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For supercritical multitype branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population…
The first chapter concerns monotype population models. We first study general birth and death processes and we give non-explosion and extinction criteria, moment computations and a pathwise representation. We then show how different scales…
In this paper, we consider contact processes on locally compact separable metric spaces with birth and death rates heterogeneous in space. Conditions on the rates that ensure the existence of invariant measures of contact processes are…
In the system we study, 1's and 0's represent occupied and vacant sites in the contact process with births at rate $\lambda$ and deaths at rate 1. $-1$'s are sterile individuals that do not reproduce but appear spontaneously on vacant sites…
This paper is an attempt to formalize analytically the question raised in "World Population Explained: Do Dead People Outnumber Living, Or Vice Versa?" Huffington Post, \cite{HJ}. We start developing simple deterministic Malthusian growth…
We discuss the criticality in the stochastic SIR model for infectious diseases. We adopt the path-integral formalism for the propagation of infections among susceptible, infectious, and removed individuals, and perform the perturbative and…
This paper applies the theory of continuous phase transitions of statistical mechanics to a slider-block model. The slider-block model is chosen as a representative of systems with avalanches. Similar behavior can be observed in a…
The parameters of many classes of birth-death processes cannot be inferred uniquely from phylogenetic trees: infinitely many parameter combinations yield the same distribution of phylogenetic trees. Here, we show that parameter…
In this review, we discuss the applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described.…
The stacked contact process is a stochastic model for the spread of an infection within a population of hosts located on the $d$-dimensional integer lattice. Regardless of whether they are healthy or infected, hosts give birth and die at…
I present three models of plant--pathogen interactions. The models are stochastic and spatially explicit at the scale of individual plants. For each model, I use a version of pair approximation or moment closure along with a separation of…
We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by…
This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered:…
We present a generic epidemic model with stochastic parameters, in which the dynamics self-organize to a critical state with suppressed exponential growth. More precisely, the dynamics evolve into a quasi-steady-state, where the effective…
In this paper we explore the life expectancy limits by based on the stochastic modeling of mortality and applying the first exit or hitting time theory of a stochastic process. The main assumption is that the health state or the "vitality",…
Our understanding of past evolutionary change is often based on reconstructions based on incomplete data, raising fundamental questions about the degree to which we can make reliable inferences about past evolutionary processes. This was…
We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new…
A branching process in varying environment with generation-dependent immigration is a modification of the standard branching process in which immigration is allowed and the reproduction and immigration laws may vary over the generations.…
Stochastic kinetic models are often used to describe complex biological processes. Typically these models are analytically intractable and have unknown parameters which need to be estimated from observed data. Ideally we would have…
We establish connections between the absorption probabilities of a class of birth-death processes with killing, and the stationary tail of a related class of birth-death processes with catastrophes. The major ingredients of the proofs are a…