Biostochastic Workshop 2023

Speaker: Laurence Reboul (I2M, ECODEP) (Joint work with M. Boutahar and I. Kchaou, Institut de Mathématiques de Marseille)
Abstract: Modeling dependence structures of multivariate extremes, defined as random vectors which jointly exceed large thresholds, is of great interest in many application fields such as Earth Sciences (e.g., hydrology, coastal and offshore engineering, volcanology, …). A well-known way to model these structures is to use an extreme-value copula. Such copula is totally characterized by the so called Pickands dependence function. The problem of estimating Pickands dependence function by nonparametric methods has been extensively studied in the literature. The underlying sequence of extreme observations is always assumed to be i.i.d., which excludes a possible serial correlation. In practical situation however, the temporal independence of extremes is usually an unrealistic assumption. In this work, we propose to revisit the properties of a classical estimator of the Pickands dependence function, the CFG estimator, when it is based on some kind of weakly dependent strictly stationary sequence of extremes. We then use these properties to build a test of independence of the multivariate extreme vectographs' margins. We illustrate our results by studying the dependence structure of a series of extreme waves and wave heights recorded at Newlyn, a coastal town in southwest England.

Speaker: Guillaume Franchi (ENSAI, ECODEP)
Abstract: Motivated by the dynamic modeling of relative abundance data in ecology, we introduce a general approach to model time series on the simplex. Our approach is based on a general construction of infinite memory models, called chains with complete connections. Simple conditions ensuring the existence of stationary paths are given for the transition kernel that defines the dynamic. We then study in detail two specific examples with a Dirichlet and a multivariate logistic-normal conditional distribution. Inference methods can be based on either likelihood maximization or on some convex criteria that can be used to initialize likelihood optimization. We also give an interpretation of our models in term of additive perturbations on the simplex and relative risk ratios which are useful to analyze abundance data in ecosystems. An illustration concerning the evolution of the distribution of three species of Scandinavian birds is provided.

Speaker: Lionel Truquet (ENSAI-CREST)
Joint work with Joel E. Cohen (Rockefeller and Columbia University) and Paul Doukhan (Cergy Paris University)
Abstract: Taylor’s power law (TL) is an important relation between the population mean and the population variance, when these moments exist, and between the sample mean and the sample variance even when the moments may not exist, which has been verified empirically for the abundances of many species in many ecosystems. TL assumes that the variance is directly proportional to a power of the mean, exactly for population moments and approximately for sample moments. In many ecological papers, the population means and variances are estimated over time or across space from arrays of abundance data collected at different locations and time points. Linear regression of log variance as a function of log mean is then used to estimate TL’s two parameters, the exponent of the mean and the coefficient of proportionality. We provide some statistical guarantees for this kind of inference under general conditions, and we derive confidence bands for the parameters. When the ratio between the time-series length and the number of spatial points converges to a constant as both become large, the usual normalized statistics is asymptotically biased, which prevents practitioners from using standard results in this context. We then provide a correction of this We illustrate our methods on CPUE for fisheries data in the North Sea.

Speaker: Paul Doukhan (Université Cergy-Pontoise, ECODEP)
Abstract: The project ECODEP resulted in the fact that time series naturally appear in ecology which was pointed by many prevous talks. Here I will first recall some of the notions used for application to Time Series Analysis as well as a rapid tour on building such time series.
I will consider 2 applications
- The Taylor law under dependence as well as a dynamic counterpart of Taylor law in which the individual variance is replaced by the series of covariances which occurs naturally under any type of weak dependence condition as the limiting variance in the CLT for such random processes. For this a block technique allows the corresponding change. A paper providing tests of hypothesis for this Taylor law problem has been developed in a paper with Victor de la Pena and Yahia Salhi (JAP2022) and an ongoing work in pendant with the same authors to develop a two scales estimation of the involved coefficients.
- The estimation of convex envelops for the support of a probability density is also developed with Catherine Aaron and Laurence Reboul under adapted dependence conditions. This is a way to describe the territory of some individuals or species and such problems are developed for ecological purpose. A possible multivariate extension may take a sense in the classical prey-predator paradigm of ecology.
Both questions have ongoing applications to population of voles in the island of Okaïdo already considered in papers by Joel Cohen. This question is more natural that it first seems since this involves a closed system and in that case, they may be no competitions problems.

Speaker: Nicolas Champagnat (Université de Nancy, Institut Élie Cartan)
Abstract: Starting from an individual-based model of adaptive dynamics with three scaling parameters (population size, mutation rate, mutation steps size), we will describe several scaling limits that can be applied to this model to obtain macroscopic models of different natures (deterministic partial differential equations, stochastic adaptive walks, ordinary differential equations describing the evolution of the dominant phenotype, stochastic differential equations with genetic drift...). We will discuss the different time-scales of evolution observed under these parameter scalings and the properties of the models in terms of possible (local) extinction of (sub)populations.
This presentation will gather results obtained with several collaborators: Régis Ferrière, Sylvie Méléard, Amaury Lambert, Viet Chi Tran, Sepideh Mirrahimi, Vincent Hass.

Speaker: Cristóbal Quiñinao (UOH)
Abstract: In this talk, we consider a (control) optimization problem that involves a stochastic dynamic. The model proposes selecting the best control function that keeps a bounded stochastic process over time with a high probability level. This setting becomes a chance-constrained control optimization problem, where the constraint is given by the probability level of infinitely many random inequalities. Since such a model is challenging, we discretize the dynamic and restrict the space of control functions to piecewise mappings. Finally, the results are illustrated with numerical results, where the classical models for the growth of a population are considered.

Speaker: Mauricio Tejo (Instituto de Estadística, Universidad de Valparaíso)
Abstract: We propose a stochastic model for interacting species in a spatially structured system. The stochastic component is fundamentally associ- ated to migration and is represented by Poisson random measures. This stochastic model includes two dynamic scales: a continuous one, which refers to the interactions among species, and a low frequency one, referring to migrations following a Poisson scheme. We show the well-posedness of the model and that we can also study its long-term behavior using Lyapunov exponents. To exemplify our approach we use a Lotka-Volterra type model with stochastic migrations in a spatially structured system represented by an archipelago containing J species and I islands. We show numerically that the model capture the existence of competition/dispersal tradeoffs, whereby competing species can coexist or even reverse their competitive status due to migration. The extinction of a species is also associated with a negative slope in the time trajectory of the Lyapunov exponent; otherwise, it is equal to zero.

Speaker: Leonardo Videla (Universidad de Santiago de Chile, DMCC)
Abstract: In this talk I discuss some preliminary results regarding the time-asymptotics of an ensemble of interacting particles obeying a replicator-like stochastic dynamics. We prove the propagation of chaos property and establish conditions for the strong persistence of the N -replicator system. For the related hydrodynamic limit, I discuss some problems arising in the context of the coexistence of the associated McKean-Vlasov type replicator.

Speaker: Milan Stehlik (Instituto de Estadística, Universidad de Valparaíso)
Abstract: Biological and ecological systems offer a lot of complexity which should be well understood before we can make valuable regulations. We will address both complex and extreme measurements from given systems. There is a necessity to classify appropriate learning mechanisms and define transfer functions and statistics. I will address learning mechanisms of data transformation and aggregation.
In particular, I will introduce SPOCU transfer function and provide some of its unique properties for processing of complex data.
Statistical learning will be discussed and tuning of parameters of SPOCU based neural networks will be given. Attractive applications to biological systems e.g. mass balance in the ecosystem of glaciers in Patagonia, or methane emissions from wetlands will be addressed.

Speaker: M. Isidora Ávila Thieme (Pontificia Universidad Católica de Chile)
Abstract: This talk will present a complex socio-ecological problem associated with non-compliance on fishing recommendations, which strongly threatens the sustainability of natural resources and coastal ecosystems.
Kelps are a valuable resource for fishers and the broader Chilean economy. But also, they are key species for maintaining marine coastal ecosystem biodiversity and stability. However, Kelp fishery faces a high level of non-compliance, the kelp abundance is being strongly threatened, and their population structure has been juvenilized toward less valuable plants to fishers and with different ecological roles. Based on the socio-ecological role of kelps, non-compliance behaviors could propagate to the entire ecosystem through the complex network of species interactions. Here, I will present a socio-ecological framework of the kelps fishery and assess the feedback between (non-) compliance decision-making of fishers, the abundance and structure of kelps (social dimension), and its propagation through the ecosystem (ecological dimension). Preliminary results are associated with the social dimension, in which we have used empirical and survey information to parametrize the (non-) compliance demographic kelp model. These preliminary results allow us to understand the relative importance of each fishing recommendation on the kelp abundance and structure and the role that fishers play.
I expect this study case inspire the discussion about the importance of connect biostochastic approaches to address complex socio-ecological problems.
This is a collaborative project with Sergio A. Navarrete, Nicole Maturana, C. Josh Donlan & Stefan Gelcich