weak and classical solutions to predator prey system with cross diffusion abstract in this paper we are concerned with a system of nonlinear partial differential equations modeling a predator prey system with cross diffusion in heterogeneous habitats predators are assumed to feed on preys with a holling type ii functional response to prey density and preys are assumed to follow a logistic growth in the absence of predation the mobility of each classes is assumed to be influenced by the gradient of other classes the existence result is proved by means of an approximation system the faedo galerkin method and the compactness method the global existence of classical solutions is proved under certain restrictions on the coefficients

global solutions to a class of multi species reaction diffusion systems with cross diffusions arising in population dynamics in this paper an n species strongly coupled cooperating diffusive system is considered in a bounded smooth domain subject to homogeneous neumann boundary conditions employing the method of energy estimates we obtain some conditions on the diffusion matrix and inter specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution the globally asymptotical stability of the constant positive steady state is also discussed as a consequence all the results hold true for multi species lotka volterra type competition model and prey predator model

analysis of a parabolic cross diffusion population model without self diffusion abstract the global existence of non negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown the cross diffusion terms are allowed to be arbitrarily large whereas the self diffusion terms are assumed to disappear the last assumption complicates the analysis since these terms usually provide h 1 estimates of the solutions the existence proof is based on a positivity preserving backward euler galerkin approximation discrete entropy estimates and l 1 weak compactness arguments furthermore employing the entropy entropy production method we show for special stationary solutions that the transient solution converges exponentially fast to its steady state as a by product we prove that only constant steady states exist if the inter specific competition parameters disappear no matter how strong the cross diffusion constants are

smooth solutions to a quasi linear system of diffusion equations for a certain population model abstract the system of diffusion equations proposed by kawasaki shigesada and teramoto describes a population model of two competing species with self and cross population pressures the densities of the two species are denoted by u and v in this paper the author studies the initial boundary value problem associated with the neumann boundary condition corresponds to the case where the flux is zero at the boundary many investigators have considered nonlinear diffusion systems arising from various physical and biological problems these equations however have a special structure the highest order derivatives are not coupled or at least the coefficient matrix for the highest order derivatives is positive definite this is not the case for the system and hence some of the techniques which are effective for the system are no longer applicable to our case nevertheless we can still use sobolevski s method see reference 2 to establish the local existence of solutions under the special assumption c1 c2 in we can also prove the global existence of solutions by energy estimates the unusual structure of seems to make it difficult to settle the question of asymptotic stability of solutions

error estimates for discrete time approximations of nonlinear cross diffusion systems this paper is concerned with nonlinear and linear discrete time algorithms for cross diffusion systems the nonlinear scheme corresponds to backward differences in time and the linear algorithm is a very easy to implement scheme proposed by the author esaim math model numer anal 45 2011 pp 1141 1161 the main purpose of this paper is to derive convergence rates of the discrete time schemes we obtain the same orders for both the nonlinear and the linear scheme moreover these orders are optimal we also establish uniqueness and regularity results of weak solutions of the cross diffusion systems

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

on a cross diffusion population model deduced from mutation and splitting of a single species we deduce a particular case of the population cross diffusion model introduced by shigesada et al 1979 1 by using the ideas of mutation and splitting from a single species as described by sanchez palencia for ode s systems sanchez palencia 2011 21 the resulting equations of the pde system only differ in the cross diffusion terms the corresponding diffusion matrix being self diffusion dominated which implies that the well known population segregation patterns of the shigesada et al model do not appear in this case we prove existence and uniqueness of solutions of the pde system and use a finite element approximation to discuss numerically stability properties of solutions with respect to the parameters in comparison with related models

modeling spatial adaptation of populations by a time non local convection cross diffusion evolution problem abstract in 19 sighesada et al presented a system of partial differential equations for modeling spatial segregation of interacting species apart from competitive lotka volterra reaction and population pressure cross diffusion terms a convective term modeling the populations attraction to more favorable environmental regions is included in this article we introduce a modification of their convective term to take account for the notion of spatial adaptation of populations after describing the model we briefly discuss its well possedness and propose a numerical discretization in terms of a mass preserving time semi implicit finite differences scheme finally we provide the results of two biologically inspired numerical experiments showing qualitative differences between the original model of 19 and the model proposed in this article

a method to estimate the parameters of an arma model a method of estimating the parameters of an arma model is proposed in the transient case we show that the formulas recently given by li and dickinson 7 for the ar parameters are easily obtained by considering the formulation of the one step predictor in the steady state case we can get separately the ar parameters and the ma ones

ld sup 2 arma identification algorithm the authors present the ld sup 2 arma identifier a novel algorithm that solves the essentially nonlinear autoregressive moving averaged arma identification problem with a linear procedure in two steps an order selection algorithm followed by an arma parameter estimator the determination of the ar and ma coefficients involves the solution of two dual systems of linear equations these systems decouple the estimation of the autoregressive component from the estimation of the moving average component the selection of the number of poles and of the number of zeros is accomplished by a scheme that minimizes the mismatch of the data to each proposed model simulated experiments on the proposed order selection procedure are presented

on a cross diffusion system arising in image denoising abstract we study a generalization of a cross diffusion problem deduced from a nonlinear complex variable diffusion model for signal and image denoising we prove the existence of weak solutions of the time independent problem with fidelity terms under mild conditions on the data problem then we show that this translates on the well posedness of a quasi steady state approximation of the evolution problem and also prove the existence of weak solutions of the latter under more restrictive hypothesis we finally perform some numerical simulations for image denoising comparing the performance of the cross diffusion model to other well known denoising methods

a comparison of two linear methods of estimating the parameters of arma models a finite order stationary and minimum phase arma autoregressive moving average p q model is equivalent to an infinite order ar autoregressive model two methods of estimating the parameters of the arma p q model by solving only linear equations are based on or closely related to this equivalence relation one method was derived directly from the equivalence relation by d graupe et al ibid vol ac 20 p 104 107 feb 1975 the other was derived by s li and b w dickinson ibid vol ac 31 p 275 278 mar 1986 and ieee trans acoust speech signal process vol assp 36 p 502 512 apr 1988 based on an iterated least squares regression approach the end results bear close resemblance to those of graupe et al the two methods are compared and ways to improve the parameter estimates are suggested

optimal instrumental variable estimates of the ar parameters of an arma process the modified yule walker myw equations for estimating the ar parameters of an arma process are presented as a special case of an instrumental variable iv method the consistency and accuracy of the ar parameter estimates are studied it is shown that estimation accuracy increases monotonically with the number of myw equations for an optimal choice of the weighting matrix used in the least squares solution of these equations the asymptotic error covariance of the optimal iv method equals that of the prediction error method the results of this paper verify experimental results reported in the literature regarding the performance of the myw method and provide the necessary accuracy analysis furthermore they suggest several simple asymptotically efficient multistep algorithms for estimating the ar parameters which are presented in a companion paper

global existence analysis of cross diffusion population systems for multiple species the existence of global in time weak solutions to reaction cross diffusion systems for an arbitrary number of competing population species is proved the equations can be derived from an on lattice random walk model with general transition rates in the case of linear transition rates it extends the two species population model of shigesada kawasaki and teramoto the equations are considered in a bounded domain with homogeneous neumann boundary conditions the existence proof is based on a refined entropy method and a new approximation scheme global existence follows under a detailed balance or weak cross diffusion condition the detailed balance condition is related to the symmetry of the mobility matrix which mirrors onsager s principle in thermodynamics under detailed balance and without reaction the entropy is nonincreasing in time but counter examples show that the entropy may increase initially if detailed balance does not hold

qualitative behavior of solutions to cross diffusion systems from population dynamics abstract a general class of cross diffusion systems for two population species in a bounded domain with no flux boundary conditions and lotka volterra type source terms is analyzed although the diffusion coefficients are assumed to depend linearly on the population densities the equations are strongly coupled generally the diffusion matrix is neither symmetric nor positive definite three main results are proved the existence of global uniformly bounded weak solutions their convergence to the constant steady state in the weak competition case and the uniqueness of weak solutions the results hold under appropriate conditions on the diffusion parameters which are made explicit and which contain simplified shigesada kawasaki teramoto population models as a special case the proofs are based on entropy methods which rely on convexity properties of suitable lyapunov functionals

analysis of a splitting differentiation population model leading to cross diffusion starting from the dynamical system model capturing the splitting differentiation process of populations we extend this notion to show how the speciation mechanism from a single species leads to the consideration of several well known evolution cross diffusion partial differential equations among the different alternatives for the diffusion terms we study the model introduced by busenberg and travis for which we prove the existence of solutions in the one dimensional spatial case using a direct parabolic regularization technique we show that the problem is well posed in the space of bounded variation functions and demonstrate with a simple example that this is the best regularity expected for solutions we numerically compare our approach to other alternative regularizations previously introduced in the literature for the particular case of the contact inhibition problem simulation experiments indicate that the numerical scheme arising from the approximation introduced in this article outperforms those of the existent models from the stability point of view

entropic structure and duality for multiple species cross diffusion systems this paper deals with the existence of global weak solutions for a wide class of multiple species cross diffusions systems the existence is based on two different ingredients an entropy estimate giving some gradient control and a duality estimate that gives naturally l 2 control the proof relies on a semi implicit scheme tailored for cross diffusion systems firstly defined by the two authors and collaborators these results are applied to models having an entropy relying on the detailed balance condition recently exhibited by chen et al

positive steady state solutions of a competing reaction diffusion system with large cross diffusion coefficients abstract this paper deals with a coupled system of reaction diffusion equations modeling the competition of two species with cross diffusion we discuss the existence of positive steady state solutions in relation to the intrinsic birth rates a 1 a 2 of species and the cross diffusion coefficients 12 21 our results show that positive steady state solutions exist if a 1 and a 2 lie in certain range or if 12 and 21 are sufficiently large

system identification of the speech production process based on a state space representation in this paper a new method for analysis of speech signals based on a representation in terms of an autoregressive moving average arma process with variable order is presented the process of speech production is approximated by an arma process described by a state equation and an observation equation an algorithm for the minimal realization of the model of speech production process is proposed the state transition matrix that characterizes the autoregressive ar scheme is determined to minimize the mean squared error in the estimation of the speech signal by kalman filtering the state transition matrix and the autocorrelation function of the signal are used to determine parameters of the moving average ma scheme application to spectral density estimation of speech corrupted by additive noise is also discussed

an efficient method to compute consistent estimates of the ar parameters of an arma model a computationally efficient method for consistent estimation of the ar parameters of an arma model is obtained based on an iterated least squares approach proposed by tsay and tiao the method is derived from the lattice filter structure which generates an orthogonal basis for the hilbert space of interest it is shown that for the arma p q model we can get consistent estimates of the ar parameters by using a least squares lattice estimation algorithm and solving a system of q linear equations

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

a convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross diffusion in this paper we are concerned with a model of the indirect transmission of an epidemic disease between two spatially distributed host populations having non coincident spatial domains with nonlocal and cross diffusion the epidemic disease transmission occurring through a contaminated environment the mobility of each class is assumed to be influenced by the gradient of the other classes we address the questions of existence of weak solutions and existence and uniqueness of classical solution by using respectively a regularization method and an interpolation result between banach spaces moreover we propose a finite volume scheme and proved the well posedness nonnegativity and convergence of the discrete solution the convergence proof is based on deriving a series of a priori estimates and by using a general l p compactness criterion finally the numerical scheme is illustrated by some examples

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

cepstral coefficients covariance lags and pole zero models for finite data strings one of the most widely used methods of spectral estimation in signal and speech processing is linear predictive coding lpc lpc has some attractive features which account for its popularity including the properties that the resulting modeling filter i matches a finite window of n 1 covariance lags ii is rational of degree at most n and iii has stable zeros and poles the only limiting factor of this methodology is that the modeling filter is all pole i e an autoregressive ar model in this paper we present a systematic description of all autoregressive moving average arma models of processes that have properties i iii in the context of cepstral analysis and homomorphic filtering we show that each such an arma model determines and is completely determined by its finite windows of cepstral coefficients and covariance lags we show that these nth order windows form local coordinates for all arma models of degree n and that the pole zero model can be determined from the windows as the unique minimum of a convex objective function we refine this optimization method by first noting that the maximum entropy design of an lpc filter is obtained by maximizing the zeroth cepstral coefficient subject to the constraint i more generally we modify this scheme to a more well posed optimization problem where the covariance data enter as a constraint and the linear weights of the cepstral coefficients are positive in a sense that a certain pseudo polynomial is positive rather succinctly generalizing the maximum entropy method this new problem is a homomorphic filter generalization of the maximum entropy method providing a procedure for the design of any stable minimum phase modeling filter of degree less or equal to n that interpolates the given covariance window we present an algorithm for realizing these filters in a lattice ladder form given the covariance window and the moving average part of the model while we also show how to determine the moving average part using cepstral smoothing one can make use of any good a priori estimate for the system zeros to initialize the algorithm we conclude the paper with an example of this method incorporating an example from the literature on arma modeling

semi discretization in time and numerical convergence of solutions of a nonlinear cross diffusion population model a positivity preserving numerical scheme for a strongly coupled cross diffusion model for two competing species is presented based on a semi discretization in time the variables are the population densities of the species existence of strictly positive weak solutions to the semidiscrete problem is proved moreover it is shown that the semidiscrete solutions converge to a non negative solution of the continuous system in one space dimension the proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional

on instrumental variable estimation of sinusoid frequencies and the parsimony principle multiple sinusoids in noise can be modeled as an arma process with the ar parameters satisfying certain symmetry relations according to the parsimony principle the constraints on the ar parameters should he taken into account to get improved estimation accuracy it is shown in this note that when estimating the ar parameters by a general instrumental variable method such a parsimony does not necessarily apply however the parsimony principle does hold when an optimal instrumental variable method is used

an initial boundary value problem for a certain density dependent diffusion system on considere le systeme parabolique quasilineaire u t x c 1 d 1 v v e 1 a 1 v b 1 u u v t x c 2 d 2 u v e 2 a 2 u b 2 o v dans o t u x s 0 v x s sur o t u x o w 1 x v x o w 2 x sur ou r n est un domaine borne a frontiere lisse