Jean-Francois Arnoldi

Resilience, reactivity and variability: A mathematical comparison of ecological stability measures.

In theoretical studies, the most commonly used measure of ecological stability is resilience: ecosystems asymptotic rate of return to equilibrium after a pulse-perturbation -or shock. A complementary notion of growing popularity is reactivity: the strongest initial response to shocks. On the other hand, empirical stability is often quantified as the inverse of temporal variability, directly estimated on data, and reflecting ecosystems response to persistent and erratic environmental disturbances. It is unclear whether and how this empirical measure is related to resilience and reactivity. Here, we establish a connection by introducing two variability-based stability measures belonging to the theoretical realm of resilience and reactivity. We call them intrinsic, stochastic and deterministic invariability; respectively defined as the inverse of the strongest stationary response to white-noise and to single-frequency perturbations. We prove that they predict ecosystems worst response to broad classes of disturbances, including realistic models of environmental fluctuations. We show that they are intermediate measures between resilience and reactivity and that, although defined with respect to persistent perturbations, they can be related to the whole transient regime following a shock, making them more integrative notions than reactivity and resilience. We argue that invariability measures constitute a stepping stone, and discuss the challenges ahead to further unify theoretical and empirical approaches to stability.

An invariability-area relationship sheds new light on the spatial scaling of ecological stability

The spatial scaling of stability is key to understanding ecological sustainability across scales and the sensitivity of ecosystems to habitat destruction. Here we propose the invariability–area relationship (IAR) as a novel approach to investigate the spatial scaling of stability. The shape and slope of IAR are largely determined by patterns of spatial synchrony across scales. When synchrony decays exponentially with distance, IARs exhibit three phases, characterized by steeper increases in invariability at both small and large scales. Such triphasic IARs are observed for primary productivity from plot to continental scales. When synchrony decays as a power law with distance, IARs are quasilinear on a log–log scale. Such quasilinear IARs are observed for North American bird biomass at both species and community levels. The IAR provides a quantitative tool to predict the effects of habitat loss on population and ecosystem stability and to detect regime shifts in spatial ecological systems, which are goals of relevance to conservation and policy.

Generic assembly patterns in complex ecological communities

Significance Biodiversity may lead to the emergence of simple and robust relationships between ecosystem properties. Here we show that a wide range of models of species dynamics, in the limit of high diversity, exhibit generic behavior predictable from a few emergent parameters, which control ecosystem functioning and stability. Our work points toward ways to tackle the staggering complexity of ecological systems without relying on empirically unavailable details of their structure. The study of ecological communities often involves detailed simulations of complex networks. However, our empirical knowledge of these networks is typically incomplete and the space of simulation models and parameters is vast, leaving room for uncertainty in theoretical predictions. Here we show that a large fraction of this space of possibilities exhibits generic behaviors that are robust to modeling choices. We consider a wide array of model features, including interaction types and community structures, known to generate different dynamics for a few species. We combine these features in large simulated communities, and show that equilibrium diversity, functioning, and stability can be predicted analytically using a random model parameterized by a few statistical properties of the community. We give an ecological interpretation of this “disordered” limit where structure fails to emerge from complexity. We also demonstrate that some well-studied interaction patterns remain relevant in large ecosystems, but their impact can be encapsulated in a minimal number of additional parameters. Our approach provides a powerful framework for predicting the outcomes of ecosystem assembly and quantifying the added value of more detailed models and measurements.

How ecosystems recover from pulse perturbations: A theory of short- to long-term responses

Quantifying stability properties of ecosystems is an important problem in ecology. A common approach is based on the recovery from pulse perturbations, and posits that the faster an ecosystem return to its pre-perturbation state, the more stable it is. Theoretical studies often collapse the recovery dynamics into a single quantity: the long-term rate of return, called asymptotic resilience. However, empirical studies typically measure the recovery dynamics at much shorter time scales. In this paper we explain why asymptotic resilience is rarely representative of the short-term recovery. First, we show that, in contrast to asymptotic resilience, short-term return rates depend on features of the perturbation, in particular on the way its intensity is distributed over species. We argue that empirically relevant predictions can be obtained by considering the median response over a set of perturbations, for which we provide explicit formulas. Next, we show that the recovery dynamics are controlled through time by different species: abundant species tend to govern the short-term recovery, while rare species often dominate the long-term recovery. This shift from abundant to rare species typically causes short-term return rates to be unrelated to asymptotic resilience. We illustrate that asymptotic resilience can be determined by rare species that have almost no effect on the observable part of the recovery dynamics. Finally, we discuss how these findings can help to better connect empirical observations and theoretical predictions.

Unifying dynamical and structural stability of equilibria

We exhibit a fundamental relationship between measures of dynamical and structural stability of linear dynamical systems—e.g. linearized models in the vicinity of equilibria. We show that dynamical stability, quantified via the response to external perturbations (i.e. perturbation of dynamical variables), coincides with the minimal internal perturbation (i.e. perturbations of interactions between variables) able to render the system unstable. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian’s coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a linear system’s response to white-noise perturbations directly reflects the intensity of internal white-noise disturbance that it can accommodate before becoming stochastically unstable.

Resilience, invariability, and ecological stability across levels of organization

Ecological stability is a bewildering broad concept. The most common stability measures are asymptotic resilience, widely used in theoretical studies, and measures based on temporal variability, commonly used in empirical studies. We construct measures of invariability, defined as the inverse of variability, that can be directly compared with asymptotic resilience. We show that asymptotic resilience behaves like the invariability of the most variable species, which is often a rare species close to its extinction boundary. Therefore, asymptotic resilience displays complete loss of stability with changes in community composition. In contrast, mean population invariability and ecosystem invariability are insensitive to rare species and quantify stability consistently whether details of species composition are considered or not. Invariability provides a consistent framework to predict diversity-stability relationships that agree with empirical data at population and ecosystem levels. Our findings can enhance the dialogue between theoretical and empirical stability studies.

The cavity method for community ecology

This article is addressed to researchers and students in theoretical ecology, as an introduction to “disordered systems” approaches from statistical physics, and how they can help understand large ecological communities. We discuss the relevance of these approaches, and how they fit within the broader landscape of models in community ecology. We focus on a remarkably simple technique, the cavity method, which allows to derive the equilibrium properties of Lotka-Volterra systems. We present its predictions, the new intuitions it suggests, and its technical underpinnings. We also discuss a number of new results concerning possible extensions, including different functional responses and community structures.

Generic assembly patterns in large ecological communities

Ecological communities have mainly been investigated theoretically in two ways: piecewise, a few species at a time; or as complex networks, simulated in exhaustive detail. But our empirical knowledge of networks is limited, and the space of simulation models and parameters is mindbogglingly vast. We show that a large fraction of that space of possibilities exhibits generic dynamics, which can be predicted from a single minimal model. To demonstrate this, we consider a wide array of ecological models, from resource competition to predation and mutualism, known to display very different behaviors for a few species. We simulate large communities, and show that equilibrium diversity, functioning and stability can often be predicted analytically from only four broad statistical properties of the community. Our approach provides a convenient framework for exploring generic patterns in ecosystem assembly and quantifying the added value of detailed models and measurements.

Particularity of “Universal resilience patterns in complex networks”

In a recent Letter to Nature,Gao, Barzel and Barabási 1 describe an elegant procedure to reduce the dimensionality of complex dynamical networks, which they claim reveals “universal patterns of network resilience”, offering “ways to prevent the collapse of ecological, biological or economic systems, and guiding the design of technological systems resilient to both internal failures and environmental changes”. However, Gao et al restrict their attention to systems for which all interactions between nodes are mutualistic. Since antagonism is ubiquitous in natural and social networks, we clarify why this stringent hypothesis is necessary and what happens when it is relaxed. By analyzing broad classes of competitive and predator-prey networks we provide novel insights into the underlying mechanisms at work in Gao et al’s theory, and novel predictions for dynamical systems that are not purely mutualistic.

The variability spectrum of ecological communities: How common and rare species shape stability patterns

Empirical knowledge of ecosystem stability and diversity-stability relationships is mostly based on the analysis of temporal variability of population and ecosystem properties. Variability, however, often depends on external factors that act as disturbances, making it difficult to compare its value across systems and relate it to other stability concepts. Here we show how variability, when viewed as a response to stochastic perturbations, can reveal inherent stability properties of ecological communities, with clear connections with other stability notions. This requires abandoning one-dimensional representations, in which a single variability measurement is taken as a proxy for how stable a system is, and instead consider the whole set of variability values associated to a given community, reflecting the whole set of perturbations that can generate variability. Against the vertiginous dimensionality of the perturbation set, we show that a generic variability-abundance pattern emerges from community assembly, which relates variability to the abundance of perturbed species. As a consequence, the response to stochastic immigration is governed by rare species while common species drive the response to environmental perturbations. In particular, the contrasting contributions of different species abundance classes can lead to opposite diversity-stability patterns, which can be understood from basic statistics of the abundance distribution. Our work shows that a multidimensional perspective on variability allows one to better appreciate the dynamical richness of ecological systems and the underlying meaning of their stability patterns.Our empirical knowledge of ecosystem stability and of diversity-stability relationships is mostly built on the analysis of temporal variability of population and ecosystem properties. Variability, however, often depends on external factors that act as disturbances, making it difficult to compare its value across systems, and relate it to other stability concepts. Here we show how variability, when seen as a response to stochastic perturbations of various types, can reveal inherent stability properties of ecological communities, with clear connections with other stability measures. This requires abandoning one-dimensional representations of stability, in which a single variability measurement is taken as a proxy for how stable a system is, and instead consider the whole variability spectrum , i.e. the distribution of the system9s response to the vast set of perturbations that can generate variability. In species-rich model communities, we show that there exist generic patterns for which specific abundance classes of species govern variability. In particular, the response to stochastic immigration is typically governed by rare species while common species drive the response to environmental perturbations. We show that the contrasting contributions of different species abundance classes can be responsible for opposite diversity-stability patterns. More generally, our work proposes that a multidimensional perspective on stability allows one to better appreciate the dynamical richness of ecosystems, and to better understand the causes and consequences of their stability patterns.