arXiv subjects

Fábio P. Machado

Publications and source records attributed to Fábio P. Machado.

14 recordsLinked to original sources

arXiv2026

Catastrophe-dispersion models in random and varying environments across generations

We study a class of branching processes in which the offspring distribution is not specified directly but is induced by a cycle of internal colony growth, catastrophic reduction and structured dispersal. The parameters governing growth, survival and dispersal are allowed to vary deterministically or randomly from one generation to the next, giving rise to branching processes in varying and random environments with implicitly defined offspring laws. We show that survival and extinction are governed entirely by the associated log-mean process, exactly as in the classical theory. The paper treats four qualitatively different dispersal mechanisms and establishes a universal ordering of the induced offspring means. For Poissonian growth with binomial survival, explicit thresholds are obtained that determine extinction or survival uniformly over all four mechanisms. A series of ecologically motivated examples with Yule-Simon growth illustrates the versatility of the framework.

math.PR↗

arXiv2026

Growth Models Under Uniform Catastrophes

We consider stochastic growth models for populations organized in colonies and subject to uniform catastrophes. To assess population viability, we analyze scenarios in which individuals adopt dispersion strategies after catastrophic events. For these models, we derive explicit expressions for the survival probability and the mean time to extinction, both with and without spatial constraints. In addition, we complement this analysis by comparing uniform catastrophes with binomial and geometric catastrophes in models with dispersion and no spatial restrictions. Here, the terms uniform, binomial and geometric refer to the probability distributions governing the number of individuals that survive immediately after a catastrophe. This comparison allows us to quantify the impact of different types of catastrophic events on population persistence.

math.PR↗

arXiv2026

The Frog Model on $\mathbb{Z}$ with Discrete Weibull Lifetimes and Random Parameter $p$

We study the frog model on $\mathbb{Z}$ with particle wise discrete Weibull lifetimes. Each particle has an i.i.d. survival parameter $π\in(0,1)$; conditionally on $π=p$, its lifetime $Ξ$ satisfies \[ P(Ξ\ge k\mid π=p)=p^{k^γ},\qquad k\in\mathbb{N}_0,γ>0. \] The law of $π$ has right edge density \[ f_π(u)\sim(1-u)^{β-1},L\big((1-u)^{-1}\big)\qquad (u\uparrow 1), \] with $β>0$ and $L$ slowly varying; let $η$ denote the common law of the i.i.d. initial occupation numbers $\{η_x\}_{x\in\mathbb{Z}}$. The survival parameter distribution strictly extends the Beta family, while the lifetime distribution extends the geometric case. We prove a sharp extinction and survival dichotomy with the $γ-$dependent threshold \[ β_c:=\frac{1}{2γ}. \] If $β>β_c$ and $E(η)<\infty$, the process becomes extinct almost surely; if $β<β_c$ and $P(η=0)<1$, it survives with positive probability. At the boundary $β=β_c$ we provide explicit criteria in terms of $\limsup/\liminf$ of $L(n^{2γ})$. The case $γ=1$ (geometric lifetimes) recovers the benchmark $β_c=\frac{1}{2}$ and the critical refinements previously obtained for random geometric lifetimes.

math.PR↗

arXiv2025

The Frog Model on $\mathbb{Z}$ with General Random Survival Parameter

We study the frog model on $\mathbb{Z}$ with particle-wise random geometric lifetimes: each particle has a survival parameter $π\in(0,1)$ sampled i.i.d., whose density near $1$ satisfies $f_π(u)\sim (1-u)^{β-1}L\big((1-u)^{-1}\big)$ with $β>0$, and $L$ slowly varying. This strictly extends the $\mathrm{Beta}(α,β)$ case. Let $η$ denote the common law of the i.i.d.\ initial number of particles $\{η_x\}_{x\in\mathbb{Z}}$. Using a percolation comparison and sharp one-particle displacement tails, we obtain a universal threshold at $β=\tfrac12$. If $β>\tfrac12$ and $E(η)<\infty$, extinction occurs almost surely. If $β<\tfrac12$ and $\mathbb{P}(η=0)<1$, survival has positive probability. At the boundary $β=\tfrac12$ we give sharp criteria: extinction if $E(η)<\infty$ and $8\,\limsup_{n\to\infty}L(n^2)<1/E(η)$; survival if $\mathbb{P}(η=0)<1$ and $\sqrt{2}\,\liminf_{n\to\infty}L(n^2)>1/E(η)$. These results recover the Carvalho-Machado threshold for Beta laws and show that only the exponent $β$ governs the phase transition, while $L$ impacts the critical regime.

math.PR↗

arXiv2025

Frog model on $\mathbb{Z}$ with random survival parameter

We study the frog model on $ \mathbb{Z} $ with geometric lifetimes, introducing a random survival parameter. Active and inactive particles are placed at the vertices of $ \mathbb{Z} $. The lifetime of each active particle follows a geometric random variable with parameter $ 1-p $, where $ p $ is randomly sampled from a distribution $ π$. Each active particle performs a simple random walk on $ \mathbb{Z} $ until it dies, activating any inactive particles it encounters along its path. In contrast to the usual case where $ p $ is fixed, we show that there exist non-trivial distributions $ π$ for which the model survives with positive probability. More specifically, for $π\sim Beta(α,β)$, we establish the existence of a critical value $ β=0.5 $, that separates almost sure extinction from survival with positive probability. Furthermore, we show that the model is recurrent whenever it survives with positive probability.

math.PR↗

arXiv2024

Critical Conditions for the Coverage of Complete Graphs with the Frog Model

We consider a system of interacting random walks known as the frog model. Let $\mathcal{K}_n=(\mathcal{V}_n,\mathcal{E}_n)$ be the complete graph with $n$ vertices and $o\in\mathcal{V}_n$ be a special vertex called the root. Initially, $1+η_o$ active particles are placed at the root and $η_v$ inactive particles are placed at each other vertex $v\in\mathcal{V}_n\setminus\{o\}$, where $\{η_v\}_{v\in \mathcal{V}_n}$ are i.i.d. random variables. At each instant of time, each active particle may die with probability $1-p$. Every active particle performs a simple random walk on $\mathcal{K}_n$ until the moment it dies, activating all inactive particles it hits along its path. Let $V_\infty(\mathcal{K}_n,p)$ be the total number of visited vertices by some active particle up to the end of the process, after all active particles have died. In this paper, we show that $V_\infty(\mathcal{K}_n,p_n)\geq (1-ε)n$ with high probability for any fixed $ε>0$ whenever $p_n\rightarrow 1$. Furthermore, we establish the critical growth rate of $p_n$ so that all vertices are visited. Specifically, we show that if $p_n=1-\fracα{\log n}$, then $V_\infty(\mathcal{K}_n,p_n)=n$ with high probability whenever $0<α E(η)$.

math.PR↗

arXiv2024

Uniform dispersion in growth models on homogeneous trees

We consider the dynamics of a population spatially structured in colonies that are vulnerable to catastrophic events occurring at random times, which randomly reduce their population size and compel survivors to disperse to neighboring areas. The dispersion behavior of survivors is critically significant for the survival of the entire species. In this paper, we consider an uniform dispersion scheme, where all possible survivor groupings are equally probable. The aim of the survivors is to establish new colonies, with individuals who settle in empty sites potentially initiating a new colony by themselves. However, all other individuals succumb to the catastrophe. We consider the number of dispersal options for surviving individuals in the aftermath of a catastrophe to be a fixed value $d$ within the neighborhood. In this context, we conceptualize the evolution of population dynamics occurring over a homogeneous tree. We investigate the conditions necessary for these populations to survive, presenting pertinent bounds for survival probability, the number of colonized vertices, the extent of dispersion within the population, and the mean time to extinction for the entire population.

math.PR↗

arXiv2022

The coverage ratio of the frog model on complete graphs

The frog model is a system of interacting random walks. Initially, there is one particle at each vertex of a connected graph $\mathcal{G}$. All particles are inactive at time zero, except for the one which is placed at the root of $\mathcal{G}$, which is active. At each instant of time, each active particle may die with probability $1-p$. Once an active particle survives, it jumps on one of its nearest vertices, chosen with uniform probability, performing a discrete time simple symmetric random walk (SRW) on $\mathcal{G}$. Up to the time it dies, it activates all inactive particles it hits along its way. From the moment they are activated on, every such particle starts to walk, performing exactly the same dynamics, independent of everything else. In this paper, we take $\mathcal{G}$ as the $n-$complete graph ($\mathcal{K}_n$, a finite graph with each pair of vertices linked by an edge). We study the limit in $n$ of the coverage ratio, that is, the proportion of visited vertices by some active particle up to the end of the process, after all active particles have died.

math.PR↗

arXiv2016

Colonization and collapse on Homogeneous Trees

We investigate a basic immigration process where colonies grow, during a random time, according to a general counting process until collapse. Upon collapse a random amount of individuals survive. These survivors try independently establishing new colonies at neighbour sites. Here we consider this general process subject to two schemes, Poisson growth with geometric catastrophe and Yule growth with binomial catastrophe. Independent of everything else colonies growth, during an exponential time, as a Poisson (or Yule) process and right after that exponential time their size is reduced according to geometric (or binomial) law. Each survivor tries independently, to start a new colony at a neighbour site of a homogeneous tree. That colony will thrive until its collapse, and so on. We study conditions on the set of parameters for these processes to survive, present relevant bounds for the probability of survival, for the number of vertices that were colonized and for the reach of the colonies compared to the starting point.

math.PR↗

arXiv2016

The rumor percolation model and its variations

The study of rumor models from a percolation theory point of view has gained a few adepts in the last few years. The persistence of a rumor, which may consistently spread out throughout a population can be associated to the existence of a giant component containing the origin of a graph. That is one of the main interest in percolation theory. In this paper we present a quick review of recent results on rumor models of this type.

math.PR↗

arXiv2016

The fitness of the strongest individual in the subcritical GMS model

We derive the strongest individual fitness distribution on a variation for a species survival model proposed by Guiol, Machado and Schinazi \cite{GMS11}. We point out to the fact that this distribution relies on the Gauss hypergeometric function and when $p=\frac{1}{2}$ on the Hypergeometric function type I distribution

math.PR↗

arXiv2011

Limit theorems for a general stochastic rumour model

We study a general stochastic rumour model in which an ignorant individual has a certain probability of becoming a stifler immediately upon hearing the rumour. We refer to this special kind of stifler as an uninterested individual. Our model also includes distinct rates for meetings between two spreaders in which both become stiflers or only one does, so that particular cases are the classical Daley-Kendall and Maki-Thompson models. We prove a Law of Large Numbers and a Central Limit Theorem for the proportions of those who ultimately remain ignorant and those who have heard the rumour but become uninterested in it.

math.PR↗

arXiv2010

The cone percolation on $\bbT_d$

We study a rumour model from a percolation theory and branching process point of view. The existence of a giant component is related to the event where the rumour spreads out trough an infinite number of individuals. We present sharp lower and upper bounds for the probability of that event, according to the distribution of the random variables that defines the radius of influence of each individual.

math.PR↗

arXiv2010

On the behaviour of a rumour process with random stifling

We propose a realistic generalization of the Maki-Thompson rumour model by assuming that each spreader ceases to propagate the rumour right after being involved in a random number of stifling experiences. We consider the process with a general initial configuration and establish the asymptotic behaviour (and its fluctuation) of the ultimate proportion of ignorants as the population size grows to $\infty$. Our approach leads to explicit formulas so that the limiting proportion of ignorants and its variance can be computed.

math.PR↗