Mesoscopic derivation of predator-prey dynamics via diffusive fast-reaction asymptotics: cross-diffusion and fear effects
Abstract
We propose a kinetic derivation of a spatial predator-prey model with cross-diffusion interactions and a Beddington-DeAngelis functional response. At the mesoscopic level, we divide the two populations into distinct behavioral states: prey can be active or hidden in a refuge, while predators can be either searching or handling. The microscopic state of each subpopulation is described in terms of group size and a continuous directional velocity variable, with state transitions governed by suitable Boltzmann-type interaction operators. By combining a diffusive scaling with a slow-fast formulation of the kinetic reaction operators, we formally derive a macroscopic reaction-cross-diffusion system for the total population densities of prey and predators. The limiting model consistently recovers the Beddington-DeAngelis functional response along with its associated nonlinear diffusion terms. Crucially, the kinetic upscaling yields a predator-dependent prey growth term taking a logistic form with an effective carrying capacity. This carrying capacity depends non-monotonically on predator density because prey redistribute between active and refuge states. Since density-dependent competition acts separately within these behavioral states, intermediate predator levels reduce aggregate crowding and thereby increase the effective carrying capacity of the total prey population. Overall, our derivation provides a mechanistic link connecting predator-induced behavioral shifts, refuge use, and macroscopic cross-diffusion.
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BibTeX
@misc{bisi2026mesoscopic,
title = {{Mesoscopic derivation of predator-prey dynamics via diffusive fast-reaction asymptotics: cross-diffusion and fear effects}},
author = {Marzia Bisi and Andrea Bondesan and Maria Groppi and Cinzia Soresina},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.14132},
eprint = {2611.14132},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.14132},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mesoscopic derivation of predator-prey dynamics via diffusive fast-reaction asymptotics: cross-diffusion and fear effects
AU - Marzia Bisi
AU - Andrea Bondesan
AU - Maria Groppi
AU - Cinzia Soresina
PY - 2026
DA - 2026/11/10
PB - arXiv
DO - 10.48550/arxiv.2611.14132
UR - https://arxiv.org/abs/2611.14132
AB - We propose a kinetic derivation of a spatial predator-prey model with cross-diffusion interactions and a Beddington-DeAngelis functional response. At the mesoscopic level, we divide the two populations into distinct behavioral states: prey can be active or hidden in a refuge, while predators can be either searching or handling. The microscopic state of each subpopulation is described in terms of group size and a continuous directional velocity variable, with state transitions governed by suitable Boltzmann-type interaction operators. By combining a diffusive scaling with a slow-fast formulation of the kinetic reaction operators, we formally derive a macroscopic reaction-cross-diffusion system for the total population densities of prey and predators. The limiting model consistently recovers the Beddington-DeAngelis functional response along with its associated nonlinear diffusion terms. Crucially, the kinetic upscaling yields a predator-dependent prey growth term taking a logistic form with an effective carrying capacity. This carrying capacity depends non-monotonically on predator density because prey redistribute between active and refuge states. Since density-dependent competition acts separately within these behavioral states, intermediate predator levels reduce aggregate crowding and thereby increase the effective carrying capacity of the total prey population. Overall, our derivation provides a mechanistic link connecting predator-induced behavioral shifts, refuge use, and macroscopic cross-diffusion.
ER -