# The conditions of failing and successful plant invasions with free boundary

## TL;DR

This paper uses a mathematical model to study how invasive plants spread and whether they will successfully establish in new environments.

## Contribution

The paper introduces threshold parameters to determine the success or failure of plant invasions based on initial conditions and model dynamics.

## Key findings

- Invasive plants persist if threshold parameters exceed certain values with large initial populations.
- Failure of invasion occurs when threshold parameters are below critical values with small initial populations.
- The model uses free boundaries to represent seed-driven spreading frontiers.

## Abstract

In this paper, a reaction-diffusion model is proposed to investigate the growing process of invasive plants. Two free boundaries are introduced to describe the spreading frontiers are caused only by the seeds. The main objective is to realize the variation of the invasive plants and the free boundaries. Similar to the basic reproduction number of epidemic diseases, we define the threshold parameters \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J_0$$\end{document} and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J_0^F(t)$$\end{document} to discuss the dynamic behaviors of equilibrium solutions of invasive plants model with free boundary conditions. Based on them, we give the sufficient conditions for whether plants invade successfully or not. We show that the invasive plants will successfully persist in the new environment if \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J^F_0(0)<1<J_0$$\end{document} with large initial values or \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J_0^F(t_0)\ge 1 ~(\exists t_0\ge 0)$$\end{document}, while failing and dying out in the long run if \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J^F_0(0)<1$$\end{document} with small initial values or \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J_0\le 1$$\end{document}.