Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis

Mario Fuest

doi:10.1137/20M1344536

paper

Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis

arXiv:2004.04515 · doi:10.1137/20M1344536

Abstract

We study the system \begin{align*}\label{prob:star} \tag{ $⋆$ } \begin{cases} u_t = D_1 Δu - χ_1 \nabla \cdot (u \nabla v) + u(λ_1 - μ_1 u + a_1 v) \\ v_t = D_2 Δv + χ_2 \nabla \cdot (v \nabla u) + v(λ_2 - μ_2 v - a_2 u) \end{cases} \end{align*} (inter alia) for $D_{1}, D_{2}, χ_{1}, χ_{2}, λ_{1}, λ_{2}, μ_{1}, μ_{2}, a_{1}, a_{2} > 0$ in smooth, bounded domains $Ω \subset R^{n}$ , $n \in {1, 2, 3}$ . Without any further restrictions on these parameters, we prove that there exists a constant stable steady state $(u_{⋆}, v_{⋆}) \in [0, \infty)^{2}$ , meaning that there is $ε > 0$ such that, if $u_{0}, v_{0} \in W^{2, 2} (Ω)$ are nonnegative with $\partial_{ν} u_{0} = \partial_{ν} v_{0} = 0$ in the sense of traces and \begin{align*} \|u_0 - u_\star\|_{W^{2,2}(Ω)} + \|v_0 - v_\star\|_{W^{2,2}(Ω)} < \varepsilon, \end{align*} then there exists a global classical solution $(u, v)$ of \eqref{prob:star} with initial data $u_{0}, v_{0}$ converging to $(u_{⋆}, v_{⋆})$ in $W^{2, 2} (Ω)$ . Moreover, the convergence rate is exponential, except for the case $λ_{2} μ_{1} = λ_{1} a_{2}$ , where it is is only algebraical.

arXiv comment27 pages; changed title and incorporated referee comments

References in corpus (1)

Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system

Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis

Abstract

References in corpus (1)

Cited by in corpus (9)