Biological invasions and epidemics with nonlocal diffusion along a line

Area of Science:

• Mathematical Biology
• Reaction-Diffusion Systems
• Nonlocal Diffusion Models

Background:

• Reaction-diffusion systems are fundamental to modeling spatial spread.
• Transportation networks can influence biological invasions and epidemic propagation.
• Previous work modeled local diffusion using the Laplacian.

Purpose of the Study:

• To quantify the enhancement of reaction-diffusion processes by a line with nonlocal diffusion.
• To model the impact of transportation networks on biological spread.
• To rigorously analyze propagation speeds and their parameters.

Main Methods:

• Mathematical analysis of reaction-diffusion equations with nonlocal diffusion terms.
• Proof of existence for global propagation speed.
• Characterization of parameters influencing speed enhancement.
• Investigation of interplay between diffusion in the plane and on the line.

Main Results:

• Demonstrated significant enhancement of propagation speed by the line, even with large diffusion intensity or support size.
• Uncovered unexpected regularity properties of the model.
• Identified subtle effects from the interplay of diffusion parameters.
• Explored a transport-based model for line displacement in epidemic contexts.

Conclusions:

• Nonlocal diffusion on a line can substantially boost propagation speeds in reaction-diffusion systems.
• The model provides a more comprehensive understanding of network effects on spatial spread.
• Findings have implications for managing biological invasions and epidemics.