Asymptotic spreading speeds of a road-field reaction-diffusion predator-prey model
Abstract
This paper is devoted to studying asymptotic spreading speeds of a road-field reaction-diffusion predator-prey system, in which predators diffuse faster on the straight line R× {0} than in the half plane R× (0,+∞). We give a sharp characterization of the spreading properties of solutions of the system with compactly supported initial values by appealing to the invasion speed of the prey when predators are absent, and the invasion speed of predators when the prey is saturated. Notably, since the line may accelerate the expansion of predators in a cone of directions, the invasion speed of predators varies with the direction. To overcome the difficulty arising from the coupling between the predator-prey interaction and the road-field interaction, we first derive pointwise comparisons between predators and the prey and then reduce the components of predators in the system to a truncated field-road system depending on the invasion speed of the prey. By using known results for the single-species field-road model, we further show the propagation properties of the truncated system. This step is very crucial. Once the propagation properties of the truncated system are established, we can obtain the expected spreading properties of solutions to the original system by applying the comparison principle and contradiction arguments, and constructing appropriate Lyapunov functionals.
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BibTeX
@misc{xu2026asymptotic,
title = {{Asymptotic spreading speeds of a road-field reaction-diffusion predator-prey model}},
author = {Yaqian Xu and Shigui Ruan and Zhi-Cheng Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37652},
eprint = {2609.37652},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37652},
}
RIS
TY - UNPB
TI - Asymptotic spreading speeds of a road-field reaction-diffusion predator-prey model
AU - Yaqian Xu
AU - Shigui Ruan
AU - Zhi-Cheng Wang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37652
UR - https://arxiv.org/abs/2609.37652
AB - This paper is devoted to studying asymptotic spreading speeds of a road-field reaction-diffusion predator-prey system, in which predators diffuse faster on the straight line R× {0} than in the half plane R× (0,+∞). We give a sharp characterization of the spreading properties of solutions of the system with compactly supported initial values by appealing to the invasion speed of the prey when predators are absent, and the invasion speed of predators when the prey is saturated. Notably, since the line may accelerate the expansion of predators in a cone of directions, the invasion speed of predators varies with the direction. To overcome the difficulty arising from the coupling between the predator-prey interaction and the road-field interaction, we first derive pointwise comparisons between predators and the prey and then reduce the components of predators in the system to a truncated field-road system depending on the invasion speed of the prey. By using known results for the single-species field-road model, we further show the propagation properties of the truncated system. This step is very crucial. Once the propagation properties of the truncated system are established, we can obtain the expected spreading properties of solutions to the original system by applying the comparison principle and contradiction arguments, and constructing appropriate Lyapunov functionals.
ER -