This paper presents a heterogeneous mean-field analysis of the generalized Lotka-Volterra model on a network. The study investigates the dynamics of this model with a network structure, focusing on the role of degree heterogeneity. A mean-field theory is developed that incorporates degree heterogeneity, allowing for the description of fixed points in terms of a few order parameters. The analysis extends to cases with diverging abundances using a mapping to the replicator model, providing a unified approach for both cooperative and competitive systems. The critical degree $ g_c $ is identified as a key parameter, distinguishing high-degree nodes that are more likely to go extinct in competitive regimes and low-degree nodes that tend to go extinct in cooperative regimes. The model is defined for N interacting species with abundances $ x_i $, evolving according to a differential equation involving the adjacency matrix and interaction strengths. A statistical description is developed using tools from statistical physics, leading to a dynamical mean-field theory (DMFT) that predicts averages of p-point functions. The analysis considers both homogeneous and heterogeneous cases, with the homogeneous case corresponding to $ \sigma = 0 $, where all interactions have the same strength. The theory is applied to both cooperative and competitive systems, showing how the heterogeneity of the underlying degree distribution impacts the heterogeneity of the fixed point. The results are validated against numerical simulations, demonstrating good agreement with theoretical predictions. The study also explores the diverging regime, where the Lotka-Volterra model is better described by the replicator model. The critical degree $ g_c $ is shown to play a central role in determining the survival of species, with its value depending on the interaction strength and degree distribution. The analysis includes a linear stability study of the fixed point, revealing the conditions under which the system becomes unstable. The results highlight the importance of degree heterogeneity in shaping the dynamics of the model, with implications for understanding complex systems in economics, ecology, and other fields. The study provides a comprehensive framework for analyzing the generalized Lotka-Volterra model on a network, emphasizing the role of heterogeneity and the critical degree in determining the long-term behavior of the system.This paper presents a heterogeneous mean-field analysis of the generalized Lotka-Volterra model on a network. The study investigates the dynamics of this model with a network structure, focusing on the role of degree heterogeneity. A mean-field theory is developed that incorporates degree heterogeneity, allowing for the description of fixed points in terms of a few order parameters. The analysis extends to cases with diverging abundances using a mapping to the replicator model, providing a unified approach for both cooperative and competitive systems. The critical degree $ g_c $ is identified as a key parameter, distinguishing high-degree nodes that are more likely to go extinct in competitive regimes and low-degree nodes that tend to go extinct in cooperative regimes. The model is defined for N interacting species with abundances $ x_i $, evolving according to a differential equation involving the adjacency matrix and interaction strengths. A statistical description is developed using tools from statistical physics, leading to a dynamical mean-field theory (DMFT) that predicts averages of p-point functions. The analysis considers both homogeneous and heterogeneous cases, with the homogeneous case corresponding to $ \sigma = 0 $, where all interactions have the same strength. The theory is applied to both cooperative and competitive systems, showing how the heterogeneity of the underlying degree distribution impacts the heterogeneity of the fixed point. The results are validated against numerical simulations, demonstrating good agreement with theoretical predictions. The study also explores the diverging regime, where the Lotka-Volterra model is better described by the replicator model. The critical degree $ g_c $ is shown to play a central role in determining the survival of species, with its value depending on the interaction strength and degree distribution. The analysis includes a linear stability study of the fixed point, revealing the conditions under which the system becomes unstable. The results highlight the importance of degree heterogeneity in shaping the dynamics of the model, with implications for understanding complex systems in economics, ecology, and other fields. The study provides a comprehensive framework for analyzing the generalized Lotka-Volterra model on a network, emphasizing the role of heterogeneity and the critical degree in determining the long-term behavior of the system.