the existence bifurcation and stability of positive stationary solutions of a diffusive leslie gower predator prey model with holling type ii functional responses abstract in this paper we revisit a diffusive leslie gower predator prey model with holling type ii functional responses and dirichlet boundary condition it is shown that multiple positive steady state solutions exist under certain conditions on the parameters while for another parameter region the positive steady state solution is unique and locally asymptotically stable results are proved by using bifurcation theory fixed point index theory energy estimates and asymptotic behavior analysis

on the uniqueness and ordering of steady states of predator prey systems this paper discusses the relationship between the uniqueness and the ordering of strictly positive solutions of elliptic predator prey interacting systems if v and u v are two such solutions with u or v v then u v v when the positive solutions are numerically close to the extreme case the solution is unique

stable coexistence states in the volterra lotka competition model with diffusion we study the existence uniqueness and stability of coexistence states in the lotka volterra model with diffusion for two competing species we assume that the parameters describing the interactio

diffusive lotka volterra competition models on periodically evolving domains abstract the paper deals with the asymptotic dynamics of a diffusive lotka volterra competition model with periodic coefficients and zero flux boundary conditions acting on a spatially isotropic and temporally periodic evolving domain we determine sufficient conditions for the global stability of the spatially homogeneous positive periodic solution moreover the phenomenon of competitive exclusion is investigated

matrix completion a la dempster by the principle of parsimony dempster s covariance selection method is extended first to general nonsingular matrices and then to full rank rectangular matrices dempster observed that his completion solved a maximum entropy problem we show that our generalized completions are also solutions of a suitable entropy like variational problem

global asymptotic stability of 3 species lotka volterra models with diffusion and time delays abstract this paper is concerned with three 3 species time delayed lotka volterra reaction diffusion models with homogeneous neumann boundary condition some simple conditions are obtained for the global asymptotic stability of the nonnegative semitrivial constant steady state solutions these conditions are explicit and easily verifiable and they involve only the reaction rate constants and are independent of the diffusion and time delays the result of global asymptotic stability not only implies the nonexistence of positive steady state solution but also gives some extinction results of the models in the ecological sense the instability of some nonnegative semitrivial constant steady state solutions is also shown the conclusions for the reaction diffusion systems are directly applicable to the corresponding ordinary differential systems

a generalization of the vidav jorgens perturbation theorem for semigroups and its application to transport theory on considere des semigroupes t t engendres par l operateur de transport a k comme une perturbation du semi groupe s t engendre par a seul on montre que pour un n tous les produits de la forme ks td 1k ks tnd k t i 0 sont strictement singuliers alors s t et t t ont le meme rayon spectral essentiel

effects of a degeneracy in the competition model part i classical and generalized steady state solutions abstract we study the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of we show that there exists a critical number such that if then the model behaves similarly to the well studied classical competition model where all the coefficient functions are positive constants but when new phenomena occur our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative but can be qualitative as well in part i here we mainly study two kinds of steady state solutions which determine the dynamics of the model one consists of finite functions while the other consists of generalized functions which satisfy u v 0 on the part of the domain that b x vanishes but are positive and finite on the rest of the domain and are determined by certain boundary blow up systems the research is continued in part ii where these two kinds of steady state solutions will be used to determine the dynamics of the model

an elliptic system with bifurcation parameters on the boundary conditions in this paper we consider the elliptic system u a x upvq v b x urvs in a smooth bounded domain with boundary conditions u u v v on here and are regarded as parameters and p s 1 q r 0 verify p 1 s 1 qr we consider the case where a x 0 in and a x is allowed to vanish in an interior subdomain 0 while b x 0 in our main results include existence of nonnegative nontrivial solutions in the range 0 0 where 1 is characterized by means of an eigenvalue problem and the uniqueness of such solutions we also study their asymptotic behavior in all possible cases as both 0 as 1 for fixed respectively for fixed and when both in case 1

an optimal partition problem related to nonlinear eigenvalues we extend to the case of many competing densities the results of the paper ann inst h poincare 6 2002 more precisely we are concerned with an optimal partition problem in n dimensional domains related to the method of nonlinear eigenvalues introduced by nehari acta math 105 1961 we prove existence of the minimal partition and some extremality conditions moreover in the case of two dimensional domains we give an asymptotic formula near the multiple intersection points finally we show some connections between the variational problem and the behavior of competing species systems with large interaction

on the uniqueness of the positive steady state for lotka volterra models with diffusion abstract we give criteria for the uniqueness and stability of the componentwise positive steady state for the diffusive lotka volterra model of several competing species under dirichlet boundary conditions thereby extending known results for cases of only two species in the case of an underlying spatial domain which is one dimensional interval we obtain an estimate for a quantity occurring in the hypotheses of a number of uniqueness results the estimate sharpens those results and gives a partial negative answer to a conjecture of korman and leung

a diffusive competition model with a protection zone this paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor the weak species almost always survives but it cannot survive when the protection zone is below the critical size and its competitor is strong enough while this is the main feature of the model the actual dynamical behavior of the reaction diffusion system is more complicated the key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady state solutions our mathematical analysis shows that compared with the predator prey model discussed in yihong du junping shi a diffusive predator prey model with a protect zone j differential equations 226 2006 63 91 the protection zone has some essentially different effects on the fine dynamics of the competition model

on positive solutions of a nonlinear equilibrium boundary value problem abstract we give necessary and sufficient conditions for the existence of positive solutions to a nonlinear elliptic system arising from mathematical ecology with the dirichlet boundary data assigned by the equilibria of the corresponding ode system the main result is theorem 1 the tools are dancer fixed point index and degree theory

discrete infinite dimensional type k monotone dynamical systems and time periodic reaction diffusion systems abstract the asymptotic behavior of discrete type k monotone dynamical systems and reaction diffusion equations is investigated the studying content includes the index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one a nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits applications are made to time periodic lotka volterra systems with diffusion and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given for more general time periodic type k monotone reaction diffusion systems with spatial homogeneity a simple condition is given to guarantee the convergence of all positive solutions

optimal spectral theory of the linear boltzmann equation we give optimal compactness results in lp spaces 1 p related to spectral theory of general neutron transport equations on spatial domains with finite lebesgue measure

dynamics of a diffusive competition model in spatially heterogeneous environment abstract we discuss the effects of dispersal rate and spatial heterogeneity on population dynamics via a diffusive lotka volterra competition model under homogeneous dirichlet boundary conditions by the theory of monotone dynamical system we show that in the weak competition case for some ranges of dispersal rate of competing species 1 if dispersal rate of competing species 2 is sufficiently small there exists a unique coexistence steady state of the model which is globally asymptotically stable moreover for certain ranges of dispersal rate of competing species 1 we derive the limit behavior of the coexistence steady state as dispersal rate of competing species 2 tends to zero on the other hand we demonstrate that for some larger scopes of dispersal rate of competing species 1 if dispersal rate of competing species 2 is small enough there is no coexistence steady state whereas one of the semi trivial steady states of the model is globally asymptotically stable

numerical solutions for some coupled systems of nonlinear boundary value problems in the well known volterra lotka model concerning two competing species with diffusion the densities of the species are governed by a coupled system of reaction diffusion equations the aim of this paper is to present an iterative scheme for the steady state solutions of a finite difference system which corresponds to the coupled nonlinear boundary value problems this iterative scheme is based on the method of upper lower solutions which leads to two monotone sequences from some uncoupled linear systems it is shown that each of the two sequences converges to a nontrivial solution of the discrete equations the model under consideration may have one two or three nonzero solutions and each of these solutions can be computed by a suitable choice of initial iteration numerical results are given for these solutions under both the dirichlet boundary condition and the mixed type boundary condition

global higher bifurcations in coupled systems of nonlinear eigenvalue problems on etablit des conditions pour l existence d une bifurcation n aire globale de dimension elevee dans des systemes generaux de problemes aux valeurs propres non lineaires multiparametre qui conservent la structure de couplage des modeles de lotka volterra diffusifs etat stationnaire

instability of stationary solutions for a lotka volterra competition model with diffusion abstract we consider a lotka volterra competition model with diffusion onrwhich describes the dynamics of the population of two competing species and study the stability of positive stationary solutions of the model relative to the spacexof bounded uniformly continuous functions with the supremum norm in consideration of earlier results we shall arrive at the study of the distribution of the real eigenvalues of the linearized operator around the stationary solution and prove the existence of only one positive eigenvalue whose eigenfunction exponentially decays to 0 as this suggests that the instability of the stationary solution is not due to the spacex

a compactness result for perturbed semigroups and application to a transport model abstract in this paper we are concerned with the compactness properties of remainder terms of the dyson phillips expansion of perturbed semigroups on general banach spaces more specifically we derive conditions which ensure the compactness of the remainder term r n t for some integer n our result applies directly to discuss the time asymptotic behaviour for large times of the solution of a one dimensional transport equation with reentry boundary conditions on l 1 spaces without regularity conditions on the initial data

asymptotic behaviour of solutions of periodic competition diffusion system on considere le systeme d equations de reaction diffusion u t k 1 u a bu cv v t k 2 v v d eu fv ou k 1 k 2 a b f sont des fonctions suffisamment lisses definies sur un cylindre ou est un domaine borne lisse de r n

dynamics of lotka volterra competition reaction diffusion systems with degenerate diffusion abstract this paper is dealt with a class of lotka volterra competition reaction diffusion system and a competitor competitor mutualist system with density dependent diffusion in a bounded domain under the three basic types of dirichlet neumann and robin boundary conditions the governing equations for the competition system consist of an arbitrary number of degenerate quasilinear parabolic equations while the competitor competitor mutualist system involves three degenerate equations the goal of the paper is to show the existence of positive steady state solutions or quasi solutions the existence and uniqueness of a classical global time dependent solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions or quasi solutions the above goals are achieved under a very simple condition on the reaction rates of the reaction functions and these results yield a global attractor and the coexistence of the competing species in the case of neumann boundary condition the system has a unique constant positive steady state solution which is a global attractor of all the competing species the above conclusions lead to some interesting distinct dynamic behavior between degenerate quasilinear reaction diffusion systems and the corresponding semilinear reaction diffusion systems in which some or all of the competing species may be in extinction

asymptotic behavior of solutions of a periodic reaction diffusion system of a three species of competition model this paper concerns a three species periodic competition system which subjects to homogeneous neumann boundary conditions using a result obtained by ahmad and lazer we give some estimates and nonexistence results for the positive solutions of the system furthermore we investigate the asymptotic behavior of the solutions of the system definitely we refer to that one species dies out and the other two coexist as time t tends to infinity

asymptotic behavior of the principal eigenvalue for cooperative periodic parabolic systems and applications abstract the effects of spatial heterogeneity on population dynamics have been studied extensively however the effects of temporal periodicity on the dynamics of general periodic parabolic reaction diffusion systems remain largely unexplored as a first attempt to understand such effects we analyze the asymptotic behavior of the principal eigenvalue for linear cooperative periodic parabolic systems with small diffusion rates as an application we show that if a cooperative system of periodic ordinary differential equations has a unique positive periodic solution which is globally asymptotically stable then the corresponding reaction diffusion system with either the neumann or regular oblique derivative boundary condition also has a unique positive periodic solution which is globally asymptotically stable provided that the diffusion coefficients are sufficiently small the role of temporal periodicity spatial heterogeneity and their combined effects with diffusion will be studied in subsequent papers for further understanding and illustration

coexistence and stability of a competition diffusion system in population dynamics abstract the coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper where the effect of dispersion transportation is taken into consideration the mathematical problem involves a coupled system of lotka volterra type reaction diffusion equations together with some initial and boundary conditions including the dirichlet neumann and third type necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given for the stability problem general criteria for the stability and instability of a steady state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters also given are necessary and sufficient conditions for the existence of multiple steady state solutions and the stability or instability of each of these solutions special attention is given to the neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized it is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable the stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients