arXiv subjects

Alejandro Roldán-Correa

Publications and source records attributed to Alejandro Roldán-Correa.

15 recordsLinked to original sources

arXiv2026

Immigration Processes with Binomial Catastrophes and Random Survival Parameters

We consider immigration processes with binomial catastrophes and random survival parameters. Two sources of randomness are analyzed. In the first model, the survival parameter is independently resampled at each catastrophe. In the second model, individuals are assigned independent survival parameters at birth, which remain fixed over time. We show that the first model exhibits almost sure extinction, as in the classical case with a fixed survival parameter. In contrast, the second model exhibits a phase transition, admitting survival with positive probability, depending on the distribution of individual survival parameters. We provide explicit formulas for both the survival probability and the expected time to extinction. Finally, our proofs establish a novel methodological bridge with the Firework process, unifying population dynamics with spatial models of information spreading.

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

A stochastic model for immune response with mutations and evolution: the non-spatial setting

We consider a stochastic model for a pathogen population in the presence of an immune response, in which pathogen types are partially ordered by ancestry and the immune system must eliminate ancestor types before it can eliminate their descendants. In this model, pathogens reproduce independently at rate $λ>0$ and, at each birth, a mutation occurs with probability $r\in(0,1]$, producing a novel type that is antigenically distinct and whose elimination by the immune system is delayed relative to its ancestors. We provide an explicit characterization of the survival--extinction phase transition and compute the expected total progeny in the subcritical regime. We then extend the model by allowing mutations to be deleterious: conditional on mutation, with probability $p\in(0,1]$ the mutation is beneficial and with probability $1-p$ it is deleterious, producing a sterile offspring. For this extension, we obtain an explicit survival criterion in terms of $(λ,r,p)$ and identify parameter regimes in which survival is possible only for an intermediate range of mutation probabilities, reflecting the balance between immune escape and mutational load.

math.PR↗

arXiv2025

A contact process with stronger mutations on trees

We consider a spatial stochastic model for a pathogen population growing inside a host that attempts to eliminate the pathogens through its immune system. The pathogen population is divided into different types. A pathogen can either reproduce by generating a pathogen of its own type or produce a pathogen of a new type that does not yet exist in the population. Pathogens with living ancestral types are protected against the host's immune system as long as their progenitors are still alive. Each pathogen type without living ancestral types is eliminated by the immune system after a random period, independently of the other types. When a pathogen type is eliminated from the system, all pathogens of this type die simultaneously. In this paper, we determine the conditions on the set of model parameters that dictate the survival or extinction of the pathogen population when the dynamics unfold on graphs with an infinite tree structure.

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↗

arXiv2024

Fluctuations of the occupation density for a parking process

Consider the following simple parking process on $Λ_n := \{-n, \ldots, n\}^d,d\ge1$: at each step, a site $i$ is chosen at random in $Λ_n$ and if $i$ and all its nearest neighbor sites are empty, $i$ is occupied. Once occupied, a site remains so forever. The process continues until all sites in $Λ_n$ are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of $Λ_n$ is called the jamming limit and is denoted by $X_{Λ_n}$. Ritchie (2006) constructed a stationary random field on $\mathbb Z^d$ obtained as a (thermodynamic) limit of the $X_{Λ_n}$'s as $n$ tends to infinity. As a consequence of his construction, he proved a strong law of large numbers for the proportion of occupied sites in the box $Λ_n$ for the random field $X$. Here we prove the central limit theorem, the law of iterated logarithm, and a gaussian concentration inequality for the same statistics. A particular attention will be given to the case $d=1$, in which we also obtain new asymptotic properties for the sequence $X_{Λ_n},n\ge1$ as well as a new proof to the closed-form formula for the occupation density of the parking process.

math.PR↗

arXiv2023

Extinction time in growth models subject to geometric catastrophes

Recently, different dispersion strategies in population models subject to geometric catastrophes have been considered as strategies to improve the chance of po\-pu\-lation's survival. Such dispersion strategies have been contrasted with the strategy where there is no dispersion, comparing the probabilities of survival. In this article, we contrast survival strategies when extinction occurs almost surely, evaluating which strategy prolongs population's life span. Our results allow one to analyze what is the best strategy for survival based on parameters as the probability that each individual exposed to catastrophe survives, the growth rate of the colony, the type of dispersion and the spatial restrictions.

math.PR↗

arXiv2023

RMF accessibility percolation on oriented graphs

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology: a random number, called its fitness, is assigned to each vertex of a graph, then a path in the graph is accessible if fitnesses are strictly increasing through it. In the Rough Mount Fuji (RMF) model the fitness function is defined on the graph as $ω(v)=η(v)+θ\cdot d(v)$, where $θ$ is a positive number called the drift, $d$ is the distance to the source of the graph and $η(v)$ are i.i.d. random variables. In this paper we determine values of $θ$ for having RMF accessibility percolation on the hypercube and the two-dimensional lattices $\mathbb{L}^2$ and $\mathbb{L}^2_{alt}$.

math.PR↗

arXiv2021

Evolution with mass extinction on $\mathbb{T}_d^+$

We propose a stochastic model for evolution through mutation and natural selection of a population that evolves on a $\bbT_d^+$ tree. We think of this model as a way of describing the evolution fitness landscape of a population. We obtain sharp and distinct conditions on the set of parameters for extinction and survival.

math.PR↗

arXiv2018

Comment on "Nodal infection in Markovian susceptible-infected-susceptible and susceptible-infected-removed epidemics on networks are non-negatively correlated''

Cator and Van Mieghem [Cator E, Van Mieghem P., Phys. Rev. E 89, 052802 (2014)] stated that the correlation of infection at the same time between any pair of nodes in a network is non-negative for the Markovian SIS and SIR epidemic models. The arguments used to obtain this result rely strongly on the graphical construction of the stochastic process, as well as the FKG inequality. In this note we show that although the approach used by the authors applies to the SIS model, it cannot be used for the SIR model as stated in their work. In particular, we observe that monotonicity in the process is crucial for invoking the FKG inequality. Moreover, we provide an example of simple graph for which the nodal infection in the SIR Markovian model is negatively correlated.

physics.soc-ph↗

arXiv2018

Accessiblility Percolation with Crossing Valleys on $n$-ary Trees

In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leaf, where any $k$ consecutive vertices in the path contain at least one vertex of the subsequence. An $n$-ary tree, with height $h$, is a tree whose vertices at distance at most $h-1$ to the root have $n$ children. For the case of $n$-ary trees, we prove that, as $h$ tends to infinity the probability of having such subsequence: tends to 1, if $n$ grows significantly faster than $\sqrt[k]{h/(ek)}$; and tends to 0, if $n$ grows significantly slower than $\sqrt[k]{h/(ek)}$.

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

Colonization and Collapse

Many species live in colonies that thrive for a while and then collapse. Upon collapse very few individuals survive. The survivors start new colonies at other sites that thrive until they collapse, and so on. We introduce spatial and non-spatial stochastic processes for modeling such population dynamic. Besides testing whether dispersion helps survival in a model experiencing large fluctuations, we obtain conditions for the population to get extinct or to survive.

math.PR↗

arXiv2016

Dispersion as a survival strategy

We consider stochastic growth models to represent population subject to catastrophes. We analyze the subject from different set ups considering or not spatial restrictions, whether dispersion is a good strategy to increase the population viability. We find out it strongly depends on the effect of a catastrophic event, the spatial constraints of the environment and the probability that each exposed individual survives when a disaster strikes.

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↗