Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence and asymptotic stability of a generic Lotka-Volterra system with nonlinear spatially heterogeneous cross-diffusion

Tianxu Wang, Jiwoon Sim, Hao Wang

Abstract

This article considers a class of Lotka-Volterra systems with multiple nonlinear cross-diffusion, commonly known as prey-taxis models. The existence and stability of classic solutions for such systems with spatially homogeneous sources and taxis have been studied in one- or two-dimensional space, however, the proof is non-trivial for a more general setting with spatially heterogeneous predation functions and taxis coefficient functions in arbitrary dimensions. This study introduces a new weighted \(L_ε^p\)-norm and extends some classical inequalities within this normed space. Coupled energy estimates are employed to establish initial bounds, followed by applying heat kernel properties and an advanced bootstrap process to enhance solution regularity. For stability analysis, we extend LaSalle's invariance principle to a general \( L^\infty \) setting and utilize it alongside Lyapunov functions to analyze the stability of each possible constant equilibrium. All results are achieved without introducing an extra logistic growth term for predators or imposing smallness conditions on taxis coefficients.

Existence and asymptotic stability of a generic Lotka-Volterra system with nonlinear spatially heterogeneous cross-diffusion

Abstract

This article considers a class of Lotka-Volterra systems with multiple nonlinear cross-diffusion, commonly known as prey-taxis models. The existence and stability of classic solutions for such systems with spatially homogeneous sources and taxis have been studied in one- or two-dimensional space, however, the proof is non-trivial for a more general setting with spatially heterogeneous predation functions and taxis coefficient functions in arbitrary dimensions. This study introduces a new weighted -norm and extends some classical inequalities within this normed space. Coupled energy estimates are employed to establish initial bounds, followed by applying heat kernel properties and an advanced bootstrap process to enhance solution regularity. For stability analysis, we extend LaSalle's invariance principle to a general setting and utilize it alongside Lyapunov functions to analyze the stability of each possible constant equilibrium. All results are achieved without introducing an extra logistic growth term for predators or imposing smallness conditions on taxis coefficients.
Paper Structure (9 sections, 36 theorems, 60 equations, 1 table)

This paper contains 9 sections, 36 theorems, 60 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local existence and positivity
  4. Global existence
  5. $L^\infty$ boundedness of $Y$
  6. $L^\infty$ boundedness of $Z$
  7. Stability
  8. $W^{1,\infty}$ boundedness of U
  9. Proof of stability

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^n$$(n\geq 1)$ be a bounded domain with smooth boundary. Under Assumptions $(A_I)$ -- $(A_h)$, equation eq: general model -- eq: bc ic has a unique maximal solution $U = (X, Y, Z) \in C ( \overline{\Omega} \times [0, \infty); \mathbb R_{\geq 0}^3 ) \cap C^{2, 1} ( \overl

Theorems & Definitions (43)

  • Remark 1.1
  • Theorem 1.1: Global boundedness
  • Theorem 1.2: Stability
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.1
  • Remark 2.1
  • Proposition 2.1
  • Proposition 2.2
  • ...and 33 more