a heterogeneous diffusive logistic model of a single species population dynamics with predation and harvesting terms abstract we study existence and multiplicity of positive solutions of a heterogeneous diffusive logistic equation with predation and harvesting terms u a u b x u 2 c u 1 m u d h x in u 0 on where a c m and d are positive constants a bounded smooth domain in r n and b x is a nonnegative function on with 0 a region such that 0 and 0 x b x 0 under the strong growth rate assumption that is when a is greater than the first eigenvalue of in 0 with dirichlet boundary condition we show that the equation has at least one positive solution for 0 d d a for some d a 0 in addition in case c a we prove some multiplicity results when m m 0 with m 0 b x d x a

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

diffusive logistic equations with indefinite weights population models in disrupted environments the dynamics of a population inhabiting a strongly heterogeneous environment are modelledby diffusive logistic equations of the form u t d u m x cu u in 0 where u represents the population density c d 0 are constants describing the limiting effects of crowding and the diffusion rate of the population respectively and m x describes the local growth rate of the population if the environment is bounded and is surrounded by uninhabitable regions then u 0 on 0 the growth rate m x is positive on favourablehabitats and negative on unfavourable ones the object of the analysis is to determine how the spatial arrangement of favourable and unfavourable habitats affects the population being modelled the models are shown to possess a unique stable positive steady state implying persistence for the population provided l d where is the principle positive eigenvalue for the problem m x in 0 on analysis of how depends on m indicates that environments with favourable and unfavourable habitats closely intermingled are worse for the population than those containing large regions of uniformly favourable habitat in the limit as the diffusion rate d 0 the solutions tend toward the positive part of m x c and if m is discontinuous develop interior transition layers the analysis uses bifurcation and continuation methods the variational characterisation of eigenvalues upper and lower solution techniques and singular perturbation theory

on the positive solutions of semilinear equations δ ℎ 0 on the compact manifolds in this paper we study the existence nonexistence and uniqueness of positive solutions of semilinear equations u u hu p 0 on compact riemannian manifolds as well as on bounded smooth domains in r n with homogeneous dirichlet or neumann boundary conditions

effect of a protection zone in the diffusive leslie predator prey model abstract in this paper we consider the diffusive leslie predator prey model with large intrinsic predator growth rate and investigate the change of behavior of the model when a simple protection zone 0 for the prey is introduced as in earlier work y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 y du x liang a diffusive competition model with a protection zone j differential equations 244 2008 61 86 we show the existence of a critical patch size of the protection zone determined by the first dirichlet eigenvalue of the laplacian over 0 and the intrinsic growth rate of the prey so that there is fundamental change of the dynamical behavior of the model only when 0 is above the critical patch size however our research here reveals significant difference of the model s behavior from the predator prey model studied in y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 with the same kind of protection zone we show that the asymptotic profile of the population distribution of the leslie model is governed by a standard boundary blow up problem and classical or degenerate logistic equations

coexisting attractors in periodically modulated logistic maps we consider the logistic map wherein the nonlinearity parameter is periodically modulated for low periods there is multistability namely two or more distinct dynamical attractors coexist the case of period 2 is treated in detail and it is shown how an extension of the kneading theory for one dimensional maps can be applied in order to analyze the origin of bistability and to demarcate the principal regions of bistability in the phase space when the period of the modulation is increased char22 and here we choose periods which are the fibonacci numbers char22 the measure of multistable regions decreases the limit of quasiperiodic driving is approached in two different ways by increasing the period and keeping the drive dichotomous or by increasing the period and varying the modulation sinusoidally in the former case we find that multistability persists in small regions of the phase space while in the latter there is no evidence of multistability but strange nonchaotic attractors are created

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

exact multiplicity of positive solutions for a class of semilinear problem ii we consider the positive solutions to the semilinear problem u f u 0 in b n u 0 on b n where bn is the unit ball in rn n 1 and is a positive parameter it is well known that if is a smooth function then any positive solution to the equation is radially symmetric and all solutions can be parameterized by their maximum values we develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions u and the precise shape of the global bifurcation diagrams are rigorously proved our technique combines the bifurcation analysis stability analysis and topological methods we show that the shape of the bifurcation curve depends on the shape of the graph of function u u as well as the growth rale of

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

a modal reduction technique for the finite element formulation of biot s poroelasticity equations in acoustics recently finite element models based on displacement displacement and pressure displacement formulations of biot s poroelasticity equations have been extensively used to predict the acoustical and structural behavior of multilayer structures these models while accurate lead to large frequency dependent matrices for large finite element models and spectral analyses problems necessitating important setup time computer storage and solution time lately some authors have established criteria for low frequency approximations of viscous and thermal effects that allow for a linearization of the poroelastic eigenvalue problem complex modes for the damped system or real modes for the undamped system can then be found and used to solve the forced problem however this method is not general since it requires assumptions on the porous material physical properties to meet these criteria it is proposed in this talk to investigate another modal reduction technique the technique referred to as selective modal

realization of prescribed patterns in the competition model we demonstrate that for any prescribed set of finitely many disjoint closed subdomains d1 dm of a given spatial domain in rn if d1 d2 a1 a2 c d e are positive continuous functions on and b x is identically zero on d d1 dm and positive in the rest of then for suitable choices of the parameters and all small e 0 the competition model ut x t d1 x u x t a1 x u e 1b x 1 u2 c x uv vt x t d2 x v x t a2 x v e x v2 e 1d x uv x t 0 under natural boundary conditions on possesses an asymptotically stable positive steady state solution ue ve that has pattern d that is roughly speaking as e 0 ue converges to a positive function over d while it converges to 0 over the rest of on the other hand ve converges to 0 over d but converges to some positive function in the rest of in other words the two competing species ue and ve become spatially segregated as e 0 with ue concentrating on d and ve concentrating on d

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 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

effect of the protection zone on coexistence of the species for a ratio dependent predator prey model abstract this paper deals with the effect of the protection zone 0 on coexistence of the species for a ratio dependent predator prey model we obtain a critical value 1 n b x which is less than the well known critical value 1 d 0 obtained in the previous literatures furthermore we show that if 1 n b x then the prey persists regardless of the growth rates of the predator while if 1 n b x then there exists a real number such that the prey is ultimately extinct when as for 1 n b x and the curve of the positive steady state solutions of the model emanating from 0 c ends at a singular point 0 0 2 meantime by using the lyapunov schmidt reduction method we obtain a fine profile of its bifurcation diagrams and the uniqueness or multiplicity of its positive steady state solutions in addition as generally expected the chances of survival of the prey will increase with the size of the protection zone

diffusive logistic equation with constant yield harvesting i steady states we consider a reaction diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth we prove the existence uniqueness and stability of the maximal steady state solutions under certain conditions and we also classify all steady state solutions under more restricted conditions exact global bifurcation diagrams are obtained in the latter case our method is a combination of comparison arguments and bifurcation theory

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

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

hair triggered instability of radial steady states spread and extinction in semilinear heat equations we first study the initial value problem for a general semilinear heat equation we prove that every bounded nonconstant radial steady state is unstable if the spatial dimension is low n 10 or if the steady state is flat enough at infinity the solution of the heat equation either becomes unbounded as t approaches the lifespan or eventually stays above or below another bounded radial steady state depending on if the initial value is above or below the first steady state moreover the second steady state must be a constant if n 10 using this instability result we then prove that every nonconstant radial steady state of the generalized fisher equation is a hair trigger for two kinds of dynamical behavior extinction and spreading we also prove more criteria on initial values for these types of behavior similar results for a reaction diffusion system modeling an isothermal autocatalytic chemical reaction are also obtained

new exact multiplicity results with an application to a population model we obtain some new exact multiplicity results for the dirichlet boundary value problem on a unit ball b n in r n we consider several classes of nonlinearities f u including both positive and sign changing cases a crucial part of the proof is to establish positivity of solutions for the corresponding linearized problem as an application we obtain exact multiplicity results for the holling tanner population model

diffusive holling type ii predator prey system with harvesting of prey abstract we study diffusive heterogeneous predator prey system with holling type ii functional response in the presence of harvesting in prey equation i e u u a x u 2 b u v 1 m u c h x in v v v 2 d u v 1 m u in u v 0 on where is a smooth bounded region in r n and a x is a nonnegative continuous function on with 0 a region such that 0 and a x 0 on 0 we show that in the case of weak prey growth rate i e less than the first eigenvalue of in 0 the system has a positive solution for the values of in a bounded interval and c 0 sufficiently small we also investigate the multiplicity of positive solutions in this case when the prey growth rate is strong we will show that the system has a positive solution for large positive value of again provided that c is sufficiently small

cross diffusion induced instability and pattern formation for a holling type ii predator prey model this paper is devoted to studying a predator prey model with holling type ii functional response and cross diffusion subject to neumann boundary condition our main interest lies in the effects of cross diffusion on stability and stationary patterns more precisely the presented results show that cross diffusion can not only destabilize a uniform equilibrium which is stable for the kinetic and random diffusion reaction systems but also create spatial patterns even when the random diffusion fails to do so furthermore our results also reveal that in this kind of ecological system instability and stationary patterns can appear only when the predators rapidly move away from a large group of preys regardless of the speed that the preys keep away from the predators

a spatial sis model in advective heterogeneous environments abstract we study the effects of diffusion and advection for a susceptible infected susceptible epidemic reaction diffusion model in heterogeneous environments the definition of the basic reproduction number r 0 is given if r 0 1 the unique disease free equilibrium dfe is globally asymptotically stable asymptotic behaviors of r 0 for advection rate and mobility of the infected individuals denoted by d i are established and the existence of the endemic equilibrium when r 0 1 is studied the effects of diffusion and advection rates on the stability of the dfe are further investigated among other things we find that if the habitat is a low risk domain there may exist one critical value for the advection rate under which the dfe changes its stability at least twice as d i varies from zero to infinity while the dfe is unstable for any d i when the advection rate is larger than the critical value these results are in strong contrast with the case of no advection where the dfe changes its stability at most once as d i varies from zero to infinity

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

existence and the dynamical behaviors of the positive solutions for a ratio dependent predator prey system with the crowing term and the weak growth abstract this paper deals with a ratio dependent predator prey system with the crowing term and the weak growth in the prey equation under the condition that the coefficient is less than a critical value 1 d 0 we obtain existence of multiple positive steady state solutions of the predator prey system and the dynamical behaviors of its positive solutions our results show that the predator and the prey possess not only the common coexistence but also the very weak coexistence which both of the predator and the prey are very low meantime the persistence of the positive solutions for the corresponding parabolic type system sometime depends strictly on the ratio of its initial data therefore our results may be used to explain some special phenomena which under some bad environment the predator and the prey may still coexist

cross diffusion induced turing instability for a three species food chain model abstract in this paper we study a strongly coupled reaction diffusion system describing three interacting species in a food chain model where the third species preys on the second one and simultaneously the second species preys on the first one we first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ode system the positive equilibrium solution remains linearly stable for the reaction diffusion system without cross diffusion hence it does not belong to the classical turing instability scheme we further proved that the positive equilibrium solution is globally asymptotically stable for the reaction diffusion system without cross diffusion by constructing a lyapunov function but it becomes linearly unstable only when cross diffusion also plays a role in the reaction diffusion system hence the instability is driven solely from the effect of cross diffusion our results also exhibit some interesting combining effects of cross diffusion intra species competitions and inter species interactions

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

non existence of non constant positive steady states of two holling type ii predator prey systems strong interaction case abstract we prove the non existence of non constant positive steady state solutions of two reaction diffusion predator prey models with holling type ii functional response when the interaction between the predator and the prey is strong the result implies that the global bifurcating branches of steady state solutions are bounded loops

the role of protection zone on species spreading governed by a reaction diffusion model with strong allee effect abstract it is known that a species dies out in the long run for small initial data if its evolution obeys a reaction of bistable nonlinearity such a phenomenon which is termed as the strong allee effect is well supported by numerous evidence from ecosystems mainly due to the environmental pollution as well as unregulated harvesting and hunting to save an endangered species in this paper we introduce a protection zone that is governed by a fisher kpp nonlinearity and examine the dynamics of a reaction diffusion model with strong allee effect and protection zone we show the existence of two critical values 0 l l and prove that a vanishing transition spreading trichotomy result holds when the length of protection zone is smaller than l a transition spreading dichotomy result holds when the length of protection zone is between l and l only spreading happens when the length of protection zone is larger than l this suggests that the protection zone works when its length is larger than the critical value l furthermore we compare two types of protection zone with the same length a connected one and a separate one and our results reveal that the former is better for species spreading than the latter

a diffusive predator prey model with a protection zone in this paper we study the effects of a protection zone 0 for the prey on a diffusive predator prey model with holling type ii response and no flux boundary condition we show the existence of a critical patch size described by the principal eigenvalue d 0 of the laplacian operator over 0 with homogeneous dirichlet boundary conditions if the protection zone is over the critical patch size i e if d 0 is less than the prey growth rate then the dynamics of the model is fundamentally changed from the usual predator prey dynamics in such a case the prey population persists regardless of the growth rate of its predator and if the predator is strong then the two populations stabilize at a unique coexistence state if the protection zone is below the critical patch size then the dynamics of the model is qualitatively similar to the case without protection zone but the chances of survival of the prey species increase with the size of the protection zone as generally expected our mathematical approach is based on bifurcation theory topological degree theory the comparison principles for elliptic and parabolic equations and various elliptic estimates

effects of a degeneracy in the competition model part ii perturbation and dynamical behaviour this is the second part of our study on 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 in part i we mainly discussed the existence of two kinds of steady state solutions of this system namely the classical steady states and the generalized steady states here we use these solutions to determine the dynamics of the model we do this with the help of the perturbed model where b x is replaced by b x e which itself is a classical competition model this approach also reveals the interesting relationship between the steady state solutions both classical and generalized of the above system and that of the perturbed system

dynamics for a non autonomous predator prey system with generalist predator abstract in this paper we study complex dynamics of a non autonomous predator prey system with a holling type ii functional response and predator being generalist we first studied basic dynamics such as boundedness positive invariance permanence non persistence and globally asymptotic stability we also provide sufficient conditions for the existence uniqueness and globally asymptotic stability of positive periodic solutions and boundary periodic solutions of the proposed model when parameters are periodic we give the integral conditions to prove the extinction of prey and predator and globally asymptotic stability of boundary periodic solutions and we show that these conditions have more reasonable biological interpretation than those expressed by supremum and infimum of parameters in some literature we also perform numerical simulations to complement our analytical results and obtain more insights some of our numerical simulations indicate that periodic system may promote or suppress the permanence of its autonomous version with parameters being the averages of periodic parameters

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

asymptotic behavior of large solution for boundary blowup problems with non linear gradient terms let email protected r n n 3 be a bounded domain with smooth boundary we show the asymptotic behavior of boundary blowup solutions to non linear elliptic equation du u q b x f u in w subject to the singular boundary condition u x as dist x email protected 0 f is c varying at our analysis is based on the karamata regular variation theory combined with the method of lower and supper solution