Department of Mathematics, The University of Iowa, Iowa City IA 52242, USA.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong S.A.R., China.
Math Biosci Eng. 2022 Jun 6;19(8):8107-8131. doi: 10.3934/mbe.2022379.
This paper is concerned with the traveling wave solutions of a singular Keller-Segel system modeling chemotactic movement of biological species with logistic growth. We first show the existence of traveling wave solutions with zero chemical diffusion in $ \mathbb{R} $. We then show the existence of traveling wave solutions with small chemical diffusion by the geometric singular perturbation theory and establish the zero diffusion limit of traveling wave solutions. Furthermore, we show that the traveling wave solutions are linearly unstable in the Sobolev space $ H^1(\mathbb{R}) \times H^2(\mathbb{R}) $ by the spectral analysis. Finally we use numerical simulations to illustrate the stabilization of traveling wave profiles with fast decay initial data and numerically demonstrate the effect of system parameters on the wave propagation dynamics.
这篇论文研究了一个奇异的 Keller-Segel 系统的行波解，该系统用于描述具有逻辑增长的生物物种的趋化运动。我们首先证明了在(\mathbb{R})中存在零化学扩散的行波解。然后，我们通过几何奇异摄动理论证明了小化学扩散的行波解的存在，并建立了行波解的零扩散极限。此外，我们通过谱分析证明了行波解在 Sobolev 空间(H^1(\mathbb{R}) \times H^2(\mathbb{R}))中是线性不稳定的。最后，我们通过数值模拟说明了快速衰减初始数据的行波轮廓的稳定性，并数值演示了系统参数对波传播动力学的影响。
