Conference Agenda

{\it (I)} Individual recruitment success, or the offspring number distribution of a given population, is a fundamental element in ecology and evolution. Randomly increased recruitment refers to an increased chance of producing many offspring without natural selection being involved, and may apply to individuals in broadcast spawning populations characterised by Type III survivorship. We consider an extension of the (Schweinsberg, 2003) model of randomly increased recruitment for a haploid panmictic population of constant size $N$; the extension also works as an alternative to the Wright-Fisher model. Our model incorporates a fixed upper bound on the random number of potential offspring produced by an arbitrary individual. Depending on how the bound behaves relative to $N$ as $N$ increases, we obtain the Kingman-coalescent, an incomplete Beta-coalescent, or the original (complete) Beta-coalescent of (Schweinsberg, 2003). We argue that applying such an upper bound is biologically reasonable. Moreover, we estimate the error of the coalescent approximation. The error estimates reveal that convergence can be slow, and small sample size can be sufficient to invalidate onvergence, for example if the stated bound is of the form $N/\log N$. We use simulations to investigate the effect of increasing sample size on the site-frequency spectrum. When the model is in the domain of attraction of a Beta-coalescent, the site frequency spectrum will be as predicted by the limiting tree even though the full coalescent tree may deviate from the limiting one. When the model is in the domain of attraction of the Kingman-coalescent the effect of increasing sample size depends on the effective population size as has been noted in the case of the Wright-Fisher model. This is joint work with JA Chetwyn-Diggle.

{\it (II)} When estimates of $\alpha$ of the Beta-coalescents derived in {\it (I)} are close to 1 recovering the mutations used to estimate $\alpha$ may require strong assumptions regarding the population size and/or the mutation rate. With this in mind we consider population genetic models of randomly increased recruitment in haploid panmictic populations of constant size ($N$) evolving in a random environment. Our main results are {\it (i)} continuous-time coalescents that are either the Kingman-coalescent or specific families of Beta or Poisson-Dirichlet coalescents; when combining the results the parameter $\alpha$ of the Beta-coalescent ranges from 0 to 2, and the Beta-coalescents may be incomplete due to an upper bound on the number of potential offspring an arbitrary individual may produce; {\it (ii)} in large populations we measure time in units proportional to at least $ N/\log N$ generations; {\it (iii)} using simulations we show that in some cases estimates of the site-frequency spectra as predicted by a given coalescent do not match the estimates predicted by the corresponding pre-limiting model; {\it (iv)} estimates of the site-frequency spectra obtained by conditioning on the (random) complete sample gene genealogy are broadly similar (for the models considered here) to the estimates obtained without conditioning on the sample tree.

We discuss a probabilistic interpretation of a specific deterministic reaction-diffusion system [1,2] arising in literature for the modelling of the marble sulphation phenomenon. This phenomenon occurs when sulphur dioxide, which is present in the atmosphere due to human emissions, reacts with the calcium carbonate rock (like marble or limestone), causing a chemical reaction which facilitates the degradation of the material. Since this degradation phenomenon affects both modern and historical buildings, as well as artistic artifacts exposed to open air, a better understanding of it through mathematical modelling is needed.

We derive a single regularised non-conservative and path-dependent nonlinear partial differential equation and propose a probabilistic interpretation using a non-Markovian McKean-Vlasov-type stochastic differential equation, through a Feynman-Kac approach [3]. We discuss the well-posedness of such a stochastic model.

Then, we provide a deeper understanding of the deterministic reaction-diffusion system as a mean field approximation of a system of interacting Brownian particles subject to a non-conservative evolution law. Propagation of chaos holds for such a particle system. Therefore, the proposed McKean-Vlasov-Feynman-Kac SDE can be interpreted as the dynamics of sulfur dioxide molecules during the reaction, at the microscale.

This is a joint work with Daniela Morale and Stefania Ugolini (Universita’ degli Studi di Milano) [4].

REFERENCES

[1] D. Aregba-Driollet, F. Diele, and R. Natalini. “A Mathematical Model for the Sulphur Dioxide Aggression to Calcium Carbonate Stones: Numerical Approximation and Asymptotic Analysis”. In: SIAM Journal on Applied Mathematics 64.5 (2004), pp. 1636–1667.

[2] F. R. Guarguaglini and R. Natalini. “Fast reaction limit and large time behavior of solutions to a nonlinear model of sulphation phenomena”. In: Comm. Partial Differential Equations 32.1-3 (2007), pp. 163–189.

[3] A. Lecavil, N. Oudjane, and F. G. Russo. “Probabilistic representation of a class of non conservative nonlinear Partial Differential Equations”. In: ALEA, Lat. Am. J. Probab. Math. Stat. (2016), pp. 1189–1233.

[4] D. Morale, L. Tarquini and S. Ugolini. “A probabilistic interpretation of a non-conservative and path-dependent nonlinear reaction-diffusion system for the marble sulphation in Cultural Heritage”. 2024. arXiv:2407.19301

We identify the fluctuations of the partition function of the complex continuous random energy model (CREM) on a Galton-Watson tree in the full high-temperature regime. We show that in the strongly correlated regime a third high-temperature phase emerges as conjectured by Kabluchko and one of us. This phase is not present in the regime of weak correlations and the complex REM. A key ingredient is a Lindeberg-Feller type central limit theorem.

Stochastic reaction networks (SRNs) model stochastic effects for various applications, including intracellular chemical/biological processes, economy, and epidemiology. These models represent the dynamics of systems with several interacting species with low copy numbers. Essentially, SRNs are continuous-time Markov chains, with state being the copy number of each species (multivariate vector) and transitions representing randomly occurring reactions.

In many applications, a challenge arises when only a few of the species can be observed. This leads to the stochastic filtering problem, which is to estimate the distribution of unobserved (hidden) species counts conditional on the given observed trajectories. Despite the non-linearity, various numerical methods for this problem have been developed in recent years. However, these methods have limited applicability due to the curse of dimensionality, which means that their computational complexity grows exponentially with respect to the number of species.

We propose a novel Markovian projection based approach that reduces the dimensionality of the filtering problem for SRNs. This significantly enhances the efficiency of numerical methods for solving the filtering problem without imposing additional assumptions on the shape of the underlying distribution. Our analysis and empirical results highlight the superior computational efficiency of our method compared to state-of-the-art methods in the high-dimensional setting.