Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on \(\mathbb{R}^N\)
The authors consider the parabolic-parabolic chemotaxis model with logistic source \[ \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla b) + u(a - bu), \\ \tau v_t = \Delta v - \lambda v + \mu v \end{cases} \] in \(\mathbb R^N\) with \(\tau = 1\) and positive parameters \(\chi, a, b, \lambda, \mu\), and prove that that the spreading speed for solutions to nontrivial initial data with finite support is \(2\sqrt a\), provided \(\chi < \frac{4b}{N\mu}\). Since the spreading speed for the Fisher-KPP equation, i.e.\ for the first equation in the system above with \(\chi = 0\), is also \(2\sqrt a\), this shows that (at least when \(\chi\) is small or when \(b\) is large) chemotaxis has no effect on the spreading speed. Thereby, the authors transfer results such as [\textit{R. B. Salako} et al., J. Math. Biol. 79, No. 4, 1455--1490 (2019; Zbl 1479.35116)] for the parabolic-elliptic version, i.e\ the system above with \(\tau = 0\), to the fully parabolic setting. Upper bounds on the spreading speed are obtained by applying the comparison principle to \(w = u + \frac{\chi}{2\mu} |\nabla v|^2\) which fulfills \(w_t \le \Delta w + a w\). This technique makes essential use of the assumption \(\tau=1\).
scientific article