The Fisher-KPP equation over simple graphs: Varied persistence states in river networks

TL;DR

This paper analyzes the long-term behavior of species spread in river networks modeled by the Fisher-KPP equation on simple graphs, revealing new persistence phenomena not seen in previous models with finite or single-line networks.

Contribution

It provides a comprehensive description of species persistence states in river networks modeled by Fisher-KPP equations on simple graphs, introducing the novel phenomenon of persistence below carrying capacity.

Findings

Identification of a trichotomy in long-term dynamics.

Discovery of persistence below carrying capacity.

Differences from models with finite or single-line river networks.

Abstract

In this article, we study the growth and spread of a new species in a river network with two or three branches via the Fisher-KPP advection-diffusion equation over some simple graphs with every edge a half infinite line. We obtain a rather complete description of the long-time dynamical behavior for every case under consideration, which can be loosely described by a trichotomy (see Remark 1.7), including two different kinds of persistence states as parameters vary. The phenomenon of "persistence below carrying capacity" revealed here appears new, which does not occur in related models of the existing literature where the river network is represented by graphs with finite-lengthed edges, or the river network is simplified to a single infinite line.