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

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Ecologists can now unify dispersal models using the Whittle-Matérn-Yasuda (WMY) kernel, derived from fractal diffusion. This new framework improves predictions for organism dispersal patterns, like the mountain pine beetle.

Keywords:
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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.