Invariant Measures for Interacting Continuous-State Population Models
Abstract
We study invariant probabilities for nonlinear two-population branching systems with competition, cooperation, or predator--prey interaction, driven by Brownian noise and spectrally positive stable jumps. Without immigration, every invariant probability of a conservative competitive or predator--prey system is concentrated on the boundary under either boundary convention. With nontrivial axis dynamics, coordinatewise continuation admits only the point mass at the origin. For cooperative systems, we give criteria for boundary concentration and interior stationarity, including a sharp criterion for bilinear cooperation with a common relative noise exponent. With immigration, explicit parameter conditions yield existence and stationary moment bounds. For square-root diffusions with positive immigration, these conditions ensure interior concentration and full support on the closed quadrant. Further lower bounds on immigration rates give uniqueness and convergence of time-averaged transition laws.
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BibTeX
@misc{ji2026invariant,
title = {{Invariant Measures for Interacting Continuous-State Population Models}},
author = {Lina Ji and Jie Xiong and Wen Xu and Xu Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15352},
eprint = {2610.15352},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15352},
note = {CC0},
}
RIS
TY - UNPB
TI - Invariant Measures for Interacting Continuous-State Population Models
AU - Lina Ji
AU - Jie Xiong
AU - Wen Xu
AU - Xu Yang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15352
UR - https://arxiv.org/abs/2610.15352
AB - We study invariant probabilities for nonlinear two-population branching systems with competition, cooperation, or predator--prey interaction, driven by Brownian noise and spectrally positive stable jumps. Without immigration, every invariant probability of a conservative competitive or predator--prey system is concentrated on the boundary under either boundary convention. With nontrivial axis dynamics, coordinatewise continuation admits only the point mass at the origin. For cooperative systems, we give criteria for boundary concentration and interior stationarity, including a sharp criterion for bilinear cooperation with a common relative noise exponent. With immigration, explicit parameter conditions yield existence and stationary moment bounds. For square-root diffusions with positive immigration, these conditions ensure interior concentration and full support on the closed quadrant. Further lower bounds on immigration rates give uniqueness and convergence of time-averaged transition laws.
ER -