Marco Formentin

Sample and population exponents of generalized Taylor’s law

Significance Taylor’s law (TL) has been verified very widely in the natural sciences, information technology, and finance. The widespread observation of TL suggests that a context-independent mechanism may be at work and stimulated the search for processes affecting the scaling of population fluctuations with population abundance. We show that limited sampling may explain why TL is often observed to have exponent b=2. Abrupt transitions in the TL exponent associated with smooth changes in the environment were recently discovered theoretically and comparable real-world transitions could harm fish populations, forests, and public health. Our study shows that limited sampling hinders the anticipation of such transitions and provides estimates for the number of samples required to reveal early warning signals of abrupt biotic change. Taylor’s law (TL) states that the variance V of a nonnegative random variable is a power function of its mean M; i.e., V=aMb. TL has been verified extensively in ecology, where it applies to population abundance, physics, and other natural sciences. Its ubiquitous empirical verification suggests a context-independent mechanism. Sample exponents b measured empirically via the scaling of sample mean and variance typically cluster around the value b=2. Some theoretical models of population growth, however, predict a broad range of values for the population exponent b pertaining to the mean and variance of population density, depending on details of the growth process. Is the widely reported sample exponent b≃2 the result of ecological processes or could it be a statistical artifact? Here, we apply large deviations theory and finite-sample arguments to show exactly that in a broad class of growth models the sample exponent is b≃2 regardless of the underlying population exponent. We derive a generalized TL in terms of sample and population exponents bjk for the scaling of the kth vs. the jth cumulants. The sample exponent bjk depends predictably on the number of samples and for finite samples we obtain bjk≃k/j asymptotically in time, a prediction that we verify in two empirical examples. Thus, the sample exponent b≃2 may indeed be a statistical artifact and not dependent on population dynamics under conditions that we specify exactly. Given the broad class of models investigated, our results apply to many fields where TL is used although inadequately understood.

Time to Absorption for a Heterogeneous Neutral Competition Model

Neutral models aspire to explain biodiversity patterns in ecosystems where species difference can be neglected and perfect symmetry is assumed between species. Voter-like models capture the essential ingredients of the neutral hypothesis and represent a paradigm for other disciplines like social studies and chemical reactions. In a system where each individual can interact with all the other members of the community, the typical time to reach an absorbing state with a single species scales linearly with the community size. Here we show, by using a rigorous approach based on a large deviation principle and confirming previous approximate and numerical results, that in a mean-field heterogeneous voter model the typical time to reach an absorbing state scales exponentially with the system size.

Rhythmic behavior in a two-population mean-field Ising model

Many real systems composed of a large number of interacting components, as, for instance, neural networks, may exhibit collective periodic behavior even though single components have no natural tendency to behave periodically. Macroscopic oscillations are indeed one of the most common self-organized behavior observed in living systems. In the present paper we study some dynamical features of a two-population generalization of the mean-field Ising model with the scope of investigating simple mechanisms capable to generate rhythms in large groups of interacting individuals. We show that the system may undergo a transition from a disordered phase, where the magnetization of each population fluctuates closely around zero, to a phase in which they both display a macroscopic regular rhythm. In particular, there exists a region in the parameter space where having two groups of spins with inter- and intrapopulation interactions of different strengths suffices for the emergence of a robust periodic behavior.

Collective periodicity in mean-field models of cooperative behavior

We propose a way to break symmetry in stochastic dynamics by introducing a dissipation term. We show in a specific mean-field model, that if the reversible model undergoes a phase transition of ferromagnetic type, then its dissipative counterpart exhibits periodic orbits in the thermodynamic limit.

New activity pattern in human interactive dynamics

We investigate the response function of human agents as demonstrated by written correspondence, uncovering a new universal pattern for how the reactive dynamics of individuals is distributed across the set of each agents contacts. In long-term empirical data on email, we find that the set of response times considered separately for the messages to each different correspondent of a given writer, generate a family of heavy-tailed distributions, which have largely the same features for all agents, and whose characteristic times grow exponentially with the rank of each correspondent. We show this universal behavioral pattern emerges robustly by considering weighted moving averages of the priority-conditioned response-time probabilities generated by a basic prioritization model. Our findings clarify how the range of priorities in the inputs from ones environment underpin and shape the dynamics of agents embedded in a net of reactive relations. These newly revealed activity patterns constrain future models of communication and interaction networks, affecting their architecture and evolution.

Upscaling species richness and abundances in tropical forests

We provide a framework to upscale biodiversity in tropical forests from local samples of species richness and abundances. The quantification of tropical tree biodiversity worldwide remains an open and challenging problem. More than two-fifths of the number of worldwide trees can be found either in tropical or in subtropical forests, but only ≈0.000067% of species identities are known. We introduce an analytical framework that provides robust and accurate estimates of species richness and abundances in biodiversity-rich ecosystems, as confirmed by tests performed on both in silico–generated and real forests. Our analysis shows that the approach outperforms other methods. In particular, we find that upscaling methods based on the log-series species distribution systematically overestimate the number of species and abundances of the rare species. We finally apply our new framework on 15 empirical tropical forest plots and quantify the minimum percentage cover that should be sampled to achieve a given average confidence interval in the upscaled estimate of biodiversity. Our theoretical framework confirms that the forests studied are comprised of a large number of rare or hyper-rare species. This is a signature of critical-like behavior of species-rich ecosystems and can provide a buffer against extinction.

Strong existence and uniqueness of the stationary distribution for a stochastic inviscid dyadic model

We consider an inviscid stochastically forced dyadic model, where the additive noise acts only on the first component. We prove that a strong solution for this problem exists and is unique by means of uniform energy estimates. Moreover, we exploit these results to establish strong existence and uniqueness of the stationary distribution.

Metastates in mean-field models with random external fields generated by Markov chains

We extend the construction by Külske and Iacobelli of metastates in finite-state mean-field models in independent disorder to situations where the local disorder terms are a sample of an external ergodic Markov chain in equilibrium. We show that for non-degenerate Markov chains, the structure of the theorems is analogous to the case of i.i.d. variables when the limiting weights in the metastate are expressed with the aid of a CLT for the occupation time measure of the chain.As a new phenomenon we also show in a Potts example that for a degenerate non-reversible chain this CLT approximation is not enough, and that the metastate can have less symmetry than the symmetry of the interaction and a Gaussian approximation of disorder fluctuations would suggest.

Inferring macro-ecological patterns from local species\' occurrences

Biodiversity provides support for life, vital provisions, regulating services and has positive cultural impacts. It is therefore important to have accurate methods to measure biodiversity, in order to safeguard it when we discover it to be threatened. For practical reasons, biodiversity is usually measured at fine scales whereas diversity issues (e.g. conservation) interest regional or global scales. Moreover, biodiversity may change across spatial scales. It is therefore a key challenge to be able to translate local information on biodiversity into global patterns. Many databases give no information about the abundances of a species within an area, but only its occurrence in each of the surveyed plots. In this paper, we introduce an analytical framework to infer species richness and abundances at large spatial scales in biodiversity-rich ecosystems when species presence/absence information is available on various scattered samples (i.e. upscaling). This framework is based on the scale-invariance property of the negative binomial. Our approach allows to infer and link within a unique framework important and well-known biodiversity patterns of ecological theory, such as the Species Accumulation Curve (SAC) and the Relative Species Abundance (RSA) as well as a new emergent pattern, which is the Relative Species Occupancy (RSO). Our estimates are robust and accurate, as confirmed by tests performed on both in silico-generated and real forests. We demonstrate the accuracy of our predictions using data from two well-studied forest stands. Moreover, we compared our results with other popular methods proposed in the literature to infer species richness from presence-absence data and we showed that our framework gives better estimates. It has thus important applications to biodiversity research and conservation practice.

Biodiversity of Tropical Forests

The quantification of tropical tree biodiversity worldwide remains an open and challenging problem. In fact, more than two-fifths of the global tree population can be found either in tropical or sub-tropical forests1, but species identities are known only for ≈ 0.000067% of the individuals in all tropical forests2. For practical reasons, biodiversity is typically measured or monitored at fine spatial scales. However, important drivers of ecological change tend to act at large scales. Conservation issues, for example, apply to diversity at global, national or regional scales. Extrapolating species richness from the local to the global scale is not straightforward. Indeed, a vast number of different biodiversity estimators have been developed under different statistical sampling frameworks3–7, but most of them have been designed for local/regional-scale extrapolations, and they tend to be sensitive to the spatial distribution of trees8, sample coverage and sampling methods9. Here, we introduce an analytical framework that provides robust and accurate estimates of species richness and abundances in biodiversity-rich ecosystems, as confirmed by tests performed on various in silico-generated forests. The new framework quantifies the minimum percentage cover that should be sampled to achieve a given average confidence in the upscaled estimate of biodiversity. Our analysis of 15 empirical forest plots shows that previous methods10,11 have systematically overestimated the total number of species and leads to new estimates of hyper-rarity10 at the global scale11, known as Fisher’s paradox2. We show that hyper-rarity is a signature of critical-like behavior12 in tropical forests13–15, and it provides a buffer against mass extinctions16. When biotic factors or environmental conditions change, some of these rare species are more able than others to maintain the ecosystem’s functions, thus underscoring the importance of rare species.