On survival of all agents in a network with cooperative and competitive interactions

This technical note investigates the survival problem for the generalized Verhulst–Lotka–Volterra (VLV) model with cooperative and competitive interactions. First, we consider the generalized VLV model in which every agent has an individual capacity. The cooperative and competitive interactions are allowed to coexist in this model. We obtain some sufficient conditions such that the VLV model is survival. Then, we consider this generalized VLV model with a special network structure in which cooperative and competitive interactions coexist. For this special model, we show that the VLV model is survival for any admissible parameters. Second, we consider the generalized VLV model in which there exists a market capacity and there exist no competitive interactions. For this model, some sufficient conditions are derived for survival of all agents. Finally, the obtained results are confirmed through three numerical examples.