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

some uniqueness and exact multiplicity results for a predator prey model in this paper we consider positive solutions of a predator prey model with diffusion and under homogeneous dirichlet boundary conditions it turns out that a certain parameter m in this model plays a very important role a good understanding of the existence stability and number of positive solutions is gained when m is large in particular we obtain various results on the exact number of positive solutions our results for large m reveal interesting contrast with that for the well studied casem 0 i e the classical lotka volterra predator prey model

stability and hopf bifurcation of the stationary solutions to an epidemic model with cross diffusion in the previous paper cai and wang 2015 we investigated the stationary solutions of a cross diffusion epidemic model with vertical transmission in a spatially heterogeneous environment with neumann boundary condition and proved that the set of positive stationary solutions forms a bounded bifurcation branch which is monotone s or fish hook shaped with respect to the bifurcation parameter in the present paper we give some criteria on the stability of solutions on we prove that the stability of the solutions changes only at every turning point of while in a different case that a k and x are sufficiently large original stable positive stationary solutions at certain point may lose their stability and hopf bifurcation can occur these results are very different from those of the spatially homogeneous or without cross diffusion cases

counterexamples to some conjectures on the number of solutions of nonlinear equations on construit trois contre exemples resolvant par la negative certains problemes d analyse non lineaire

on positive solutions of some pairs of differential equations this paper is a sequel to the author s paper 6 which studied positive solutions of a pair of differential equations introduced by conway gardner and smoller 4 their equations were equations for population problems in biology their significance is discussed further in 141 the purpose of the present paper is to obtain further results for the conwat gardner smoller equations and to show how the methods in 6 can be modified to decide exactly when the classical predator prey system as in blat and brown 2 has a strictly positive solution we also show how the methods can be applied to a number of other problems in particular our results answer some questions in conway s secondly we show how the asymptotic methods developed in 6 can be used to study the existence and uniqueness of strictly positive solutions of competing species problems our results suggest that this is a much more complicated problem than the predator prey problem we do obtain a condition which is except for a few special cases necessary and sufficient for the existence of a strictly positive solution however the condition is complicated and rather implicit thirdly we prove the uniqueness of the strictly positive solution in the conway smoller system if d is small and n 1 that is the case of ordinary differential equations this partially answers a question in 4 our proof is a local result and can be used in other situations for example it can be used to show that in certain cases of asocial nonlinearities where our notation follows 4 sect 4 there are exactly two strictly positive solutions for small d we also show how our method applies to more general predator prey problems in section 1 we find necessary and sufficient conditions for the existence of strictly positive solutions of predator prey systems in section 2 we study competing species models by iteration and asymptotic methods finally in section 3 we prove our uniqueness results in an appendix we briefly discuss how the methods of section 3 apply to more general models

on meromorphic functions that share three values of finite weights abstract a uniqueness theorem for two distinct non constant meromorphic functions that share three values of finite weights is proved which generalizes two previous results by h x yi and x m li and h x yi as applications of it many known results by h x yi and p li etc could be improved furthermore with the concept of finite weight sharing extensions on osgood yang s conjecture and mues conjecture and a generalization of some prevenient results by m ozawa and h ueda ect could be obtained

s shaped global bifurcation curve and hopf bifurcation of positive solutions to a predator prey model where d is a bounded domain in r n 1 with smooth boundary d and a b c d m are positive constants problem 1 0 is known as the predator prey model with holling tanner type interactions where u and v represent the densities of the prey and predator respectively hence we are only interested in positive solutions by which we mean that both u and v are positive in d it is easy to see that a 1 is necessary for the existence of positive solutions to 1 0 and we shall also assume that d 1 the case d 1 is rather different and will not be discussed here we are mainly interested in studying the number of positive solutions of 1 0 and the stability of these solutions in particular we shall show that when b d c and m fall into certain range the solution set u v a of 1 0 forms a s shaped smooth curve and that hopf bifurcation occurs along this curve this not only confirms rigorously similar numerical observations on 1 0 made in 4 but also shows that the corresponding parabolic system

a general monotone scheme for elliptic systems with applications to ecological models we consider weakly coupled elliptic systems of the type with each f i being either an increasing or a decreasing function of each u j assuming the existence of coupled super and subsolutions we prove the existence of solutions and provide a constructive iteration scheme to approximate the solutions we then apply our results to study the steady states of two species interaction in the volterra lotka model with diffusion

on positive solutions of some pairs of differential equations ii on etudie l existence de solutions strictement positives de systemes predateur proie on demontre l unicite de la solution strictement positive dans le systeme de conway smoller

on the indices of fixed points of mappings in cones and applications abstract assume that k is a cone in a banach space and a k k is completely continuous we obtain a formula for the index in k of a fixed point of a under the assumption that a linearization exists and satisfies an invertibility condition we then use this formula to generalize some results of amann on the number of fixed points of a to the case where k has empty interior

multiple coexistence states for a prey predator system with cross diffusion abstract we study the multiple existence of positive solutions for the following strongly coupled elliptic system 1 v u u a u cv 0 in 1 u v v b du v 0 in u v 0 on where a b c d are positive constants and is a bounded domain in r n this is the steady state problem associated with a prey predator model with cross diffusion effects and u resp v denotes the population density of preys resp predators in particular the presence of represents the tendency of predators to move away from a large group of preys assuming that is small and that is large we show that the system admits a branch of positive solutions which is s or shaped with respect to a bifurcation parameter so that the system has two or three positive solutions for suitable range of parameters our method of analysis uses the idea developed by du lou j differential equations 144 1998 390 and is based on the bifurcation theory and the lyapunov schmidt procedure

global attractivity in a periodic chemostat with general uptake functions a system of ode s is considered as a model of two populations competing for a nutrient where nutrient inputs and chemostat washouts are periodic as well general nutrient uptake functions which may also be time dependent and periodic are considered criteria are derived for there to exist a globally attracting positive periodic solution and a theorem on global monotonicity of attracting solutions between different systems is proved

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

characterization and computation of the solution to the optimal l sub spl infin approximation problem the characterization of the solution to the problem of approximating a given stable proper rational transfer function in l sub spl infin norm sense by a rational function with prescribed number of stable and unstable modes is developed a simple state space suboptimal computational algorithm for the solution is presented

bifurcation of positive solutions for a three species food chain model with diffusion abstract in this paper we consider a reaction diffusion system describing a three species lotka volterra food chain model with homogeneous dirichlet boundary conditions by regarding the birth rate of prey r 1 as a bifurcation parameter the global bifurcation of positive steady state solutions from the semi trivial solution set is obtained via the bifurcation theory the results show that if the birth rate of mid level predator and top predator are located in the regions 0 r 2 1 a 23 u 3 r 3 and r 3 1 respectively then the three species can co exist provided the birth rate of prey exceeds a critical value moreover an explicit expression of coexistence steady state solutions is constructed by applying the implicit function theorem it is demonstrated that the explicit coexistence steady state solutions is locally asymptotically stable

uniqueness multiplicity and stability for positive solutions of a pair of reaction diffusion equations we study the number and stability of the positive solutions of a reaction diffusion equation pair when certain parameters in the equations are large the equation pair can be viewed as singular or regular perturbations of some single or essentially single equation problems for which the number and stability of their solutions can be well understood with the help of these simpler equations we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large in particular we obtain a fairly satisfactory description of the positive solution set of the equation pair

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

nontrivial solutions of predator prey systems with small diffusion on considere le systeme elliptique non lineaire u x au bu 2 cuv pour xed d v x ev fv 2 guv d region bornee de r n on etudie comment la famille des solutions non negatives de ce systeme change quand les parametres du systeme changent

the effect of mutual interference between predators on a predator prey model with diffusion abstract we consider a diffusive predator prey model with beddington deangelis functional response under homogeneous dirichlet boundary conditions the effect of large k which represents the extent of mutual interference between predators is extensively studied by making use of the fixed point index theory we obtain a complete understanding of the existence uniqueness and stability of positive steady states when k is sufficiently large moreover we present some numerical simulations that supplement the analytic results in one dimension

coexistence theorems of steady states for predator prey interacting systems on donne des conditions necessaires et suffisantes pour l existence des solutions positives des etats stationnaires pour des systemes predateur proie sous des conditions aux limites de dirichlet sur r n