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.

中文翻译：

---

简单图形上的Fisher-KPP方程：河网中的各种持久状态。

在本文中，我们基于广泛使用的Fisher-KPP对流扩散方程，研究了从河网中两个或三个分支相遇的位置扩散的新物种的动力学行为。这个局部河流系统由一些简单的图表示，每个边的一半为无限长的线，在一个顶点相交。我们对所考虑的每种情况均获得了相当完整的长期动力学行为描述，根据河流支流中的水流速度，可以将其分为三种不同类型（称为三分法），这主要取决于拓扑结构。代表当地河流系统的图形结构以及分支的横截面区域。三分法包括两种不同的持久状态，这种状态称为“