The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system,where χ, a, b, λ, and µ are positive constants. Assume b > χµ. Then if in addition, 1 + *

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The current paper is concerned with the forced waves of Keller-Segel chemoattraction systems in shifting environments of the form, u t = u xx − χ(uv x) x + u(r(x − ct) − bu), x ∈ R 0 = v xx − νv + µu, x ∈ R, (0.1) where χ, b, ν, and µ are positive constants, c ∈ R, the resource function r(x) is globally Hölder continuous, bounded, r * = sup x∈R r(x) > 0, r(±∞) := lim x→±∞ r(x) exist, and either r(−∞) < 0 < r(∞), or r(±∞) < 0. Assume that b > 2χµ. In the case that r(−∞) < 0 < r(∞), it is shown that (0.1) has a forced wave solution connecting (r * b , µ ν r * b) and (0, 0) with speed c provided that c > χµr * 2 √ ν(b−χµ) − 2 r * (b−2χµ) b−χµ. In the case that r(±∞) < 0, it is shown that (0.1) has a forced wave solution connecting (0, 0) and (0, 0) with speed c provided that χ is sufficiently small and λ ∞ > 0, where λ ∞ is the generalized principal eigenvalue of the operator u(•) → u xx (•) + cu x (•) + r(•)u(•) on R in certain sense. Some numerical simulations are also carried out. The simulations indicate the existence of forced wave solutions in some parameter regions which are not covered in the theoretical results, induce several problems to be further studied, and also provide some illustration of the theoretical results.

<p style='text-indent:20px;'>In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in{{\mathbb R}}^N,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document}</tex-math></inline-formula> are positive constants and <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula> is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption <inline-formula><tex-math id="M5">\begin{document}$ b&gt;\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, the global existence of a unique classical solution <inline-formula><tex-math id="M6">\begin{document}$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $\end{document}</tex-math></inline-formula> of (1) with <inline-formula><tex-math id="M7">\begin{document}$ u(x,0;u_0, v_0) = u_0(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ v(x,0;u_0, v_0) = v_0(x) $\end{document}</tex-math></inline-formula> for every nonnegative, bounded, and uniformly continuous function <inline-formula><tex-math id="M9">\begin{document}$ u_0(x) $\end{document}</tex-math></inline-formula>, and every nonnegative, bounded, uniformly continuous, and differentiable function <inline-formula><tex-math id="M10">\begin{document}$ v_0(x) $\end{document}</tex-math></inline-formula>. Next, under the same assumption <inline-formula><tex-math id="M11">\begin{document}$ b&gt;\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function <inline-formula><tex-math id="M12">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula> is bounded below by a positive constant independent of <inline-formula><tex-math id="M13">\begin{document}$ (u_0, v_0) $\end{document}</tex-math></inline-formula> when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function <inline-formula><tex-math id="M14">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>. We show that there is <inline-formula><tex-math id="M15">\begin{document}$ K = K(a,\lambda,N)&gt;\frac{N}{4} $\end{document}</tex-math></inline-formula> such that if <inline-formula><tex-math id="M16">\begin{document}$ b&gt;K \chi\mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ \lambda\geq \frac{a}{2} $\end{document}</tex-math></inline-formula>, then for every strictly positive initial function <inline-formula><tex-math id="M18">\begin{document}$ u_0(\cdot) $\end{document}</tex-math></inline-formula>, it holds that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \lim\limits_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big] = 0. $\end{document} </tex-math></disp-formula></p>