This paper is concerned with the following spatiotemporal population-toxicant model with toxicant-taxis in a bounded domain \(\Omega \subset \mathbb {R}^n(n\ge 1)\) with inhomogeneous Robin boundary conditions \(\begin{aligned} {\left\{ \begin{array}{ll} u_t=d\Delta u+\chi \nabla \cdot ( u\nabla w)+u(1-u)-\sigma uw,& x\in \Omega ,t>0\\ w_t=\varepsilon \Delta w-\mu w-\lambda uw,& x\in \Omega ,t>0,\\ (d\nabla u+\chi u\nabla w)\cdot \nu =0,\ \ \nabla w\cdot \nu =\xi (h(x,t)-w), & x\in \partial \Omega ,t>0\\ u(x,0)=u_0(x),\ \ w(x,0)=w_0(x),& x\in \Omega , \end{array}\right. } \end{aligned}\) where \(u=u(x,t)\) and \(w=w(x,t)\) denote the population density and toxicant concentration at location x and time t, respectively. Here the toxicant enters the environment through the boundary with a temporally and spatially heterogeneous ambient toxicant density h(x, t). Under suitable assumptions on h(x, t), we first establish the global existence of classical solutions in two-dimensional spaces ( \(n=2\) ). Moreover, we show that every solution (u, w) converges to (1, 0) uniformly if h(x, t) decays to zero as \(t \rightarrow \infty \) with a mild rate satisfying \(\begin{aligned} \lim \limits _{t\rightarrow \infty }\int _t^{t+1}\Vert h(\cdot ,\tau )\Vert _{L^1(\partial \Omega )}d\tau =0. \end{aligned}\) If \(h(x,t)\equiv h(x)\gneqq 0\) with \(0<h_0=\sup _{x\in \partial \Omega }h(x)\) , we establish the existence of non-constant positive steady states in all dimensional spaces ( \(n\ge 1\) ) under the condition \(0<h_0< h^*:=\min \Big \{\frac{1}{\sigma },\frac{d}{\chi }\Big \}.\) We further show that this non-constant steady state is unique and globally asymptotically stable if \(h_0\) is sufficiently small. On the other hand, we prove that the species u is uniformly persistent if \(\sigma <1/h_0\) , while the toxicant-only steady state is globally asymptotically stable if \(\sigma >1/M_h\) with some constant \(M_h>0\) smaller than \(h_0\) .
