Hyperbolic limit for a biological invasion

Danielle Hilhorst; Yongjung Kim; Thanh Nam Nguyen; Hyunjoon Park

doi:10.3934/dcdsb.2023070

Article Contents

2023, Volume 28, Issue 12: 6142-6158. Doi: 10.3934/dcdsb.2023070

This issue Previous Article Spreading dynamics for a predator-prey system with two predators and one prey in a shifting habitat Next Article Cross-diffusion predator–prey model derived from the dichotomy between two behavioral predator states

Hyperbolic limit for a biological invasion

1.
CNRS and Laboratoire de Mathématiques, University Paris-Saclay, France
2.
Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Korea
3.
Sorbonne Université, Université Paris-Diderot, CNRS, Laboratoire Jacques-Louis Lions and Laboratoire de, Biologie Computationnelle et Quantitative, France

^*Corresponding author: Hyunjoon Park
^*Corresponding author: Hyunjoon Park

Dedicated to the memory of Professor Masayasu Mimura

Received: August 2022

Revised: March 2023

Early access: April 2023

Published: December 2023

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Abstract

In a spatially heterogeneous environment the propagation speed of a biological invasion varies in space. The traveling wave theory in a homogeneous case is not extended to a heterogeneous case. Taking a singular limit in a hyperbolic scale is a good way to study such a wave propagation with constant speed. The goal of this project is to understand the effect of biological diffusion on the wave speed in a spatial heterogeneous environment. For this purpose, we consider

$ U_t = {\varepsilon}(\gamma(s)U)_{xx}+{1\over{\varepsilon}}U(1-{U/m(x)}), $

where $ m $ is a nonconstant carrying capacity, $ s = {U\over m} $ is a starvation measure and $ \gamma(s) = s^{\tilde{k}}, \tilde{k} \ge 1 $. The diffusion is a starvation driven diffusion. We show that the diffusion speed is constant even if $ m $ is nonconstant.

Keywords:

Singular perturbation,
Nonlinear PDE,
Reaction diffusion equations,
Interface problems; Population dynamics.

Mathematics Subject Classification: Primary: 35B25, 35G31, 35K57, 82C24; Secondary: 92D25.

Citation:

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Figure 1. A linearly connected patch system is viewed as a general diffusion in $ {{\bf R}}^1 $

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Figure 2. Numerical simulation of (7) and (IP). The black line indicates the curve which moves according to (IP) and other colors indicates the values of the function $ u^{\varepsilon} = U^{\varepsilon}/m $. The yellow color indicates that the solution is close to 1 and the blue color indicates that the solution is equal to 0

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