A unifying theory for two-dimensional spatial redistribution kernels with applications in population spread modelling

Area of Science:

• Ecology
• Mathematical Biology
• Theoretical Ecology

Background:

• Ecological dispersal models commonly use isotropic redistribution kernels (e.g., Gaussian, Laplace, Bessel) to represent organism movement.
• These kernels are often special cases of the Whittle-Matérn-Yasuda (WMY) kernel, originally derived from Fickian diffusion assumptions.
• A unified framework for these kernels is lacking, hindering flexible ecological modeling.

Purpose of the Study:

• To derive a novel, unified kernel family for organism dispersal based on fractal diffusion.
• To connect existing redistribution kernels under a single, flexible modeling framework.
• To improve the predictive performance of dispersal models.

Main Methods:

• Derived the Whittle-Matérn-Yasuda (WMY) kernel family using a non-Fickian 2D diffusion equation with fractal media and constant settling hazard.
• Unified established redistribution kernels within this novel framework.
• Replaced the Gaussian kernel with the WMY kernel in a mountain pine beetle dispersal model.

Main Results:

• Successfully derived and unified several redistribution kernels under the WMY framework.
• Demonstrated improved predictive performance in a mountain pine beetle dispersal model by using the WMY kernel instead of the Gaussian kernel.
• Introduced the product-Whittle-Matérn-Yasuda approximation, which significantly accelerates computations for large datasets.

Conclusions:

• The WMY kernel provides a more flexible and accurate framework for modeling organism dispersal patterns.
• The novel derivation based on fractal diffusion offers new insights into movement mechanisms in heterogeneous environments.
• The product-WMY approximation enables efficient application of these advanced models to large-scale ecological data.