Recolonisation by diffusion can generate increasing rates of spread.

Area of Science:

• Mathematical Biology
• Ecology
• Population Dynamics

Background:

• Dispersal is often modeled using diffusion, but this typically results in speeds too slow to match observed ecological spread rates.
• Long-distance dispersal events, modeled by integro-differential equations (IDEs) with exponentially unbounded (EU) kernels, are commonly invoked to explain rapid colonization.
• Classical models like Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) equations are generally considered too slow for observed spread dynamics.

Purpose of the Study:

• To demonstrate that classical reaction-diffusion models can replicate rapid colonization patterns typically associated with IDEs.
• To show that exponentially unbounded (EU) initial population distributions can drive fast and accelerating spread in reaction-diffusion models.
• To establish a mathematical framework for understanding reaction-diffusion models with EU initial data.

Main Methods:

• Analysis of classical reaction-diffusion models (e.g., FKPP type).
• Investigation of the impact of exponentially unbounded (EU) initial population distributions.
• Comparison of solutions from reaction-diffusion models with EU initial data to solutions from IDEs with EU kernels.

Main Results:

• Reaction-diffusion models with EU initial data produce colonization patterns highly similar to those generated by IDEs with EU kernels.
• Both model types exhibit comparable accelerating rates of spread and solution flattening.
• EU initial data, potentially arising from colonization-retraction events, can explain rapid spread without invoking long-distance dispersal.

Conclusions:

• Classical reaction-diffusion models are more versatile for explaining rapid ecological spread than previously thought.
• Exponentially unbounded initial conditions provide a viable alternative mechanism for fast dispersal, simplifying modeling.
• The findings bridge a gap in the mathematical theory of reaction-diffusion systems with non-standard initial conditions.