Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Morse index of steady-states to the SKT model with Dirichlet boundary conditions

Kousuke Kuto, Homare Sato

TL;DR

This work analyzes linear stability of steady-states for the SKT competition model with equal cross-diffusion under Dirichlet boundary conditions in a bounded domain. By examining the full cross-diffusion limit as $\alpha\to\infty$ and perturbing the limiting bifurcation curves, the authors determine the Morse index of steady-states on the main positive branch $\mathcal{C}_{\alpha,\Lambda}$ and on segregation branches $\mathcal{S}^{\pm}_{j,\alpha,\Lambda}$ via implicit-function arguments and eigenvalue counting. They prove that, for large $\alpha$, the Morse index on $\mathcal{C}_{\alpha,\Lambda}$ equals $j$ on the interval $(\beta_{j,\alpha},\beta_{j+1,\alpha}]$, while on $\mathcal{S}^{\pm}_{j,\alpha,\Lambda}$ it equals $j-1$ in the one-dimensional setting. Numerical bifurcation analysis using $\text{pde2path}$ corroborates the theoretical Morse-index structure and links instability to the number of segregation points.

Abstract

This paper deals with the stability analysis for steady-states perturbed by the full cross-diffusion limit of the SKT model with Dirichlet boundary conditions. Our previous result showed that positive steady-states consist of the branch of small coexistence type bifurcating from the trivial solution and the branches of segregation type bifurcating from points on the branch of small coexistence type. This paper shows the Morse index of steady-states on the branches and constructs the local unstable manifold around each steady-state of which the dimension is equal to the Morse index.

Morse index of steady-states to the SKT model with Dirichlet boundary conditions

TL;DR

This work analyzes linear stability of steady-states for the SKT competition model with equal cross-diffusion under Dirichlet boundary conditions in a bounded domain. By examining the full cross-diffusion limit as and perturbing the limiting bifurcation curves, the authors determine the Morse index of steady-states on the main positive branch and on segregation branches via implicit-function arguments and eigenvalue counting. They prove that, for large , the Morse index on equals on the interval , while on it equals in the one-dimensional setting. Numerical bifurcation analysis using corroborates the theoretical Morse-index structure and links instability to the number of segregation points.

Abstract

This paper deals with the stability analysis for steady-states perturbed by the full cross-diffusion limit of the SKT model with Dirichlet boundary conditions. Our previous result showed that positive steady-states consist of the branch of small coexistence type bifurcating from the trivial solution and the branches of segregation type bifurcating from points on the branch of small coexistence type. This paper shows the Morse index of steady-states on the branches and constructs the local unstable manifold around each steady-state of which the dimension is equal to the Morse index.
Paper Structure (5 sections, 11 theorems, 123 equations, 4 figures)

This paper contains 5 sections, 11 theorems, 123 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Morse index of small coexistence states
  4. Morse index of segregation states
  5. Numerical results

Key Result

Theorem 1.1

Assume $u_{0}$, $v_{0}\in W^{1,p}_{0}(\Omega )$ with $p\in (N, \infty)$. Then para has a unique solution $(u,v)$ satisfying where $T_{m}$ is a maximal existence time. Moreover, if $T_{m}<\infty$, then

Figures (4)

  • Figure 1: Bifurcation diagram of solutions of \ref{['SP']}.
  • Figure 2: Profile of a solution on blue curve at $\lambda=59.8286$.
  • Figure 3: Profiles of solutions on red and purple pitchfork bifurcation branches in Figure 1.
  • Figure 4: Morse index of solutions of \ref{['SP']}.

Theorems & Definitions (21)

  • Theorem 1.1: Am1Am2Am3
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 11 more