We introduce a new population dynamics model for mutualistic communities. The new model preserves the original logistic formulation. We perform an analytical stability analysis to study the model behavior. We perform numerical simulations to test the model behavior. The model shows as much richness or even more than other mutualistic models. a r t i c l e i n f o t r a c tMutualistic communities have an internal structure that makes them resilient to external perturbations. Late research has focused on their stability and the topology of the relations between the different organisms to explain the reasons of the system robustness. Much less attention has been invested in analyzing the systems dynamics. The main population models in use are modifications of the r À K formulation of logistic equation with additional terms to account for the benefits produced by the interspecific interactions. These models have shortcomings as the so-called r À K formulation diverges under some conditions. In this work, we introduce a model for population dynamics under mutualism that preserves the original logistic formulation. It is mathematically simpler than the widely used type II models, although it shows similar complexity in terms of fixed points and stability of the dynamics. We perform an analytical stability analysis and numerical simulations to study the model behavior in general interaction scenarios including tests of the resilience of its dynamics under external perturbations. Despite its simplicity, our results indicate that the model dynamics shows an important richness that can be used to gain further insights in the dynamics of mutualistic communities.

Population dynamics has been modelled using differential equations almost since Malthus times, more than two centuries ago. Basic ingredients of population dynamics models are typically a growth rate, a saturation term in the form of Verhulst’s logistic brake, and a functional response accounting for interspecific interactions. However, intraspecific interactions are not usually included in the equations. The simplest models use linear terms to represent a simple picture of the nature; meanwhile, to represent more complex landscapes, it is necessary to include more terms with a higher order or that are analytically more complex. The problem to use a simpler or more complex model depends on many factors: mathematical, ecological, or computational. To address it, here we discuss a new model based on a previous logistic-mutualistic model. We have generalized the interspecific terms (for antagonistic and competitive relationships), and we have also included new polynomial terms to explain any intraspecific interaction. We show that, by adding simple intraspecific terms, new free-equilibrium solutions appear driving a much richer dynamics. These new solutions could represent more realistic ecological landscapes by including a new higher order term.

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BackgroundNetwork analysis has become a relevant approach to analyze cascading species extinctions resulting from perturbations on mutualistic interactions as a result of environmental change. In this context, it is essential to be able to point out key species, whose stability would prevent cascading extinctions, and the consequent loss of ecosystem function. In this study, we aim to explain how the k-core decomposition sheds light on the understanding the robustness of bipartite mutualistic networks.MethodsWe defined three k-magnitudes based on the k-core decomposition: k-radius, k-degree, and k-risk. The first one, k-radius, quantifies the distance from a node to the innermost shell of the partner guild, while k-degree provides a measure of centrality in the k-shell based decomposition. k-risk is a way to measure the vulnerability of a network to the loss of a particular species. Using these magnitudes we analyzed 89 mutualistic networks involving plant pollinators or seed dispersers. Two static extinction procedures were implemented in which k-degree and k-risk were compared against other commonly used ranking indexes, as for example MusRank, explained in detail in Material and Methods.ResultsWhen extinctions take place in both guilds, k-risk is the best ranking index if the goal is to identify the key species to preserve the giant component. When species are removed only in the primary class and cascading extinctions are measured in the secondary class, the most effective ranking index to identify the key species to preserve the giant component is k-degree. However, MusRank index was more effective when the goal is to identify the key species to preserve the greatest species richness in the second class.DiscussionThe k-core decomposition offers a new topological view of the structure of mutualistic networks. The new k-radius, k-degree and k-risk magnitudes take advantage of its properties and provide new insight into the structure of mutualistic networks. The k-risk and k-degree ranking indexes are especially effective approaches to identify key species to preserve when conservation practitioners focus on the preservation of ecosystem functionality over species richness.

Prediction is one of the last frontiers in ecology. Indeed, predicting fine-scale species composition in natural systems is a complex challenge as multiple abiotic and biotic processes operate simultaneously to determine local species abundances. On the one hand, species intrinsic performance and their tolerance limits to different abiotic pressures modulate species abundances. On the other hand there is growing recognition that species interactions play an equally important role in limiting or promoting such abundances within ecological communities. Here, we present a joint effort between ecologists and data scientists to use data-driven models to predict species abundances using reasonably easy to obtain data. We propose a sequential data-driven modeling approach that in a first step predicts the potential species abundances based on abiotic variables, and in a second step uses these predictions to model the realized abundances once accounting for species competition. Using a curated data set over five years we predict fine-scale species abundances in a highly diverse annual plant community. Our models show a remarkable spatial predictive accuracy using only easy-to-measure variables in the field, yet such predictive power is lost when temporal dynamics are taken into account. This result suggests that predicting future abundances requires longer time series analysis to capture enough variability. In addition, we show that these data-driven models can also suggest how to improve mechanistic models by adding missing variables that affect species performance such as particular soil conditions (e.g. carbonate availability in our case). Robust models for predicting fine-scale species composition informed by the mechanistic understanding of the underlying abiotic and biotic processes can be a pivotal tool for conservation, especially given the human-induced rapid environmental changes we are experiencing. This objective can be achieved by promoting the knowledge gained with classic modelling approaches in ecology and recently developed data-driven models.