loops and branches of coexistence states in a lotka volterra competition model abstract a two species lotka volterra competition diffusion model with spatially inhomogeneous reaction terms is investigated the two species are assumed to be identical except for their interspecific competition coefficients viewing their common diffusion rate as a parameter we describe the bifurcation diagram of the steady states including stability in terms of two real functions of we also show that the bifurcation diagram can be rather complicated namely given any two positive integers l and b the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi trivial steady states of the same type they vanish at the same component and at least b other bifurcating branches of positive stable steady states that connect semi trivial steady states of different types

an automated multilevel substructuring method for eigenspace computation in linear elastodynamics we present an automated multilevel substructuring amls method for eigenvalue computations in linear elastodynamics in a variational and algebraic setting amls first recursively partitions the domain of the pde into a hierarchy of subdomains then amls recursively generates a subspace for approximating the eigenvectors associated with the smallest eigenvalues by computing partial eigensolutions associated with the subdomains and the interfaces between them we remark that although we present amls for linear elastodynamics our formulation is abstract and applies to generic h1 elliptic bilinear forms in the variational formulation we define an interface mass operator that is consistent with the treatment of elastic properties by the familiar steklov poincare operator with this interface mass operator all of the subdomain and interface eigenvalue problems in amls become orthogonal projections of the global eigenvalue problem onto a hierarchy of subspaces convergence of amls is determined in the continuous setting by the truncation of these eigenspaces independent of other discretization schemes the goal of amls in the algebraic setting is to achieve a high level of dimensional reduction locally and inexpensively while balancing the errors associated with truncation and the finite element discretization this is accomplished by matching the mesh independent amls truncation error with the finite element discretization error our report ends with numerical experiments that demonstrate the effectiveness of amls on a model problem and an industrial problem

a general model reduction with primal assembly in structural dynamics abstract a new formulation of general model reduction in structural dynamics is presented the proposed method consists of four attributes a hybrid method combining dynamic condensation and component mode synthesis that are arguably the most popular model reduction techniques a precise model reduction taking the inertia effect into consideration primal assembly at the boundary interface of substructures and general formulations of model reduction with primal assembly comparisons of the proposed model reduction method with the conventional methods are provided as well as an efficient model reduction procedure of the proposed method in particular its advantages and performance of the proposed method are thoroughly investigated with various numerical examples

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

model order reduction with oblique projections for large scale wave propagation there are many applications in which it is necessary to solve large scale parametrized wave propagation problems repeatedly this is still quite a challenging task even with the largest available computer clusters in this paper we will discuss the application of model order reduction mor to problems in seismic petroleum exploration with the aim of diminishing the necessary computing time by a significant factor we consider pod and some variants pod is a model order reduction technique that uses snapshots of a few simulations in order to quickly compute related problems with similar accuracy the method of lines via a petrov galerkin approximation that uses the snapshots as basis functions is the considered approach the order reduction comes from projecting the wave equation discretized in space to the subspace spanned by the snapshots this has been shown earlier to work well in two dimensions the challenge in three dimensions comes from the size of the spatial meshes required and the fact that the method usually requires a number of snapshots that do not fit in fast memory even for current high end multicore machines parallelization is not an option since it is already used for other aspects of this massive problem

global analysis in a predator prey system with nonmonotonic functional response a predator prey system with nonmonotonic functional response is considered global qualitative and bifurcation analyses are combined to determine the global dynamics of the model the bifurcation a

stationary patterns of a prey predator system with a protection zone and cross diffusion of fractional type abstract this paper is concerned with the stationary patterns of a prey predator model with a protection zone and fractional type cross diffusion for the prey it is shown that the fractional type cross diffusion has negative effects on the survival of the prey when the intrinsic growth rate of the predator is positive moreover our mathematical analysis shows that compared with the results obtained in k oeda 2011 and k oeda 2012 the large cross diffusion coefficient and large growth rate of the predator species have some essentially different effects on the profiles of the solutions

steady state bifurcation and hopf bifurcation for a diffusive leslie gower predator prey model we consider a diffusive leslie gower predator prey model subject to the homogeneous neumann boundary condition treating the diffusion coefficient d as a parameter the hopf bifurcation and steady state bifurcation from the positive constant solution branch are investigated moreover the global structure of the steady state bifurcations from simple eigenvalues is established by bifurcation theory in particular the local structure of the steady state bifurcations from double eigenvalues is also obtained by the techniques of space decomposition and implicit function theorem

spatiotemporal patterns in a reaction diffusion model with the degn harrison reaction scheme abstract spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades in this work we are concerned with a reaction diffusion model with the degn harrison reaction scheme which accounts for the qualitative feature of the respiratory process in a klebsiella aerogenes bacterial culture we study the global stability of the constant steady state existence and nonexistence of nonconstant steady states as well as the hopf and steady state bifurcations in particular our results show the existence of turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous these results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns

novel modal methods for transient analysis with a reduced order model based on enhanced craig bampton formulation abstract for transient analysis of structural dynamics with reduced order model rom data recovery procedures that use modal methods such as the classical mode acceleration ma and modal displacement md methods are an important step in order to increase the convergence and accuracy of the solution in this work we propose novel ma and md methods for highly accurate transient analysis with a reduced order model based on enhanced craig bampton ecb formulation which is an extension of the classical craig bampton cb method that is a way to reduce the size of a finite element fe model the performance of the proposed data recovery approach is demonstrated with two numerical examples we also investigate the relation between the proposed and classical ma and md methods

a component mode selection method based on a consistent perturbation expansion of interface displacement abstract a mode selection method is presented for the reduced order modeling rom of structural systems in conjunction with the craig bampton component mode synthesis technique the proposed method is derived by using a consistent expansion of the interface displacement in terms of a frequency dependent small parameter as applied to a craig bampton like rom formulation it is found that this procedure yields a coupling mechanism of the modes of the full model to those of substructures the present mode selection method employs this coupling mechanism as an indicator labeled as the cms method for the substructural modal contributions to the full model the performance of the proposed method is demonstrated by various numerical examples and compared favorably with existing method such as the cms method

asymptotic estimation of gaussian quadrature error for a nonsingular integral in potential theory this paper considers the n point gauss jacobi approximation of nonsingular integrals of the form 11 t t log z t dt with jacobi weight and polynomial and derives an estimate for the quadrature error that is asymptotic as n the approach follows that previously described by donaldson and elliott a numerical example illustrating the accuracy of the asymptotic estimate is presented the extension of the theory to similar integrals defined on more general analytic arcs is outlined

a reduced order method for allen cahn equations in this article we present a reduced order method for modeling and computing allen cahn equations a global basis method is used in the discretized system of the allen cahn equations and proper orthogonal decomposition pod method is utilized to reduce the global basis to treat the difficulty of nonlinearity for allen cahn equations we apply discrete empirical interpolation method deim to the nonlinear term from the discretization system a reduced order method is developed by integrating pod and deim it is well known that the allen cahn equations have a nonlinear stability property i e the free energy functional decreases with respect to time the discretized allen cahn system modeled by the pod deim reduced order method can inherit the nonlinear stability of the continuous model the computation efficiency is significantly enhanced by using the reduced order method a few numerical results are presented to illustrate the performance of the reduced order method for deterministic allen cahn equations and stochastic allen cahn equations

a family of craig bampton methods considering residual mode compensation abstract in this paper we investigate the formulation of a family of craig bampton cb methods considering residual modes by o callahan s approximation and adding generalized coordinate vectors containing unknown eigenvalues in addition we propose an nth order higher order cb hcbn method for compensating the n 1 th residual flexibility in the nth order hcb hcbn method by o callahan s approach therefore various cb methods with improved performance such as the enhanced craig bampton ecb method which uses o callahan s approximation and the higher order craig bampton hcb method which adds generalized coordinate vectors and hcb are employed for the comparison of performance with the aid of multiprecision computing through three benchmark examples it is revealed that the hcb1 method a modified version of the first order hcb method hcb1 with the aid of o callahan s approximation proposed shows better performance than hcb1 with the same number of retained modes however hcb2 and hcb3 modified versions of the second and third order hcb method respectively cannot be improved further from the results we concluded that this was due to the limitation of o callahan s approach which many researchers have fundamentally questioned

continuum design sensitivity of transient responses using ritz and mode acceleration methods in this paper a unified continuum based sizing design sensitivity analysis method is developed for the transient dynamic response of built up structures using the direct differentiation method the variational equation of the built up structure is differentiate d with respect to the design variable to obtain a variational design sensitivity equation of the transient response the same finite element analysis model that is used to obtain an approximate solution of the variational equation of the built up structure is also used to obtain an approximate solution of the variational design sensitivity equation for large size built up structures the same superposition method is used both to reduce the dimension of the matrix equation of the built up structure and also to reduce the dimension of the design sensitivity equation the continuum based design sensitivity analysis method can be implemented outside established finite element analysis codes using postprocessing data since it does not require derivatives of the stiffness damping and mass matrices moreover the method is efficient since it does not require derivatives of basis vectors either the ritz or mode acceleration method is used to further improve accuracy and efficiency of both analysis and sensitivity results an example with damping shows that accurate design sensitivity of pointwise transient displacements is obtained using only 10 of basis vectors this example indicates that in the continuum based design sensitivity analysis method the same basis vectors that are used for analysis of the built up structure are suitable for analysis of the design sensitivity equation

model order reduction for nonlinear schrodinger equation we apply the proper orthogonal decomposition pod to the nonlinear schrodinger nls equation to derive a reduced order model the nls equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid point rule a priori error estimates are derived for the pod reduced dynamical system numerical results for one and two dimensional nls equations coupled nls equation with soliton solutions show that the low dimensional approximations obtained by pod reproduce very well the characteristic dynamics of the system such as preservation of energy and the solutions

comparative numerical analysis using reduced order modeling strategies for nonlinear large scale systems we perform a comparative analysis using three reduced order strategies missing point estimation mpe method gappy pod method and discrete empirical interpolation method deim applied to a biological model describing the spatio temporal dynamics of a predator prey community the comparative study is focused on the efficiency of the reduced order approximations and the complexity reduction of the nonlinear terms different variants are discussed related to the projection based model reduction framework combined with selective spatial sampling to efficiently perform the online computations numerical results are presented

qualitative analysis for a diffusive predator prey model in this paper we study the qualitative properties of a diffusive predator prey model subject to the homogeneous neumann boundary condition by a comparison argument and iteration technique under some hypotheses we prove that the positive constant steady state is globally asymptotically stable we also establish the existence and non existence of non constant positive steady states stationary patterns by use of the degree theory and the a priori estimates

bifurcation and spatiotemporal patterns in a homogeneous diffusive predator prey system abstract a diffusive predator prey system with holling type ii predator functional response subject to neumann boundary conditions is considered hopf and steady state bifurcation analysis are carried out in details in particular we show the existence of multiple spatially non homogeneous periodic orbits while the system parameters are all spatially homogeneous our results and global bifurcation theory also suggest the existence of loops of spatially non homogeneous periodic orbits and steady state solutions these results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation

partitioned component mode synthesis via a flexibility approach a new flexibility based component mode synthesis method is presented which is derived by approximating the partitioned equations of motion that employ a localized method of lagrange multipliers the use of the localized lagrange multipliers leads to unlike the classical lagrange multipliers a linearly independent set of interface forces without any redundancies at multiply connected interface nodes hence the resulting interface flexibility matrices are uniquely determined and well suited for the present method development an attractive feature of the present method is its substructural mode selection criterion that is independent of loading conditions numerical experiments show that the present modal selection criterion is reliable and that the proposed flexibility based component mode synthesis method performs well relative to the classical craig bampton method in terms of accuracy and of its ability to include dominant substructural modes for simple plates and a relatively complex solid ring

hopf bifurcation and steady state bifurcation for an autocatalysis reaction diffusion model abstract this paper is concerned with an autocatalysis model subject to no flux boundary conditions the existence of hopf bifurcation are firstly obtained then by the center manifold theory and the normal form method the direction of hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established on the other hand the steady state bifurcations from simple and double eigenvalues are intensively studied the techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues finally some numerical simulations are shown to verify the analytical results

a diffusive predator prey model in heterogeneous environment abstract in this paper we demonstrate some special behavior of steady state solutions to a predator prey model due to the introduction of spatial heterogeneity we show that positive steady state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably moreover we observe some essential differences of the behavior of our model from that of the classical lotka volterra model that seem to arise only in the heterogeneous case

gauss legendre quadrature for the evaluation of integrals involving the hankel function the boundary integral method for the two dimensional helmholtz equation requires the approximate evaluation of the integral 1 1g x h 0 1 l x a 2 b 2 dx where g is a polynomial in particular gauss legendre quadrature is considered when the source point is close to the interval of integration that is 1

non constant positive steady states of a diffusive predator prey system in homogeneous environment abstract in this paper we investigate the existence and non existence of non constant positive steady states of a diffusive predator prey interaction system under homogeneous neumann boundary condition in homogeneous environment we show that the predator prey model with leslie gower functional response has no non constant positive solution but the system with a general functional response may have at least one non constant positive steady state under some conditions

a simple approach for reducing the order of equations with higher order nonlinearity abstract in this paper we introduce a simple approach to reduce the order of equations with higher order nonlinearity we choose the generalized forms of the kdv equation the camassa holm equation and the zakharov kuznetsov equation to illustrate the validity and advantages of the approach then the explicit solutions with different physical structures of the reduced equations can be discussed further the approach used here can be applied to some other nonlinear equations with higher order nonlinearity and is also suitable for some higher dimensional models

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

construction of energy stable projection based reduced order models an approach for building energy stable galerkin reduced order models roms for linear hyperbolic or incompletely parabolic systems of partial differential equations pdes using continuous projection is developed this method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow the key idea is to apply to the pdes a transformation induced by the lyapunov function for the system and to build the rom in the transformed variables for linear problems the desired transformation is induced by a special inner product termed the symmetry inner product which is derived herein for several systems of physical interest connections are established between the proposed approach and other stability preserving model reduction methods giving the paper a review flavor more specifically it is shown that a discrete counterpart of this inner product is a weighted l 2 inner product obtained by solving a lyapunov equation first proposed by rowley et al and termed herein the lyapunov inner product comparisons between the symmetry inner product and the lyapunov inner product are made and the performance of roms constructed using these inner products is evaluated on several benchmark test cases

hopf bifurcation analysis in the 1 d lengyel epstein reaction diffusion model abstract the lengyel epstein model with diffusion and homogeneous neumann boundary condition is considered in this paper we give the existence of multiple spatially non homogeneous periodic solutions though all the parameters of the system are spatially homogeneous

high fidelity flexibility based component mode synthesis method with interface degrees of freedom reduction the partitioned structural equations of motion via the method of localized lagrange multipliers palm consist of internal and interface variables the model reduction achieved based on palm is called flexibility based component mode synthesis f cms and it thus far has concentrated on reducing the internal degrees of freedom dofs or reducing the internal and interface dofs at once in this work well defined three level reduction formulations of the f cms method are proposed for better precision internal dofs the localized lagrange multipliers and the interface boundary displacement the proposed formulations include three different reduction techniques for the interface boundary displacement for better computational efficiency numerical experiments indicate that the proposed formulations can substantially reduce the overall reduced order models further without compromising the accuracy thus achieving a substantial additional reduction of computational cost compared with the previous works in a

comparisons between reduced order models and full 3d models for fluid structure interaction problems in haemodynamics when modelling the cardiovascular system the effect of the vessel wall on the blood flow has great relevance arterial vessels are complex living tissues and three dimensional specific models have been proposed to represent their behaviour the numerical simulation of the 3d 3d fluid structure interaction fsi coupled problem has high computational costs in terms of required time and memory storage even if many possible solutions have been explored to speed up the resolution of such problem we are far from having a 3d 3d fsi model that can be solved quickly in 3d 3d fsi models two of the main sources of complexity are represented by the domain motion and the coupling between the fluid and the structural part nevertheless in many cases we are interested in the blood flow dynamics in compliant vessels whereas the displacement of the domain is small and the structure dynamics is less relevant in these situations techniques to reduce the complexity of the problem can be used one consists in using transpiration conditions for the fluid model as surrogate for the wall displacement thus allowing problem s solution on a fixed domain another strategy consists in modelling the arterial wall as a thin membrane under specific assumptions figueroa et al 2006 nobile and vergara 2008 instead of using a more realistic but more computationally intensive 3d elastodynamic model using this strategy the dynamics of the vessel motion is embedded in the equation for the blood flow combining the transpiration conditions with the membrane model assumption we obtain an attractive formulation in fact instead of solving two different models on two moving physical domains we solve only a navier stokes system in a fixed fluid domain where the structure model is integrated as a generalized robin condition in this paper we present a general formulation in the boundary conditions which is independent of the time discretization scheme choice and on the stress strain constitutive relation adopted for the vessel wall structure our aim is first to write a formulation of a reduced order model with zero order transpiration conditions for a generic time discretization scheme then to compare a 3d 3d fsi model and a reduced fsi one in two realistic patient specific cases a femoropopliteal bypass and an aorta in particular we are interested in comparing the wall shear stresses in fact this quantity can be used as a risk factor for some pathologies such as atherosclerosis or thrombogenesis more in general we want to assess the accuracy and the computational convenience to use simpler formulations based on reduced order models in particular we show that in the case of small displacements using a 3d 3d fsi linear elastic model or the correspondent reduced order one yields many similar results

asymptotic behaviour of positive steady states to a predator prey model to understand the heterogeneous spatial effect on predator prey models we study the behaviour of the positive steady states of a predator prey model as certain parameters are small or large we compare the case when the model has a spatial degeneracy with the case when it does not have such a degeneracy our results show that the effect of the degeneracy can be clearly observed in one limiting case but not in the others

design of quadrature rules for muntz and muntz logarithmic polynomials using monomial transformation a method for constructing the exact quadratures for muntz and muntz logarithmic polynomials is presented the algorithm does permit to anticipate the precision machine precision of the numerical integration of muntz logarithmic polynomials in terms of the number of gauss legendre gl quadrature samples and monomial transformation order to investigate in depth the properties of classical gl quadrature we present new optimal asymptotic estimates for the remainder in boundary element integrals this quadrature rule can be applied to evaluate singular functions with end point singularity singular kernel as well as smooth functions the method is numerically stable efficient easy to be implemented the rule has been fully tested and several numerical examples are included the proposed quadrature method is more efficient in run time evaluation than the existing methods for muntz polynomials