A self-regulating and patch subdivided population

Methodology and Computing in Applied Probability

We consider a continuous-time symmetric branching random walk on the ddimensional lattice, d ≥ 1, and assume that at the initial moment there is one particle at every lattice point. Moreover, we assume that the underlying random walk has a finite variance of jumps and the reproduction law is described by a critical Bienamye-Galton-Watson process at every lattice point. We study the structure of the particle subpopulation generated by the initial particle situated at a lattice point x. We answer why vanishing of the majority of subpopulations does not affect the convergence to the steady state and leads to clusterization for lattice dimensions d = 1 and d = 2.

Probability Theory and Related Fields, 1992

arXiv: Probability, 2015

We consider a spatial multi-type branching model in which individuals migrate in geographic space according to random walks and reproduce according to a state-dependent branching mechanism which can be sub-, super- or critical depending on the local intensity of individuals of the different types. The model is a Lotka-Volterra type model with a spatial component and is related to two models studied in \cite{BlathEtheridgeMeredith2007} as well as to earlier work in \cite{Etheridge2004} and in \cite{NeuhauserPacala1999}. Our main focus is on the diffusion limit of small mass, locally many individuals and rapid reproduction. This system differs from spatial critical branching systems since it is not density preserving and the densities for large times do not depend on the initial distribution but mainly on the carrying capacities. We prove existence of the infinite particle model and the system of interacting diffusions as solutions of martingale problems or systems of stochastic equat...

2008

A continuous time branching random walk on the lattice is considered in which individuals may produce children at the origin only. Assuming that the underlying random walk is symmetric and the offspring reproduction law is critical we prove a conditional limit theorem for the number of individuals at the origin. Keywords: catalytic branching random walk; critical two-dimensional Bellman-Harris process 1 Statement of problem and main results We consider the following modification of a standard branching random walk on ¡. Consider a population of individuals evolving as follows. The population is initiated at time t ¢ 0 by a single particle. Being outside the origin the particle performs a continuous time random walk on ¡ with infinitesimal transition matrix A ¢¤ £ a ¥ x ¦ y§¨ £ x © y�� � ¦ a ¥ 0 ¦ 0§� � 0¦ until the moment when it hits the origin. At the origin it spends an exponentially distributed time with parameter 1 and then either jumps to � ¢ a point y �� ¥ 0 � α § a ¥ 0 ¦ y §...

Stochastic Processes and their Applications, 2015

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power p, for p ∈ [0, 2). The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.

Siberian Mathematical Journal - SIB MATH J-ENGL TR, 1988

Let us introduce the notation: R(t) is the total number of particles at $(u), having appeared on the interval (0, t); P(t)=P(~(t)=O); Q(t)=l'P(t); N=limN(t); A=MN; F(s, t)= ~-~oo t For the counting function N(t), the notation ~IfdN(~)(U) and ~ f(u)dN(u) denote the finite U~O tt=O product and sum of random components over u such that dN(u) = N(u) - N(u - O) ~ O. The al- most certain finiteness of N(t) for all t follows from MN < ~. In this paper, for critical processes (A = I)we formally describe existence conditions as t § ~ of the limits for an expression of the type (t, x) = ,1 (t) P (~ (t) r (t) > z; ~ (t) = 0). Using the simplest properties of Laplace transforms and the generalized continuity theorem (see (3)), we will reduce the original problem to an equivalent -the study of the limit U (t,:)~) = ~ e-Z~u (t, x) dx = (P (t) -- F (e-~r (t), t)) ~-~ (t),~ t-~- 0% o which is the Laplace transform for u(x)=limu(t,x). t-~oo In actuality, the presence of the limit u(x)=lim...

Journal de Physique I, 1995

In this paper we study a simple deterministic tree structure: an initial individual generates a finite number of offspring, each of which has given integer valued lifetime, iterating the same procedure when dying. Three asymptotic distributions of this asynchronous deterministic branching procedure are considered: the generation distribution, the ability of individuals to generate offspring and the age distribution. Thermodynamic formalism is then developped to reveal the multifractal nature of the mass splitting associated to our process. On considère l'itération d'une structure déterministe arborescente selon laquelle un ancêtre engendre un nombre fini de descendants dont la durée de vie (à valeurs entières) est donnée. Dans un premier temps on s'intéresse aux trois distributions asymptotiques suivantes : répartition des générations, aptitude à engendrer des descendants et répartition selon l'âge. Ensuite nous développons le formalisme thermodynamique pour mettre en évidence le caractère multifractal de la scission d'une masse unitaire associée à cette arborescence.

arXiv (Cornell University), 2022

Motivated by applications to COVID dynamics, we describe a branching process in random environments model {Zn} whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points {(τj, νj)}-specifically the values of the process at crossing times, viz., {(Zτ j , Zν j)}-along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distribution of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.

2017

We study the Markov dynamics of an infinite birth-and-death system of point entities placed in $\mathbb{R}^d$, in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are just continuous and integrable we prove that the evolution of states of this model preserves their sub-Poissonicity, and hence the local self-regulation (suppression of clustering) takes place. Upper bounds for the correlation functions of all orders are also obtained for both long and short dispersals, and for all values of the intrinsic mortality rate $m\geq 0$.

Physica A: Statistical Mechanics and its Applications, 1999

We study a plant population model introduced recently by J. Wallinga [OIKOS 74, 377 (1995)]. It is similar to the contact process ('simple epidemic', 'directed percolation'), but instead of using an infection or recovery rate as control parameter, the population size is controlled directly and globally by removing excess plants. We show that the model is very closely related to directed percolation (DP). Anomalous scaling laws appear in the limit of large populations, small densities, and long times. These laws, associated critical exponents, and even some non-universal parameters, can be related to those of DP. As in invasion percolation and in other models where the rôles of control and order parameters are interchanged, the critical value p c of the wetting probability p is obtained in the scaling limit as singular point in the distribution of infection rates. We show that a mean field type approximation leads to a model studied by Y.C. Zhang et al. [J. Stat. Phys. 58, 849 (1990)]. Finally, we verify the claim of Wallinga that family extinction in a marginally surviving population is governed by DP scaling laws, and speculate on applications to human mitochondrial DNA.