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Invasion dynamics of super invaders: Elimination of Allee effects by a strategy at the range boundary

Yihong Du, Ling Li, Wenjie Ni, Narges Shabgard

TL;DR

This paper analyzes a free-boundary reaction-diffusion model for an invading species with Allee-type growth and demonstrates that maintaining a boundary density $\delta$ in the interval $(\theta_f^*,1)$ while moving the range edges eliminates Allee effects and yields robust invasion. The analysis reduces to a monostable regime and leverages semi-wave constructions to establish global spreading with asymptotic front speed $c^*$ and density $u\to1$, with the front profiles governed by $q^*$. The authors extend the framework to heterogeneous environments, prove local and global well-posedness, and provide numerical experiments that reveal transitions at $\delta=1$ and various outcomes across different $f$-types, highlighting a mechanism for “super invader” dynamics. These results have potential implications for understanding real-world invasions and trait evolution near invasion fronts, and offer a rigorous bridge between boundary-density strategies and classical spreading theory.

Abstract

Using a reaction-diffusion model with free boundaries in one space dimension for a single population species with density $u(t,x)$ and population range $[g(t), h(t)]$, we demonstrate that the Allee effects can be eliminated if the species maintains its population density at a suitable level at the range boundary by advancing or retreating the fronts. It is proved that with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density level at the range boundary is maintained at other levels. If the invading cane toads in Australia used this strategy at the range boundary to become a super invader, then our results may explain why toads near the invading front evolve to have longer legs and run faster.

Invasion dynamics of super invaders: Elimination of Allee effects by a strategy at the range boundary

TL;DR

This paper analyzes a free-boundary reaction-diffusion model for an invading species with Allee-type growth and demonstrates that maintaining a boundary density in the interval while moving the range edges eliminates Allee effects and yields robust invasion. The analysis reduces to a monostable regime and leverages semi-wave constructions to establish global spreading with asymptotic front speed and density , with the front profiles governed by . The authors extend the framework to heterogeneous environments, prove local and global well-posedness, and provide numerical experiments that reveal transitions at and various outcomes across different -types, highlighting a mechanism for “super invader” dynamics. These results have potential implications for understanding real-world invasions and trait evolution near invasion fronts, and offer a rigorous bridge between boundary-density strategies and classical spreading theory.

Abstract

Using a reaction-diffusion model with free boundaries in one space dimension for a single population species with density and population range , we demonstrate that the Allee effects can be eliminated if the species maintains its population density at a suitable level at the range boundary by advancing or retreating the fronts. It is proved that with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density level at the range boundary is maintained at other levels. If the invading cane toads in Australia used this strategy at the range boundary to become a super invader, then our results may explain why toads near the invading front evolve to have longer legs and run faster.

Paper Structure

This paper contains 21 sections, 11 theorems, 226 equations, 1 table.

Table of Contents

  1. Introduction
  2. Fisher-KPP type reaction-diffusion models
  3. Allee effects
  4. Reaction-diffusion models with a Stefan type free boundary
  5. Super invaders
  6. Reaction-diffusion models with traits
  7. A free boundary model for super invaders
  8. Main results and structure of the paper
  9. Longtime dynamics of \ref{['a']}
  10. Proof of Theorem \ref{['th1.3']}
  11. Numerical simulation and conjectures
  12. Extensions to heterogeneous environment and proof of some conjectures
  13. Local Existence and Uniqueness
  14. Comparison principles and a priori bounds
  15. Global existence
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose that $f$ is of Allee type ${\bf (f_A)}$ and $\delta\in (\theta_f^*, 1)$. Then for every initial function $u_0 \in X(h_0)$, a has a unique solution $(u(t,x), g(t), h(t))$ defined for all $t>0$. Moreover, as $t \to \infty$, Furthermore, there exist some constants $\hat{h}, \hat{g} \in \mathbb{R}$ such that where $(c^*, q^*)$ is the unique solution pair $(c,q)$ of

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • Lemma 3.5: Comparison Principle 1
  • ...and 10 more