Hopf bifurcation in a predator-prey model with diffusion and spatial heterogeneity
Abstract
In this paper, we investigate how spatial heterogeneity affects the stability of positive equilibria in one-dimensional predator--prey reaction--diffusion systems with Neumann boundary conditions. Our examples show that spatial heterogeneity can lead to instability even when the positive equilibrium is unique. For the smooth, spatially heterogeneous coefficient families constructed here, this equilibrium loses linear stability as a parameter in the reaction coefficients varies. At the critical parameter value, a simple conjugate pair of eigenvalues crosses the imaginary axis transversely, and a Hopf bifurcation produces a local branch of strictly positive, nonstationary classical time-periodic solutions. The examples include two families with spatially concentrated prey, one with an everywhere positive and the other with an everywhere negative predator intrinsic growth rate. A third family has a spatially concentrated predator population and an everywhere negative predator intrinsic growth rate.
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BibTeX
@misc{lou2026hopf,
title = {{Hopf bifurcation in a predator-prey model with diffusion and spatial heterogeneity}},
author = {Yuan Lou and Ming-Zhen Xin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02630},
eprint = {2610.02630},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02630},
}
RIS
TY - UNPB
TI - Hopf bifurcation in a predator-prey model with diffusion and spatial heterogeneity
AU - Yuan Lou
AU - Ming-Zhen Xin
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02630
UR - https://arxiv.org/abs/2610.02630
AB - In this paper, we investigate how spatial heterogeneity affects the stability of positive equilibria in one-dimensional predator--prey reaction--diffusion systems with Neumann boundary conditions. Our examples show that spatial heterogeneity can lead to instability even when the positive equilibrium is unique. For the smooth, spatially heterogeneous coefficient families constructed here, this equilibrium loses linear stability as a parameter in the reaction coefficients varies. At the critical parameter value, a simple conjugate pair of eigenvalues crosses the imaginary axis transversely, and a Hopf bifurcation produces a local branch of strictly positive, nonstationary classical time-periodic solutions. The examples include two families with spatially concentrated prey, one with an everywhere positive and the other with an everywhere negative predator intrinsic growth rate. A third family has a spatially concentrated predator population and an everywhere negative predator intrinsic growth rate.
ER -