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

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts abstract this paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts after making a central rescaling the problem reduces to a perturbation problem of a one parameter family of quadratic reversible systems as the parameter a 1 1 0 is rational except the case a 2 3 based on the chebyshev criterion for abelian integrals and a rationalizing transformation the problem could be solved theoretically to illustrate our approaches two particular cases corresponding to nilpotent codimension 3 saddle and elliptic case respectively are proved where the upper bound of the number of limit cycles is two

bifurcation analysis of a predator prey system involving group defence a class of odes of generalized gause type modeling predator prey interaction is considered the prey are assumed to exhibit a phenomenon called group defence that is predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase using the carrying capacity of the environment as the bifurcation parameter it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a hopf bifurcation conditions that hold even in the case of no group defence that ensure a subcritical hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair one stable and one unstable of periodic orbits are given ecological ramifications are considered unlike the classical model sufficient enrichment of the environment combined with group defence leads to extinction of the predator deterministically for almost all initial conditions providing strong support fo

a stochastic representation for backward incompressible navier stokes equations by reversing the time variable we derive a stochastic representation for backward incompressible navier stokes equations in terms of stochastic lagrangian paths which is similar to constantin and iyer s forward formulations in constantin and iyer comm pure appl math lxi 330 345 2008 using this representation a self contained proof of local existence of solutions in sobolev spaces are provided for incompressible navier stokes equations in the whole space in two dimensions or large viscosity an alternative proof to the global existence is also given moreover a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero

stochastic cascades and 3 dimensional navier stokes equations in this article we study the incompressible navier stokes equations in 3 the non linear integral equation satisfied by the fourier transform of the laplacian of the velocity field can be interpreted in terms of a branching process and a composition rule along the associated tree we derive from this representation new classes where global existence and uniqueness can be proven

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

bounding the number of zeros of certain abelian integrals abstract in this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n abelian integrals to have at most n k 1 zeros counted with multiplicities this condition involves the functions in the integrand of the abelian integrals and it can be checked in many cases in a purely algebraic way

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

bifurcation analysis in a host generalist parasitoid model with holling ii functional response abstract in this paper we study a host generalist parasitoid model with holling ii functional response where the generalist parasitoids are introduced to control the invasion of the hosts it is shown that the model can undergo a sequence of bifurcations including cusp focus and elliptic types degenerate bogdanov takens bifurcations of codimension three and a degenerate hopf bifurcation of codimension at most two as the parameters vary and the model exhibits rich dynamics such as the existence of multiple coexistent steady states multiple coexistent periodic orbits homoclinic orbits etc moreover there exists a critical value for the carrying capacity of generalist parasitoids such that i when the carrying capacity of the generalist parasitoids is smaller than the critical value the invading hosts can always persist despite of the predation by the generalist parasitoids i e the generalist parasitoids cannot control the invasion of hosts ii when the carrying capacity of the generalist parasitoids is larger than the critical value the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations i e whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations iii in both cases the generalist parasitoids always persist numerical simulations are presented to illustrate the theoretical results

on the existence of global analytic conjugations for polynomial mappings of yagzhev type abstract consider a mappingf c n c nof the form identity plus a term with polynomial components that are homogeneous of the third degree and suppose that the jacobian determinant of this mapping is constant throughout c n polynomial mapping of yagzhev type as a stronger version of the classical jacobian conjecture the question has been posed whether for some values of c 0 there exists a global change of variables conjugation on c nthrough which the mapping fbecomes its linear part at the origin van den essen has recently produced a simple yagzhev mapping for which no suchpolynomialconjugation exists we show here that van den essen s example still admitsglobal analyticconjugations the question on the existence of global conjugations for general yagzhev maps is then still open

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

a stochastic lagrangian representation of the three dimensional incompressible navier stokes equations in this paper we derive a probabilistic representation of the deterministic three dimensional navier stokes equations based on stochastic lagrangian paths the particle trajectories obey sdes driven by a uniform wiener process the inviscid weber formula for the euler equations of ideal fluids is used to recover the velocity field this method admits a self contained proof of local existence for the nonlinear stochastic system and can be extended to formulate stochastic representations of related hydrodynamic type equations including viscous burgers equations and lagrangian averaged navier stokes alpha models 2007 wiley periodicals inc

global dynamics of a predator prey model abstract this paper deals with the dynamics of a predator prey model with hassell varley holling functional response first we show that the predator coexists with prey if and only if predator s growth ability is greater than its death rate second using a blow up technique we prove that the origin equilibrium point is repelling and extinction of both predator and prey populations is impossible third the local and global stability of the positive steady state coincide when the predator interference is large finally for a typical biological case we show instability of the positive equilibrium implies global stability of the limit cycle numerical simulations are carried out for a hypothetical set of parameter values to substantiate our analytical findings

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

multiple bifurcation in a predator prey system with nonmonotonic predator response a model of a predator prey system showing group defence on the part of the prey is formulated and reduced to a three parameter family of quartic polynomial systems of equations mathematically this system contains the volterra lotka system and yields numerous kinds of bifurcation phenomena including a codimension two singularity of cusp type in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation biologically the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant so that the system exhibits an enrichment paradox and for certain choices of parameters coexistence of stable oscillation and a stable equilibrium

a forward backward probabilistic algorithm for the incompressible navier stokes equations abstract a novel probabilistic scheme for solving the incompressible navier stokes equations is studied in which we approximate a generalized nonlinear feyman kac formula the velocity field is interpreted as the mean value of a stochastic process ruled by forward backward stochastic differential equations fbsdes driven by brownian motion following an approach by delbaen qiu and tang introduced in 2015 the pressure term is obtained from the velocity by solving a poisson problem as computing the expectation of an integral functional associated to an extra bsde the fbsdes components are numerically solved by using a forward backward algorithm based on euler type schemes for the local time integration and the quantization of the increments of brownian motion following the algorithm proposed by delarue and menozzi in 2006 numerical results are reported on spatially periodic analytic solutions of the navier stokes equations for incompressible fluids we illustrate the proposed algorithm on a two dimensional taylor green vortex and three dimensional beltrami flows

new family of abelian integrals satisfying chebyshev property abstract this paper is motivated by the chebyshev property of a family of abelian integrals along non algebraic level ovals a new criterion is established to study the chebyshev property of these abelian integrals as its applications we give three examples to show that the ordered sets defined by abelian integrals are ect systems moreover the number of the abelian integrals can be an arbitrary positive integer

a probabilistic approach to the two dimensional navier stokes equations we turn the navier stokes equations for a 2 dimensional viscous incompressible fluid into a system of functional integrals in the trajectory space of a suitable diffusion process using probabilistic techniques as girsanov s transformation and bismut elworthy formula we prove the existence of a unique global solution of this system in a constructive way

dynamical behavior of an epidemic model with a nonlinear incidence rate abstract in this paper we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action by carrying out global qualitative and bifurcation analyses it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region when such a region exists it is shown that the model undergoes a bogdanov takens bifurcation i e it exhibits a saddle node bifurcation hopf bifurcations and a homoclinic bifurcation existence of none one or two limit cycles is also discussed

explicit transversality conditions and local bifurcation diagrams for bogdanov takens bifurcation on center manifolds abstract this application oriented study is concerned with the derivation of parameter dependent normal forms for the codimension two bogdanov takens bifurcation of n dimensional m parameterized systems on the basis of the homological method in the case of an enduring equilibrium simple formulas are obtained for the transformation of parameters enabling the formulation of explicit transversality conditions and bifurcation diagrams to at most the second order moreover in z 2 symmetric systems the calculation can be further limited within certain subspaces in the general case existing results are re derived and a revision necessary for determining the bifurcation diagrams to the second order is then provided these results facilitate the derivation of normal forms check of transversality and depiction of bifurcation diagrams for the bogdanov takens bifurcation

a local solution to the navier stokes equations on manifolds via stochastic representation abstract we obtain a c k local existence result for the navier stokes equations on compact riemannian manifolds using the representation formula for the solution obtained by fang and luo

flow decomposition and large deviations we study large deviations properties related to the behavior asegoes to 0 of diffusion processes generated bye2l1 l2 wherel1andl2are two second order differential operators extending recent results of doss and stroock and rabeherimanana the main tool is the decomposition theorem for flows of stochastic differential equations proved by bismut and kunita we give another application of flow decomposition in a nonlinear filtering problem

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

a system with three limit cycles appearing in a hopf bifurcation and dying in a homoclinic bifurcation the cusp of order 4 abstract in this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a hopf bifurcation h of order 3 and vanishing simultaneously in a homoclinic loop bifurcation hl of order 3 the region with three limit cycles is a topological 3 simplex the system is a generalization of bogdanov s system at the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4 this bifurcation diagram is a cone it contains a cone on the bifurcation diagram with the three limit cycles inside a 3 simplex region plus saddle node and cusp bifurcations of lower order

two examples on the property of the noise in the systems of equations of fluid mechanics abstract in this manuscript we discuss two examples of the results on the systems of partial differential equations in fluid mechanics obtained via stochastic analysis for which deterministic case has no counterpart firstly it is well known that for a deterministic system of equations in fluid mechanics a lack of diffusion creates a significant obstacle in proving its global well posedness for a sufficiently small initial data nevertheless it may be proven that for the hall magnetohydrodynamics system forced by l e vy noise such a result may be obtained even with zero viscous diffusion secondly we may extend the classical kelvin s conservation of circulation flow as well as the cauchy s vorticity transport formula on the euler equations to show that the solution to the damped euler equations enjoys the exact same identities except with an exponential decay in time such identities are known to completely fail once we add diffusion and consider the damped navier stokes equations nevertheless discovering a stochastic lagrangian formulation we are able to show that such identities indeed do continue to hold informally on average i e via a mathematical expectation over the random ensemble at earlier time the stochastic lagrangian formulation and these identities have also been obtained for the solution to the boussinesq system these results are rigorously proven in mohan and yamazaki 0000 and yamazaki 0000 in this manuscript we will not pursue complete proofs which were in particular mostly inspired by kim 2011 and iyer 2006 but rather elaborate on motivations difficulty and remaining open problems for future works

navier stokes equation and diffusions on the group of homeomorphisms of the torus a stochastic variational principle for the two dimensional navier stokes equation is established the velocity field can be considered as a generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold navier stokes equation is reinterpreted as a perturbed equation of geodesics for the l 2 norm the method described here should hold as well in higher dimensions

bifurcations in a predator prey system of leslie type with generalized holling type iii functional response abstract we consider a predator prey system of leslie type with generalized holling type iii functional response p x m x 2 a x 2 b x 1 by allowing b to be negative b 2 a p x is monotonic for b 0 and nonmonotonic for b 0 when x 0 the model has two non hyperbolic positive equilibria one is a multiple focus of multiplicity one and the other is a cusp of codimension 2 for some values of parameters and a degenerate bogdanov takens singularity focus or center case of codimension 3 for other values of parameters when there exist a multiple focus of multiplicity one and a cusp of codimension 2 we show that the model exhibits subcritical hopf bifurcation and bogdanov takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria respectively different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have i a stable limit cycle enclosing two non hyperbolic positive equilibria ii a stable limit cycle enclosing an unstable homoclinic loop iii two limit cycles enclosing a hyperbolic positive equilibrium iv one stable limit cycle enclosing three hyperbolic positive equilibria or v the coexistence of three stable states two stable equilibria and a stable limit cycle when the model has a bogdanov takens singularity of codimension 3 we prove that the model exhibits degenerate focus type bogdanov takens bifurcation of codimension 3 these results not only demonstrate that the dynamics of this model when b 2 a are much more complex and far richer than the case when b 0 but also provide new bifurcation phenomena for predator prey systems

stochastic lagrangian particle approach to fractal navier stokes equations in this article we study the fractal navier stokes equations by using the stochastic lagrangian particle path approach in constantin and iyer comm pure appl math lxi 330 345 2008 more precisely a stochastic representation for the fractal navier stokes equations is given in terms of stochastic differential equations driven by levy processes based on this representation a self contained proof for the existence of a local unique solution for the fractal navier stokes equation with initial data in mathbb w 1 p is provided and in the case of two dimensions or large viscosity the existence of global solutions is also obtained in order to obtain the global existence in any dimensions for large viscosity the gradient estimates for levy processes with time dependent and discontinuous drifts are proved

three limit cycles in a leslie gower predator prey model with additive allee effect in this work a bidimensional continuous time differential equations system is analyzed which is derived of leslie type predator prey schemes by considering a nonmonotonic functional response and allee effect on population prey for the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant the only quadrant of interest for the sake of realism we show that under certain conditions over the parameters the system allows the existence of three limit cycles the first two cycles are infinitesimal ones generated by hopf bifurcation the third one arises from a homoclinic bifurcation furthermore we give conditions over the parameters such that the model allows long term extinction or survival of both populations in particular the presence of a weak allee effect does not imply extinction of populations necessarily for our model

local and global bifurcations in an sirs epidemic model this paper studies the existence and stability of the disease free equilibrium and endemic equilibria for the sirs epidemic model with the saturated incidence rate considering the factor of population dynamics such as the disease related the natural mortality and the constant recruitment of population analytical techniques are used to show for some parameter values the periodic solutions can arise through the hopf bifurcation which is important to carry different strategies for the controlling disease then the codimension two bifurcation i e bt bifurcation is investigated by using a global qualitative method and the curves of saddle node bifurcation hopf bifurcation and homoclinic bifurcation are obtained at the degenerate equilibrium moreover several numerical simulations are given to support the theoretical analysis

bifurcation analysis of a predator prey system with nonmonotonic functional response we consider a predator prey system with nonmonotonic functional response p x frac mx ax 2 bx 1 by allowing b to be negative b 2 sqrt a p x is concave up for small values of x 0 a

a stochastic perturbation of inviscid flows we prove existence and regularity of the stochastic flows used in the stochastic lagrangian formulation of the incompressible navier stokes equations with periodic boundary conditions and consequently obtain a ck local existence result for the navier stokes equations our estimates are independent of viscosity allowing us to consider the inviscid limit we show that as 0 solutions of the stochastic lagrangian formulation with periodic boundary conditions converge to solutions of the euler equations at the rate of o sqrt nu t

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

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

computer assisted techniques for the verification of the chebyshev property of abelian integrals abstract we develop techniques for the verification of the chebyshev property of abelian integrals these techniques are a combination of theoretical results analysis of asymptotic behavior of wronskians and rigorous computations based on interval arithmetic we apply this approach to tackle a conjecture formulated by dumortier and roussarie in f dumortier r roussarie birth of canard cycles discrete contin dyn syst 2 2009 723 781 which we are able to prove for q 2

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

the cusp of order n we study the bifurcation diagram of the family x y y x2 0 1 y 2xy 3x3y xly where l 3 n 1 2 n 2 and i is small when l 1 it corresponds to the bogdanov takens bifurcation

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets