Long time dynamics of the Cauchy problem for the predator-prey model with cross-diffusion

TL;DR

This work analyzes the Cauchy problem for a cross-diffusion predator-prey model with prey-taxis and predator-taxis in $\mathbb{R}^N$ ($N=1,2,3$). It develops a framework based on scaling, heat-semigroup estimates, and a Leray-Schauder fixed-point approach to establish global strong solutions for small data and to analyze long-time behavior. It proves that strong solutions decay in $L^p$ and converge to the Gaussian heat-kernel profile, achieving optimal rates for $N=2,3$ and a slower rate for $N=1$, with convergence in $L^p$ to $(\int u_0) G(\cdot,t)$ and $(\int v_0) G(\cdot,t)$. These results advance the understanding of cross-diffusion systems by showing diffusion-dominated asymptotics and providing a rigorous global-existence theory under small initial data.

Abstract

This paper is concerned with a predator-prey model in $N$-dimensional spaces ($N=1, 2, 3$), given by \begin{align*}\left\{\begin{aligned} &\frac{\partial u}{\partial t}=Δu-χ\nabla\cdot(u\nabla v),\\ &\frac{\partial v}{\partial t}=Δv+ξ\nabla\cdot(v\nabla u), \end{aligned}\right. \end{align*} which describes random movement of both predator and prey species, as well as the spatial dynamics involving predators pursuing prey and prey attempting to evade predators. It is shown that any global strong solutions of the corresponding Cauchy problem converge to zero in the sense of $L^p$-norm for any $1<p\le \infty$, and also converge to the heat kernel with respect to $L^p$-norm for any $1\le p\le \infty$. In particular, the decay rate thereof is optimal in the sense that it is consistent with that of the heat equation in $\mathbb R^N$ ($N=2, 3$). Undoubtedly, the global existence of solutions appears to be among the most challenging topic in the analysis of this model. Indeed even in the one-dimensional setting, only global weak solutions in a bounded domain have been successfully constructed by far. Nevertheless, to provide a comprehensive understanding of the main results, we append the conclusion on the global existence and asymptotic behavior of strong solutions, although certain smallness conditions on the initial data are required.