear differential systems of the form y r y where is diagonal and r is small more specifically we allow for different scales of asymptotic behaviour of the individual matrix elements of and r we also discuss an application to a self adjoint fourth order equation

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

chemotactic and diffusive migration on a nonuniformly growing domain numerical algorithm development and applications a numerical algorithm to simulate chemotactic and or diffusive migration on a one dimensional growing domain is developed the domain growth can be spatially nonuniform and the growth derived advection term must be discretised the hyperbolic terms in the conservation equations associated with chemotactic migration and domain growth are accurately discretised using an explicit central scheme generality of the algorithm is maintained using an operator split technique to simulate diffusive migration implicitly the resulting algorithm is applicable for any combination of diffusive and or chemotactic migration on a growing domain with a general growth induced velocity field the accuracy of the algorithm is demonstrated by testing the results against some simple analytical solutions and in an inter code comparison the new algorithm demonstrates that the form of nonuniform growth plays a critical role in determining whether a population of migratory cells is able to overcome the domain growth and fully colonise the domain

existence of solutions of the hyperbolic keller segel model we are concerned with the hyperbolic keller segel model with quorum sensing a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities this is a first order quasilinear equation its flux depends on space and time via the solution to an elliptic pde in which the right hand side is the solution to the hyperbolic equation this model lacks strong compactness or contraction properties our purpose is to prove the existence of an entropy solution obtained as usual in passing to the limit in a sequence of solutions to the parabolic approximation the method consists in the derivation of a kinetic formulation for the weak limit the specific structure of the limiting kinetic equation allows for a rigidity theorem which identifies some property of the solution which might be nonunique to this kinetic equation this is enough to deduce a posteriori the strong convergence of a subsequence

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

non blowup of a generalized boussinesq burgers system with nonlinear dispersion relation and large data abstract we study the qualitative behavior of classical solutions to the cauchy problem of a generalized boussinesq burgers system in one space dimension assuming initial data belong to h 2 r and utilizing energy methods we show that there exist unique global in time classical solutions to the cauchy problem of the model and the solutions converge to constant equilibrium states as time goes to infinity regardless of the magnitude of the initial data moreover it is shown that the viscous and inviscid models are consistent in the process of vanishing viscosity limit

global dynamics of a hyperbolic parabolic model arising from chemotaxis we prove global existence and qualitative behavior of classical solutions for a hyperbolic parabolic system describing chemotaxis on bounded domains it is shown that classical solutions to the initial boundary value problem of the one dimensional model exist globally in time for large initial data and the solutions converge to constant equilibrium states exponentially in time which rigorously demonstrates the collapsing of cell populations in chemotaxis moreover similar results are established for the multidimensional model when the initial data are small

lattice free models of directed cell motility directed cell migration often occurs when individual cells move in response to an external chemical stimulus cells can respond by moving in either the direction of increasing chemoattraction or decreasing chemorepulsion concentration many previous models of directed cell migration use a lattice based framework where agents undergo a lattice based random walk and the direction of nearest neighbour motility events is biased in a preferred direction such lattice based models can lead to unrealistic configurations of agents since the agents always move on an artificial lattice structure which is never observed experimentally we present a lattice free model of directed cell migration that incorporates two key features first agents move on a continuous domain with the possibility that there is some preferred direction of motion second to be consistent with experimental observations we enforce a crowding mechanism so that motility events that would lead to agent overlap are not permitted we compare simulation data from the new lattice free model with a more traditional lattice based model to provide additional insight into the lattice free model we construct an approximate conservation statement which corresponds to a nonlinear advection diffusion equation in the continuum limit the solution of this mean field model compares well with averaged data from the individual based model

well posedness of a 3d parabolic hyperbolic keller segel system in the sobolev space framework abstract we investigate global strong solution to a 3 dimensional parabolic hyperbolic system arising from the keller segel model we establish the global well posedness and asymptotic behavior in the energy functional setting precisely speaking if the initial difference between cell density and its mean is small in l 2 and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in h 1 then they remain to be small in l 2 h 1 for all time moreover if the mean value of the initial cell density is smaller than some constant then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity the proof relies on an application of fourier analysis to a linearized parabolic hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

asymptotic profile of a parabolic hyperbolic system with boundary effect arising from tumor angiogenesis this paper concerns a parabolic hyperbolic system on the half space r with boundary effect the system is derived from a singular chemotaxis model describing the initiation of tumor angiogenesis we show that the solution of the system subject to appropriate boundary conditions converges to a traveling wave profile as time tends to infinity if the initial data is a small perturbation around the wave which is shifted far away from the boundary but its amplitude can be arbitrarily large

the solvability of some chemotaxis systems abstract in this paper we study the local existence and uniqueness of classical solutions to a wide class of systems of chemotaxis equations these systems are essentially quasi linear strongly coupled partial differential equations we also study the maximal interval of existence in time of solutions the results are illustrated in application to a number of partial differential equation models arising in biology

blow up criterion for a hyperbolic parabolic system arising from chemotaxis abstract in this paper we investigate a hyperbolic parabolic system arising from chemotaxis a blow up criterion of local strong solutions to the model in r d is established based on the scaling invariance of the system as a by product we prove a global in time well posedness result when d 1 for large initial data

interplay of chemotaxis and chemokinesis mechanisms in bacterial dynamics motivated by observations of the dynamics of myxococcus xanthus we present a self interacting random walk model that describes the competition between chemokinesis and chemotaxis cells are constrained to move in one dimension but release a chemical chemoattractant at a steady state the bacteria sense the chemical that they produce the probability of direction reversals is modeled as a function of both the absolute level of chemoattractant sensed directly under each cell as well as the gradient sensed across the length of the cell if the chemical does not degrade or diffuse rapidly the one dimensional trajectory depends on the entire past history of the trajectory we derive the corresponding fokker planck equations use an iterative mean field approach that we solve numerically for short times and perform extensive monte carlo simulations of the model cell positional distributions and the associated moments are computed in this feedback system average drift and mean squared displacements are found crossover behaviors among different diffusion regimes are found

stability of constant steady states and existence of unbounded solutions in time to a reaction diffusion equation modelling chemotaxis abstract we consider the initial boundary value problem to a reaction diffusion equation modelling chemotaxis in one space dimension under neumann boundary conditions we study the stability of constant steady states and show that there exists an unbounded solution in time induced by the instability of constant steady states

a tau approach for solution of the space fractional diffusion equation fractional differentials provide more accurate models of systems under consideration in this paper approximation techniques based on the shifted legendre tau idea are presented to solve a class of initial boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain the fractional derivatives are described in the caputo sense the technique is derived by expanding the required approximate solution as the elements of shifted legendre polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from the computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use

persistent random walk of cells involving anomalous effects and random death the purpose of this paper is to implement a random death process into a persistent random walk model which produces sub ballistic superdiffusion levy walk we develop a stochastic two velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction it is assumed that the switching rate is a decreasing function of residence running time this assumption leads to the power law for the velocity switching time distribution this describes the anomalous persistence of cell motility the longer the cell moves in one direction the smaller the switching probability to another direction becomes we derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives we show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process in the long time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a levy jump distribution we find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death rate dependent diffusion coefficient monte carlo simulations confirm these bounds

modified chebyshev wavelet methods for fractional delay type equations in this article we develop the chebyshev wavelet method for solving the fractional delay differential equations and integro differential equations according to the development we approximate the delay unknown functions by the chebyshev wavelets series at delay time which we call the delay chebyshev wavelet series we also proposed a technique by combining the method of steps and chebyshev wavelet method for solving fractional delay differential equations this technique converts the fractional delay differential equation on a given interval to a fractional non delay differential equation over that interval by using the function defined on previous interval and then apply the chebyshev wavelet method on the obtained fractional non delay differential equation to find the solution numerical examples will be presented to demonstrate the benefits of computing with these approaches

the spectra of well posed operators in this paper the general ordinary quasidifferential expression m of n th order with complex coefficients and its formal adjoint m are considered it is shown in the case of two singular endpomts and when all solutions of the equation and the adjoint equation are in the limit circle case that all well posed extensions of the minimal operator t 0 m have resolvents which are hilbert schmidt integral operators and consequently have a wholly discrete spectrum this implies that all the regularly solvable operators have all of the standard essential spectra to be empty these results extend those for the formally symmetric expression m studied in 1 and 14 and also extend those proved in 8 for one singular endpoint

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

continuum models of cohesive stochastic swarms the effect of motility on aggregation patterns abstract mathematical models of swarms of moving agents with non local interactions have many applications and have been the subject of considerable recent interest for modest numbers of agents cellular automata or related algorithms can be used to study such systems but in the present work instead of considering discrete agents we discuss a class of one dimensional continuum models in which the agents possess a density x t at location x at time t the agents are subject to a stochastic motility mechanism and to a global cohesive inter agent force the motility mechanisms covered include classical diffusion nonlinear diffusion which may be used to model in a phenomenological way volume exclusion or other short range local interactions and a family of linear redistribution operators related to fractional diffusion equations a variety of exact analytic results are discussed including equilibrium solutions and criteria for unimodality of equilibrium distributions full time dependent solutions and transitions between asymptotic collapse and asymptotic escape we address the behaviour of the system for diffusive motility in the low diffusivity limit for both smooth and singular interaction potentials and show how this elucidates puzzling behaviour in fully deterministic non local particle interaction models we conclude with speculative remarks about extensions and applications of the models

on a parabolic elliptic system with chemotaxis and logistic type growth we consider a nonlinear pdes system of two equations of parabolic elliptic type with chemotactic terms the system models the movement of a biological population u towards a higher concentration of a chemical agent w in a bounded and regular domain rn for arbitrary n n after normalization the system is as follows ut u div um w u 1 u in t 0 t w w u in t for some positive constants m and with positive initial datum u0 and neumann boundary conditions we study the range of parameters and constrains for which the solution exists globally in time if either m 1 or m 1 and n 22 m 1 n the solution exists globally in time moreover if m 1 and 2 and there exist positive constants u 0 and u 0 such that 0 u 0 u0 u 0 we have that u 1 l w 1 l 0 as t

a fluctuation theorem for moderately interacting diffusion processes we study the asymptotic behaviour of the fluctuations of the empirical processes of the second example of moderately interacting diffusion processes of 3 and show that in the limit as the number of different processes tends to infinity these fluctuations behave gaussian

traveling waves for time delayed reaction diffusion equations with degenerate diffusion abstract this paper is concerned with time delayed reaction diffusion equations with degenerate diffusion when the term for birth rate is a nonlocal integral with a heat kernel the family of minimum wave speeds corresponding to all the degenerate diffusion coefficients is proved to admit a uniform positive infimum however when the term for birth rate is local there is no positive infimum of all the minimum wave speeds this difference indicates that the nonlocal effect plays a role as laplacian such that a positive lower bound independent of the degenerate diffusion exists for the minimum wave speeds the approach adopted for the proof is the monotone technique with the viscosity vanishing method the degeneracy of diffusion for the equation causes us essential difficulty in the proof a number of numerical simulations are also carried out at the end of the paper which further numerically confirm our theoretical results