This paper explores the dynamics of the generalized Lotka-Volterra model on a network with degree heterogeneity. The authors develop a heterogeneous mean-field theory (HDMFT) to describe the fixed points of the model, which incorporates both degree heterogeneity and interaction strengths. They focus on the regime where there is typically a unique fixed point and analyze how the heterogeneity of the degree distribution impacts the fixed point. The paper presents a unified approach for both cooperative and competitive systems, using a mapping to the replicator model to extend the analysis to diverging abundances. The central role of an order parameter called the critical degree, \( q_c \), is highlighted, which distinguishes high-degree nodes that are more likely to go extinct in the competitive regime and low-degree nodes that tend to go extinct in the cooperative regime. The authors also discuss the linear stability analysis and the regime of validity of the mean-field assumption. The results are validated against numerical simulations, showing good agreement.This paper explores the dynamics of the generalized Lotka-Volterra model on a network with degree heterogeneity. The authors develop a heterogeneous mean-field theory (HDMFT) to describe the fixed points of the model, which incorporates both degree heterogeneity and interaction strengths. They focus on the regime where there is typically a unique fixed point and analyze how the heterogeneity of the degree distribution impacts the fixed point. The paper presents a unified approach for both cooperative and competitive systems, using a mapping to the replicator model to extend the analysis to diverging abundances. The central role of an order parameter called the critical degree, \( q_c \), is highlighted, which distinguishes high-degree nodes that are more likely to go extinct in the competitive regime and low-degree nodes that tend to go extinct in the cooperative regime. The authors also discuss the linear stability analysis and the regime of validity of the mean-field assumption. The results are validated against numerical simulations, showing good agreement.