local well posedness for a lotka volterra system in besov spaces this paper is devoted to study local existence uniqueness and continuous dependence upon the initial data of mild solutions for a diffusive non autonomous lotka volterra system initial data are taken in the besov space b p q n

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

traveling shock waves arising in a model of malignant invasion members of a family of traveling wave solutions for a simple two variable model of malignant invasion driven by haptotaxis are shown to have shocks which satisfy an entropy condition and to be computationally stable by seeking traveling wave solutions a pde model equivalent to three first order pdes defined on the real space and time axes may be reduced to an ordinary differential equation system which can be studied in a two dimensional phase plane this phase plane is shown to contain a singular barrier and a hole in this barrier which admits two singular phase trajectories thus orbits are able to smoothly cross this barrier along either of the two singular trajectories once this has occurred the barrier seems to prevent any heteroclinic steady state connections however we will show that shocks admitted by the pdes allow the orbit to jump back over the singular barrier this process leads to a previously unnoticed family of solutions for malignant invasion which have slower wavespeeds than

soliton like phenomena in one dimensional cross diffusion systems a predator prey pursuit and evasion example abstract we have studied properties of non linear waves in a mathematical model of a predator prey system with pursuit and evasion we demonstrate a new type of propagating wave in this system the mechanism of propagation of these waves essentially depends on the taxis represented by non linear cross diffusion terms in the mathematical formulation we have shown that the dependence of the velocity of wave propagation on the taxis has two distinct forms parabolic and linear transition from one form to the other correlates with changes in the shape of the wave profile dependence of the propagation velocity on diffusion in this system differs from the square root dependence typical of reaction diffusion waves we also demonstrate that for systems with negative and positive taxis for example pursuit and evasion there typically exists a large region in the parameter space where the waves demonstrate quasi soliton interaction colliding waves can penetrate through each other and waves can also reflect from impermeable boundaries

positivity preserving nonstandard finite difference schemes for cross diffusion equations in biosciences we design nonstandard finite difference nsfd schemes which are unconditionally dynamically consistent with respect to the positivity property of solutions of cross diffusion equations in biosciences this settles a problem that was open for quite some time the study is done in the setting of three concrete and highly relevant cross diffusion systems regarding tumor growth convective predator prey pursuit and evasion model and reaction diffusion chemotaxis model it is shown that nsfd schemes used for classical reaction diffusion equations such as the fisher equation for which the solutions enjoy the positivity property are not appropriate for cross diffusion systems the reliable nsfd schemes are therefore obtained by considering a suitable implementation on the cross diffusive term of mickens rule of nonlocal approximation of nonlinear terms apart from his rule of complex denominator function of discrete derivatives we provide numerical experiments that support the theory as well as the power of the nsfd schemes over the standard ones in the case of the cancer growth model we demonstrate computationally that our nsfd schemes replicate the property of traveling wave solutions of developing shocks observed in marchant et al 2000

traveling wave solutions in a diffusive system with two preys and one predator abstract this paper is concerned with the traveling wave solutions in a diffusive system with two preys and one predator by constructing upper and lower solutions the existence of nontrivial traveling wave solutions is established the asymptotic behavior of traveling wave solutions is also confirmed by combining the asymptotic spreading with the contracting rectangles applying the theory of asymptotic spreading the nonexistence of traveling wave solutions is proved

traveling wave solutions for gause type predator prey systems with density dependence a heteroclinic orbit in r4 abstract this paper investigates the existence of traveling wave solutions for a class of diffusive gause type predator prey systems with linear density dependence for predator using the shooting method we establish the existence of a non negative traveling wave solution connecting a boundary equilibrium point to the coexistence equilibrium point with the help of topological method of wazewski theorem and lasalle s invariance principle

tactically driven nonmonotone travelling waves models of cell invasion incorporating directed cell movement up a gradient of an external substance and carrying capacity limited proliferation give rise to travelling wave solutions travelling wave profiles with various shapes including smooth monotonically decreasing shock fronted monotonically decreasing and shock fronted nonmonotone shapes have been reported previously in the literature the existence of tactically driven shock fronted nonmonotone travelling wave solutions is analysed for the first time we develop a necessary condition for nonmonotone shock fronted solutions this condition shows that some of the previously reported shock fronted nonmonotone solutions are genuine while others are a consequence of numerical error our results demonstrate that for certain conditions travelling wave solutions can be either smooth and monotone smooth and nonmonotone or discontinuous and nonmonotone these different shapes correspond to different invasion speeds a necessary and sufficient condition for the travelling wave with minimum wave speed to be nonmonotone is presented several common forms of the tactic sensitivity function have the potential to satisfy the newly developed condition for nonmonotone shock fronted solutions developed in this work

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

a geometric approach in the study of traveling waves for some classes of non monotone reaction diffusion systems abstract in this paper we further extend a recently developed method to investigate the existence of traveling waves solutions and their minimum wave speed for non monotone reaction diffusion systems our approach consists of two steps first we develop a geometrical shooting argument with the aid of the theorem of homotopy invariance on the fundamental group to obtain the positive semi traveling wave solutions for a large class of reaction diffusion systems including the models of predator prey interaction for both predator independent dependent functional responses the models of combustion belousov zhabotinskii reaction si type of disease transmission and the model of biological flow reactor in chemostat next we apply the results obtained from the first step to some models such as the beddinton deangelis model and the model of biolocal flow reactor to show the convergence of these semi traveling wave solutions to an interior equilibrium point by the construction of a lyapunov type function or the convergence of semi traveling waves to another boundary equilibrium point by the further analysis of the asymptotical behavior of semi traveling wave solutions

how good are interior point methods klee minty cubes tighten iteration complexity bounds by refining a variant of the klee minty example that forces the central path to visit all the vertices of the klee minty n cube we exhibit a nearly worst case example for path following interior point methods namely while the theoretical iteration complexity upper bound is o 2 n n frac 5 2 we prove that solving this n dimensional linear optimization problem requires at least 2 n 1 iterations

diffusive and chemotactic cellular migration smooth and discontinuous traveling wave solutions a mathematical model describing cell migration by diffusion and chemotaxis is considered the model is examined using phase plane numerical and perturbation techniques for a proliferative cell population travelingwave solutions are observed regardless of whether the migration is driven by diffusion chemotaxis or a combination of the two mechanisms for pure chemotactic migration both smooth and discontinuous solutions with shocks are shown to exist using phase plane analysis involving a curve of singularities and identical results are obtained numerically alternatively pure diffusive migration and combinations of diffusive and chemotactic migration yield smooth solutions only for all cases the wave speed depends on the exponential decay rate of the initial cell density and it is bounded by a minimum value which is numerically observed whenever the initial cell distribution has compact support the minimum wave speed c min is proportional to sqrt chi or sqrt d for pure chemotaxis

spreading speeds of a lotka volterra predator prey system the role of the predator abstract this paper is concerned with the spreading speeds of a modified lotka volterra predator prey system the intention is to describe the propagation mode whereby the two species coinvade a new habitat under some assumptions it is proved that the spreading speed of the prey is slower than in the case where the predator vanishes and the density of the prey on the coexistence domain is also smaller compared with the case where interspecies actions disappear therefore the predator has a negative effect on the evolution of the prey by slowing the spreading speed and decreasing the population density of the prey

global stability and wavefronts in a cooperation model with state dependent time delay abstract this article deals with a diffusive cooperative model with state dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds for the cooperative dde system the positivity and boundedness of solutions are firstly given using the comparison principle of the state dependent delay equations obtained the stability criterion of model is analyzed both from local and global points of view when the diffusion is properly introduced the existence of traveling waves is obtained by constructing a pair of upper lower solutions and schauder s fixed point theorem calculating the minimum wave speed shows that the wave is slowed down by the state dependent delay finally the traveling wavefront solutions for large wave speed are also discussed and the fronts appear to be all monotone regardless of the state dependent time delay this is an interesting property since many findings are frequently reported that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

classification of coupled systems with two component nonlinear diffusion equations by the invariant subspace method the invariant subspace method is developed to perform classification of systems with two component nonlinear diffusion equations which was carried out with respect to the invariant subspaces defined by linear ordinary differential equations as a result the corresponding exact solutions generated by invariant subspaces to the resulting systems are obtained in most cases two components of these exact solutions belong to different scalar subspaces behaviour to several exact solutions of the systems is described

building macroscale models from microscale probabilistic models a general probabilistic approach for nonlinear diffusion and multispecies phenomena a discrete agent based model on a periodic lattice of arbitrary dimension is considered agents move to nearest neighbor sites by a motility mechanism accounting for general interactions which may include volume exclusion the partial differential equation describing the average occupancy of the agent population is derived systematically a diffusion equation arises for all types of interactions and is nonlinear except for the simplest interactions in addition multiple species of interacting subpopulations give rise to an advection diffusion equation for each subpopulation this work extends and generalizes previous specific results providing a construction method for determining the transport coefficients in terms of a single conditional transition probability which depends on the occupancy of sites in an influence region these coefficients characterize the diffusion of agents in a crowded environment in biological and physical processes

a two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion abstract invasive cells variously show changes in adhesion protease production and motility in this paper the authors develop and analyse a model for malignant invasion brought about by a combination of proteolysis and haptotaxis a common feature of these two mechanisms is that they can be produced by contact with the extracellular matrix through the mediation of a class of surface receptors called integrins an unusual feature of the model is the absence of cell diffusion by seeking travelling wave solutions the model is reduced to a system of ordinary differential equations which can be studied using phase plane analysis the authors demonstrate the presence of a singular barrier in the phase plane and a hole in this singular barrier which admits a phase trajectory the model admits a family of travelling waves which depend on two parameters i e the tissue concentration of connective tissue and the rate of decay of the initial spatial profile of the invading cells the slowest member of this family corresponds to the phase trajectory which goes through the hole in the singular barrier using a power series method the authors derive an expression relating the minimum wavespeed to the tissue concentration of the extracellular matrix which is arbitrary the model is applicable in a wide variety of biological settings which combine haptotaxis with proteolysis by considering various functional forms the authors show that the key mathematical features of the particular model studied in the early parts of the paper are exhibited by a wider class of models which characterise the behaviour of invading cells

erratum to on wave fronts propagation in multicomponent media trans amer math soc 276 1983 no 1 181 191 mr0684501 84a 35027

turing pattern formation in the brusselator system with nonlinear diffusion in this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the brusselator system through linear stability analysis of the basic solution we determine the turing and the oscillatory instability boundaries a comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of turing pattern formation we study the process of pattern formation both in one dimensional and two dimensional spatial domains through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns the analysis of the amplitude equations shows the occurrence of a number of different phenomena including stable supercritical and subcritical turing patterns with multiple branches of stable solutions leading to hysteresis moreover we consider traveling patterning waves when the domain size is large the pattern forms sequentially and traveling wave fronts are the precursors to patterning we derive the ginzburg landau equation and describe the traveling front enveloping a pattern which invades the domain we show the emergence of radially symmetric target patterns and through a matching procedure we construct the outer amplitude equation and the inner core solution

wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations linear diffusion is an established model for spatial spread in biological systems including movement of cell populations however for interacting closely packed cell populations simple diffusion is inappropriate because different cell populations will not move through one another rather a cell will stop moving when it encounters another cell in this paper i introduce a nonlinear diffusion term that reflects this phenomenon known as contact inhibition of migration i study this term in the context of two competing cell populations one of which has a proliferative advantage over the other this is motivated by the very early stages of solid tumour growth i focus in particular on travelling wave solutions corresponding to moving interfaces between the two cell populations numerical simulations indicate that there are wavefront solutions for wave speeds above a critical minimum value and i present linear analysis that explains the selection of wave speeds by initial conditions i obtain an approximation to the shape of these waves for high speeds and show that the minimum speed arises via quite new behaviour in the travelling wave equations with the proportion of cells of each type approaching a step function as the wave speed decreases towards the minimum exploiting this structure i use singular perturbation theory to investigate the wave shape for speeds close to the minimum

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase space

traveling wave solutions for diffusive predator prey type systems with nonlinear density dependence abstract traveling wave solution for a class of diffusive predator prey system with nonlinear density dependence is considered using methods of topological shooting we show the existence of a non negative traveling wave solution connecting a boundary equilibrium to the co existence steady state with the help of a wazewski like set together with lyapunov function constructed elaborately this means that the traveling wave solution established by huang 2012 can be preserved in the presence of the nonlinear density dependence for the predator and the results of huang 2012 are generalized

wavefronts and global stability in a time delayed population model with stage structure we formulate and study a one dimensional single species diffusive delay population model the time delay is the time taken from birth to maturity without diffusion the delay differential model extends the well known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes this delayed model is known to have simple global dynamics similar to that of the logistic equation through the use of a sub supersolution pair method we show that the diffusive delay model continues to generate simple global dynamics this has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics we also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population connecting the zero solution of that equation with the positive steady state our main finding here is that our fronts appear to be all monotone regardless of the size of the delay this is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

prediction of traveling front behavior in a lattice gas cellular automaton model for tumor invasion cancer invasion is the process of cells detaching from a primary tumor and infiltrating the healthy tissue cancer invasion has been recognized as a complex system since a tumor s invasive behavior emerges from the combined effect of tumor cell proliferation tumor cell migration and cell microenvironment interactions cellular automata ca provide simple models of self organizing complex systems in which collective behavior can emerge out of an ensemble of many interacting simple components here we introduce a lattice gas cellular automaton lgca model of tumor cell proliferation necrosis and tumor cell migration the impact of the tumor environment on tumor cells has been investigated in a previous study our analysis aims at predicting the velocity of the traveling invasion front which depends upon fluctuations that arise from the motion of the discrete cells at the front we find an excellent agreement between the velocities measured in simulations of the lgca and an analytical estimate derived in the cut off mean field approximation via the discrete lattice boltzmann equation and its linearization in particular we predict the front velocity to scale with the square root of the product of probabilities for mitosis and the migration coefficient finally we calculate the width of the traveling front which is found to be proportional to the front velocity

minimal wave speed for a class of non cooperative reaction diffusion systems of three equations abstract in this paper we study the traveling wave solutions and minimal wave speed for a class of non cooperative reaction diffusion systems consisting of three equations based on the eigenvalues a pair of upper lower solutions connecting only the invasion free equilibrium are constructed and the schauder s fixed point theorem is applied to show the existence of traveling semi fronts for an auxiliary system then the existence of traveling semi fronts of original system is obtained by limit arguments the traveling semi fronts are proved to connect another equilibrium if natural birth and death rates are not considered and to be persistent if these rates are incorporated then non existence of bounded traveling semi fronts is obtained by two sided laplace transform then the above results are applied to some disease transmission models and a predator prey model

multi species simple exclusion processes abstract a motility mechanism based on a simple exclusion process where the movement of discrete agents on a lattice is either unbiased symmetric or biased asymmetric is considered estimates of diffusivities from tracking data do not describe the population level response of the system this mismatch between the individual level and population level behaviour can be resolved by averaging the individual level mechanism in terms of an expected site occupancy new insight into simple exclusion processes is obtained by representing the system as a series of interacting subpopulations this formalism leads to a system of nonlinear advection diffusion equations which can be interpreted in terms of the agent fluxes these interactions have consequences for both agent based modelling and continuum modelling in cell biology such as tracking subpopulations of cells within a total cell population

persistence of wavefronts in delayed nonlocal reaction diffusion equations we develop a perturbation argument based on existing results on asymptotic autonomous systems and the fredholm alternative theory that yields the persistence of traveling wavefronts for reaction diffusion equations with nonlocal and delayed nonlinearities when the time lag is relatively small this persistence result holds when the nonlinearity of the corresponding ordinary reaction diffusion system is either monostable or bistable we then illustrate this general result using five different models from population biology epidemiology and bio reactors

using krylov subspace and spectral methods for solving complementarity problems in many body contact dynamics simulation summary many body dynamics problems are expected to handle millions of unknowns when for instance investigating the three dimensional flow of granular material unfortunately the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds this is typically the case when the equations of motion embed a differential variational inequality problem that captures contact and possibly frictional interactions between rigid and or flexible bodies as the size of the physical system increases the speed and or the quality of the numerical solution decreases this paper describes three methods the gradient projected minimum residual method the preconditioned spectral projected gradient with fallback method and the modified proportioning with reduced gradient projection method that demonstrate better scalability than the projected jacobi and gauss seidel methods commonly used to solve contact problems that draw on a differential variational inequality based modeling approach copyright 2013 john wiley sons ltd

fourier expansion of eisenstein series on the hilbert modular group and hilbert class fields in this paper we consider the eisenstein series for the hilbert modular group of a general number field we compute the fourier expansion at each cusp explicitly the fourier coefficients are given in terms of completed partial hecke l series and from their functional equations we get the functional equation for the eisenstein vector that is we identify the scattering matrix when we compute the determinant of the scattering matrix in the principal case the dedekind function of the hilbert class field shows up a proof in the imaginary quadratic case was given in efrat and sarnak and for totally real fields with class number one a proof was given in efrat

a variational approach to symmetry monotonicity and comparison for doubly nonlinear equations abstract we advance a variational method to prove qualitative properties such as symmetries monotonicity upper and lower bounds sign properties and comparison principles for a large class of doubly nonlinear evolutionary problems including gradient flows some nonlocal problems and systems of nonlinear parabolic equations our method is based on the so called weighted energy dissipation wed variational approach this consists in defining a global parameter dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter goes to zero qualitative properties and comparison principles can be easily proved for minimizers of the wed functional and by passing to the limit for the limiting problem several applications of the abstract results to systems of nonlinear pdes and to fractional nonlocal problems are presented eventually we present some extensions of this approach in order to deal with rate independent systems and hyperbolic problems

nonlinear diffusion and exclusion processes with contact interactions exclusion processes on a regular lattice are used to model many biological and physical systems at a discrete level the average properties of an exclusion process may be described by a continuum model given by a partial differential equation we combine a general class of contact interactions with an exclusion process we determine that many different types of contact interactions at the agent level always give rise to a nonlinear diffusion equation with a vast variety of diffusion functions d c we find that these functions may be dependent on the chosen lattice and the defined neighborhood of the contact interactions mild to moderate contact interaction strength generally results in good agreement between discrete and continuum models while strong interactions often show discrepancies between the two particularly when d c takes on negative values we present a measure to predict the goodness of fit between the discrete and continuous model and thus the validity of the continuum description of a motile contact interacting population of agents this work has implications for modeling cell motility and interpreting cell motility assays giving the ability to incorporate biologically realistic cell cell interactions and develop global measures of discrete microscopic data

existence of travelling wave solutions of predator prey systems via the connection index we consider a reaction diffusion system which arises as a model of predator prey interactions in mathematical ecology the system admits four constant equilibria two of which are stable the main result is an existence theorem for travelling wave solutions which connect the two stable equilibria most previous efforts with regard to travelling waves for systems employ singular perturbations to locate the connecting orbit the new aspect of the present discussion lies in devising some novel methods for locating the connecting solution in phase space which do not require that the equations be singularly perturbed in certain cases the main result holds for an arbitrary choice of diffusion coefficients

traveling waves and competitive exclusion in models of resource competition and mating interference abstract in this article we study reaction diffusion models for two closed related biological species u and v under resource competition and mating interference from v species to u species the effect of one sided sexual competition makes the trivial state and u dominance state both unstable while the v dominance state is shown to be asymptotically stable with attraction regions and convergent rates depending on the biological parameters using the upper lower solution method we further prove that for a family of wave speeds there exist traveling wave solutions connecting the u dominance state and the v dominance state at this confirms an earlier conjecture that unbalanced mating interference will lead to competitive exclusion these results can also be obtained on an extended model with instantaneous effects of resource competition and temporal delay on mating interference through a transformation into three equation system we prove that the temporal delay does not affect the stability of the steady states and the existence of the traveling waves but causes changes on the attraction regions and convergence rates finally numerical simulations are also presented to illustrate the theoretical results

on discontinuous travelling wave solutions for a class of hyperbolic reaction diffusion models abstract a class of hyperbolic reaction diffusion models is derived within the context of extended thermodynamics this kind of models avoids the unphysical features concerning the instantaneous diffusive effects typical of parabolic equations and it results to be suitable for describing invasive phenomena with a well defined boundary under suitable assumptions the non existence of smooth travelling waves is proved within a region in the state space as an illustrative example of such a theoretical analysis a model describing the infiltration of rain water in semi arid environment is derived and both continuous and discontinuous travelling waves are investigated