Persistence and stability of solutions of Hamilton-Jacobi equations

Given a convergent sequence of Hamiltonians (Hn) and a convergent sequence of initial data (gn) for the first-order evolutionary Hamilton-Jacobi equation, we look for conditions ensuring that the sequences (un) and (vn) of Lax solutions and Hopf solutions respectively converge. The convergences we deal with are variational convergences. We take advantage of several recent results giving criteria for the continuity of usual operations.     

Determination of nonlinear stability for low order resonances by a geometric criterion

We consider the problem of stability of equilibrium points in Hamiltonian systems of two degrees of freedom under low order resonances. For resonances of order bigger than two there are several results giving stability conditions, in particular one based on the geometry of the phase flow and a set of invariants. In this paper we show that this geometric criterion is still valid for low order resonances, that is, resonances of order two and resonances of order one. This approach provides necessary stability conditions for both the semisimple and non-semisimple cases, with an appropriate choice of invariants.     

Topological indices of spencer complexes associated with geometric structures

Translated from Matematicheskie Zametki, Vol. 45, No. 4, pp. 59–68, April, 1989.     

Persistence and stability for a two-species ratio-dependent predator-prey system with distributed time delay

A two-species ratio-dependent predator-prey model with distributed time delay is investigated. It is shown that the system is persistent under some appropriate conditions, and sufficient conditions are obtained for both the local and global stability of the positive equilibrium of the system.     

Persistence and global stability in discrete models of Lotka-Volterra type

In this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka-Volterra type:

     

一类具有无穷时滞和功能反应的捕食扩散系统的持久性与稳定性

研究一类非自治的具有HollingⅡ类功能性反应且包含时变时滞与多个无穷时滞的两种群n斑块捕食扩散系统的持久性与稳定性.利用比较原理,结合构造Lyapunov泛函的方法,得到了保证该系统永久持续生存和任意正解全局渐近稳定的充分性条件.     

Persistence and global stability for n-species ratio-dependent predator-prey system with time delays

A n-species ratio-dependent predator-prey food-chain model with time delays is investigated. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are obtained for the global stability of the positive equilibrium of the system.     

Persistence and stability in general nonautonomous single-species Kolmogorov systems with delays

This paper studies the general nonautonomous single-species Kolmogorov systems with delays. The sufficient conditions on the persistence and permanence of species, global asymptotic stability and the existence of positive periodic solutions are established. As applications of these results, the permanence, global asymptotic stability and the existence of positive periodic solutions for a series of special single-species growth systems with delays are discussed.     

带时滞的比率型的食物链系统的持久性和稳定性

A delayed three-species ratio-dependent predator-prey food-chain model without dominating instantaneous negative feedback is investigated. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are obtained for the local asymptotic stability of a positive equilibrium of the system.     

Bounded stability for strongly coupled critical elliptic systems below the geometric threshold of the conformal Laplacian

We prove bounded stability for strongly coupled critical elliptic systems in the inhomogeneous context of a compact Riemannian manifold when the potential of the operator is less, in the sense of bilinear forms, than the geometric threshold potential of the conformal Laplacian.     

Persistence and global stability in a delayed Leslie-Gower type three species food chain

Our investigation concerns the three-dimensional delayed continuous time dynamical system which models a predator-prey food chain. This model is based on the Holling-type II and a Leslie-Gower modified functional response. This model can be considered as a first step towards a tritrophic model (of Leslie-Gower and Holling-Tanner type) with inverse trophic relation and time delay. That is when a certain species that is usually eaten can consume immature predators. It is proved that the system is uniformly persistent under some appropriate conditions. By constructing a proper Lyapunov function, we obtain a sufficient condition for global stability of the positive equilibrium.     

Persistence, contractivity and global stability in logistic equations with piecewise constant delays

We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N^*=1/(a+∑_i=0^mb_i) of the following differential equation with piecewise constant arguments:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑_i=0^mb_i>0, b_i0, i=0,1,2,…,m, and a+∑_i=0^mb_i>0. These new conditions depend on a,b₀ and ∑_i=1^mb_i, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms:

where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x^*>0 and g(x₀,x₁,…,x_m)C¹[(0,+∞)×(0,+∞)××(0,+∞)].     

Persistence and global stability in discrete models of pure-delay nonautonomous Lotka-Volterra type

Consider the persistence and the global asymptotic stability of the following discrete model of pure-delay nonautonomous Lotka-Volterra type:

     

Gaps for geometric genera

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in \({{\mathbb{P}}^3}\), the initial gap interval was found by Xu (see J Differ Geom 39:139–172, 1994) and the next one in our previous paper (Int Math Res Not 22:12177–12205, 2015), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach.     

Homological Algebra for Persistence Modules

Foundations of Computational Mathematics - We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either...     

Topological Persistence for Circle-Valued Maps

We study circle-valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real-valued maps, circle-valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle-valued map on an input simplicial complex.