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Turing instability analysis of a singular cross-diffusion problem

Gonzalo Galiano, Víctor González-Tabernero

Abstract

The population model of Busenberg and Travis is a paradigmatic model in ecology and tumour modelling due to its ability to capture interesting phenomena like the segregation of populations. Its singular mathematical structure enforces the consideration of regularized problems to deduce properties as fundamental as the existence of solutions. In this article we perform a weakly nonlinear stability analisys of a general class of regularized problems to study the convergence of the instability modes in the limit of the regularization parameter. We demonstrate with some specific examples that the pattern formation observed in the regularized problems, with unbounded wave numbers, is not present in the limit problem due to the amplitude decay of the oscillations. We also check the results of the stability analysis with direct finite element simulations of the problem.

Turing instability analysis of a singular cross-diffusion problem

Abstract

The population model of Busenberg and Travis is a paradigmatic model in ecology and tumour modelling due to its ability to capture interesting phenomena like the segregation of populations. Its singular mathematical structure enforces the consideration of regularized problems to deduce properties as fundamental as the existence of solutions. In this article we perform a weakly nonlinear stability analisys of a general class of regularized problems to study the convergence of the instability modes in the limit of the regularization parameter. We demonstrate with some specific examples that the pattern formation observed in the regularized problems, with unbounded wave numbers, is not present in the limit problem due to the amplitude decay of the oscillations. We also check the results of the stability analysis with direct finite element simulations of the problem.

Paper Structure

This paper contains 5 sections, 6 theorems, 88 equations, 3 figures, 2 tables.

Table of Contents

  1. Main results
  2. Numerical experiments
  3. Experiment 1
  4. Experiment 2
  5. Proofs

Key Result

Theorem 1

Assume $H_D$, with $b\geq0$. Let $\mathbf{u}^*$ be the coexistence equilibrium defined by (equilibrium). If then there exists $\delta_c>0$ such that if $\delta<\delta_c$ then $\mathbf{u}^*$ is a linearly unstable equilibrium for problem (eq:u1g)-(eq:idg). In such situation, the wave number of the main instability mode tends to infinity as $\delta\to0$.

Figures (3)

  • Figure 1: Typical evolution of disturbances
  • Figure 2: Experiment 1. WNA and FEM approximations corresponding to Simulations 1 to 3 (left to right). Notice the different scales in the ordinates axis showing the decreasing amplitude of the oscillations.
  • Figure 3: Experiment 1. Behaviour of the patterns as $\delta\to0$.

Theorems & Definitions (6)

  • Theorem 1: Linear instability
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3