Global structural stability and the role of cooperation in mutualistic system

Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this concept, defined as Global Structural Stability. This approach would fit with the proper mathematical concept of structural stability, in which we find a full description of the complex dynamics on the phase space due to nonlinear dynamics. This fact can be analyzed as an informational field grounded in a global attractor whose structure can be completely characterized. These attractors are stable under perturbation and suppose the minimal structurally stable sets. We also study in detail, mathematically and computationally, the zones characterizing different levels of biodiversity in bipartite graphs describing mutualistic antagonistic systems of population dynamics. In particular, we investigate the dependence of the region of maximal biodiversity of a system on its connectivity matrix. On the other hand, as the network topology does not completely determine the robustness of the dynamics of a complex network, we study the correlation between structural stability and several graph measures. A systematic study on synthetic and biological graphs is presented, including 10 mutualistic networks of plants and seed-dispersal and 1000 random synthetic networks. We compare the role of centrality measures and modularity, concluding the importance of just cooperation strength among nodes when describing areas of maximal biodiversity. Indeed, we show that cooperation parameters are the central role for biodiversity while other measures act as secondary supporting functions. Author summaryWe introduce the concept of Global Structural Stability as a proper mathematical concept to fully understand biodiversity in some ecological systems. Our concept retakes the definitions in the classical works of R. Thom [1] and Andronov-Pontryagin [2]. Moreover, there exists a close relation between the structure of a complex network, described as a graph, and its associated dynamics. Mutualistic networks introduce cooperation links between two groups of species, as plant and pollinators or seed-dispersal. The understanding of organizational aspects leading to maximizing biodiversity is one of the more important research areas in Theoretical and Applied Ecology. In this work we introduce a systematic study on different graph measures in order to identify optimal organization for maximal biodiversity (defined as structural stability). Our results conclude that, for mutualistic systems, the strength in cooperation parameters are the core fact, i.e., cooperation is the real fact optimizing biodiversity among other possible structural configurations.