arXiv subjects

João Luiz de Oliveira Madeira

Publications and source records attributed to João Luiz de Oliveira Madeira.

4 recordsLinked to original sources

arXiv2026

A functional law of large numbers for a spatial model of Muller's ratchet

The spatial Muller's ratchet is a model introduced by Foutel-Rodier and Etheridge to study the impact of cooperation and competition on the fitness of an expanding asexual population. The model is an interacting particle system consisting of particles performing symmetric random walks that reproduce and die with rates that depend on the local number of particles. For each particle, we keep track of the number of deleterious mutations that it carries, and after each birth event, with some positive probability, the offspring particle can acquire an additional mutation that gives it a lower reproduction rate than its parent. We show that, under an appropriate scaling, the process converges weakly to the solution of an infinite system of partial differential equations (PDEs), confirming non-rigorous computations of Foutel-Rodier and Etheridge. Combining the weak convergence with analytical results for the limiting PDE system, we derive quantitative lower and upper bounds on the proportion of particles with mutations that hold with high probability for the particle system. A key obstacle is the absence of uniform bounds on the number of particles per site, together with the presence of infinitely many types of particle and the nonlinear interactions. To address this, we establish a new tightness criterion for interacting particle systems in general Lp spaces based only on local properties of the dynamics.

math.PR↗

arXiv2026

Existence, uniqueness and moment bounds for a spatial model of Muller's ratchet

In this article, we consider a generalisation of the spatial Muller's ratchet introduced by Foutel-Rodier and Etheridge. This particle system is a spatial model of an asexual population, with birth and death rates that depend on the local population density. Particles live in discrete demes and migrate to neighbouring demes. Each particle carries some number of mutations (its `type'), and additional mutations can occur during birth events. Mutations are assumed to be deleterious, i.e.~carrying a higher number of mutations results in a lower birth rate. Our main result shows that this interacting particle system can be constructed even when the total initial number of particles is infinite. We also prove moment bounds on the local density of particles; these bounds are a crucial ingredient of the proof of a law of large numbers result for the particle system in the companion article. The construction of the particle system uses a sequence of approximating processes. Proving weak convergence of this sequence of processes is non-trivial because the particle system is non-monotone and interactions are non-local in type space. The uniqueness of the limit relies on a delicate coupling argument.

math.PR↗

arXiv2026

How seed banks evolve in plants: a stochastic dynamical system subject to a strong drift

We study how changes in population size and fluctuating environmental conditions influence the establishment of seed banks in plants. Our model is a modification of the Wright-Fisher model with seed bank, introduced by Kaj, Krone and Lascoux. We distinguish between wild type individuals, producing only nondormant seeds, and mutants, producing seeds with dormancy. To understand how changing population size shapes the establishment of seed banks, we analyse the process under a diffusive scaling. The results support the biological insight that seed banks are favoured in a declining population, and disfavoured if population size is constant or increasing. The surprise is that this is true even when population sizes are changing very slowly -- over evolutionary timescales. We also investigate the influence of short-term fluctuations, such as annual variations in rainfall or temperature. Mathematically, our analysis reduces to a stochastic dynamical system forced onto a manifold by a large drift, which converges under scaling to a diffusion on the manifold. Inspired by the Lyapunov--Schmidt reduction, we derive an explicit formula for the limiting diffusion coefficients by projecting the system onto its linear counterpart. This provides a general framework for deriving diffusion approximations in models with strong drift and nonlinear constraints.

math.PR↗

arXiv2022

Homeostasis in Networks with Multiple Input Nodes and Robustness in Bacterial Chemotaxis

A biological system achieve homeostasis when there is a regulated quantity that is maintained within a narrow range of values. Here we consider homeostasis as a phenomenon of network dynamics. In this context, we improve a general theory for the analysis of homeostasis in network dynamical systems with distinguished input and output nodes, called `input-output networks'. The theory allows one to define `homeostasis types' of a given network in a `model independent' fashion, in the sense that the classification depends on the network topology rather than on the specific model equations. Each `homeostasis type' represents a possible mechanism for generating homeostasis and is associated with a suitable `subnetwork motif' of the original network. Our contribution is an extension of the theory to the case of networks with multiple input nodes. To showcase our theory, we apply it to bacterial chemotaxis, a paradigm for homeostasis in biochemical systems. By considering a representative model of Escherichia coli chemotaxis, we verify that the corresponding abstract network has multiple input nodes. Thus showing that our extension of the theory allows for the inclusion of an important class of models that were previously out of reach. Moreover, from our abstract point of view, the occurrence of homeostasis in the studied model is caused by a new mechanism, called input counterweight homeostasis. This new homeostasis mechanism was discovered in the course of our investigation and is generated by a balancing between the several input nodes of the network -- therefore, it requires the existence of at least two input nodes to occur. Finally, the framework developed here allows one to formalize a notion of `robustness' of homeostasis based on the concept of `genericity' from the theory dynamical systems. We discuss how this kind of robustness of homeostasis appears in the chemotaxis model.

q-bio.MN↗