the general properties of discrete time competitive dynamical systems abstract for strongly competitive discrete time dynamical systems on a strongly ordered topological vector space it is proved that any or limit set is unordered and lies on some invariant hypersurface with codimension one which generalizes hirsch s results for competitive autonomous systems of ordinary differential equations to competitive maps in a very general framework and implies the sarkovskii s theorem for planar strongly competitive and monotone maps we also show that the definition of competitiveness gives more restriction on homeomorphic property

bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii abstract in this paper we study a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 the goal of our study is to give the bifurcation diagram of the model for this we need to study saddle node bifurcations hopf bifurcation of codimension 1 and 2 heteroclinic bifurcation and nilpotent saddle bifurcation of codimension 2 and 3 the nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram the hopf bifurcation is studied by means of a generalized lienard system and for b 0 we discuss the potential integrability of the system the nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4 but because it occurs with an invariant line the effective highest codimension is 3 we develop normal forms in which the invariant line is preserved for studying of the nilpotent saddle bifurcation for b 0 the reversibility of the nilpotent saddle is discussed we study the type of the heteroclinic loop and its cyclicity the phase portraits of the bifurcations diagram partially conjectured via the results obtained allow us to give a biological interpretation of the behavior of the two species

dynamical behavior of lotka volterra competition systems non autonomous bistable case and the effect of telegraph noise this article is concerned with the study of trajectory behavior of lotka volterra competition bistable systems and systems with telegraph noises we proved that for bistable systems there exists a unique solution bounded above and below by positive constants the oscillatory situation of systems with telegraph noises is pointed out

profile of the unique limit cycle in a class of general predator prey systems many predator prey systems with oscillatory behavior possess a unique limit cycle which is globally asymptotically stable for a class of general predator prey system we show that the solution orbit of the limit cycle exhibits the temporal pattern of a relaxation oscillator when a certain parameter is small

global attractivity of positive periodic solution to periodic lotka volterra competition systems with pure delay abstract we consider a periodic lotka volterra competition system without instantaneous negative feedbacks i e pure delay systems x i t x i t r i t j 1 n a i j t x j t i j t i 1 2 n we establish some 3 2 type criteria for global attractivity of a positive periodic solution of the system which generalize the well known wright s 3 2 criteria for the autonomous delay logistic equation and thereby address the open problem proposed by both kuang y kuang global stability in delayed nonautonomous lotka volterra type systems without saturated equilibria differential integral equations 9 1996 557 567 and teng z teng nonautonomous lotka volterra systems with delays j differential equations 179 2002 538 561

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

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme abstract here we present well posedness results for first order stochastic differential inclusions more precisely for sweeping process with a stochastic perturbation these results are provided in combining both deterministic sweeping process theory recently developed in edmond and thibault 2005 2006 18 19 and methods concerning the reflection of a brownian motion lions and sznitman 1984 23 and saisho 1987 31 in addition we prove convergence results for an euler scheme discretizing these stochastic differential inclusions

hopf bifurcation of a predator prey system with stage structure and harvesting abstract in this paper a two species predator prey system with stage structure and harvesting is investigated the existence of hopf bifurcations of the system is given and the stability and directions of hopf bifurcations are determined by applying the normal form theory and the center manifold theorem

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

stochastic persistence and stationary distribution in a holling tanner type prey predator model abstract in this paper we study a stochastic predator prey model with beddington deangelis type functional response and logistic growth for predators the deterministic model is already well studied and we recall some important results here we construct the stochastic model from the deterministic model by introducing multiplicative noise terms into the growth equations of prey and predator populations for the stochastic model we show that the system admits unique positive global solution starting from the positive initial value then we prove that the system is strongly persistent in mean when the intensity of environmental forcing is less than some threshold magnitudes finally we show that the system has a stationary distribution under certain parametric restrictions numerical simulations are carried out to substantiate the analytical results

chaotic dynamics of a three species prey predator competition model with noise in ecology in this paper we investigate a three species ecosystem consisting of a prey a predator and a top predator the predator survives on the prey and the top predator survives on both the prey and the predator the mathematical model consists of non linear simultaneous differential equations all the eight equilibrium points of the model are identified and by using routh hurwitz criteria as well as lyapunov function we derive the criteria for local and global stabilities also the population intensities of fluctuations variances around the positive equilibrium due to noise are computed their stability is also analyzed with computer simulation using matlab some numerical simulations for justifying the theoretical analysis are also provided the main conclusions are supplied in conclusion

global analysis in a predator prey system with nonmonotonic functional response a predator prey system with nonmonotonic functional response is considered global qualitative and bifurcation analyses are combined to determine the global dynamics of the model the bifurcation a

bounded solutions and wavefronts for discrete dynamics this paper deals with the second order nonlinear difference equation dd r k dd u k q kg u k 1 0 where r k and q k are positive real sequences defined on n cup 0 and the nonlinearity g r to r is nonnegative and nontrivial sufficient and necessary conditions are given for the existence of bounded solutions starting from a fixed initial condition u 0 the same dynamic with f instead of g such that uf u 0 for u not 0 was recently extensively investigated on the contrary our nonlinearity g is of a small appearance in the discrete case its introduction is motivated by the analysis of wavefront profiles in biological and chemical models the paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f some applications in the study of wavefronts complete this work

on the stability of non autonomous perturbed lotka volterra models the paper is devoted to an extended lotka volterra system of differential equations of predator prey model the extension is proposed with perturbation terms which are null for the positive equilibrium state in the original lotka volterra system the equilibrium state is not asymptotically stable due to the fact that perturbations are periodic in time the aim of the paper is to characterize a form of perturbation terms guaranteeing the asymptotic stability or instability of equilibrium state the reason of the proposed model is that for large time scale the lotka volterra model is too simple to be realistic in the paper the non autonomous perturbations do not change the equilibrium state but introduce functions of time as well as for additional perturbed terms as for the main part of the equations modified from lotka volterra model theorems are proposed in a renormalized form of the differential equations for time and the two variables the key point of the paper comes from the use of a liapunov function introduced in section 2 which allows to obtain conditions for the asymptotic stability section 3 and instability section 4 by using a cetaiev instability theorem following conditions on the renormalized coefficients in time of system 6 an appendix recalls the main results of the liapunov direct method for non autonomous binary systems of ordinary differential equations

resistance scaling and the number of spanning trees in self similar lattices we consider the problem of enumerating spanning trees in self similar lattices motivated by recent work of chang chen and yang who determined explicit formulae in the case of sierpinski graphs and some of their generalizations the aim of this note is to show that their results hold in more generality and that there is a strong relation between this enumeration problem and resistance scaling on self similar lattices

topological degree in the generalized gause prey predator model abstract we consider a generalized gause prey predator model with t periodic continuous coefficients in the case where the poincare map p over time t is well defined the result of the paper can be explained as follows we locate a subset u of r 2 such that the topological degree d i p u equals to 1 the novelty of the paper is that the later is done under only continuity and some monotonicity assumptions for the coefficients of the model a suitable integral operator is used in place of the poincare map to cope with possible nonuniqueness of solutions the paper therefore provides a new framework for studying the generalized gause model with functional differential perturbations and multi valued ingredients

the spherical model on graphs we extend the lattice spherical model of berlin and kac to infinite graphs describing inhomogeneous structures such as fractals polymers and amorphous materials we analytically calculate the exact values of the critical exponents which turn out to depend only on the vibrational spectral dimension of the graph this functional dependence coincides with the analytic continuation in d of the corresponding exponents for the lattice model this result provides an example of geometrical universality classes for non translationally invariant systems and strongly suggests considering as the natural generalization of the euclidean dimension d for critical phenomena

competing subcommunities of mutualist and a generalized kamke theorem a model of a biological community of species is considered the community consists of two competing subcommunities each of which has the property that each pair of species of the subcommunity interact in a mutually beneficial or noninterfering manner sufficient conditions are given that the two subcommunities can coexist the coexistence state being a stable equilibrium the domains of attraction of stable equilibria are considered the sharpest results are for the special case of lotka volterra systems mathematically the fundamental tool is a generalized kamke theorem

dynamics of a class of nonautonomous semi ratio dependent predator prey systems with functional responses abstract in this paper we investigate the dynamics of a class of the so called semi ratio dependent predator prey interaction models with functional responses based on systems of nonautonomous differential equations with time dependent parameters the functional responses are classified into five types and typical examples of each type are provided then we establish sufficient criteria for the boundedness of solutions the permanence of system and the existence uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution some conclusive discussion is presented at the end of this paper

a non convex setup for multivalued differential equations driven by oblique subgradients abstract the paper proves the existence and uniqueness of the solution for the following multivalued deterministic variational inequality with oblique subgradients considered in a non convex domain x t h t x t x t g t x t t 0 x 0 x 0 where stands for the frechet subdifferential of the semiconvex function and the matrix application x h x is a lipschitz mapping the presence of the oblique reflection brought by the term h leads to the use of different techniques comparing to the cases of standard reflection in non convex domains or oblique reflection in convex domains the last section of the article is focused on the qualitative analysis of a non convex skorohod problem with generalized reflection and as applications stochastic variational inequalities driven by oblique frechet subgradients are addressed

stability analysis on a predator prey system with distributed delays abstract this paper concerns the local and global dynamical properties of the nonnegative and positive equilibria of a lotka volterra predator prey system with distributed delays it is shown that while the positive equilibrium does not exist the nonnegative equilibrium is globally asymptotically stable or globally attractive as long as the delays are small enough if the positive equilibrium exists it is shown that it is locally asymptotically stable when the delays are suitably small furthermore an explicit asymptotic stability region for the positive equilibrium is also obtained based on a liapunov functional

the concept of force in population dynamics abstract the area of population dynamics has a rich history of the development and analysis of models of biological and social phenomena using ordinary differential equations this paper describes a method for understanding the influence one variable exerts on another in such models as a force with the relative effects of these forces providing a narrative explanation of the curvature in variable behaviour using the stock flow form of a model a symbolic notation is developed that identifies the forces with the causal pathways of the model s feedback loops a force is measured by its impact defined as the ratio of acceleration to rate of change computed by differentiation along its associated pathway between variables different phases of force dominance are determined to enhance the standard stability analysis of the models providing an explanation of model behaviour in newtonian mechanical terms the concepts developed are applied to well known models from mathematical biology the spruce budworm model where force dominance identifies scenarios that give clarity to intervention points and the lotka volterra predator prey model where the analysis highlights the importance of dissipative forces in achieving stability conclusions are drawn on the explanatory power of this approach with suggestions made for future work

stability in a predator prey model with crowley martin function and stage structure for prey abstract this paper is concerned with a stage structured predator prey system with crowley martin functional response by analyzing the characteristic equation the local stability of the equilibria is investigated the persistence of system and global asymptotic stability of the positive equilibrium are carried out finally several numerical simulations to illustrate the theoretical analysis are also given

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

global existence of periodic solutions in a class of delayed gause type predator prey systems where x t y t denote the population density of prey and predator at time t respectively g e p e and z s are assumed to satisfy appropriate conditions v a positive constant stands for the death rate of predator y in the absence of prey x we may think of this system as herbivores y grazing upon vegetation x which take time r to recover in view of the fact that many predator prey systems display sustained fluctuations for examples see freedman l it is thus desirable to construct predator prey models capable of producing nonconstant periodic solutions when r 0 1 1 reduces to the well known undelayed gause type or kolmogorov predator prey systems again see freedman l and the references cited theorein under some assumptions the resulting systems can have a unique limit cycle in the positive cone see kuang and freedman 2 and the references cited therein to the best of our knowledge about delayed population interaction models the existing literature establishes the existence of a periodic solution by hopf bifurcation argument or reduces the delayed systems to undelayed systems of higher dimensions therefore these existence results are generally local most of the existing results on the global existence of nonconstant periodic solutions in delayed equations deal with scalar equations with single discrete delay such as 13 101

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

finite element approximation of a spatially structured metapopulation pde model abstract we present a new fully spatially structured pde metapopulation model for predator prey dynamics in d 3 space dimensions a nonlinear reaction diffusion system of rosenzweig macarthur form models predator prey dynamics in two high quality patches embedded in a low quality subdomain where species can diffuse convect and die our model substantially generalizes and improves earlier fully structured metapopulation models after a nondimensionalization procedure in order to approximate the metapopulation model we present a fully discrete galerkin finite element method in two space dimensions which is a generalization of the finite element method analyzed in a previous single patch predator prey model the numerical solutions are illustrated for some test cases using matlab numerical experiments demonstrate that the initial local extinction of predators in one patch leads to waves of recolonization from another patch in an appendix we also give an outline for the proof of the well posedness of the model

periodic solutions for a delayed predator prey system with stage structured predator on time scales in this paper a modified delayed predator prey system with stage structure for predator is proposed and studied which generalizes and incorporates as special cases some known models with the help of continuation theorem based on the gaines and mawhin coincidence degree theory we investigate the existence of periodic solutions for the proposed delayed predator prey system with stage structured predator and two kinds of functional responses called holling iii functional response and beddington deangelis functional response on time scales in particular when the time scale is chosen as the set of the real numbers or the integers the existence of the periodic solutions of the corresponding continuous time and discrete time models follows

bifurcations in persistence theory for models of population growth represented by systems of autonomous ordinary differential equations the question of whether persistence outcomes may be changed due to violation of one of the standard assumptions by means of a perturbation leading to bifurcations is investigated the results are illustrated by several examples

potts models on hierarchical lattices and renormalization group dynamics we prove that the generator of the renormalization group of potts models on hierarchical lattices can be represented by a rational map acting on a finite dimensional product of complex projective spaces in this framework we can also consider models with an applied external magnetic field and multiple spin interactions we use recent results regarding iteration of rational maps in several complex variables to show that for some class of hierarchical lattices lee yang and fisher zeros belong to the unstable set of the renormalization map

persistence of periodic patterns for perturbed biological oscillators abstract we apply the perturbation method and the implicit function theorem to study the persistence of periodic patterns for two perturbed general biological oscillators one weakly coupled ordinary differential equation and one reaction diffusion system the proof relies also on the formal adjoint equation theory and the lyapunov schmidt method for the perturbed reaction diffusion model spatial temporal patterns such as rotating waves are shown to exist numerical simulations are presented to illustrate our analytical results

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

convergence to equilibrium on invariant d hypersurfaces for strongly increasing discrete time semigroups abstract convergence to equilibrium for every positive semi orbit of strongly increasing discrete time semigroups on strongly ordered spaces is proved the domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered banach space which is continuously imbedded into a banach lattice the semigroup is assumed to be order compact and every positive semi orbit is assumed to be order bounded the crucial hypothesis is the ljapunov stability of all equilibria it is also proved that the set of equilibria is a simply ordered arc the key tools are invariant d hypersurfaces

convergence en loi des suites d integrales stochastiques sur l espace mathbb d 1 de skorokhod for sequences of stochastic integrals k n x n functional limit theorems are presented these results hold under simple natural conditions

complex temperature properties of the two dimensional ising model for nonzero magnetic field we study the complex temperature phase diagram of the square lattice ising model for nonzero external magnetic field h i e for 0 le mu le infty where mu e 2 beta h we also carry out a similar analysis for infty le mu le 0 the results for the interval 1 le mu le 1 provide a new way of continuously connecting the two known exact solutions of this model viz at mu 1 onsager yang and mu 1 lee and yang our methods include calculations of complex temperature zeros of the partition function and analysis of low temperature series expansions for real nonzero h the inner branch of a lima c c on bounding the fm phase breaks and forms two complex conjugate arcs we study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs for mu 0 there are two line segments of singularities on the negative and positive u axis and we carry out a similar study of the behavior at the inner endpoints of these arcs which constitute the nearest singularities to the origin in this case finally we also determine the exact complex temperature phase diagrams at mu 1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero field phase diagrams

dynamics of a general prey predator model with prey stage structure and diffusive effects in this paper we propose and study the dynamics of a diffusive prey predator model with general functional response and stage structure for the prey firstly we consider the asymptotical stability of equilibrium points and hopf bifurcation for the reduced ode system secondly the existence and uniform boundedness of global solutions and stability of equilibrium points for the corresponding reaction diffusion system are discussed finally we establish the existence and the nonexistence of nonconstant positive steady states of this reaction diffusion system which indicates the effect of large diffusivity our results shows the importance of the diffusion rate of the predator species i e d 3 the large diffusion rate of the predator alone will help the generation of patterns however a large diffusion rate of the immature prey species or a large diffusion rate of the mature prey species can lead to the nonexistence of spatial patterns

lee yang zeros and the ising model on the sierpinski gasket we study the distribution of the complex temperature zeros for the partition function of the ising model on a sierpinski gasket using an exact recursive relation although the zeros arrange on a curve pinching the real axis at t 0 in the thermodynamic limit their density vanishes asymptotically along the curve approaching the origin this phenomenon explains the coincidence of the low temperature regime on the sierpinski gasket and on the linear chain

existence of traveling wave solutions for gause type models of predator prey systems this paper deals with the existence of three types of traveling waves for a general predator prey systems of gause type traveling wave train solution point to point and point to periodic traveling wave solutions applying the methods of wazewski theorem lasalle s invariance principle and hopf bifurcation theorem we obtain the existence results also the minimal wave speed for biological invasion is obtained furthermore some applications are given to illustrate our results

probleme de skorohod multivoque on prouve un resultat d existence et d unicite pour une generalisation par introduction d un operateur maximal monotone multivoque du probleme de skorohod avec reflexion normale deterministe associe a un convexe ferme d de r d la formulation a partir des operateurs maximaux monotones multivoques permet de considerer des drifts singuliers explosant au bord du domaine cette approche multivoque clarifie la connexion entre probleme de skorohod et semigroupes nonlineaires en guise d application on considere ensuite le cas stochastique les equations differentielles stochastiques multivoques sont ainsi revisitees par consequent cette contribution fournit une nouvelle methode de construction de diffusions reflechies normalement possedant un coefficient de derive discontinu explosif

qualitative analysis of a stochastic ratio dependent predator prey system a stochastic ratio dependent predator prey model is investigated in this paper by the comparison theorem of stochastic equations and ito s formula we obtain the global existence of a positive unique solution of the ratio dependent model besides a condition for species to be extinct is given and a persistent condition is established we also conclude that both the prey population and the ratio dependent function are stable in time average in the end numerical simulations are carried out to confirm our findings

cadlag skorokhod problem driven by a maximal monotone operator abstract the article deals with the existence and uniqueness of the solution of the following differential equation a cadlag skorokhod problem driven by a maximal monotone operator and with singular input generated by the cadlag function m d x t a x t d t d k t d d m t t 0 x 0 m 0 where k d is a pure jump function the jumps outside of the constrained domain d a are counteracted through the generalized projection by taking x t x t m t whenever x t m t d a approximations of the solution based on discretization and yosida penalization are also considered

analysis of autonomous lotka volterra competition systems with random perturbation abstract this paper discusses a randomized n species lotka volterra competition system we show that this system is stable in time average under certain conditions furthermore there is a stationary distribution of this system if extra conditions are satisfied also we give the extinction condition of this system finally numerical simulations are carried out to support our results

dynamics of age structured predator prey population models a predator prey model is investigated in which the prey population is assumed to have age structure and is governed by the mckendrick von foerster partial differential equation and the predator population is described by the classical volterra lotka ordinary differential equation quite general hypotheses are assumed for the mortality function the fertility function and the functional responses of predation existence and stability of three biologically meaningful equilibria corresponding to extinction of both species persistence of one species prey and coexistence of two species are studied a particular example and some numerical results are given

uniqueness of the limit cycle for gause type predator prey systems abstract in this paper we derive some results to ensure that the number of the limit cycle of a generalized gause type predator prey system does not exceed one two examples are presented in the final section

potts models on hierarchical lattices and renormalization group dynamics ii examples and numerical results we obtain the exact renormalization map and plots of lee yang and fisher zeros distributions for potts models on a number of hierarchical lattices the diamond hierarchical lattice a lattice we call spider web the sierpinski gasket and cylinders such models are only examples among those we can study in the general framework of hierarchical lattices developed in a previous paper

neimark sacker bifurcations in a non standard numerical scheme for a class of positivity preserving odes we discuss the nature of neimark sacker bifurcations occurring in a non standard numerical scheme for a class of positivity preserving systems of ordinary differential equations odes which undergoes a corresponding hopf bifurcation extending previous work alexander moghadas 2005a electron j diff eqn conf 12 9 19 it is shown that the type of neimark sacker bifurcation supercritical or subcritical may be affected by the integration time step the general form of the scheme in the vicinity of a fixed point is given from which the expression for the first lyapunov coefficient for two dimensional systems valid for arbitrary time step is explicitly derived the analysis shows that this coefficient undergoes an o shift with respect to the corresponding coefficient of the original ode this could lead to a type of bifurcation which differs from the corresponding hopf bifurcation in the ode due to changes in the sign of the first lyapunov coefficient as varies this is especially problematic in the vicinity of certain types of degenerate hopf bifurcation at which this coefficient may vanish we also present a general method to eliminate the possible o shift in the bifurcation parameter of the scheme however the first lyapunov coefficient may still be subjected to an o shift leading to a possibly erroneous type of bifurcation examples are given to illustrate the theoretical results of the paper with applications to mathematical biology

bv solutions of nonconvex sweeping process differential inclusion with perturbation abstract this paper is devoted to the study of differential inclusions particularly discontinuous perturbed sweeping processes in the infinite dimensional setting on the one hand the sets involved are assumed to be prox regular and to have a variation given by a function which is of bounded variation and right continuous on the other hand the perturbation satisfies a linear growth condition with respect to a fixed compact subset finally the case where the sets move in an absolutely continuous way is recovered as a consequence

stability and bifurcation in a delayed predator prey system with beddington deangelis functional response we consider a delayed predator prey system with beddington deangelis functional response the stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation by choosing the delay as a bifurcation parameter we show that hopf bifurcation can occur as the delay crosses some critical values the direction and stability of the hopf bifurcation are investigated by following the procedure of deriving normal form given by faria and magalhaes an example is given and numerical simulations are performed to illustrate the obtained results

dynamics of a stochastic holling tanner predator prey model abstract in the paper we study a stochastic holling tanner predator prey system first by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions to the system after that sufficient causes for extinction of the predator population is obtained meanwhile we also obtain the threshold between persistence and extinction of the predator population at last some examples and numerical simulations are provided to illustrate our theoretical results

homoclinic bifurcation of prey predator model with impulsive state feedback control abstract in this paper homoclinic bifurcations of a prey predator system with state impulse are investigated firstly the existence of order 1 homoclinic cycle of system 1 2 with 0 is investigated secondly choosing q as a control parameter the sufficient conditions of existence and stability of order 1 periodic solution of system 1 2 with 0 are obtained by means of the geometry theory of semi continuous dynamic systems thirdly on the basis of the theory of rotated vector fields homoclinic bifurcation of system 1 2 about parameter are also studied finally some simulations are provided to prove the main results the methods used in this paper are intuitive to prove the existence of order 1 homoclinic cycle and homoclinic bifurcations

limit cycles for two families of cubic systems abstract in this paper we study the number of limit cycles of two families of cubic systems introduced in previous papers to model real phenomena the first one is motivated by a model of star formation histories in giant spiral galaxies and the second one comes from a model of volterra type to prove our results we develop a new criterion on the non existence of periodic orbits and we extend a well known criterion on the uniqueness of limit cycles due to kuang and freedman both results allow to reduce the problem to the control of the sign of certain functions that are treated by algebraic tools moreover in both cases we prove that when the limit cycles exist they are non algebraic

periodic solutions and permanence for a delayed nonautonomous ratio dependent predator prey model with holling type functional response by using the continuation theorem of coincidence degree theory the existence of positive periodic solutions for a delayed ratio dependent predator prey model with holling type iii functional response x t x t a t b t t k t s x s ds c t x2 t y t m2y2 t x2 t y t y t e t x2 t m2y2 t x2 t d t is established where a t b t c t e t and d t are all positive periodic continuous functions with period 0 m 0 and k s is a measurable function with period is a nonnegative constant the permanence of the system is also considered in particular if k s 0 s where 0 s is the dirac delta function at s 0 our results show that the permanence of the above system is equivalent to the existence of positive periodic solution

existence of a polycycle in non lipschitz gause type predator prey models abstract we study gause type predator prey models when the interaction between predator and prey is not locally lipschitz continuous in the absence of one of them we shall show that in this case there appears a polycycle which affects the existence of limit cycles for the system we apply the results to study the existence of limit cycles for a classical gause system

dynamical and numerical analyses of a generalized food chain model this paper focuses on the qualitative and quantitative analysis of a generalized three species food chain model the conditions for the existence and stability of the boundary and positive equilibria of the model are established by constructing an appropriate bifurcation function it is shown that the model undergoes a hopf bifurcation from the positive equilibrium for certain parameter values the root of this bifurcation function gives the bifurcation value a robust non standard numerical method is constructed and used to obtain the solution of the model numerical simulations using this method show that the hopf bifurcation is supercritical with a stable limit cycle in the positive octant

iterative scheme for the observation of a competitive lotka volterra system abstract in this work in terms of the model parameters sufficient conditions are established to construct a sequence of approximate observers for a two species competitive lotka volterra system this iterative approach makes it possible to localize the solution of the system and reveal its long term behaviour the main results are also illustrated by numerical simulations

singular perturbations of the holling i predator prey system with a focus abstract we study the occurrence of limit cyles in a model for two species predator prey interaction referred to as holling i this model is based on the phenomenological fitting of the so called functional response function to observed data in nature by a continuous piecewise differentiable function of the original models presented by holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet we extend the work of liu who showed that for certain values of parameters of the holling i system at least two nested limit cycles will occur surrounding a focus his case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities i e a singular perturbation problem with slow fast solutions we prove that exactly two hyperbolic limit cycles occur after perturbation

positive periodic solutions in a non selective harvesting predator prey model with multiple delays abstract this paper is concerned with a predator prey model with hassell varley type functional response non selective harvesting and multiple delays some new sufficient conditions are obtained for the existence of positive periodic solutions by applying the coincidence degree theorem in addition the results are delay dependent which is different from the previous work finally simulations illustrate the effectiveness of our results

existence of solutions to the nonconvex sweeping process abstract in this note we prove the existence of solutions for the sweeping process problem x t nc t x t a e x t c t x 0 x0 c 0 where c is an arbitrary hausdorff lipschitzean multifunction from i 0 t onto the set of nonempty closed subsets of r d this generalizes a well known result of j j moreau 1971 in sem d analyse convexe montpelier exp 15 in the convex case

stability and bifurcation analysis on a ratio dependent predator prey model with time delay a ratio dependent predator prey model with time delay due to the gestation of the predator is investigated by analyzing the corresponding characteristic equations the local stability of a positive equilibrium and a semi trivial boundary equilibrium is discussed respectively further it is proved that the system undergoes a hopf bifurcation at the positive equilibrium using the normal form theory and the center manifold reduction explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions by means of an iteration technique sufficient conditions are obtained for the global attractiveness of the positive equilibrium by comparison arguments the global stability of the semi trivial equilibrium is also addressed numerical simulations are carried out to illustrate the main results

the basins of attraction in a modified may holling tanner predator prey model with allee affect abstract i analyse a modified may holling tanner predator prey model considering an allee effect in the prey and alternative food sources for predator additionally the predation functional response or predation consumption rate is linear the extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation co existence and extinction of the predator prey population we also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle node hopf and bogdanov takens bifurcations we use simulations to illustrate the behaviour of the model

permanence and stability in non autonomous predator prey lotka volterra systems with feedback controls the main purpose of this article is considering whether or not the feedback controls have an influence on a non autonomous predator prey lotka volterra type system general criteria on permanence are established which is described by an integral form and independent of some feedback controls by constructing suitable lyapunov functionals a set of easily verifiable sufficient conditions are derived for the global stability of any positive solution to the model

existence and global stability of positive periodic solutions of a predator prey system with delays this paper studies the existence global stability and uniform persistence of positive periodic solutions of a periodic predator prey system with holling type iii functional response by using the continuation theorem of coincidence degree theory and liapunov functional some sufficient conditions are obtained

rich dynamics of a stochastic michaelis menten type ratio dependent predator prey system abstract stochastic predator prey systems with different functional responses have been studied but the dynamics of a michaelis menten type ratio dependent predator prey system with stochastic perturbation is not investigated systematically in this paper we give the asymptotic behavior of this system sufficient criteria for the existence of a stationary distribution and ergodicity are obtained which means the species are permanent besides we show the situations in which the species are non persistence finally examples and simulations are carried on to verify these results

dynamic analysis of fractional order singular holling type ii predator prey system abstract in this paper a fractional order singular fos predator prey model with holling type ii functional response has been introduced and the mathematical behavior of the model from the aspect of local stability is investigated through the fractional calculus and economic theory a new and more realistic predator prey model has been extended and the solvability condition is presented besides numerical simulations are considered to illustrate the effectiveness of the numerical method and confirm the theoretical results to explore the impacts of fractional order and economic interest on the presented system in biological context it is found that the presence of fractional order in the differential model can improve the stability of the solutions and enrich the dynamics of system in addition singular models exhibit more complicated dynamics rather than standard models especially the bifurcation phenomena which can reveal the instability mechanism of systems

permanence and periodicity of a delayed ratio dependent predator prey model with holling type functional response and stage structure a periodic and delayed ratio dependent predator prey system with holling type iii functional response and stage structure for both prey and predator is investigated it is assumed that immature predator and mature individuals of each species are divided by a fixed age and immature predator do not have the ability to attack prey sufficient conditions are derived for the permanence and existence of positive periodic solution of the model numerical simulations are presented to illustrate the feasibility of our main results

finite element approximation of spatially extended predator prey interactions with the holling type ii functional response we study the numerical approximation of the solutions of a class of nonlinear reaction diffusion systems modelling predator prey interactions where the local growth of prey is logistic and the predator displays the holling type ii functional response the fully discrete scheme results from a finite element discretisation in space with lumped mass and a semi implicit discretisation in time we establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions the class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics expressed in nondimensional form the theoretical techniques were adapted from a previous study of an idealised reaction diffusion system garvie and blowey in eur j appl math 16 5 621 646 2005

persistence in dynamical systems soit un espace metrique localement compact de metrique d et soit e un sous ensemble clos de a frontiere e et interieur e on suppose que l on a un flot continu 7b f defini sur e tel que e est invariant par 7b f on considere la notion de persistance et on donne un critere de persistance

sweeping process with regular and nonregular sets abstract differential inclusions involving the normal cone to a moving set are investigated a special attention is paid to the sweeping process associated with sets for which no regularity assumption is required

multiple periodic solutions for a non autonomous delayed predator prey model with harvesting terms abstract by using the generalized continuation theorem the existence of at least two positive periodic solutions for a non autonomous delayed periodic predator prey model with harvesting x t x t r t a t x t b t x t m x t y t h 1 t y t y t c t x t t m x t t d t h 2 t is established the method used here is more convenient than mawhin s continuation theorem and the main results show that the harvesting terms consult in the additional periodic solution