Methods and Applications of Analysis

Volume 28 (2021)

Number 4

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part III

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Ecological and evolutionary dynamics in periodic and advective habitats

Pages: 423 – 452



Shuang Liu (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China)

Yuan Lou (Department of Mathematics, Ohio State University, Columbus, Oh., U.S.A.; and School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, China)


We study the dynamics of reaction-diffusion-advection models for single and two competing species in one-dimensional periodic habitats, where the individuals are subject to both diffusion and advection. We investigate the monotone dependence of the principal eigenvalue on diffusion and drift rates. As applications, we first consider the persistence and spatial spreading of a single species and establish the critical threshold for the persistence as well as the monotone dependence of the minimal wave speed on the drift rate. We also consider two competing species model and study the local and global stability of semi-trivial steady states. Furthermore, the existence of evolutionarily singular strategies is established, which helps gain deeper insight into the evolution of dispersal in advective environments.


reaction-diffusion-advection, persistence, spreading, competition, singular strategy, periodic habitat

2010 Mathematics Subject Classification

35K57, 35P15, 35Q92, 92D25, 92-xx

Shuang Liu was partially supported by the Outstanding Innovative Talents Cultivation Funded Programs 2018 of Renmin Univertity of China and the NSFC grant No. 11571364.

Yyan Lou was partially supported by the NSF grant DMS-1853561.

Received 6 August 2020

Accepted 8 December 2020

Published 10 June 2022