qualitative analysis for a diffusive predator prey model with hunting cooperative abstract this paper investigates a predator prey model that adds a cooperation term to the attack rate of the predator population for the non spatial model the existence and stability of non negative equilibrium points and bifurcations are studied by choosing cooperation coefficient as control parameter our analytical results show that hunting cooperation can be beneficial to the predator population however as the cooperation coefficient increases hunting cooperation can also destabilize the model and promote a sudden collapse of the predator population for the spatial model the stability of positive constant steady state solution hopf bifurcation and turing instability are discussed it is obtained that when the predation diffusion is not smaller than the prey diffusion the spatial model can reserve the stable stability of the positive constant steady state solution it is noted that the model without hunting cooperative does not generate turing instability while the model with hunting cooperative may generate turing instability

linear free resolutions and minimal multiplicity let s k x x be a polynomial ring over a field and let a4 a m be a finitely generated graded module in the most interesting case a4 is an ideal of s for a given natural number p there is a great interest in the question can m be generated by homogeneous elements of degree p no simple answer say in terms of the local cohomology of m is known but somewhat surprisingly the stronger question can the jth syzygy of m be generated by elements of degree qp j for all j 0 l n does admit a simple response the following is neither new nor difficult to prove though it seems not well known

multivariate elliptically contoured distributions for repeated measurements summary the multivariate power exponential distribution a member of the multivariate elliptically contoured family provides a useful generalization of the multivariate normal distribution for the modeling of repeated measurements both light and heavy tailed distributions are included the covariance matrix retains its interpretation so that it can easily be structured for serial dependence and several levels of variance components a crossover trial on insulin applied to rabbits with a series of repeated measurements within each period is analyzed by means of this distribution using autocorrelation and two levels of variance components

complex dynamics of a time periodic nonlocal and time delayed model of reaction diffusion equations for modeling cd4 t cells decline abstract pyroptosis an intensively inflammatory form of programmed cell death triggered during abortive hiv infection is associated with the release of inflammatory cytokines the inflammatory cytokines can attract more cd4 t cells to be infected based on the new biological perspective a time periodic reaction diffusion equation model with spatial heterogeneity and latent period is developed to investigate whether or not pyroptosis can explain cd4 t cells decline during hiv infection threshold dynamics is explored in terms of the basic reproduction number 0 it is shown that the infection free periodic solution is globally attractive if 0 1 while there exists positive infection periodic solution and the virus is uniformly persistent if 0 1 which might be a new finding for viral infection dynamical models theoretical analyses and simulations for the spatially homogeneous model demonstrate rich dynamics including the occurrence of turing instability which may result in hopf bifurcation and spatially inhomogeneous pattern formations it turns out that the new model issues some new challenges due to the enhancement infection term in the global stability problem and turing instability analysis for the high dimensional system our results reveal that the inflammatory cytokines can make the ctls level increase which is a new phenomenon not presented in the existing literature

the geometry of sdp exactness in quadratic optimization consider the problem of minimizing a quadratic objective subject to quadratic equations we study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation for the euclidean distance problem this is a bundle of spectrahedral shadows surrounding the given variety we characterize the algebraic boundary of this region and we derive a formula for its degree

dynamics in a diffusive predator prey system with ratio dependent predator influence abstract a diffusive predator prey system with predator functional response subject to neumann boundary conditions is considered existence nonexistence and boundedness of positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation hopf and steady state bifurcation analyses are carried out in detail these results provide theoretical evidences to the complex spatio temporal dynamics found by numerical simulation

towards characterization of all 3 3 extremal quasiconvex quadratic forms given a d d quasiconvex quadratic form d 3 we prove that if the determinant of its acoustic tensor is an irreducible extremal polynomial that is not identically zero then the form itself is an extremal quasiconvex quadratic form i e it loses its quasiconvexity whenever a convex quadratic form is subtracted from it in the special case d 3 we slightly weaken the condition namely we prove that if the determinant of the acoustic tensor of the quadratic form is an extremal polynomial that is not a perfect square then the form itself is an extremal quadratic form in the case d 3 we also prove that if the determinant of the acoustic tensor of the form is identically zero then the form is either extremal or polyconvex also if the determinant of the acoustic tensor of the form is a perfect square then the form is either extremal or polycovex or is a sum of a rank 1 form and an extremal whose acoustic tensor determinant is identically zero here we use the notion of extremality introduced by milton in 1990 2017 wiley periodicals inc

hopf bifurcation in spatially homogeneous and inhomogeneous autocatalysis models this paper is concerned with spatially homogeneous and inhomogeneous autocatalysis models with arbitrary order for the spatially homogeneous model the existence and stability of hopf bifurcation surrounding the interior equilibrium are considered for the model subject to zero flux boundary conditions turing instability of the interior equilibrium is firstly discussed and turing instability region regarding the diffusion coefficients is established then by the center manifold theory and normal form method the existence of hopf bifurcation and the stability of bifurcating periodic solutions are obtained which is much more difficult and complicated than dealing with the spatially homogeneous case finally to verify our analytical results some examples of numerical simulations are also included

bifurcation analysis in a diffusive segel jackson model abstract bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction diffusion system modeling predator prey interaction the existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns

modelling irregularly sampled profiles of non negative dog triglyceride responses under different distributional assumptions general methodology for modelling series of non negative data observed at unequally spaced times is developed the parameterization enables both the importance of the serial association as well the order of this dependence to be expressed an example is given where the effects of three fibre based diets on dog triglyceride profiles are analysed and compared many different types of models based on common distributions such as the normal exponential gamma weibull and log normal observations are presented comparison of possibly non nested models fitted on the same data set is made using the akaike criterion

steady states of a predator prey model with prey taxis abstract we study the steady states of a predator prey model with prey taxis incorporating holling type ii functional response under the homogeneous neumann boundary condition the stability of equilibrium points and the existence of non constant steady states are investigated we obtain that the prey tactic sensitivity coefficient delays the stability of the unique positive constant solution but for other equilibrium points stability the prey tactic sensitivity coefficient does not influence on it furthermore we derive some sufficient conditions relative to the prey tactic sensitivity coefficient which confines the existence of steady states and find that even if the interaction coefficient is sufficiently large there also exist non constant positive steady states under some conditions

bifurcation analysis of an enzyme catalyzed reaction diffusion system abstract in this paper we concern about the dynamics of a diffusive enzyme catalyzed system arising from glycolysis describing a biochemical reaction in which a substrate is converted into a product with positive feedback and into a branched sink the temporal and spatiotemporal dynamics of the system under homogeneous neumann boundary conditions are studied preliminary analysis on the local asymptotic stability and hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented for the reaction diffusion model firstly the parameter regions for the stability or instability of the unique constant steady state are discussed finally bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied numerical simulations are presented to verify and illustrate the theoretical results

a delayed diffusive predator prey system with predator cannibalism abstract a diffusive predator prey system with cannibalism and maturation delay in predator subject to neumann boundary condition is studied in this paper instability and hopf bifurcation induced by time delay are investigated by the theory of normal form and center manifold method the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived some numerical simulations are given to support our results

steady state bifurcation and hopf bifurcation for a diffusive leslie gower predator prey model we consider a diffusive leslie gower predator prey model subject to the homogeneous neumann boundary condition treating the diffusion coefficient d as a parameter the hopf bifurcation and steady state bifurcation from the positive constant solution branch are investigated moreover the global structure of the steady state bifurcations from simple eigenvalues is established by bifurcation theory in particular the local structure of the steady state bifurcations from double eigenvalues is also obtained by the techniques of space decomposition and implicit function theorem

spatiotemporal patterns in a reaction diffusion model with the degn harrison reaction scheme abstract spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades in this work we are concerned with a reaction diffusion model with the degn harrison reaction scheme which accounts for the qualitative feature of the respiratory process in a klebsiella aerogenes bacterial culture we study the global stability of the constant steady state existence and nonexistence of nonconstant steady states as well as the hopf and steady state bifurcations in particular our results show the existence of turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous these results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns

dynamics in a diffusive predator prey system with strong allee effect and ivlev type functional response abstract the dynamics of a kind of reaction diffusion predator prey system with strong allee effect in the prey population is considered we prove the existence and uniqueness of the solution and give a priori bound hopf bifurcation and steady state bifurcation are studied results show that the allee effect has significant impact on the dynamics

coexistence states of a holling type ii predator prey system abstract this paper characterize the existence of coexistence states to a reaction diffusion predator prey model with holling type ii functional response subject to dirichlet boundary conditions we find the necessary and sufficient conditions for existence of coexistence states by fixed point index theory and bifurcation theory

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

stability hopf bifurcations and spatial patterns in a delayed diffusive predator prey model with herd behavior in this paper we consider a delayed diffusive predator prey model with herd behavior firstly by choosing the appropriate bifurcation parameter the stability of the positive equilibria and the existence of hopf bifurcations induced by diffusion and delay respectively are investigated by analyzing the corresponding characteristic equation then applying the normal form theory and the center manifold argument for partial functional differential equations the formula determining the properties of the hopf bifurcation are obtained furthermore the instability of the hopf bifurcation leads to the emergence of spatial patterns finally some numerical simulations are also carried out to illustrate and expand the theoretical results

spatiotemporal dynamics induced by delay and diffusion in a predator prey model with mutual interference among the predator abstract in this work we study a reaction diffusion predator prey model with mutual interference among the predators while searching for food we prove that the model exhibits bistability which indicates that there are no patterns for our model when time delay is incorporated into the model multiple stability switches phenomenon of positive constant steady state emerged by taking delay as a bifurcation parameter the hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay the algorithm for determining the direction and the stability of the bifurcating periodic solutions is also derived the delay diffusion driven turing instability of the positive constant steady state is investigated our results show that delay and diffusion can create periodic oscillatory patterns of spatially homogeneous and inhomogeneous and turing patterns

a general family of distributions for longitudinal dependence with special reference to event histories event histories play an increasingly important role in medical studies examples include times between recurrences of tumours as with bladder cancer and between repeated infections as with chronic granulotomous disease a general method for generating new distributions is proposed by introducing an intensity function into a density this procedure yields as special cases several distributions already proposed in the literature the families of distributions based on the pareto distribution are of particular interest for event history analysis because of their relationship to the laplace transform of a gamma distribution they can yield multivariate distributions with longitudinal serial dependence by a procedure similar to updating in the kalman filter and with uniform dependence in a similar way to copulas for longitudinal dependence several such updating procedures are proposed

spatiotemporal pattern formation of a diffusive bimolecular model with autocatalysis and saturation law in this paper a reaction diffusion bimolecular model with autocatalysis and saturation law is investigated firstly we provide some conditions for the stability turing instability of the constant positive solution then we mainly consider hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive solution of the system our results suggest the existence of spatially non homogeneous periodic orbit and non constant positive steady state solutions which implies the possibility of rich spatiotemporal patterns in this diffusive bimolecular model numerical examples are presented to support our theoretical analysis furthermore non existence of non constant steady state is investigated in terms of parameters

dynamics of a diffusion reaction prey predator model with delay in prey effects of delay and spatial components abstract we study the complex dynamics of a monod haldane type predator prey interaction model that incorporates 1 a constant time delay in the prey growth and 2 diffusion in both prey and predator we provide the rigorous results of our system including the asymptotic stability of a positive equilibrium hopf bifurcation and the direction of hopf bifurcation and the stability of bifurcated periodic solutions we also perform numerical simulations on the effects of diffusion or and delay when the corresponding ode model has either a unique interior equilibrium with two interior attractors or two interior equilibria our theoretical and numerical results show that diffusion can either stabilize or destabilize the system large delay could destabilize the system and the combination of diffusion and delay could intensify the instability of the system moreover when the corresponding ode system has two interior equilibria diffusion or time delay in prey or both could lead to the extinction of predator our results may provide us useful biological insights on population managements for prey predator interaction systems

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

lyapunov coefficients for degenerate hopf bifurcations and an application in a model of competing populations abstract this paper focuses on the study of codimension one and two hopf bifurcations and the pertinent lyapunov stability coefficients for a general reaction diffusion system we display algebraic expressions for the first and second lyapunov coefficients for the infinite dimensional system subject to neumann boundary conditions as an application a special subset of a three dimensional lotka volterra dynamical system with diffusions subject to neumann boundary conditions is analyzed the main goal is to perform a detailed local stability analysis for the proposed predator prey model to show the existence of multiple spatially homogeneous and non homogeneous periodic orbits due to the occurrence of codimension one hopf bifurcation

formulation of the normal form of turing hopf bifurcation in partial functional differential equations abstract turing hopf bifurcation is considered as an important mechanism for generating complex spatio temporal patterns in dynamical systems in this work the normal form up to the third order for the hopf steady state bifurcation which includes the turing hopf bifurcation of a general system of partial functional differential equations pfdes in a bounded spatial region of any dimension is derived based on the center manifold and the normal form theories of pfdes the explicit formula for the coefficients in the normal form of hopf steady state bifurcation are presented concisely in a matrix form which makes it more convenient in not only symbolic derivation but also numerical implementation this provides an approach of showing the existence and stability of stationary and time periodic solutions with spatial heterogeneity the derived formulas are also applicable to partial differential equations and functional differential equations

bifurcation and spatiotemporal patterns in a homogeneous diffusive predator prey system abstract a diffusive predator prey system with holling type ii predator functional response subject to neumann boundary conditions is considered hopf and steady state bifurcation analysis are carried out in details in particular we show the existence of multiple spatially non homogeneous periodic orbits while the system parameters are all spatially homogeneous our results and global bifurcation theory also suggest the existence of loops of spatially non homogeneous periodic orbits and steady state solutions these results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation

qualitative analysis of a predator prey model with holling type ii functional response incorporating a prey refuge abstract we study a predator prey model with holling type ii functional response incorporating a prey refuge under homogeneous neumann boundary condition we show the existence and non existence of non constant positive steady state solutions depending on the constant m 0 1 which provides a condition for protecting 1 m u of prey u from predation moreover we investigate the asymptotic behavior of spacially inhomogeneous solutions and the local existence of periodic solutions

dynamics in a diffusive plant herbivore model with toxin determined functional response abstract a diffusive plant herbivore model with the toxin determined functional response and homogeneous neumann boundary condition is considered the stability and hopf bifurcation of the constant positive steady state are investigated by using the linearization method

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

hopf bifurcation and global stability of a diffusive gause type predator prey models abstract this paper mainly provides hopf bifurcation formulas for a general gause type predator prey system with diffusion and neumann boundary condition by using the center manifold theory and normal form method where the spectral and stability analysis around an equilibrium is addressed and our results can be applied to the case without diffusion as an application of these results we give a complete and rigorous analysis of the global dynamics of a diffusive predator prey model with herd behavior especially the hopf bifurcation and its direction and the stability of the bifurcating periodic solutions

an index for global hopf bifurcation in parabolic systems on considere le systeme non lineaire d t u a u f u 0 et on cherche des solutions periodiques en temps u t qui bifurquent a partir d une solution stationnaire

projective invariants of quadratic embeddings x sx s2x sx sk1 1x s1x p where k1 k1 x min k skx pk and sk 1x 6 skx for 1 k k1 for example if r a 1 b 1 1 pr is the projectivization of the linear space of a 1 b 1 matrices and x pa pb pr is the segre variety corresponding to the a 1 b 1 matrices of rank one then skx corresponds to the matrices whose rank does not exceed k 1 and k1 x min a b the number k1 and the dimensions of the secant varieties sk x dimsk x yield a natural collection of invariants of the projective variety x there are however two serious drawbacks

hopf bifurcation and steady state bifurcation for an autocatalysis reaction diffusion model abstract this paper is concerned with an autocatalysis model subject to no flux boundary conditions the existence of hopf bifurcation are firstly obtained then by the center manifold theory and the normal form method the direction of hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established on the other hand the steady state bifurcations from simple and double eigenvalues are intensively studied the techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues finally some numerical simulations are shown to verify the analytical results

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

dynamics of a diffusive predator prey system with a nonlinear growth rate for the predator abstract in this paper we investigate a diffusive predator prey system with holling type ii functional response and a nonlinear growth rate for the predator our results include the global attractivity of constant equilibria and non existence of non constant positive steady states these results give some ranges for the model parameters within which spatiotemporal pattern formation is impossible

global stability of prey taxis systems abstract in this paper we prove the global boundedness and stability of the predator prey system with prey taxis in a two dimensional bounded domain with neumann boundary conditions by deriving an entropy like equality and a boundedness criterion we show that the intrinsic interaction between predators and preys is sufficient to prevent the population overcrowding even the prey taxis is included and strong furthermore by constructing appropriate lyapunov functionals we show that prey only steady state is globally asymptotically stable if the predation is weak and the co existence steady state is globally asymptotically stable under some conditions like the prey taxis is weak or the prey diffuses fast if the predation is strong the convergence rates of solutions to the steady states are derived in the paper

global stability and hopf bifurcation of a three component model for cell production systems abstract the dynamics of a three component model for hierarchical cell production systems regulated by the level of mature cells is considered our model is based on a multi component model a collection of ordinary differential equations which rely on a mechanism of external signals feedback we analyze the existence and global stability of non negative equilibria we obtain the existence of hopf bifurcation and the direction and stability of bifurcating periodic orbits are calculated numerical simulations are included to show the rich spatiotemporal dynamics

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

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

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

positive solutions of a lotka volterra competition model with cross diffusion abstract this paper concerns a lotka volterra competition reaction diffusion system with nonlinear diffusion effects we first briefly discuss the stability of trivial solutions by spectrum analysis based on the boundedness of the solutions the existence of the positive steady state solution is also investigated by the monotone iteration method

analysis on bifurcation and stability of a generalized gray scott chemical reaction model abstract in numerous chemical reactions the autocatalytic reaction is a very important and characteristic catalytic reaction the gray scott model is an extremely representative autocatalytic reaction model among many catalytic chemical reaction models this paper deals with a generalized gray scott chemical reaction model firstly we give the a priori estimates of positive solutions secondly we discuss the conditions which ensure the model permitting a local bifurcating solution thirdly we investigate the stable property of the bifurcating solution by using a new idea finally the numerical simulations are carried out to illustrate the rich dynamical behaviors of the model including the existence and stability of bifurcating solution the result shows that the system exhibits abundant dynamical behaviors and the chemical reaction in reactor will be in balance in the end under certain conditions

dynamics and pattern formation in a diffusive predator prey system abstract this paper is devoted to study a general diffusive predator prey system with strong allee effect in the prey a priori estimate and nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation the hopf bifurcation and steady state bifurcation from the positive constant solution branch are investigated these results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulations

bifurcation from a degenerate simple eigenvalue it is proved that a symmetry breaking bifurcation occurs at a simple eigenvalue despite the usual transversality condition fails and this bifurcation from a degenerate simple eigenvalue result complements the classical one with the transversality condition the new result is applied to an imperfect pitchfork bifurcation in which a forward transcritical bifurcation changes to a backward one when the perturbation parameter changes several applications in ecological and genetics models are shown

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

optimality for a diffusive predator prey system in a spatially heterogeneous environment incorporating a prey refuge abstract in this paper we consider the optimal harvesting controls for a diffusive predator prey system which is formulated by prey refuge and modified leslie gower functional response the locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous neumann boundary condition the nonexistence and existence of nonconstant steady states are obtained by leray schauder degree theory also the optimal harvesting strategies to get maximum sustainable yield msy and maximum economic yield mey are correspondingly given

on global bifurcation for quasilinear elliptic systems on bounded domains general second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated into nonlinear mappings between sobolev spaces it is shown that the linearized mapping is a fredholm operator of index zero this and the abstract global bifurcation theorem of jacobo pejsachowicz patrick j rabier degree theory for c1 fredholm mappings of index 0 j anal math 76 1998 289 319 allow us to carry out bifurcation analysis directly on these elliptic systems at the abstract level we establish a unilateral global bifurcation result that is needed when studying positive solutions finally we supply two examples of cross diffusion population model and chemotaxis model to demonstrate how the theory can be applied

inequalities and ehrhart delta vectors for any lattice polytope p we consider an associated polynomial p t and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions as a consequence we improve upon known inequalities satisfied by the coefficients of the ehrhart vector of a lattice polytope we also provide combinatorial proofs of two results of stanley that were previously established using techniques from commutative algebra finally we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope

diffusion driven instability and hopf bifurcation in spatial predator prey model on a circular domain in this paper we investigate theoretically and numerically a 2 d spatio temporal dynamics of a predator prey mathematical model which incorporates the holling type ii and a modified leslie gower functional response and logistic growth of the prey this system is modeled by a reaction diffusion equations defined on a disc domain x y r 2 x 2 y 2 r 2 with dirichlet initial conditions and neumann boundary conditions we study the local and global stability of the positive equilibrium point we show that the diffusion can induce instability of the uniform equilibrium point which is stable with respect to a constant perturbation as shown by turing in 1950s and derive the conditions for hopf and turing bifurcation in the spatial domain numerical results are given in order to illustrate how biological processes affect spatiotemporal pattern formation in a spatial domain we perform the computations and generalize on a circular domain the results presented in camara and aziz alaoui 6

stability and bifurcation of a ratio dependent prey predator system with cross diffusion abstract this paper is purported to investigate a ratio dependent prey predator system with cross diffusion in a bounded domain under no flux boundary condition the asymptotical stabilities of nonnegative constant solutions are investigated to this system moreover without estimating the lower bounds of positive solutions the existence multiplicity of positive steady states are considered by using fixed points index theory bifurcation theory energy estimates and the differential method of implicit function and inverse function

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

complex dynamics of a diffusive predator prey model with strong allee effect and threshold harvesting abstract in this paper complex dynamics of a diffusive predator prey model is investigated where the prey is subject to strong allee effect and threshold harvesting the existence and stability of nonnegative constant steady state solutions are discussed the existence and nonexistence of nonconstant positive steady state solutions are analyzed to identify the ranges of parameters of pattern formation spatially homogeneous and nonhomogeneous hopf bifurcation and discontinuous hopf bifurcation are proved these results show that the introduction of strong allee effect and threshold harvesting increases the system spatiotemporal complexity finally numerical simulations are presented to validate the theoretical results

two parameter bifurcation study of the regularized long wave equation we perform a two parameter bifurcation study of the driven damped regularized long wave equation by varying the amplitude and phase of the driver increasing the amplitude of the driver brings the system to the regime of spatiotemporal chaos stc a chaotic state with a large number of degrees of freedom several global bifurcations are found including codimension two bifurcations and homoclinic bifurcations involving three tori and the manifolds of steady waves leading to the formation of chaotic saddles in the phase space we identify four distinct routes to stc they depend on the phase of the driver and involve boundary and interior crises intermittency the ruelle takens scenario the feigenbaum cascade an embedded saddle node homoclinic and other bifurcations this study elucidates some of the recently reported dynamical phenomena

a review of some extensions to generalized linear models although generalized linear models are reasonably well known they are not as widely used in medical statistics as might be appropriate with the exception of logistic log linear and some survival models at the same time the generalized linear modelling methodology is decidedly outdated in that more powerful methods involving wider classes of distributions non linear regression censoring and dependence among responses are required limitations of the generalized linear modelling approach include the need for the iterated weighted least squares iwls procedure for estimation and deviances for inferences these restrict the class of models that can be used and do not allow direct comparisons among models from different distributions powerful non linear optimization routines are now available and comparisons can more fruitfully be made using the complete likelihood function the link function is an artefact necessary for iwls to function with linear models but that disappears once the class is extended to truly non linear models restricting comparisons of responses under different treatments to differences in means can be extremely misleading if the shape of the distribution is changing this may involve changes in dispersion or of other shape related parameters such as the skewness in a stable distribution with the treatments or covariates any exact likelihood function defined as the probability of the observed data takes into account the fact that all observable data are interval censored thus directly encompassing the various types of censoring possible with duration type data in most situations this can now be as easily used as the traditional approximate likelihood based on densities finally methods are required for incorporating dependencies among responses in models including conditioning on previous history and on random effects one important procedure for constructing such likelihoods is based on kalman filtering

hopf bifurcation analysis of a reaction diffusion sel kov system abstract a reaction diffusion system known as the sel kov model subject to the homogeneous neumann boundary condition is investigated where detailed hopf bifurcation analysis is performed we not only show the existence of the spatially homogeneous non homogeneous periodic solutions of the system but also derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution

global existence of solutions and uniform persistence of a diffusive predator prey model with prey taxis this paper proves the global existence and boundedness of solutions to a general reaction diffusion predator prey system with prey taxis defined on a smooth bounded domain with no flux boundary condition the result holds for domains in arbitrary spatial dimension and small prey taxis sensitivity coefficient this paper also proves the existence of a global attractor and the uniform persistence of the system under some additional conditions applications to models from ecology and chemotaxis are discussed

global hopf bifurcation of a general predator prey system with diffusion and stage structures abstract in this paper the dynamics of a general predator prey system with diffusion and stage structures are investigated under some assumptions the stability of positive steady state and the existence of local hopf bifurcation are obtained by analyzing the distribution of eigenvalues moreover the global existence of positive periodic solution is established by using the global hopf bifurcation result of wu finally an example and some numerical simulations are carried out for illustrating the theoretical results

dynamics and pattern formation in a diffusive predator prey system with strong allee effect in prey abstract the dynamics of a reaction diffusion predator prey system with strong allee effect in the prey population is considered nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied these results show that the impact of the allee effect essentially increases the system spatiotemporal complexity

sums of squares and varieties of minimal degree the study of nonnegativity and its relation with sums of squares is a basic challenge in real algebraic geometry the classification of varieties of minimal degree is one of the milestones of classical complex algebraic geometry the goal of this paper is to establish the deep connection between these apparently separate topics to achieve this let x p be an embedded real projective variety with homogeneous coordinate ring r the variety x has minimal degree if it is nondegenerate not contained in a hyperplane and deg x 1 codim x a homogeneous element f r of even degree is nonnegative if its evaluation at each real point of x is at least zero our main theorem is a broad generalization of hilbert s 1888 classification of nonnegative forms and provides a tight connection between real and complex algebraic geometry

global spectral analysis for convection diffusion reaction equation in one and two dimensions effects of numerical anti diffusion and dispersion abstract convection diffusion reaction cdr equation plays a central role in many disciplines of engineering science and finances as a consequence importance of analysis of numerical methods for the accurate solution of cdr equation has motivated the present research we have used the global spectral analysis to characterize all the three important physical processes in terms of the non dimensional numerical parameters namely the non dimensional wavenumber kh courant friedrich lewy cfl number n c the peclet number pe and the damkohler number da for the purpose of illustration we have focused on two space time discretization schemes known for accuracy and robustness the basic properties relate to numerical issues arising for the numerical amplification factor numerical diffusion coefficient numerical phase speed and numerical group velocity with the help of model one dimensional 1d and two dimensional 2d cdr equations we have reported the numerical property charts for the cases i when all the processes of convection diffusion and reaction are of same order with critical numerical behaviour enforcing low values of da for the 1d cdr equation studied here ii the 2d cdr equation considered is diffusion reaction dominated and as a consequence this enforces da to be larger we have thoroughly analyzed these cases to identify the essential roles of anti diffusion on the critical n c and pe values which in turn decides admissible space and time steps to be used with the discretization schemes the property charts have been used to calibrate the analysis with two model equations one of which has an exact solution for a 1d cdr equation and the second case for the 2d cdr equation has numerical solution available in the literature these cases help to identify the importance of such analysis in explaining the utility of the choice one can exercise in fixing the numerical parameters this also identifies and explains some hitherto unknown numerical problems for cdr equation and their alleviation techniques

hopf bifurcation analysis in the 1 d lengyel epstein reaction diffusion model abstract the lengyel epstein model with diffusion and homogeneous neumann boundary condition is considered in this paper we give the existence of multiple spatially non homogeneous periodic solutions though all the parameters of the system are spatially homogeneous

conditional full stability of positivity preserving finite difference scheme for diffusion advection reaction models this work has been partially supported by the ministerio de economia y competitividad spanish grant mtm2017 89664 p

turing patterns in a reaction diffusion model with the degn harrison reaction scheme abstract in this paper we consider a reaction diffusion model with degn harrison reaction scheme some fundamental analytic properties of nonconstant positive solutions are first investigated we next study the stability of constant steady state solution to both ode and pde models our result also indicates that if either the size of the reactor or the effective diffusion rate is large enough then the system does not admit nonconstant positive solutions finally we establish the global structure of steady state bifurcations from simple eigenvalues by bifurcation theory and the local structure of the steady state bifurcations from double eigenvalues by the techniques of space decomposition and implicit function theorem

spatiotemporal dynamics in a diffusive ratio dependent predator prey model near a hopf turing bifurcation point abstract spatiotemporal dynamics in a ratio dependent predator prey model with diffusion is studied by analytical methods normal forms associated with codimension two hopf turing bifurcation are derived which can be used to understand and classify the spatiotemporal dynamics of the model for values of parameters close to the hopf turing bifurcation point in the vicinity of this degenerate point a wealth of complex spatiotemporal dynamics are observed our theoretical results are confirmed by numerical simulations

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

global bifurcation analysis and pattern formation in homogeneous diffusive predator prey systems abstract the dynamics of a general diffusive predator prey system is considered existence and nonexistence of non constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non constant steady state solutions are studied

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

imperfect transcritical and pitchfork bifurcations imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on banach spaces in particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined and two parameter unfoldings of singularities are rigorously established applications include semilinear elliptic equations imperfect euler buckling beam problem and perturbed diffusive logistic equation

bifurcation branch of stationary solutions in a general predator prey system with prey taxis abstract in this paper a general reaction diffusive predator prey system with prey taxis subject to the homogeneous neumann boundary condition is considered firstly we investigate the local stability of the unique positive equilibrium by analyzing the characteristic equation and study a priori estimates of positive solutions by the iterative technique and then choosing the prey tactic sensitivity coefficient as bifurcation parameter we proved that a branch of nonconstant solutions can bifurcate from the unique positive equilibrium when the prey tactic sensitivity is repulsive moreover we find the stable bifurcating solutions near the bifurcation point by the spectrum theory under some suitable conditions our results show that prey taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns

stability and bifurcation in a diffusive lotka volterra system with delay in this paper we investigate the dynamics of a class of diffusive lotka volterra equation with time delay subject to the homogeneous dirichlet boundary condition in a bounded domain the existence of spatially nonhomogeneous steady state solution is investigated by applying lyapunov schmidt reduction the stability and nonexistence of hopf bifurcation at the spatially nonhomogeneous steady state solution with the changes of a specific parameter are obtained by analyzing the distribution of the eigenvalues moreover we illustrate our general results by applications to models with a single delay and one dimensional spatial domain

positive steady states in an epidemic model with nonlinear incidence rate abstract in this paper we investigate a diffusive epidemic model with nonlinear incidence rate under homogeneous neumann boundary condition the value of this study lies in two aspects mathematically we show the stability of the constant positive steady state solution the existence and nonexistence the local and global structure of nonconstant positive steady states and epidemiologically we find that the system exhibits turing patterns controlled by diffusion

new examples of extremal positive linear maps abstract new families of nonnegative biquadratic forms that have 8 9 or 10 real zeros in p 2 p 2 are constructed these are the first examples with 8 9 or 10 real zeros it is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros our examples show that the upper bound is obtained such biquadratic forms define positive linear maps on real symmetric 3 3 matrices that are not completely positive our constructions are explicit and moreover we are able to determine which of the examples are extremal we extend the examples to positive maps on complex matrices and find families of extreme rays in the cone of positive maps

bifurcation solutions in the diffusive minimal sediment abstract this paper is concerned with the diffusive minimal sediment model with no flux boundary conditions the steady state bifurcation with two dimensional kernel is firstly obtained with respect to this model by the space decomposition and the implicit function theorem the existence of the hopf bifurcation is next attained and the direction and the stability of the hopf bifurcation are analyzed in detail numerical simulations are done to support and complement the theoretical results

a complete analysis of the global dynamics of a diffusive predator and toxic prey model considering many species can release toxic substances to protect themselves against predators a diffusive predator and toxic prey system with spatial heterogeneity in predator and prey populations has been investigated for this system we give a complete and rigorous analysis of the global dynamics with the boundedness globally asymptotical stability transcritical bifurcation hopf bifurcation and its direction and the stability of the bifurcating periodic solutions meanwhile we consider the effects of toxins produced by the prey on the dynamic behavior the consequence of the global stability of the coexistence equilibrium is that the toxin s intrinsic characteristic will not change the stability of the system irreversibly our results show that the toxin produced by the prey phytoplankton may be used as a bio control agent for the harmful algal bloom problems