Propagation dynamics for nonlocal dispersal predator-prey systems in shifting habitats: A Hamilton-Jacobi approach

TL;DR

This study analyzes the spreading speeds of predator-prey systems with nonlocal dispersal in shifting habitats using Hamilton-Jacobi equations, revealing complex interactions among spreading speeds, habitat shifts, and initial conditions.

Contribution

It introduces a Hamilton-Jacobi framework to classify spreading speeds and provides explicit formulas, highlighting the role of nonlocal dispersal in propagation dynamics.

Findings

Complete classification of prey spreading speeds under various conditions

Derived upper bounds for predator spreading speeds considering habitat shift

Identified distinct mechanisms leading to nonlocal determinacy results

Abstract

This paper is concerned with the spreading speeds of nonlocal dispersal predator-prey systems in shifting habitats under general initial conditions. By employing geometric optics techniques and theory of viscosity solutions, we reformulate the problem into the study of Hamilton-Jacobi equations. Through a detailed analysis of the structure of viscosity solutions, we provide a complete classification of explicit formulas for the spreading speed of the prey population, especially in cases where it invades the habitat more rapidly than predators, yielding two fundamentally distinct ``nonlocal determinacy'' results derived by different mechanisms. We also obtain an upper bound for spreading speed of the predators, incorporating the decay rate of the initial data and the speed of shifting habitats. These findings demonstrate that there are complex connections among spreading speeds, habitat shifting speed and initial conditions, and emphasize the significance of nonlocal dispersal in determining the propagation dynamics of predator-prey systems.