Influence of a Depensatory Growth Function for a Renewable Resource in an Open-Access Fisheries Model

The main purpose of this study is to analyze the dynamics of a simple open-access fishery model, considering the influence of the Allee effect (or depensation) on the exploited resource. The model incorporates continuous variation in both the biomass population growth and the nominal fishing effort over time. This framework modifies the classical Smith model, a foundational bioeconomic model in renewable resource management. A notable advantage of the proposed approach is its ability to address scenarios where the exploited resource faces extinction risk, threatening the sustainability of the fishery. Additionally, the inclusion of an asymmetric growth function enhances the model’s adaptability to real-world situations where empirical data deviate from logistic growth assumptions. Our findings reveal that the mathematical properties of the original model undergo significant changes. Specifically, the equilibrium point \(\left ( 0,0\right )\) becomes an attractor for all parameter values due to the introduction of a separator curve defined by the stable manifold of the equilibrium point \(\left ( m,0\right )\) , which emerges as a consequence of depensation. This result underscores the critical risk of resource extinction when the biomass-to-effort ratio is low, ultimately leading to fishery collapse. We conclude that the dynamic implications of depensation are highly sensitive to the specific algebraic form used to represent this phenomenon in the resource’s natural growth equation.