positive solutions of a predator prey model with cross diffusion abstract in this paper we consider the positive solutions for a predator prey model with cross diffusion and holling type ii functional response in particular the existence of positive solutions can be established by the bifurcation theory moreover the uniqueness and the exact number of positive solutions is studied when the parameter m is large furthermore for large cross diffusion rate with the spatial dimension is less than 5 we can derive the corresponding limit systems and also study the asymptotic behavior of positive solutions finally some numerical simulations are presented to supplement the analysis results in one dimension case this results give us some important information on the structure of positive solutions

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

some uniqueness multiplicity and complete dynamics for a cooperative model abstract in this paper a two species cooperative model with diffusion and under homogeneous dirichlet boundary conditions is investigated it is shown the existence stability uniqueness and multiplicity of positive solutions in particular we study the global asymptotical stability of the unique positive solution when a 1 b d 1 d and is large our method of analysis is based on perturbation technique the lyapunov schmidt procedure and the bifurcation theory

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

on the number of positive solutions of weakly non linear elliptic equations when a parameter is large on etudie les solutions positives du probleme u f u dans u 0 sur ou est un domaine borne lisse de r n f r r est c 1 et est grand

positive solutions of a prey predator model with predator saturation and competition abstract in this paper we study the existence multiplicity bifurcation and stability of positive solutions to a prey predator model with predator saturation and competition u u a u b v f u v x v v c v d u f u v x u v 0 x where f u v 1 1 u 1 v and parameters are all positive constants and u and v are the densities of the prey and predator respectively

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

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

non constant positive steady states of a predator prey system with non monotonic functional response and diffusion this paper deals with non constant positive steady state solutions of a predator prey system with non monotonic functional response also called holling type iv interaction terms and diffusion under the homogeneous neumann boundary condition we first establish positive upper and lower bounds for such solutions and then study their non existence global existence and bifurcation

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

multiplicity and uniqueness of positive solutions for a predator prey model with b d functional response abstract in this paper we consider positive solutions of a predator prey model with beddington deangelis functional response first we establish the existence of positive solutions and present some sufficient and necessary conditions mainly using the standard fixed point index theory in cone second the multiplicity uniqueness and stability of positive solutions are obtained by means of bifurcation theory perturbation theory of eigenvalues and the fixed point index theory the parameters k and m play very important roles in deciding the number of positive solutions which are showed explicitly in this paper

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

stationary patterns of strongly coupled prey predator models abstract we study some elliptic systems arising from 3 component predator prey models where cross diffusions are included in such a way that predator chases the prey and the prey runs away from the predator we establish the existence and non existence of non constant positive solutions our results show that the cross diffusions can create the stationary patterns

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

non constant positive steady state of one resource and two consumers model with diffusion abstract in this paper a predator prey reaction diffusion system with one resource and two consumers is considered assume that one consumer species exhibits holling ii functional response while the other consumer species exhibits beddington deangelis functional response and they compete for the common resource first it is proved that the unique positive constant steady state is stable for the ode system and the reaction diffusion system second a prior estimates of positive steady state is given finally the non existence of non constant positive steady state the existence and bifurcation of non constant positive steady state are studied

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

the effect of predator competition on positive solutions for a predator prey model with diffusion abstract a diffusive predator prey model with predator competition is considered under dirichlet boundary conditions some existence and non existence results are firstly obtained then by investigating the bifurcation of positive solutions the multiplicity of positive solutions is established for suitably large m furthermore by meticulously analyzing the asymptotic behaviors of positive solutions when k goes to we find that there is at most a positive solution for any c r when k is sufficiently large at last some numerical simulations are presented to supplement the analytic results in one dimension

uniqueness and stability of a predator prey model with c m functional response in this paper we are concerned with positive solutions of a predator prey model with crowley martin functional response under dirichlet boundary conditions first we establish the existence of positive solutions when a c is close to 1 1 next the stability properties of nonnegative solutions are discussed by spectral analysis in addition we obtain a complete understanding of the uniqueness and non uniqueness of positive solutions by the lyapunov schmidt procedure and the perturbation technique at last some numerical simulations are presented to supplement the analytic results in one dimension

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 multiplicity and stability of positive solutions of a diffusive prey predator model abstract in the paper we consider the following diffusive prey predator model u u a u v 1 m u in u 0 on v v b m v m u in v 0 on we are mainly concerned with the positive solutions of the system in the case that the parameter m is large and obtain a complete understanding for the existence multiplicity and stability of positive solutions

stationary problem of a predator prey system with nonlinear diffusion effects in this paper we are concerned with the positive solution set of the following quasilinear elliptic system 1 v u u a u c v 1 m u x 1 1 u v v b v d u 1 m u x u v 0 x where is a bounded smooth domain in r n are nonnegative constants a c d m are positive constants b is a real constant this elliptic system is the stationary problem of a predator prey model in which u and v denote the population densities of the prey and predator respectively regarding b as the bifurcation parameter the global bifurcation structure of the positive solution set is shown then the nonlinear effect of either large or on the bifurcation point is deduced moreover as is large we show that the positive solution set is only of one type whereas as is large the positive solution set is of two types additionally in certain circumstances we also show that which one of the two types can characterize the positive solution set

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

asymptotic behavior and multiplicity for a diffusive leslie gower predator prey system with crowley martin functional response abstract a diffusive predator prey system with modified leslie gower and crowley martin functional response is considered the extinction and permanence of the time dependent system are determined by virtue of the comparison principle then the sufficient and necessary conditions for the existence of coexistence states are obtained furthermore the stability uniqueness and exact multiplicity of coexistence states are investigated by means of the combination of the perturbation theory bifurcation theory and degree theory our results indicate that c 1 have an effect on the stability and exact multiplicity of coexistence states finally some numerical simulations are presented to verify and complement the theoretical results

uniqueness and multiplicity of positive solutions for a diffusive predator prey model in the heterogeneous environment this is the second part of our study on the spatially heterogeneous predator prey model where the interaction is governed by a crowley martin type functional response in part i we have proved that when the predator competition is strong i e k is large the model has at most one positive steady state solution for any this part is denoted to study the effect of saturation our result shows that the large saturation coefficient i e large m can not only lead to the uniqueness of positive solutions but also lead to the multiplicity of positive solutions moreover the stability of the corresponding positive solutions is also completely obtained this work can be regarded as a supplement of ref 10

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

proof of a conjecture for the perturbed gelfand equation from combustion theory abstract in this paper we prove that when the domain b is a two dimensional ball for all small positive e the solution curve u of the perturbed gelfand equation u eu 1 eu with u b 0 is exactly s shaped this settles a long standing conjecture on the exact number of solutions to this equation

qualitative analysis on a diffusive predator prey model with toxins abstract this paper is concerned with the diffusive predator prey model with toxins subject to dirichlet boundary conditions the uniform persistence of positive solution is given under certain conditions in addition by the liapunov schmidt method the existence and stability of the bifurcation solution from a double eigenvalues are investigated moreover by the fixed point index theory and perturbation theory of eigenvalues the uniqueness stability and multiplicity of coexistence states are analyzed when some key parameter changes finally some numerical simulations are presented to verify the theoretical conclusions and further to reflect the importance of parameters to the number of coexistence states

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

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

on a class of representations of the yangian and moduli space of monopoles a new class of infinite dimensional representations of the yangians yopen image in new window and yopen image in new window corresponding to a complex semisimple algebra open image in new window and its borel subalgebra open image in new window is constructed it is based on the generalization of the drinfeld realization of open image in new window in terms of quantum minors to the case of an arbitrary semisimple lie algebra open image in new window the poisson geometry associated with the constructed representations is described in particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of g monopoles defined as the components of the space of based maps of 1 into the generalized flag manifold open image in new window thus the constructed representations of the yangian may be considered as a quantization of the moduli space of the monopoles

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

positive solutions of a diffusive predator prey model with modified leslie gower and holling type ii schemes abstract in this paper we investigate the existence multiplicity and stability of positive solutions to a prey predator model with modified leslie gower and holling type ii schemes p u u a 1 b u c 1 v u k 1 in v v a 2 c 2 v u k 2 in u 0 v 0 in u v 0 on where r n n 1 is a bounded domain with a smooth boundary the parameters a i b c i k i i 1 2 are positive numbers u and v are the respective populations of prey and predator here we say u v with u v 0 is a positive solution of problem p if u v is a solution of p and u v 0 in

a qualitative study on general gause type predator prey models with constant diffusion rates abstract in this paper we study the qualitative behavior of non constant positive solutions on a general gause type predator prey model with constant diffusion rates under homogeneous neumann boundary condition we show the existence and non existence of non constant positive steady state solutions by the effects of the induced diffusion rates in addition we investigate the asymptotic behavior of spatially inhomogeneous solutions local existence of periodic solutions and diffusion driven instability in some eigenmode

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

allee effect and bistability in a spatially heterogeneous predator prey model a spatially heterogeneous reaction diffusion system modelling predator prey interaction is studied where the interaction is governed by a holling type ii functional response existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior it is found that while the predator population is not far from a constant level the prey population could be extinguished persist or blow up depending on the initial population distributions the various parameters in the system and the heterogeneous environment in particular our results show that when the prey growth is strong the spatial heterogeneity of the environment can play a dominant role for the presence of the allee effect our mathematical analysis relies on bifurcation theory topological methods various comparison principles and elliptic estimates we combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations though the system itself does not keep an order structure as the competition system does among other things this allows us to obtain partial descriptions of the dynamical behavior of the system

non constant positive steady states of the sel kov model abstract this paper deals with the reaction diffusion system known as the sel kov model with the homogeneous neumann boundary condition this model has been applied to various problems in chemistry and biology we first give a priori estimates positive upper and lower bounds of positive steady states and then study the non existence bifurcation and global existence of non constant positive steady states as the parameters and are varied

stability and bifurcation of a predator prey model with disease in the prey and temporal spatial nonlocal effect in this paper we consider the dynamics of a predator prey model with disease in the prey and ratio dependent michaelis menten functional response the model is a reaction diffusion system with a nonlocal term representing the temporal spatial weighted average for the prey density the limiting case of the system reduces to the lotka volterra diffusive system with logistic growth of the prey we study the linear stability of the two non trivial steady states either with or without nonlocal term the bifurcations to three types of periodic solutions occurring from the coexistence steady state are investigated for two particular kernels which reveal the important significance of temporal spatial nonlocal effects