Global existence and stability in a class of chemotaxis systems with lethal interactions, nonlinear diffusion and production
Gnanasekaran Shanmugasundaram, Jitraj Saha
TL;DR
This work analyzes a chemotaxis system with lethal interactions, nonlinear diffusion $D(u)$, production $a u^m$, and toxin-mediated mortality in a bounded domain under Neumann boundaries, across fully parabolic ($\tau=1$) and parabolic-elliptic ($\tau=0$) regimes. The authors establish global existence and uniform boundedness of smooth solutions under dimension-dependent diffusion-production balance conditions, and they construct Lyapunov functionals to derive exponential convergence toward biologically relevant steady states. Specifically, they prove convergence to a coexistence state when $\chi^2<\dfrac{4 d_1 d_2 \mu}{a u^*}$ and $2\beta\le\alpha$, or to a semi-coexistence state when $f\mu\ge br$ and $\kappa=2$. The results provide rigorous insight into long-time behavior of lethal-chemotaxis systems with nonlinear diffusion/production and external input, with implications for microbial self-toxicity dynamics and ecological balance.
Abstract
This paper investigates a class of chemotaxis systems modeling lethal interactions in a smooth, bounded domain $Ω\subset \mathbb{R}^n$ with homogeneous Neumann boundary conditions. We examine two distinct cases: (i) a fully parabolic system where both equations exhibit parabolic dynamics, and (ii) a parabolic-elliptic system featuring a parabolic first equation coupled with an elliptic second equation. Under appropriate parameter constraints, we establish the existence of unique globally bounded classical solutions for arbitrary spatial dimensions $n \geq 1$. Additionally, we employ carefully constructed Lyapunov functionals to analyze the long-term behavior of solutions, obtaining rigorous asymptotic stability results.