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Global dynamics of planar discrete type-K competitive systems

Zhanyuan Hou

TL;DR

The paper investigates the global dynamics of planar discrete type-K competitive systems, modeled as $x_i(n+1)=x_i(n) f_i(x(n))$ on the nonnegative orthant. It introduces type-K retrotone maps to capture backward-monotone behavior and shows that, under dissipativity and origin repeller assumptions, the system's dynamics on a bounded region can be described by a type-K retrotone map, yielding a compact global attractor $\Sigma$ in $C\setminus\{\mathbf{0}\}$. In the planar case ($N=2$), the global attractor decomposes into two monotone invariant curves on the axes, $\Sigma_H$ and $\Sigma_V$, plus a central monotone curve $\Sigma_0$ connecting interior fixed points, and every forward orbit converges to a fixed point. The paper provides a detailed planar proof, illustrates the theory with a concrete example, and outlines open problems for higher dimensions, including the geometry of $\Sigma$ and conditions guaranteeing retrotone behavior.

Abstract

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.

Global dynamics of planar discrete type-K competitive systems

TL;DR

The paper investigates the global dynamics of planar discrete type-K competitive systems, modeled as on the nonnegative orthant. It introduces type-K retrotone maps to capture backward-monotone behavior and shows that, under dissipativity and origin repeller assumptions, the system's dynamics on a bounded region can be described by a type-K retrotone map, yielding a compact global attractor in . In the planar case (), the global attractor decomposes into two monotone invariant curves on the axes, and , plus a central monotone curve connecting interior fixed points, and every forward orbit converges to a fixed point. The paper provides a detailed planar proof, illustrates the theory with a concrete example, and outlines open problems for higher dimensions, including the geometry of and conditions guaranteeing retrotone behavior.

Abstract

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.

Paper Structure

This paper contains 6 sections, 7 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Type-K retrotone maps and preliminaries
  3. Global dynamics of planar type-K competitive systems
  4. Proof of Proposition \ref{['pro2.5']}
  5. Proof of Proposition \ref{['pro3.1']} and Theorem \ref{['the3.2']}
  6. Conclusion

Key Result

Proposition 1

Assume (A1) and the existence of continuous $u: \mathbb{R}_+\to C$ such that $\mathbf{0} \ll u(s)\ll u(t)$ for all $t>s\geq 0$ and $u_i(t)\to\infty$ as $t\to \infty$ for all $i\in I_N$. If $\forall t\geq 0, \forall i\in H, \forall j\in V$, then (e1) is dissipative and for $r=u(0)$, $T([\mathbf{0}, r])\subset [\mathbf{0}, r)$ and $T^n(x)\in [\mathbf{0}, r]$ for each $x\in C$ and some $n\geq 0$.

Figures (4)

  • Figure 1: Illustration of $\Sigma = \Sigma_H\cup\Sigma_0\cup\Sigma_V$ when $N=2$: (a) $\Sigma_0$ is a monotone curve in $\ll$, (b) $\Sigma_0 =\{p_0\}$.
  • Figure 2: Illustration of $\overline{xy}$, $\overline{pq}$, $\overline{T(x)T(y)}$ and $\overline{T(p)T(q)}$: (a) & (b) The region $T([\mathbf{0}, r])\setminus \overline{T(x)T(y)}$ is completely on one side of $\overline{T(x)T(y)}$, (c) & (d) $\overline{xy}\subset \overline{pq}\subset\ell_1$.
  • Figure 3: Illustration of $\overline{uv}$, $\overline{pq}$, $\overline{T(u)T(v)}$ and $\overline{T(p)T(q)}$: (a) & (b) The region $R$ bounded by $T^{-1}(\overline{T(u)T(v)}), T^{-1}(\overline{T(p)T(q)}), \ell_3$ and $\ell_4$. (c) & (d) The region $T(R)$ bounded by $\overline{uv}, T(\ell_3), \overline{pq}$ and $T(\ell_4)$.
  • Figure 4: Illustration of the regions divided by $\ell_1$ and $\ell_2$: (a) Only four regions $R_1$--$R_4$, (b) Five or more regions.

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • ...and 7 more