Heteroclinic cycles and long-time behavior in a singularly perturbed three-species Lotka--Volterra system
Abstract
We study a three-species competitive Lotka–Volterra system with cyclic dominance, motivated by rock–paper–scissors interactions in ecological communities. The model contains a singularly strong interaction coefficient of order 1/ε, representing a regime in which one species exerts a much stronger competitive pressure on another. We show that, despite this strong asymmetry, the cyclic dominance mechanism persists. Our main result is a quantitative description of one complete dominance pattern in the singular regime 0<ε≪1. Starting near the state where the third species dominates, we prove that the solution successively passes near the dominance states of the second and first species before returning to a comparable configuration. We obtain logarithmic estimates for the return time and for the population densities at the return section. In particular, although one species becomes extremely small at the beginning and at the end of each pattern, it recovers to an order-one density during the cycle. We then prove that this pattern repeats indefinitely and that the omega-limit set is composed by three heteroclinic orbits on the boundary of the positive orthant, connecting the three single-species equilibria. Thus the long-time behavior is cyclic but aperiodic, with increasingly long excursions near successive single-species states.
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BibTeX
@misc{ducrot2026heteroclinic,
title = {{Heteroclinic cycles and long-time behavior in a singularly perturbed three-species Lotka--Volterra system}},
author = {Arnaud Ducrot and Quentin Griette and Quang-Vinh Tran},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06908},
eprint = {2610.06908},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05785138v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Heteroclinic cycles and long-time behavior in a singularly perturbed three-species Lotka--Volterra system
AU - Arnaud Ducrot
AU - Quentin Griette
AU - Quang-Vinh Tran
PY - 2026
DA - 2026/10/01
PB - HAL
DO - 10.48550/arxiv.2610.06908
UR - https://hal.science/hal-05785138v1
AB - We study a three-species competitive Lotka–Volterra system with cyclic dominance, motivated by rock–paper–scissors interactions in ecological communities. The model contains a singularly strong interaction coefficient of order 1/ε, representing a regime in which one species exerts a much stronger competitive pressure on another. We show that, despite this strong asymmetry, the cyclic dominance mechanism persists. Our main result is a quantitative description of one complete dominance pattern in the singular regime 0<ε≪1. Starting near the state where the third species dominates, we prove that the solution successively passes near the dominance states of the second and first species before returning to a comparable configuration. We obtain logarithmic estimates for the return time and for the population densities at the return section. In particular, although one species becomes extremely small at the beginning and at the end of each pattern, it recovers to an order-one density during the cycle. We then prove that this pattern repeats indefinitely and that the omega-limit set is composed by three heteroclinic orbits on the boundary of the positive orthant, connecting the three single-species equilibria. Thus the long-time behavior is cyclic but aperiodic, with increasingly long excursions near successive single-species states.
ER -