Related papers: A random walk with catastrophes
Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the…
We study the extreme events taking place on complex networks. The transport on networks is modelled using random walks and we compute the probability for the occurance and recurrence of extreme events on the network. We show that the nodes…
Consider a uniformly mixing population which grows as a super-critical linear birth and death process. At some time an infectious disease (of SIR or SEIR type) is introduced by one individual being infected from outside. It is shown that…
An important challenge in several disciplines is to understand how sudden changes can propagate among coupled systems. Examples include the synchronization of business cycles, population collapse in patchy ecosystems, markets shifting to a…
We study the ergodic behaviour of a discrete-time process $X$ which is a Markov chain in a stationary random environment. The laws of $X_t$ are shown to converge to a limiting law in (weighted) total variation distance as $t\to\infty$.…
We introduce a counting process to model the random occurrence in time of car traffic accidents, taking into account some aspects of the self-excitation typical of this phenomenon. By combining methods from probability and differential…
The question of whether biological populations survive or are eventually driven to extinction has long been examined using mathematical models. In this work we study population survival or extinction using a stochastic, discrete…
We study a single-lane traffic model that is based on human driving behavior. The outflow from a traffic jam self-organizes to a critical state of maximum throughput. Small perturbations of the outflow far downstream create emergent traffic…
Simulation results for Mobile Ad-Hoc Networks (MANETs) are fundamentally governed by the underlying Mobility Model. Thus it is imperative to find whether events functionally dependent on the mobility model 'converge' to well defined…
We consider a trait-structured population subject to mutation, birth and competition of logistic type, where the number of coexisting types may fluctuate. Applying a limit of rare mutations to this population while keeping the population…
We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a…
We consider an asexually reproducing population on a finite type space whose evolution is driven by exponential birth, death and competition rates, as well as the possibility of mutation at a birth event. On the individual-based level this…
The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model (VM) dynamics)…
We investigate a simple quantitative genetics model subjet to a gradual environmental change from the viewpoint of the phylogenies of the living individuals. We aim to understand better how the past traits of their ancestors are shaped by…
How many shuffles are needed to mix up a deck of cards? This question may be answered in the language of a random walk on the symmetric group, $S_{52}$. This generalises neatly to the study of random walks on finite groups, themselves a…
We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on ${\mathbb Z}^d$ , and its jump rate at $({\mathbf x},t)$ is given by a fixed function $\varphi$ of the…
Theoretical ecologists have long sought to understand how the persistence of populations depends on biotic and abiotic factors. Classical work showed that demographic stochasticity causes the mean time to extinction to increase…
Life is on the razor's edge as resulting from competitive birth and death random forces. We illustrate this aphorism in the context of three Markov chain population models where systematic random immigration events promoting growth are…
In this paper, we propose a mathematical framework that governs the evolution of epidemic dynamics, encompassing both intra-population dynamics and inter-population mobility within a metapopulation network. By linearizing this dynamical…
The dynamics of one dimensional iterative maps in the regime of fully developed chaos is studied in detail. Motivated by the observation of dynamical structures around the unstable fixed point we introduce the geometrical concept of a…