A jump-growth model for predator-prey dynamics: derivation and application to marine ecosystems

This paper investigates the dynamics of biomass in a marine ecosystem. A stochastic process is defined in which organisms undergo jumps in body size as they catch and eat smaller organisms. Using a systematic expansion of the master equation, we derive a deterministic equation for the macroscopic dynamics, which we call the deterministic jump-growth equation, and a linear Fokker-Planck equation for the stochastic fluctuations. The McKendrick--von Foerster equation, used in previous studies, is shown to be a first-order approximation, appropriate in equilibrium systems where predators are much larger than their prey. The model has a power-law steady state consistent with the approximate constancy of mass density in logarithmic intervals of body mass often observed in marine ecosystems. The behaviours of the stochastic process, the deterministic jump-growth equation and the McKendrick--von Foerster equation are compared using numerical methods. The numerical analysis shows two classes of attractors: steady states and travelling waves.Comment: 27 pages, 4 figures. Final version as published. Only minor change

Jump-diffusion unravelling of a non Markovian generalized Lindblad master equation

The "correlated-projection technique" has been successfully applied to derive a large class of highly non Markovian dynamics, the so called non Markovian generalized Lindblad type equations or Lindblad rate equations. In this article, general unravellings are presented for these equations, described in terms of jump-diffusion stochastic differential equations for wave functions. We show also that the proposed unravelling can be interpreted in terms of measurements continuous in time, but with some conceptual restrictions. The main point in the measurement interpretation is that the structure itself of the underlying mathematical theory poses restrictions on what can be considered as observable and what is not; such restrictions can be seen as the effect of some kind of superselection rule. Finally, we develop a concrete example and we discuss possible effects on the heterodyne spectrum of a two-level system due to a structured thermal-like bath with memory.Comment: 23 page

Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion

We study the voter model and related random-copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random-copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales, and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are hard to access by existing theoretical approaches.Comment: 37 pages, 10 figures. Revised version with additional discussion and simulation results to appear in J Phys

Sub-diffusion in External Potential: Anomalous hiding behind Normal

We propose a model of sub-diffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model. Constant force leads to the transition from non-ergodic sub-diffusion to seemingly ergodic diffusive behavior. However, we show it remains anomalous in a sense that the diffusion coefficient depends on the force and the anomalous exponent. For the quadratic potential we find that the anomalous exponent defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.Comment: 6 pages, 3 figure

Non-homogeneous random walks, subdiffusive migration of cells and anomalous chemotaxis

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.Comment: 18 pages, accepted for publicatio

Some stochastic models for structured populations : scaling limits and long time behavior

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 may lead to different qualitative approximations, either ODEs or SDEs. The prototypes of these equations are the logistic (deterministic) equation and the logistic Feller diffusion process. The convergence in law of the sequence of processes is proved by tightness-uniqueness argument. In these large population approximations, the competition between individuals leads to nonlinear drift terms. We then focus on models without interaction but including exceptional events due either to demographic stochasticity or to environmental stochasticity. In the first case, an individual may have a large number of offspring and we introduce the class of continuous state branching processes. In the second case, catastrophes may occur and kill a random fraction of the population and the process enjoys a quenched branching property. We emphasize on the study of the Laplace transform, which allows us to classify the long time behavior of these processes. In the second chapter, we model structured populations by measure-valued stochastic differential equations. Our approach is based on the individual dynamics. The individuals are characterized by parameters which have an influence on their survival or reproduction ability. Some of these parameters can be genetic and are inheritable except when mutations occur, but they can also be a space location or a quantity of parasites. The individuals compete for resources or other environmental constraints. We describe the population by a point measure-valued Markov process. We study macroscopic approximations of this process depending on the interplay between different scalings and obtain in the limit either integro-differential equations or reaction-diffusion equations or nonlinear super-processes. In each case, we insist on the specific techniques for the proof of convergence and for the study of the limiting model. The limiting processes offer different models of mutation-selection dynamics. Then, we study two-level models motivated by cell division dynamics, where the cell population is discrete and characterized by a trait, which may be continuous. In 1 particular, we finely study a process for parasite infection and the trait is the parasite load. The latter grows following a Feller diffusion and is randomly shared in the two daughter cells when the cell divides. Finally, we focus on the neutral case when the rate of division of cells is constant but the trait evolves following a general Markov process and may split in a random number of cells. The long time behavior of the structured population is then linked and derived from the behavior a well chosen SDE (monotype population)