Moderate intervention paradox in threshold-intensity regulated rock–paper–scissors competition systems

The cyclic competition governed by the rock–paper–scissors (RPS) game plays a crucial role in maintaining biodiversity and ecological stability in ecosystems. As an ecological management tool, human intervention often regulates population density to prevent species extinction or overpopulation. However, the synergistic effects of intervention thresholds and intensities on the stability of cyclic competition systems have not been sufficiently studied. In this paper, we extend the classical RPS dynamical framework by introducing a threshold-intensity-based intervention mechanism targeting a single species and construct a three-type nonlinear dynamical model. We systematically analyze the resulting dynamic behavior through linear stability analysis and numerical simulations at the mean-field level. As a result, we find the existence of four types of equilibria, demonstrating that only two- and three-species coexistence can achieve asymptotic stability within specific parameter ranges. While increasing intervention intensity can promote multi-species coexistence, excessive intervention intensity in environments with high competition-reproductive rates can lead to the dominance of a single species. This study reveals the “paradox of moderate intervention.” Interventions can prevent the extinction of target species, but two-way regulation can limit density growth, potentially weakening long-term competitive advantage. Our study provides a theoretical basis for quantifying the relationship between the threshold and intensity of ecological interventions and guides for designing intervention strategies that balance conservation benefits and management costs.