Abstract
In this article, we study the dynamical behaviour of a new species spreading from a location in a river network where two or three branches meet, based on the widely used Fisher-KPP advection–diffusion equation. This local river system is represented by some simple graphs with every edge a half infinite line, meeting at a single vertex. We obtain a rather complete description of the long-time dynamical behaviour for every case under consideration, which can be classified into three different types (called a trichotomy), according to the water flow speeds in the river branches, which depend crucially on the topological structure of the graph representing the local river system and on the cross section areas of the branches. The trichotomy includes two different kinds of persistence states, and the state called “persistence below carrying capacity” here appears new.
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Notes
It is also well known that (1.1) has two positive steady-states \(\phi _\beta \) and 1 if \(\beta \ge c_*\), and 1 is the only positive steady-state if \(0<\beta <c_*\).
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We thank the two anonymous referees for their suggestions and comments which helped to improve the presentation of the paper. This research was partially supported by the Australian Research Council (No. DP190103757), the NSF of China (No. 11671262, 11671175, 11571200), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (No. PPZY2015A013) and Qing Lan Project of Jiangsu Province.
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Du, Y., Lou, B., Peng, R. et al. The Fisher-KPP equation over simple graphs: varied persistence states in river networks. J. Math. Biol. 80, 1559–1616 (2020). https://doi.org/10.1007/s00285-020-01474-1
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DOI: https://doi.org/10.1007/s00285-020-01474-1