Morris W. Hirsch
Certain dynamical models of competition have a unique invariant hypersurface to whichevery nonzero tractory is asymptotic, having simple geometry and topology.
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We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for th…
Existence of the carrying simplex for a retrotone map
提出基于径向表示与 Harnack、Hausdorff 度量的动力学方法，证明了 retrotone 映射无序流形的 Kuratowski 收敛性，确立了 carrying simplex 的局部 Lipschitz 径向表示。
Permanence for continuous-time competitive Kolmogorov systems via the carrying simplex
本文通过承载单形研究连续时间竞争Kolmogorov系统的持久性，拓展了相关排斥子存在性结果，提出边界稳定性判据，完成三维线性零倾线系统的持久性分类并应用于多种经典种群动力学模型。
A folklore result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In particular, the classical Grobman-Hartman theorem can not be used on the carrying simplex. In this paper, we prove that the restriction of the map to the carrying simplex in a neighborhood of an in…
For a $C^1$ map $T$ from $C =[0, +\infty)^N$ to $C$ of the form $T_i(x) = x_if_i(x)$, the dynamical system $x(n) =T^n(x)$ as a population model is competitive if $\frac{\partial f_i}{\partial x_j}\leq 0$ $(i\not= j)$. A well know theorem for competitive systems, presented by Hirsch (J. Bio. Dyn. 2 (2008) 169--179) and proved by Ruiz-Herrera (J. Differ. Equ. Appl. 19 (2013) 96--113) with various versions by others, states that, under certain conditions, the system has a compact invariant surface $Σ\subset C$ that is homeomorphic to $Δ^{N-1} =\{x\in C: x_1+ \cdots + x_N=1\}$, attracting all the …
Time-periodic carrying simplex for a competitive system of Carathéodory ODEs
本文研究了时间周期性竞争系统的承载简单集，通过Carathéodory条件放宽了增长率的正则性要求，并提出数值近似方法，证明了系统在扩展承载简单集上与低一维系统拓扑共轭。
There is a recent development in the carrying simplex theory for competitive maps: under some weaker conditions a map has a modified carrying simplex (one of the author's latest publications). In this paper, as one of the applications of the modified carrying simplex theory, a criterion is established for a three dimensional Kolmogorov map to have a globally repelling (attracting) heteroclinic limit cycle. As a concrete example, a discrete competitive model is investigated to illustrate the above criteria for global repulsion (attraction) of a hetericlinic limit cycle.
For a class of competitive maps there is an invariant one-codimensional manifold (the carrying simplex) attracting all non-trivial orbits. In the present paper it is shown that its convexity implies that it is a $C^1$ submanifold-with-corners, neatly embedded in the non-negative orthant. The proof uses the characterization of neat embedding in terms of inequalities between Lyapunov exponents for ergodic invariant measures supported on the boundary of the carrying simplex.
It is proved that a convex carrying simplex for a three-dimensional competitive map is a $C^1$ submanifold-with-corners neatly embedded in the non-negative octant.
We consider dissipative strongly competitive systems $\dot{x}_{i}=x_{i}f_{i}(x)$ of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface $Σ$, called the carrying simplex. In this note we give an amenable condition for $Σ$ to be a $C^{1}$ submanifold-with-corners. We also provide conditions, based on a recent work of M. Bena\"ım "On invariant hypersurfaces of strongly monotone maps", J. Differential Equations 137 (1997), 302-319, guaranteeing that $Σ$ is of class $C^{k+1}$.
It is very well known that periodic orbits of autonomous Hamiltonian systems are generically organized into smooth one-parameter families (the parameter being just the energy value). We present a simple example of an integrable Hamiltonian system (with an arbitrary number of degrees of freedom greater than one) with a unique periodic orbit in the phase space (which is not compact). Similar examples are given for Hamiltonian systems with a unique invariant torus (of any prescribed dimension) carrying conditionally periodic motions. Parallel examples for Hamiltonian systems with a compact phase …
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