Persistence Theorem Research Articles

Spreading dynamics for a predator-prey system with two predators and one prey in a shifting habitat

<p style='text-indent:20px;'>We study the spreading dynamics for a three-species predator-prey system with two weak competing predators and one prey in a shifting habitat. First, we derive some extinction results for each species. Then we provide some persistence theorems for each species with moving speeds exceed the shifting speed, but less than some certain quantities. Finally, the convergence to a certain constant state is proven in each persistent regime.</p>

Persistence of two weak competing species in a shifting environment

In this paper, we are concerned with the persistence of both competing species in a weak competition model under a shifting environment. We first prove a Liouville type theorem on the entire solution of the corresponding weak competition system without the shifting effect. Then we prove a persistence theorem for both competing species with general intrinsic growth rates. This theorem not only extends a known result by Yuan et al. (2019) with equal growth rates, but also relaxes the assumptions on the initial data for a persistence theorem by Zhang et al. (2017).

Dynamics of an HIV Infection Model with Two Infection Routes and Evolutionary Competition between Two Viral Strains

The existing combination therapy of HIV antiretroviral drugs can lead to the emergence of drug-resistant viruses, and cannot effectively block direct cell-to-cell infections, these factors results in incomplete virus suppression and increased risk of disease progression. In this paper, we formulate an HIV model with two strains representing a drug-sensitive virus and a drug-resistant virus to study the joint mechanism of drug resistance. We first reduce the infection-age model to a system of integro-differential equations with infinite delays. Then the stability of the equilibria and the dynamics of competition between two viruses are studied to illuminate the joint effects of infection-age and two infection routes on the evolution of both drug-sensitive and drug-resistant strains before and during drug treatment. Applying a persistence theorem for infinite dimensional systems, we obtain that the disease is always present when the basic reproduction number is larger than unity. Numerical simulations confirm that the basic reproduction numbers and mutation coefficient are the key threshold parameters for determining the competition results of the two viral strains and indicate the cell-to-cell transmission increases the likelihood that HIV breaks out within the host. Finally, sensitivity analyses suggest that the available combination therapy should be taken once symptoms of resistance appear during drug treatment, and demonstrate that the presence of cell-to-cell transmission attenuates the efficacy of the existing antiretroviral drug treatments.

Selection of a stochastic Landau-Lifshitz equation and the stochastic persistence problem

In this article, we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms: external and internal. The first case corresponds to add a noise term to a given equation using the framework of Itô or Stratonovich stochastic differential equations. The second case corresponds to consider parameter dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations. Our main concerns for the preservation of properties are stability/instability of equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different. We then apply our results to develop a stochastic version of the Landau-Lifshitz equation. We discuss, in particular, previous results obtained by Étoré et al. [J. Differ. Equations 257, 2115–2135 (2014)] and we finally propose a new family of stochastic Landau-Lifshitz equations.

Extremal behavior in sectional matrices

In this paper, we recall the object sectional matrix which encodes the Hilbert functions of successive hyperplane sections of a homogeneous ideal. We translate and/or reprove recent results in this language. Moreover, some new results are shown about their maximal growth, in particular, a new generalization of Gotzmann’s Persistence Theorem, the presence of a GCD for a truncation of the ideal, and applications to saturated ideals.

The effect of parameters on positive solutions and asymptotic behavior of an unstirred chemostat model with B\u2013D functional response

This paper deals with the effect of parameters on properties of positive solutions and asymptotic behavior of an unstirred chemostat model with the Beddington–DeAngelis (denote by B–D) functional response under the Robin boundary condition. Firstly, we establish some a priori estimates and a sufficient condition for the existence of positive solutions (see (Feng et al. in J. Inequal. Appl. 2016(1):294, 2016)). Secondly, we study the effect of the small parameter k_{1} and sufficiently large k_{2} in B–D functional response, which shows that the model has at least two positive solutions. Thirdly, we investigate the case of sufficiently large k_{1}. The results show that if k_{1} is sufficiently large, then the positive solution of this model is determined by a limiting equation. Finally, we present an asymptotic behavior of solutions depending on time. The main methods used in this paper include the fixed point index theory, bifurcation theory, perturbation technique, comparison principle, and persistence theorem.

Asymptotic behavior of solutions to a class of diffusive predator–prey systems

We study the large time behavior of a class of diffusive predator–prey systems posed on the whole Euclidean space. By studying a family of similar problems with all possible spatial translations, we first prove the asymptotic persistence of the prey for the spatially heterogeneous case under certain assumptions on the coefficients. Then, applying this persistence theorem, we prove the convergence of the solution to the unique positive equilibrium for the spatially homogeneous case, under certain restrictions on the space dimension and the predation coefficient.

Gotzmann's persistence theorem for finite modules

We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal.

Discrete and continuous fractional persistence problems – the positivity property and applications

In this article, we study the continuous and discrete fractional persistence problem which looks for the persistence of properties of a given classical (α=1) differential equation in the fractional case (here using fractional Caputo’s derivatives) and the numerical scheme which are associated (here with discrete Grünwald–Letnikov derivatives). Our main concerns are positivity, order preserving ,equilibrium points and stability of these points. We formulate explicit conditions under which a fractional system preserves positivity. We deduce also sufficient conditions to ensure order preserving. We deduce from these results a fractional persistence theorem which ensures that positivity, order preserving, equilibrium points and stability is preserved under a Caputo fractional embedding of a given differential equation. At the discrete level, the problem is more complicated. Following a strategy initiated by R. Mickens dealing with non local approximations, we define a non standard finite difference scheme for fractional differential equations based on discrete Grünwald–Letnikov derivatives, which preserves positivity unconditionally on the discretization increment. We deduce a discrete version of the fractional persistence theorem for what concerns positivity and equilibrium points. We then apply our results to study a fractional prey–predator model introduced by Javidi et al.

Applications of mapping cones over Clements–Lindström rings

We prove that Gotzmann's Persistence Theorem holds over every Clements–Lindström ring. We also construct the infinite minimal free resolution of a square-free Borel ideal over such a ring.

Persistence of laminations

We present a modern proof of some extensions of the well known Hirsch-Pugh-Shub theorem on persistence of normally hyperbolic compact laminations. Our extensions consist of allowing the dynamics to be an endomorphism, the complex analytic case and of allowing the laminations to be non compact. To study the analytic case, we use the formalism of deformations of complex structures. We present various persistent complex laminations which appear in dynamics of several complex variables: Hénon maps, fibered holomorphic maps. In order to prove the persistence theorems,weconstructalaminarstructureonthestableandunstablesetofthenormally hyperbolic laminations.

Gotzmann monomial ideals

A Gotzmann monomial ideal of a polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. Let $R=K[x_1,\dots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $M^d$ the set of monomials of $R$ of degree $d$. A subset $V\subset M^d$ is said to be a Gotzmann subset if the ideal generated by $V$ is a Gotzmann monomial ideal. In the present paper, we find all integers $a > 0$ such that every Gotzmann subset $V\subset M^d$ with $|V|=a$ is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of $K[x_1,x_2,x_3]$.

A combinatorial proof of Gotzmann’s persistence theorem for monomial ideals

Gotzmann proved the persistence for minimal growth of Hilbert functions of homogeneous ideals. His theorem is called Gotzmann’s persistence theorem. In this paper, based on the combinatorics of binomial coefficients, a simple combinatorial proof of Gotzmann’s persistence theorem in the special case of monomial ideals is given.

Persistence theorems and simultaneous uniformization

One of the most intriguing problems in the theory of foliations by analytic curves is that of the persistence of complex limit cycles of a polynomial vector field, as well as related problems concerning the persistence of identity cycles and saddle connections and the global extendability of the Poincare map. It is proved that all these persistence problems have positive solutions for any foliation admitting a simultaneous uniformization of leaves. The latter means that there exists a uniformization of leaves that analytically depends on the initial condition and satisfies certain additional assumptions, called continuity and boundedness. Thus, the results obtained are conditional, but they establish a relation between very different properties of foliations.

Standard Bigraded Hilbert Functions

ABSTRACT We show Macaulay-type bounds and persistence results for bigraded Hilbert function along rays. Global behavior of the bigraded Hilbert function and very supportive sets for bigraded Hilbert polynomials are also discussed. We indicate by several results that direct generalizations of the standard graded theory are not true. In particular, we give a counterexample to an immediate generalization of Gotzmann's Regularity Theorem and Persistence Theorem. It is also made clear that maximal growth of a bigraded Hilbert function from bidegree (u, v) to bidegree (u + 1, v) usually is obtained at the expense of the growth from bidegree (u, v) to bidegree (u, v + 1).

PERSISTENCE OF RESONANT INVARIANT TORI WITH NON-HAMILTONIAN PERTURBATION

A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone's for the three dimensional non-Hamiltonian systems.

Kupka-Smale theorem for polynomial automorphisms of ℂ2 and persistence of heteroclinic intersections

A map is Kupka-Smale if all periodic points are hyperbolic and the stable and unstable manifolds of any two saddle points are transverse. Here we prove that Kupka-Smale maps form a residual set of full Lebesgue measure in the space $\mathcal{P}_d$ of polynomial automorphisms of ℂ2 of fixed dynamical degree d≥2. We also prove that a heteroclinic point of two saddle periodic orbits may be continued over (almost) the entire parameter space for this set of maps. This is one of the first persistence theorems proved in holomorphic dynamics in several variables.

On the Continuation of an Invariant Torus in a Family with Rapid Oscillations

A persistence theorem for attracting invariant tori for systems subjected to rapidly oscillating perturbations is proved. The singular nature of these perturbations prevents the direct application of the standard persistence results for normally hyperbolic invariant manifolds. However, as is illustrated in this paper, the theory of normally hyperbolic invariant manifolds, when combined with an appropriate continuation method, does apply.

Uniform persistence and global attractivity theorem for generalized prey–predator system with time delay

Uniform persistence and global attractivity theorem for generalized prey–predator system with time delay

Completely lexsegment ideals

In this paper we study ideals which are generated by lexsegments of monomials. In contrast to initial lexsegments, the shadow of an arbitrary lexsegment is in general not again a lexsegment. An ideal generated by a lexsegment is called completely lexsegment, if all iterated shadows of the set of generators are lexsegments. We characterize all completely lexsegment ideals and describe cases in which they have a linear resolution. We also prove a persistence theorem which states that all iterated shadows of a lexsegment are again lexsegments if the first shadow has this property.