Marco Favretti

Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints

This paper is concerned with the dynamics of a mechanical system subject to nonintegrable constraints. In the first part, we prove the equivalence between the classical nonholonomic equations and those derived from the nonholonomic variational formulation, proposed by Kozlov in [10–12], for a class of constrained systems with constraints transverse to a foliation. This result extends the equivalence between the two formulations, proved for holonomic constraints, to a class of linear nonintegrable ones. In the second part, we derive the nonholonomic variational reduced equations for a constrained system with symmetry and constraint transverse to a principal bundle fibration, using a reduction procedure similar to the one developed in [5]. The resulting equations are compared with the nonholonomic reduced ones through mechanical examples.

Dynamics of a Chain of Springs with Nonconvex Potential Energy

We study the dynamics of a discretized model of an elastic bar in a hard device formed by a chain of point masses connected by nonlinear springs whose total length is a controlled parameter. We compare the description of the system dynamics given by the first-order (gradient) dynamics, the second-order (Newtonian) dumped dynamics and the Relaxation Oscillation Theory. Using a technique based on Liapunovs second method, we prove a dynamic stability result concerning the above-mentioned ODEs.

The Maximum Entropy Rate Description of a Thermodynamic System in a Stationary Non-Equilibrium State

In this paper we present a simple model to describe a rather general system in a stationary non-equilibrium state, which is an open system traversed by a stationary flux. The probabilistic description is provided by a non-homogeneous Markov chain, which is not assumed on the basis of a model of the microscopic interactions but rather derived from the knowledge of the macroscopic fluxes traversing the system through a maximum entropy rate principle.

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.

Remarks on the Maximum Entropy Principle with Application to the Maximum Entropy Theory of Ecology

In the first part of the paper we work out the consequences of the fact that Jaynes’ Maximum Entropy Principle, when translated in mathematical terms, is a constrained extremum problem for an entropy function H(p) expressing the uncertainty associated with the probability distribution p. Consequently, if two observers use different independent variables p or g(p), the associated entropy functions have to be defined accordingly and they are different in the general case. In the second part we apply our findings to an analysis of the foundations of the Maximum Entropy Theory of Ecology (M.E.T.E.) a purely statistical model of an ecological community. Since the theory has received considerable attention by the scientific community, we hope to give a useful contribution to the same community by showing that the procedure of application of MEP, in the light of the theory developed in the first part, suffers from some incongruences. We exhibit an alternative formulation which is free from these limitations and that gives different results.

Replicated point processes with application to population dynamics models

In this paper we study spatially clustered distribution of individuals using point process theory. In particular we discuss the spatially explicit model of population dynamics of Shimatani (2010) which extend previous works on Malécot theory of isolation by distance. We reformulate Shimatani model of replicated Neyman-Scott process to allow for a general dispersal kernel function and we show that the random immigration hypothesis can be substituted by the long dispersal distance property of the kernel. Moreover, the extended framework presented here is fit to handle spatially explicit statistical estimators of genetic variability like Moran autocorrelation index, Sørensen similarity index, average kinship coefficient. We discuss the pivotal role of the choice of dispersal kernel for the above estimators in a toy model of dynamic population genetics theory.

The distance decay of similarity in tropical rainforests. A spatial point processes analytical formulation

In this paper we are concerned with the analytical description of the change in floristic composition (species turnover) with the distance between two plots of a tropical rainforest due to the clustering of the individuals of the different species. We describe the plant arrangement by a superposition of spatial point processes and in this framework we introduce an analytical function which represents the average spatial density of the Sørensen similarity between two infinitesimal plots at distance r. We see that the decay in similarity with the distance is essentially described by the pair correlation function of the superposed process and that it is governed by the most abundant species. We test our analytical model with empirical data obtained for the Barro Colorado Island and Pasoh rainforests. To this end we adopt the statistical estimator for the pair correlation function in Shimatani (2001) and we design a novel one for the Sørensen similarity. Furthermore, we test our analytical formula by modeling the forest study area with Neyman-Scott point processes. We conclude comparing the advantages of our approach with other ones existing in literature.

Maximum Entropy Theory of Ecology: A Reply to Harte

In a paper published in this journal, I addressed the following problem: under which conditions will two scientists, observing the same system and sharing the same initial information, reach the same probabilistic description upon the application of the Maximum Entropy inference principle (MaxEnt) independent of the probability distribution chosen to set up the MaxEnt procedure. This is a minimal objectivity requirement which is generally asked for scientific investigation. In the same paper, I applied the findings to a critical examination of the application of MaxEnt made in Harte’s Maximum Entropy Theory of Ecology (METE). Prof. Harte published a comment to my paper and this is my reply. For the sake of the reader who may be unaware of the content of the papers, I have tried to make this reply self-contained and to skip technical details. However, I invite the interested reader to consult the previously published papers.

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.

Lagrangian submanifolds generated by the Maximum Entropy principle

We show that the Maximum Entropy principle (E.T. Jaynes, [8]) has a natural description in terms of Morse Families of a Lagrangian submanifold. This geometric approach becomes useful when dealing with the M.E.P. with nonlinear constraints. Examples are presented using the Ising and Potts models of a ferromagnetic material.