global existence of solutions to a cross diffusion predator prey system with holling type ii functional response this paper is concerned with a cross diffusion system arising in a predator prey population model with holling type ii functional response in a bounded domain with no flux boundary condition employing the energy estimates and gagliardo nirenberg type inequalities to establish w 2 1 bounds uniform in time the global existence and uniform boundedness of solutions for a cross diffusion system are proved the global asymptotic stability of the constant positive steady state is also investigated

traveling waves for competitive lotka volterra systems with spatial diffusions and spatio temporal delays abstract this paper is concerned with the existence of traveling waves for reaction diffusion equations with spatio temporal delays our approach is to use the cross iteration technique and schauder s fixed point theorem the iterative technique can be established for a class of integral operators when the reaction terms satisfy new monotonicity conditions which is different from those in previous works the results can be well applied to a two species competitive lotka volterra system with more general kernel functions in order to overcome the difficulty of spatio temporal delays we need to use a basic result from a pde textbook existence of traveling waves which connect the trivial equilibrium and the positive equilibrium indicates that there is a transition zone moving the steady state with no species to the steady state with the coexistence of two species our results can extend some existing ones for two species diffusion competitive lotka volterra systems

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

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

boundary controllability of a cascade system coupling fourth and second order parabolic equations abstract a control system coupling fourth and second order parabolic equations is considered in this paper the main topic is the study of the control properties of this system when we only control the second order partial differential equation through a boundary condition depending on the choice of the diffusion coefficients we obtain positive and negative results for approximate and null controllability in particular we prove that for any given positive time t 0 we can find some diffusion coefficients such that the system is null controllable in time t if t t 0 and is not null controllable if t t 0

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

pattern formation of a spatial predator prey system abstract there are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors to investigate how these migration directed movement and diffusion random movement affect predator prey systems we have studied the spatiotemporal complexity in a predator prey system with holling tanner form a theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated at the same time we present the properties of pattern solutions the results of numerical simulations show that migration has prominent effect on the pattern formation of the population i e changing turing pattern into traveling pattern this study suggests that modelling by migration and diffusion in predator prey systems can account for the dynamical complexity of ecosystems

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

rich dynamics of a predator prey model with spatial motion one of the central issues in ecosystem is the influence of space on the stability of the populations as a result a predator prey model with spatial diffusion and cubic growth rate is investigated asymptotical properties of solutions of such spatial model are obtained based on comparison principle what is more we show the global stability of positive equilibrium by constructing lyapunov function we also show how predator diffuses has impact on the pattern dynamics based on numerical simulations the results may well enrich the findings in predator prey systems with spatial motions

stable two dimensional solitary pulses in linearly coupled dissipative kadomtsev petviashvili equations we present a two dimensional 2d generalization of the stabilized kuramoto sivashinsky system based on the kadomtsev petviashvili kp equation including dissipation of the generic newell whitehead segel nws type and gain the system directly applies to the description of gravity capillary waves on the surface of a liquid layer flowing down an inclined plane with a surfactant diffusing along the layer s surface actually the model is quite general offering a simple way to stabilize nonlinear media combining the weakly 2d dispersion of the kp type with gain and nws dissipation other applications are internal waves in multilayer fluids flowing down an inclined plane double front flames in gaseous mixtures etc parallel to this weakly 2d model we also introduce and study a semiphenomenological one whose dissipative terms are isotropic rather than of the nws type in order to check if qualitative results are sensitive to the exact form of the lossy terms the models include an additional linear equation of the advection diffusion type linearly coupled to the main kp nws equation the extra equation provides for stability of the zero background in the system thus opening a way for the existence of stable localized pulses we focus on the most interesting case when the dispersive part of the system is of the kp i type which corresponds e g to capillary waves and makes the existence of completely localized 2d pulses possible treating the losses and gain as small perturbations and making use of the balance equation for the field momentum we find that the equilibrium between the gain and losses may select two steady state solitons from their continuous family existing in the absence of the dissipative terms the latter family is found in an exact analytical form and is numerically demonstrated to be stable the selected soliton with the larger amplitude is expected to be stable direct simulations completely corroborate the analytical predictions for both the physical and phenomenological models

kernel optimum nearly analytical discretization kond algorithm applied to parabolic and hyperbolic equations abstract two applications of the kernel optimum nearly analytical discretization kond algorithm to the parabolic and the hyperbolic type equations are presented in detail to lead to novel numerical schemes with very high numerical accuracy it is demonstrated numerically that the two dimensional kond p scheme for the parabolic type yields much less numerical error by over 2 3 orders measured quantitatively by the root mean square deviation from analytical solutions and reduces the cpu time to about 1 5 for a common numerical accuracy compared with the conventional explicit scheme of reference it is also demonstrated numerically that the kond h scheme for the hyperbolic type yields much less diffusive error and has fairly high stability for both of the linear and the nonlinear wave propagations compared with other conventional schemes origins of numerical errors in data processing in general numerical schemes are discussed

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

null controllability of the stabilized kuramoto sivashinsky system with one distributed control this paper presents a control problem for a one dimensional nonlinear parabolic system which consists of a kuramoto sivashinsky korteweg de vries equation coupled to a heat equation we address the problem of controllability by means of a control supported in an interior open subset of the domain and acting on one equation only the local null controllability of the system is proved the proof is based on a carleman estimate for the linearized system around the origin a local inversion theorem is applied to get the result for the nonlinear system

stackelberg nash exact controllability for the kuramoto sivashinsky equation abstract this paper deals with a hierarchical control problem for the kuramoto sivashinsky equation following a stackelberg nash strategy we assume that there is a main control called the leader and two secondary controls called the followers the leader tries to drive the solution to a prescribed target and the followers intend to be a nash equilibrium for given functionals it is known that this problem is equivalent to a null controllability result for an optimality system consisting of three non linear equations one of the novelties is a new carleman estimate for a fourth order equation with right hand sides in sobolev spaces of negative order which allows to relax some geometric conditions for the observation sets for the followers

local exact controllability to the trajectories of the swift hohenberg equation abstract this paper is addressed to study the local exact controllability to the trajectories of the swift hohenberg equation we first establish a new global carleman estimate for the fourth order parabolic operator and we consider the exact controllability to the trajectories of the linearized system then by local inversion theorem we show that the same result holds for the swift hohenberg equation

elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport finite difference acoustic wave modeling and reverse time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well however conventional finite difference methods for solving the acoustic or elastic wave equation suffer from numerical dispersion when too few samples per wavelength are used the flux corrected transport fct algorithm adapted from hydrodynamics reduces the numerical dispersion in finite difference wavefield continuation the flux correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion incorporating the flux correction procedure in conventional finite difference modeling or reverse time migration can provide finite difference solutions with no numerical dispersion even for impulsive sources the fct correction which can be applied to finite difference approximations of any order in space and time is an efficient alternative to use for finite difference approximations of increasing order through demonstrations of modeling and migration on both synthetic and field data we show the benefits of the fct algorithm as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse

discontinuous galerkin methods for wave propagation in poroelastic media we have developed a new numerical method to solve the heterogeneous poroelastic wave equations in bounded three dimensional domains this method is a discontinuous galerkin method that achieves arbitrary high order accuracy on unstructured tetrahedral meshes for the low frequency range and the inviscid case by using biot s equations and darcy s dynamic laws we have built a scheme that can successfully model wave propagation in fluid saturated porous media when anisotropy of the pore structure is allowed zero inflow fluxes are used as absorbing boundary conditions a continuous arbitrary high order derivatives time integration is used for the high frequency inviscid case whereas a space time discontinuous scheme is applied for the low frequency case we conducted a numerical convergence test of the proposed methods we used a series of examples to quantify the quality of our numerical results comparing them to analytic solutions as well as numerical solutions obtained by other methodologies in partic

a highly accurate discontinuous galerkin method for complex interfaces between solids and moving fluids we have extended a new highly accurate numerical scheme for unstructured 2d and 3d meshes based on the discontinuous galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid solid interfaces because of the formulation of the wave equations as a unified first order hyperbolic system in velocity stress the fluid can be in movement along the interface the governing equations within the moving fluid are derived from well known first principles in fluid mechanics the discontinuous galerkin approach allows for jumps of the material parameters and the solution across element interfaces which are handled by riemann solvers or numerical fluxes the use of riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero no additional compatibility relations such as vanishing shear stress or continuity of normal stresses are necessary to couple the solid and fluid along an interface the

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

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

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

an optimal nearly analytic discrete weighted runge kutta discontinuous galerkin hybrid method for acoustic wavefield modeling abstractthe newly developed optimal nearly analytic discrete onad and the weighted runge kutta discontinuous galerkin wrkdg methods can effectively suppress the numerical dispersion caused by discretizing wave equations but it is difficult for onad to implement on flexible meshes whereas the wrkdg has high computational cost for wavefield simulations we have developed a new hybrid algorithm by combining the onad method with the wrkdg method in this hybrid algorithm the computational domain was split into several subdomains in which the subdomain for the onad method used regular cartesian grids whereas the subdomain for the wrkdg method used triangular grids the hybrid method was at least third order spatially accurate we have applied the proposed method to simulate the scalar wavefields for different models including a homogeneous model a rough topography model a fracture model and a cave model the numerical results found that the hybrid method can deal with complicated geometrical struc