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The existence and instability of blowing-up steady states for the Shigesada-Kawasaki-Teramoto competition model with cross-diffusion

Kousuke Kuto, Yaping Wu

Abstract

We investigate the existence and instability of a class of blowing-up positive steady states arising in a shadow system of the Shigesada-Kawasaki-Teramoto (SKT) two-species competition model, as well as in the corresponding perturbed SKT model with a sufficiently large cross-diffusion coefficient and bounded random diffusion parameters. In their classical work (Lou-Ni, 1999), it was shown that, under the limit where one cross-diffusion parameter tends to infinity, coexistence steady states of the SKT model are characterized by three types of shadow systems. In a previous study (Kuto, 2015), the first author analyzed one of these shadow systems in one space dimension (the second shadow system) and proved that a component of a bifurcating branch blows up as the bifurcation parameter approaches the first positive Neumann eigenvalue of $-Δ$. In the present paper, using a new approach based on suitable transformations and the Lyapunov-Schmidt reduction method, we derive the existence and detailed asymptotic structure of several branches of positive steady states near blow-up points for shadow systems with one or two cross-diffusion terms, in both one- and multi-dimensional domains. We further show that all such large-amplitude steady state branches are spectrally unstable. Moreover, by means of perturbation arguments, we establish the existence and instability of corresponding branches of perturbed positive steady states for the original SKT model when one cross-diffusion coefficient is sufficiently large.

The existence and instability of blowing-up steady states for the Shigesada-Kawasaki-Teramoto competition model with cross-diffusion

Abstract

We investigate the existence and instability of a class of blowing-up positive steady states arising in a shadow system of the Shigesada-Kawasaki-Teramoto (SKT) two-species competition model, as well as in the corresponding perturbed SKT model with a sufficiently large cross-diffusion coefficient and bounded random diffusion parameters. In their classical work (Lou-Ni, 1999), it was shown that, under the limit where one cross-diffusion parameter tends to infinity, coexistence steady states of the SKT model are characterized by three types of shadow systems. In a previous study (Kuto, 2015), the first author analyzed one of these shadow systems in one space dimension (the second shadow system) and proved that a component of a bifurcating branch blows up as the bifurcation parameter approaches the first positive Neumann eigenvalue of . In the present paper, using a new approach based on suitable transformations and the Lyapunov-Schmidt reduction method, we derive the existence and detailed asymptotic structure of several branches of positive steady states near blow-up points for shadow systems with one or two cross-diffusion terms, in both one- and multi-dimensional domains. We further show that all such large-amplitude steady state branches are spectrally unstable. Moreover, by means of perturbation arguments, we establish the existence and instability of corresponding branches of perturbed positive steady states for the original SKT model when one cross-diffusion coefficient is sufficiently large.

Paper Structure

This paper contains 10 sections, 15 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Formulation of the problem and statement of main results
  3. Existence and parameterization of the solution branches to the 2nd shadow system \ref{['LNlim2']} near the blow-up points
  4. Lyapunov-Schmidt reduction scheme
  5. The limiting system of \ref{['G=0']} as $\varepsilon\to 0$
  6. Proofs of Theorems \ref{['LSprop']} and \ref{['blthm']}
  7. Instability of steady states for the shadow system near the blow-up point
  8. Existence and instability of coexistence states of the SKT model with large cross diffusion
  9. Perturbation of positive stationary solutions of the 2nd shadow system
  10. Perturbation of unstable eigenvalues of the linearized problem for the 2nd shadow system

Key Result

Theorem 1.1

Suppose that $N\le 3$, $a_{1}/a_{2}\neq b_{1}/b_{2}$, $a_{1}/a_{2}\neq c_{1}/c_{2}$ and $a_{2}/d_{2}\neq\lambda_{j}$ for any positive eigenvalue $\lambda_{j}$ of $-\Delta$ with the homogeneous Neumann boundary condition on $\partial\Omega$. Let $(u_n, v_n)$ be a sequence of positive nontrivial solut (ii) $(u_n,\; \alpha_n v_n)$ converges uniformly to positive $(u, w)$ as $n\to +\infty$, where $(u,

Theorems & Definitions (28)

  • Theorem 1.1: LN2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1: LNY2
  • Proposition 3.2
  • Proposition 3.3
  • ...and 18 more