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

TL;DR

This paper proves the existence and asymptotic stability of solutions for a broad class of nonlinear, spatially heterogeneous Lotka-Volterra prey-taxis models in arbitrary dimensions, using advanced analytical techniques.

Contribution

It introduces a new weighted norm and extends classical inequalities to analyze complex prey-taxis models without logistic growth or smallness assumptions.

Findings

Established existence of classical solutions in general settings.

Proved asymptotic stability of constant equilibria.

Extended LaSalle's invariance principle to a new functional framework.

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 \(L_\epsilon^p\)-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.