arXiv subjects

Zhanyuan Hou

Publications and source records attributed to Zhanyuan Hou.

5 recordsLinked to original sources

arXiv2025

Global dynamics of planar discrete type-K competitive systems

For a continuously differentiable Kolmogorov map defined from the nonnegative orthant to itself, a type-K competitive system is defined. Under the assumptions that the system is dissipative and the origin is a repeller, the global dynamics of such systems is investigated. A (weakly) type-K retrotone map is defined on a bounded set, which is backward monotone in some order. Under certain conditions, the dynamics of a type-K competitive system is the dynamics of a type-K retrotone map. Under these conditions, there exists a compact invariant set A that is the global attractor of the system on the nonnegative orthant exluding the origin. Some basic properties of A are established and remaining problems are listed for further investigation for general N-dimensional systems. These problems are completely solved for planar type-K competitive systems: every forward orbit is eventually monotone and converges to a fixed point; the global attractor A consists of two monotone curves each of which is a one-dimensional compact invariant manifold. A concrete example is provided to demonstrate the results for planar systems.

math.DS↗

arXiv2022

Global attraction and repulsion of a heteroclinic limit cycle in three dimensional Kolmogorov maps

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.

math.DS↗

arXiv2021

On existence and uniqueness of a modified carrying simplex for discrete Kolmogorov systems

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 points of $C\setminus\{0\}$, and called carrying simplex. The theorem has been well accepted with a large number of citations. In this paper, we point out that one of its conditions requiring all the $N^2$ entries of the Jacobian matrix $Df = (\frac{\partial f_i}{\partial x_j})$ to be negative is unnecessarily strong and too restrictive. We prove the existence and uniqueness of a modified carrying simplex by reducing that condition to requiring every entry of $Df$ to be nonpositive and each $f_i$ is strictly decreasing in $x_i$. As an example of applications of the main result, sufficient conditions are provided for vanishing species and dominance of one species over others.

math.DS↗

arXiv2013

Permanence criteria for Kolmogorov systems with delays

In this paper, a class of Kolmogorov systems with delays are studied. Sufficient conditions are provided for a system to have a compact uniform attractor. Then Jansen's result (J. Math. Biol. Vol. 25 (1987) 411-422) for autonomous replicator and Lotka-Volterra systems has been extended to delayed nonautonomous Kolmogorov systems with periodic or autonomous Lotka-Volterra subsystems. Thus, simple algebraic conditions are obtained for partial permanence and permanence. An outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays.

math.DS↗

arXiv2007

Global Attractor in Competitive Lotka-Volterra Systems

For autonomous Lotka-Volterra systems of differential equations modelling the dynamics of n competing species, new criteria are established for the existence of a single point global attractor. Under the conditions of these criteria, some of the species will survive and stabilise at a steady state whereas the others, if any, will die out.

math.DS↗