Long-time behaviour and bifurcation analysis of a non-reciprocal two-species aggregation-diffusion system on the torus
Abstract
A two-species aggregation-diffusion system on the one-dimensional torus with antisymmetric cross-interactions modeling pursuit-evasion dynamics is studied. The self-interaction strength is used as a bifurcation parameter, and the linear stability of the homogeneous state is characterized by Fourier analysis. It is shown that non-reciprocal coupling enlarges the stability region. In a symmetric diffusion setting, exponential convergence to the homogeneous state is established for sufficiently weak self-interactions by means of an entropy argument, with a decay condition independent of the cross-interaction strength. Under suitable nondegeneracy and spectral assumptions, the first loss of stability is shown to occur through a Hopf bifurcation associated with a single spatial Fourier mode. A branch of time-periodic patterns is thereby obtained, along which the two species oscillate with a fixed phase difference. An explicit formula for the cubic Hopf coefficient is derived, and an essential dependence of the direction and stability of the bifurcating branch on the second spatial harmonic is identified. In the case of matching diffusion coefficients and interaction kernels, sufficiently strong non-reciprocity is shown to transform a subcritical stationary instability of the corresponding reciprocal system into a supercritical Hopf bifurcation with stable periodic solutions. An exchange-of-stability result is proved, and the theory is illustrated for Kuramoto, two-mode cosine, and top-hat interaction kernels.
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BibTeX
@misc{gess2026longtime,
title = {{Long-time behaviour and bifurcation analysis of a non-reciprocal two-species aggregation-diffusion system on the torus}},
author = {Benjamin Gess and Paul Nikolaev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04655},
eprint = {2611.04655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04655},
}
RIS
TY - UNPB
TI - Long-time behaviour and bifurcation analysis of a non-reciprocal two-species aggregation-diffusion system on the torus
AU - Benjamin Gess
AU - Paul Nikolaev
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04655
UR - https://arxiv.org/abs/2611.04655
AB - A two-species aggregation-diffusion system on the one-dimensional torus with antisymmetric cross-interactions modeling pursuit-evasion dynamics is studied. The self-interaction strength is used as a bifurcation parameter, and the linear stability of the homogeneous state is characterized by Fourier analysis. It is shown that non-reciprocal coupling enlarges the stability region. In a symmetric diffusion setting, exponential convergence to the homogeneous state is established for sufficiently weak self-interactions by means of an entropy argument, with a decay condition independent of the cross-interaction strength. Under suitable nondegeneracy and spectral assumptions, the first loss of stability is shown to occur through a Hopf bifurcation associated with a single spatial Fourier mode. A branch of time-periodic patterns is thereby obtained, along which the two species oscillate with a fixed phase difference. An explicit formula for the cubic Hopf coefficient is derived, and an essential dependence of the direction and stability of the bifurcating branch on the second spatial harmonic is identified. In the case of matching diffusion coefficients and interaction kernels, sufficiently strong non-reciprocity is shown to transform a subcritical stationary instability of the corresponding reciprocal system into a supercritical Hopf bifurcation with stable periodic solutions. An exchange-of-stability result is proved, and the theory is illustrated for Kuramoto, two-mode cosine, and top-hat interaction kernels.
ER -