On the structural stability of mutualistic systems

Introduction Several major developments in theoretical ecology have relied on either dynamical stability or numerical simulations, but oftentimes, they have found contradictory results. This is partly a result of not rigorously checking either the assumption that a steady state is feasible—meaning, all species have constant and positive abundances—or the dependence of results to model parameterization. Here, we extend the concept of structural stability to community ecology in order to account for these two problems. Specifically, we studied the set of conditions leading to the stable coexistence of all species within a community. This shifts the question from asking whether we can find a feasible equilibrium point for a fixed set of parameter values, to asking how large is the range of parameter values that are compatible with the stable coexistence of all species. The architecture of plant-animal mutualistic networks modulates the range of conditions leading to the stable coexistence of all species. The area of the different domains represents the structural stability of a model of mutualistic communities with a given network architecture. The nested networks observed in nature—illustrated here by the network at the bottom—lead to a maximum structural stability. Rationale We begin by disentangling the conditions of global stability from the conditions of feasibility of a steady state in ecological systems. To quantify the domain of stable coexistence, we first find its center (the structural vector of intrinsic growth rates). Next, we determine the boundaries of such a domain by quantifying the amount of variation from the structural vector tolerated before one species goes extinct. Through this two-step approach, we disentangle the effects of the size of the feasibility domain from how close a solution is to its boundary, which is at the heart of previous contradictory results. We illustrate our method by exploring how the observed architecture of mutualistic networks between plants and their pollinators or seed dispersers affects their domain of stable coexistence. Results First, we determined the network architecture that maximizes the structural stability of mutualistic systems. This corresponds to networks with a maximal level of nestedness, a small trade-off between the number and intensity of interactions a species has, and a high level of mutualistic strength within the constraints of global stability. Second, we found that the large majority of observed mutualistic networks are close to this optimum network architecture, maximizing the range of parameters that are compatible with species coexistence. Conclusion Structural stability has played a major role in several fields such as evolutionary developmental biology, in which it has brought the view that some morphological structures are more common than others because they are compatible with a wider range of developmental conditions. In community ecology, structural stability is the sort of framework needed to study the consequences of global environmental change—by definition, large and directional—on species coexistence. Structural stability will serve to assess both the range of variability a given community can withstand and why some community patterns are more widespread than others. A structural approach to species interactions What determines the stability of ecological networks? Rohr et al. devised a conceptual approach to study interactions between species that emphasizes the role of network structure (see the Perspective by Pawar). Using the example of mutualistic networks of communities of plants and their pollinator species, they show how the structure of networks can determine the persistence of the interactions. Network structures and architectures observed in nature intrinsically match the most stable solution. This approach has promise for application to questions of ecological community stability under global change. Science, this issue 10.1126/science.1253497; see also p. 383 In ecology, structural stability influences the range of perturbations mutualistic networks can withstand. [Also see Perspective by Pawar] In theoretical ecology, traditional studies based on dynamical stability and numerical simulations have not found a unified answer to the effect of network architecture on community persistence. Here, we introduce a mathematical framework based on the concept of structural stability to explain such a disparity of results. We investigated the range of conditions necessary for the stable coexistence of all species in mutualistic systems. We show that the apparently contradictory conclusions reached by previous studies arise as a consequence of overseeing either the necessary conditions for persistence or its dependence on model parameterization. We show that observed network architectures maximize the range of conditions for species coexistence. We discuss the applicability of structural stability to study other types of interspecific interactions.