strong allee effect in a diffusive predator prey system with a protection zone article history a reaction diffusion predator prey system with strong allee effect and a protection zone for the prey is considered dynamics and steady state solutions of the system are analyzed in particular it is shown that the overexploitation phenomenon can be avoided if the allee effect threshold is low and the protection zone is large

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

integral bilinear forms coxeter transformations and coxeter polynomials of finite posets abstract linear algebra technique in the study of linear representations of finite posets is developed in the paper a concept of a quadratic wandering on a class of posets i is introduced and finite posets i are studied by means of the four integral bilinear forms b i b i b i b i z i z i z 1 1 the associated coxeter transformations and the coxeter polynomials in connection with bilinear forms of dynkin diagrams extended dynkin diagrams and irreducible root systems are also studied bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of dynkin diagrams and extended dynkin diagrams are discussed a homological interpretation of the bilinear forms 1 1 is given and z bilinear equivalences between them are discussed by applying well known results of bongartz loupias and zavadskij shkabara we give several characterisations of posets i with the euler form q i x b i x x weakly positive resp with the reduced euler form q i x b i x x weakly positive and posets i with the tits form q i x b i x x weakly positive

numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

periodicity in bilinear lattices and the coxeter formalism abstract we introduce and study in detail so called circulant coxeter periodic elements and circulant families in a bilinear lattice k as well as their dual versions called anti circulant we show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the coxeter polynomial k of k and in consequence of coxeter polynomials of algebras of finite global dimension we discuss the properties of quadratic forms induced by circulant and anti circulant families moreover we interpret the results in the language of representation theory of algebras and point out applications facts concerning tubular families in auslander reiten quivers and quadratic forms of algebras abstract considerations in bilinear lattices are illustrated with a collection of non trivial examples arising from module and derived categories the results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts

effect of cross diffusion in the diffusion prey predator model with a protection zone ii abstract in the current paper we continue the mathematical analysis studied in li and wu 2017 15 and oeda 2011 22 and further study the effect of cross diffusion for the predator on the stationary problem the existence of positive solutions is first established by the bifurcation theory we next discuss the limiting behavior of positive solutions when the intrinsic growth rate of the predator species tends to infinity moreover as the prevention coefficient tends to infinity we obtain two shadow systems and give the complete limiting characterization of positive solutions

loops and branches of coexistence states in a lotka volterra competition model abstract a two species lotka volterra competition diffusion model with spatially inhomogeneous reaction terms is investigated the two species are assumed to be identical except for their interspecific competition coefficients viewing their common diffusion rate as a parameter we describe the bifurcation diagram of the steady states including stability in terms of two real functions of we also show that the bifurcation diagram can be rather complicated namely given any two positive integers l and b the interspecific competition coefficients can be chosen such that there exist at least l bifurcating branches of positive stable steady states which connect two semi trivial steady states of the same type they vanish at the same component and at least b other bifurcating branches of positive stable steady states that connect semi trivial steady states of different types

dynamics of cubic siegel polynomials we study the one dimensional parameter space of cubic polynomials in the complex plane which have a fixed siegel disk of rotation number where is a given irrational number of brjuno type the main result of this work is that when is of bounded type the boundary of the siegel disk is a quasicircle which contains one or both critical points of the cubic polynomial we also show that these boundaries vary continuously as one moves in the parameter space this is most nontrivial near the set of cubics with both critical points on the boundary of their siegel disk we prove that this locus is a jordan curve in the parameter space most of the techniques and results can be generalized to polynomials of higher degrees

positive steady states of the holling tanner prey predator model with diffusion this paper is concerned with the holling tanner prey predator model with diffusion subject to the homogeneous neumann boundary condition we obtain the existence and non existence of positive non constant steady states

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

bifurcation analysis of modified leslie gower predator prey model with michaelis menten type prey harvesting abstract in the present paper we discuss bifurcation analysis of a modified leslie gower prey predator model in the presence of nonlinear harvesting in prey we give a detailed mathematical analysis of the model to describe some significant results that may arise from the interaction of biological resources the model displays a complex dynamics in the prey predator plane the permanence stability and bifurcation saddle node bifurcation transcritical hopf andronov and bogdanov takens of this model are discussed we have analyzed the effect of prey harvesting and growth rate of predator on the proposed model by considering them as bifurcation parameters as they are important from the ecological point of view the local existence and stability of the limit cycle emerging through hopf bifurcation is given the emergence of homoclinic loops has been shown through simulation when the limit cycle arising though hopf bifurcation collides with a saddle point this work reflects that the feasible upper bound of the rate of harvesting for the coexistence of the species can be guaranteed numerical simulations using matlab are carried out to demonstrate the results obtained

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

analysis on a diffusive sis epidemic model with saturated incidence rate and linear source in a heterogeneous environment abstract in this paper we are concerned with a reaction diffusion sis epidemic model with saturated incidence rate and linear source in spatially heterogeneous environments we derive the uniform bounds of parabolic system and study the extinction and persistence of the infectious disease in terms of the basic reproduction number the existence of the endemic equilibrium ee is established when the basic reproduction number is greater than one we further investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium our study shows that the linear source can enhance persistence of infectious disease and large saturation can help speed up the elimination of disease

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

inflation algorithm for loop free non negative edge bipartite graphs of corank at least two we continue the study of finite connected loop free edge bipartite graphs with m 3m 3 vertices a class of signed graphs we started in simson 2013 48 and m g siorek et al 2016 19 by means of the non symmetric gram matrix g mm z of its symmetric gram matrix g 12 g g tr mm 12z and the gram quadratic form q zm zq zm z in particular we study connected non negative edge bipartite graphs with n r 3n r 3 vertices of corank r 2r 2 in the sense that the symmetric gram matrix g mn r z g mn r z of is positive semi definite of rank n 1n 1 the edge bipartite graphs of corank r 2r 2 are studied up to the weak gram zz congruence z z where z z means that g btr g bg btr g b for some b mn r z b mn r z such that det b 1det b 1 our main result of the paper asserts that given a connected edge bipartite graph with n r 3n r 3 vertices of corank r 2r 2 there exists a suitably chosen sequence t of the inflation operators of the form tab such that the composite operator t reduces to a connected bigraph d n r such that zd n r and d n r is one of the canonical r vertex extensions a n r n 1n 1 d n r n 4n 4 e 6 r e 7 r and e 8 r with n rn r vertices of the simply laced dynkin diagrams an dn e6 e7 e8an dn e6 e7 e8 with n 1n 1 vertices the algorithm constructs also a matrix b mn r z b mn r z with det b 1det b 1 defining the weak gram zz congruence zd n r that is satisfying the equation gd n r btr g b

periodicity and attractivity of a ratio dependent leslie system with impulses a ratio dependent leslie system with impulses is studied by using a comparison theorem continuation theorem base on coincidence degree and constructing a suitable lyapunov function we establish sufficient and necessary conditions for the existence and global attractivity of periodic solution examples show that the obtained criteria are easily verifiable

stationary patterns of a prey predator system with a protection zone and cross diffusion of fractional type abstract this paper is concerned with the stationary patterns of a prey predator model with a protection zone and fractional type cross diffusion for the prey it is shown that the fractional type cross diffusion has negative effects on the survival of the prey when the intrinsic growth rate of the predator is positive moreover our mathematical analysis shows that compared with the results obtained in k oeda 2011 and k oeda 2012 the large cross diffusion coefficient and large growth rate of the predator species have some essentially different effects on the profiles of the solutions

simulations of a fractional rate type nanofluid flow with non integer caputo time derivatives abstract unsteady developed flow of a rate type anomalous nanofluid with non integer caputo fractional derivatives is studied numerically in this article mixed convection and diffusion are taken into account while analyzing transport phenomena in the flow field thermophoresis and pedesis effects are also incorporated fluid is confined between nonisothermal parallel plates and flows by the moving lower plate variable concentration is assumed at both the plates in literature no such result exists with non integer caputo fractional derivatives boundary layer flow is modeled with the help of fractional calculus approach governing flow partial differential equations with appropriate conditions are solved by finite difference finite element scheme given scheme is flexible for the solution of non linear flow problems local nusselt and sherwood numbers are computed for the fractional model flow behavior is presented for various values of involved parameters influence of different dimensionless quantities on the nusselt and sherwood numbers is discussed by tabular results the acquired results revealed that fractional exponents have opposite effects on the velocity profiles it is also noted that thermophoresis and pedesis parameters have similar effects on heat flux while opposite effects are observed for mass flux at both the plates various stretching flows particularly in paper polymeric and food production processes can be modeled in a similar manner

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

uniqueness and global stability of positive stationary solution for a predator prey system abstract in this paper we are concerned about a predator prey model with beddington deangelis functional response in which a protection zone is set for the prey in a previous paper it has been shown that the coefficient k measuring the mutual interference between predators in the beddington deangelis functional response plays a crucial role in the predator prey system as k is large enough there exists a unique globally stable positive stationary solution except a special case by introducing an equivalent system this paper reveals that the uniqueness and global stability still holds true in the special case

numerical computations of fractional nonlinear hartmann flow with revised heat flux model abstract in this communication we have examined the magneto hydrodynamic flow of viscoelastic liquid with revise version of thermal flux thermal conductivity of the fluid is variable effects of body forces such as gravity are also examined in terms of convection non integer time derivatives are used for the better understanding of viscoelastic flow characteristics in literature these derivatives have been proved suitable for the predictions of hereditary systems such as viscoelastic flows heat transfer is analyzed in the flow field via modified fractional thermal gradient temperature gradient exists between the flow boundaries and increases with the passage of time flow is generated by the variable movement of flow boundary all the segments of numerical modeling are completely addressed in this paper i e physical system mathematical model simulation prediction validation and verification unsteady motion of in compressible fluid is directed by differential equations with fractional partial derivatives for the solutions of these equations numerical techniques are necessary to use governing equations with appropriate flow conditions have been discretized by finite difference along with finite element algorithms theoretical error appraisals are hypothesized and created numerical plan has been approved by doing error investigations numerically regarding spatial directions we have shown the estimate error and plotted the error curves in logarithmic scales of l 2 and h 1 norms by taking absolute approximations of the space discretization local nusselt number and coefficients of skin friction are determined for non integer viscoelastic model flow field is simulated for various values of physical parameters the obtained results showed that fractional exponent has opposite effects on the velocity temperature profiles for time t 2 it is also noted that temperature gradient increases by the increase of fractional number and thermal relaxation parameter viscoelastic flows in polymer and plastics industries can be addressed by similar technique

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

spatial pattern in an ecosystem of phytoplankton nutrient from remote sensing abstract in this paper we consider an ecosystem of phytoplankton nutrient which comes from pelagic zone of the ocean by using remote sensing we are concerned with the pattern formation of the model we first give the asymptotic behavior of positive solutions of parabolic dynamical system then we discuss the stability of constant positive solution and the boundedness of stationary solutions finally the non existence and existence results of the pattern are investigated by considering some parameters the result shows that under certain range of these parameters the pattern of the model occurs

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

convergence in competition models with small diffusion coefficients abstract it is well known that for reaction diffusion 2 species lotka volterra competition models with spatially independent reaction terms global stability of an equilibrium for the reaction system implies global stability for the reaction diffusion system this is not in general true for spatially inhomogeneous models we show here that for an important range of such models for small enough diffusion coefficients global convergence to an equilibrium holds for the reaction diffusion system if for each point in space the reaction system has a globally attracting hyperbolic equilibrium this work is planned as an initial step towards understanding the connection between the asymptotics of reaction diffusion systems with small diffusion coefficients and that of the corresponding reaction systems

elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation abstract we derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems this result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation in particular we characterize the bifurcation values from infinity to positive steady states the stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed

computation of isotopisms of algebras over finite fields by means of graph invariants in this paper we define a pair of faithful functors that map isomorphic and isotopic finite dimensional algebras over finite fields to isomorphic graphs these functors reduce the cost of computation that is usually required to determine whether two algebras are isomorphic in order to illustrate their efficiency we determine explicitly the classification of two and three dimensional partial quasigroup rings

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

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

all bounded type siegel disks of rational maps are quasi disks we prove that every bounded type siegel disk of a rational map must be a quasi disk with at least one critical point on its boundary this verifies douady sullivan s conjecture in the case of bounded type rotation numbers

varying total population enhances disease persistence qualitative analysis on a diffusive sis epidemic model abstract this paper performs qualitative analysis on an sis epidemic reaction diffusion system with a linear source in spatially heterogeneous environment the main feature of our model lies in that its total population number varies compared to its counterpart proposed by allen et al 2 the uniform bounds of solutions are derived based on which the threshold dynamics in terms of the basic reproduction number is established and the global stability of the unique endemic equilibrium is discussed when spatial environment is homogeneous in particular the asymptotic profile of endemic equilibria is determined if the diffusion rate of the susceptible or infected population is small or large the theoretical results show that a varying total population can enhance persistence of infectious disease and therefore the disease becomes more threatening and harder to control

renormalization group analysis of quasi periodicity in analytic maps analytic maps of the form f z e 2 pi i omega z mathcal o z 2 display quasiperiodicity when satisfies a diophantine condition quasiperiodic motion is confined to a neighborhood of the origin known as a siegel domain the boundary of this domain obeys universal scaling relations in this paper we investigate these scaling relations through a renormalization group analysis and we discuss singularities and asymptotic form of the scaling function

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

the parameter planes of λz m exp z for m 2 we consider the families of entire transcendental maps given by f m z zm exp z where m 2 all functions f m have a superattracting fixed point at z 0 and a critical point at z m in the parameter planes we focus on the capture zones i e values for which the critical point belongs to the basin of attraction of z 0 denoted by a 0 in particular we study the main capture zone parameter values for which the critical point lies in the immediate basin a 0 and prove that is bounded connected and simply connected all other capture zones are unbounded and simply connected for each parameter in the main capture zone a 0 consists of a single connected component with non locally connected boundary for all remaining values of a 0 is a quasidisk on a different approach we introduce some families of holomorphic maps of mathbb c which serve as a model for f m in the sense that they are related by means of quasiconformal surgery to f m

symbolic computation of strong gram congruences for cox regular positive edge bipartite graphs with loops abstract the paper can be viewed as a second part of the author s paper simson 2018 43 our main aim is to construct symbolic algorithms for the coxeter spectral classification of a class of signed graphs called edge bipartite graphs with n 2 vertices we started in simson 2013 37 and bocian et al 2014 6 more precisely we construct algorithms for the classification up to the strong gram z congruence z of all finite connected cox regular edge bipartite graphs with at least one loop bigraphs for short that are positive in the sense that the associated symmetric gram matrix g 1 2 g g t r m n 1 2 z is positive definite where g m n z is the non symmetric gram matrix of defined in section 1 and z means that there is a z invertible matrix b m n z such that g b t r g b we recall from 43 that every such a bigraph with a loop is strongly gram z congruent with a bigraph d that is one of the positive cox regular bigraphs b n n 2 c n n 3 f 4 m 4 g 2 presented in section 1 here by applying the geometry of mesh root system technique we construct symbolic algorithms that compute the correspondence d and the set gl n z d of all z invertible matrices b m n z defining the strong gram z congruence z d for any connected positive bigraph with n 2 vertices and at least one loop

asymptotic profile of the positive steady state for an sis epidemic reaction diffusion model effects of epidemic risk and population movement abstract identifying the epidemic risk for infectious disease is crucial in order to effectively perform control measures in a series of our work from an analytical aspect we study the effects of epidemic risk and population movement on the spatiotemporal transmission of infectious disease via an sis epidemic reaction diffusion model proposed by allen et al 2008 in 36 in allen et al 2008 36 peng 2009 37 it was assumed that the habitat of the populations consists of only the low and high risk areas the present paper concerns a more complicated heterogeneous environment where the moderate risk area occurs and deals with two cases i only the moderate and high risk areas exist ii the low moderate and high risk areas coexist in each case we rigorously determine the asymptotic profile of the positive steady state i e the endemic equilibrium as the migration rate of either the susceptible or infected population tends to zero our results show how epidemic risk and population movement affect the spatial distribution of infectious disease and thereby suggest important implications for predicting the patterns of disease occurrence and designing optimal control strategies numerical simulations are carried out to support the theoretical results

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

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

spatial heterogeneity of resources versus lotka volterra dynamics abstract the problem is motivated by a consideration of two phenotypes of a species in a strongly heterogenous spatial environment the phenotypes vary in their diffusion rates and interspecific interactions the aim is to investigate the relative strengths of the diffusion and reaction effects the problem is thus of competing species type but there are many questions which arise which are not covered by standard theory we investigate in particular the stability of the equilibria and the existence of coexistence solutions with emphasis on cases where the spatial variation of the environment becomes large we discuss briefly the implications of our results for the principle of competitive exclusions and for the question of the evolution of diffusion discussed in dockery et al j math biol 37 1998 61 83

p critical integral quadratic forms and positive unit forms an algorithmic approach abstract we introduce the class of p critical integral unit forms q z n z containing the critical forms in the sense of ovsienko 13 several characterisations of p critical forms are given in particular it is proved that q is p critical if and only if there is a uniquely determined extended dynkin diagram a n d n e 6 e 7 e 8 and a special group isomorphism t z n z n such that q t is the quadratic form q z n z n 0 of the diagram a correspondence between positive forms p z n 1 z with a sincere root and p critical forms q z n z is described and efficient linear algebra algorithms for computing p critical unit forms and positive forms are constructed

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

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

on siegel disks of a class of entire maps let f c c be an entire map of the form f z p z exp q z where p and q are polynomials of arbitrary degrees we allow the case q 0 building upon a method pioneered by m shishikura we show that if f has a siegel disk of bounded type rotation number centered at the origin then the boundary of this siegel disk is a quasicircle containing at least one critical point of f this unifies and generalizes several previously known results

dynamics and patterns of a diffusive leslie gower prey predator model with strong allee effect in prey abstract this paper is devoted to study the dynamical properties and stationary patterns of a diffusive leslie gower prey predator model with strong allee effect in the prey population we first analyze the nonnegative constant equilibrium solutions and their stabilities and then study the dynamical properties of time dependent solutions moreover we investigate the stationary patterns induced by diffusions turing pattern our results show that the impact of the strong allee effect essentially increases the system spatiotemporal complexity

effect of cross diffusion on the stationary problem of a prey predator model with a protection zone abstract this paper is concerned with the stationary problem of a prey predator cross diffusion system with a protection zone for the prey we discuss the existence and non existence of coexistence states of the two species by using the bifurcation theory as a result it is shown that the cross diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive we also study the asymptotic behavior of positive stationary solutions as the cross diffusion coefficient of the prey tends to infinity

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

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

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

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

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

quadratic like dynamics of cubic polynomials a small perturbation of a quadratic polynomial f with a non repelling fixed point gives a polynomial g with an attracting fixed point and a jordan curve julia set on which g acts like angle doubling however there are cubic polynomials with a non repelling fixed point for which no perturbation results into a polynomial with jordan curve julia set motivated by the study of the closure of the cubic principal hyperbolic domain we describe such polynomials in terms of their quadratic like restrictions

bounded type siegel disks of a family of sine families abstract we prove that for any positive integer n there is a polynomial p z with degree n such that the entire function g z p sin z has a bounded type siegel disk bounded by a quasi circle in the plane which passes through at least one critical point of g in addition let 0 1 be an irrational number of bounded type we prove that for any integer k 0 the boundary of the siegel disk of g z 0 sin z e 2 i 1 s 2 k d s centered at the origin is a quasi circle passing through exactly two critical points 2 and 2 with multiplicity k 1 respectively

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

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

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

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

julia sets in parameter spaces given a complex number of modulus 1 we show that the bifurcation locus of the one parameter family fb z z bz2 z3 b contains quasi conformal copies of the quadratic julia set j z z2 as a corollary we show that when the julia set j z z2 is not locally connected for example when z z z2 has a cremer point at 0 the bifurcation locus is not locally connected to our knowledge this is the first example of complex analytic parameter space of dimension 1 with connected but non locally connected bifurcation locus we also show that the set of complex numbers of modulus 1 for which at least one of the parameter rays has a non trivial accumulation set contains a dense g subset of s1

turing patterns of a strongly coupled predator prey system with diffusion effects abstract the main purpose of this work is to investigate the effects of cross diffusion in a strongly coupled predator prey system by a linear stability analysis we find the conditions which allow a homogeneous steady state stable for the kinetics to become unstable through a turing mechanism in particular it is shown that turing instability of the reaction diffusion system can disappear due to the presence of the cross diffusion which implies that the cross diffusion induced stability can be regarded as the cross stability of the corresponding reaction diffusion system furthermore we consider the existence and non existence results concerning non constant positive steady states patterns of the system we demonstrate that cross diffusion can create non constant positive steady state solutions these results exhibit interesting and very different roles of the cross diffusion in the formation and the disappearance of the turing instability

mating siegel quadratic polynomials 1 1 mating definitions and some history mating quadratic polynomials is a topological construction suggested by douady and hubbard do2 to partially parametrize quadratic rational maps of the riemann sphere by pairs of quadratic polynomials some results on matings of higher degree maps exist but we will not discuss them in this paper while there exist several presumably equivalent ways of describing the construction of mating the following approach is perhaps the most standard consider two monic quadratic polynomials fi and f2 whose filled julia sets k fi are locally connected for each fi let di denote the conformal isomorphism between the basin of infinity c k f and c ed with i 00 00 and v oo 1 these bottcher maps conijugate the polynomials to the squaring map

systems of subspaces of a unitary space abstract for a finite poset p p 1 p t we study systems u 1 u t u of subspaces of a unitary space u such that u i u j if p i p j two systems u 1 u t u and v 1 v t v are said to be isometric if there exists an isometry u v such that u i v i we classify such systems up to isometry if p is a semichain we prove that the problem of their classification is unitarily wild if p is not a semichain a classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space which is hopeless in a certain sense

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

on the dynamics of a family of generated renormalization transformations abstract we study the family of renormalization transformations of the generalized d dimensional diamond hierarchical potts model in statistical mechanic and prove that their julia sets and non escaping loci are always connected where d 2 in particular we prove that their julia sets can never be a sierpinski carpet if the parameter is real we show that the julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models moreover the asymptotic formula of the hausdorff dimension of the julia set is calculated as the parameter tends to infinity

asymptotic behaviour of positive steady states to a predator prey model to understand the heterogeneous spatial effect on predator prey models we study the behaviour of the positive steady states of a predator prey model as certain parameters are small or large we compare the case when the model has a spatial degeneracy with the case when it does not have such a degeneracy our results show that the effect of the degeneracy can be clearly observed in one limiting case but not in the others

fish hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross diffusion abstract in this paper we will consider the following strongly coupled cooperative system in a spatially heterogeneous environment with neumann boundary condition u u u b x v 0 x 1 k x u v v v d x u 0 x u v 0 x where is a bounded domain in r n n 1 with smooth boundary k is a positive constant and are real constants which may be non positive b x 0 and d x 0 are continuous functions in x is a smooth positive function in with x 0 is the outward unit normal vector on and for the case 0 we show that if is small and k is large a spatial segregation of x and b x can cause the positive solution curve to form an unbounded fish hook shaped curve with parameter for the case 0 if is small and k is large and the cooperative effect is strong for species u and very weak for species v then the positive solution set also forms an unbounded fish hook shaped continuum these results are quite different from those of predator prey systems and the cooperative system under dirichlet boundary condition both of which can form a bounded continuum our results deduce that the spatial heterogeneity of environments can produce multiple coexistence states our method of analysis is based on the bifurcation theory and the lyapunov schmidt procedure

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

on the periodicity of coxeter transformations and the non negativity of their euler forms abstract we show that for piecewise hereditary algebras the periodicity of the coxeter transformation implies the non negativity of the euler form contrary to previous assumptions the condition of piecewise heredity cannot be omitted even for triangular algebras as demonstrated by incidence algebras of posets we also give a simple direct proof that certain products of reflections defined for any square matrix a with 2 on its main diagonal and in particular the coxeter transformation corresponding to a generalized cartan matrix can be expressed as a 1 a t where a a are closely associated with the upper and lower triangular parts of a