Abstract:
We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we present a full kinetic framework for age-structured interacting populations undergoing birth, death and fission processes, in spatially dependent environments. We define the complete probability density for the population-size-age-chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behaviour. Our approach utilizes mixed type, multi-dimensional probability distributions similar to those employed in the study of gas kinetics, with terms that satisfy BBGKY-like equation hierarchies.

Abstract:
We study the exact entanglement dynamics of two qubits in a common structured reservoir. We demonstrate that, for certain classes of entangled states, entanglement sudden death occurs, while for certain initially factorized states, entanglement sudden birth takes place. The backaction of the non-Markovian reservoir is responsible for revivals of entanglement after sudden death has occurred, and also for periods of disentanglement following entanglement sudden birth.

Abstract:
We develop a fully stochastic theory for age-structured populations via Doi-Peliti quantum field theoretical methods. The operator formalism of Doi is first developed, whereby birth and death events are represented by creation and annihilation operators, and the complete probabilistic representation of the age-chart of a population is represented by states in a suitable Hilbert space. We then use this formalism to rederive several results in companion paper [6], including an equation describing the moments of the age-distribution, and the distribution of the population size. The functional representation of coherent states used by Peliti to analyze discrete Fock space is then adapted to incorporate the continuous age parameters, and a path integral formulation constructed. We apply these formalisms to a range of birth-death processes and show that although many of the results from Doi-Peliti formalism can be derived in a purely probabilistic way, the efficient formalism offered by second quantization methods provides a powerful technique that can manage algebraically complex birth death processes in a compact manner.

Abstract:
Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but they are structurally unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using e.g., the Bellman-Harris equation, do not resolve a population's age-structure and are unable to quantify population-size dependencies. Conversely, current theories that include size-dependent population dynamics (e.g., mathematical models that include carrying capacity such as the Logistic equation) cannot be easily extended to take into account age-dependent birth and death rates. In this paper, we present a systematic derivation of a new fully stochastic kinetic theory for interacting age-structured populations. By defining multiparticle probability density functions, we derive a hierarchy of kinetic equations for the stochastic evolution of an ageing population undergoing birth and death. We show that the fully stochastic age-dependent birth-death process precludes factorization of the corresponding probability densities, which then must be solved by using a BBGKY-like hierarchy. However, explicit solutions are derived in two simple limits and compared with their corresponding mean-field results. Our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age- and population-dependent stochastic dynamics applicable to the study of demography, stem cell dynamics, and disease evolution.

Abstract:
Spatial birth-and-death processes with a finite number of particles are obtained as unique solutions to certain stochastic equations. Conditions are given for existence and uniqueness of such solutions, as well as for continuous dependence on the initial conditions. The possibility of an explosion and connection with the heuristic generator of the process are discussed.

Abstract:
We investigate steady states of a quasilinear first order hyperbolic partial integro-differential equation. The model describes the evolution of a hierarchical structured population with distributed states at birth. Hierarchical size-structured models describe the dynamics of populations when individuals experience size-specific environment. This is the case for example in a population where individuals exhibit cannibalistic behavior and the chance to become prey (or to attack) depends on the individual's size. The other distinctive feature of the model is that individuals are recruited into the population at arbitrary size. This amounts to an infinite rank integral operator describing the recruitment process. First we establish conditions for the existence of a positive steady state of the model. Our method uses a fixed point result of nonlinear maps in conical shells of Banach spaces. Then we study stability properties of steady states for the special case of a separable growth rate using results from the theory of positive operators on Banach lattices.

Abstract:
We study the probabilistic evolution of a birth and death continuous time measure-valued process with mutations and ecological interactions. The individuals are characterized by (phenotypic) traits that take values in a compact metric space. Each individual can die or generate a new individual. The birth and death rates may depend on the environment through the action of the whole population. The offspring can have the same trait or can mutate to a randomly distributed trait. We assume that the population will be extinct almost surely. Our goal is the study, in this infinite dimensional framework, of quasi-stationary distributions when the process is conditioned on non-extinction. We firstly show in this general setting, the existence of quasi-stationary distributions. This result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators. We then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi-stationary distribution with maximal exponential decay rate. The proof of uniqueness is based on an absolute continuity property with respect to a reference measure.

Abstract:
We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least!) in three different ways. One approach is to study spectral properties of a parametrized family of unbounded operators. The alternative approach, on which we focus here, is based on the reformulation of the problem as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we briefly discuss a third approach, which is based on the finite rank approximation of the recruitment operator.

Abstract:
Spectral measures and transition probabilities of birth and death processes with 0= 0=0 are obtained as limite when 0 ￠ ’0+ of the corresponding quantities. In particular the case of finite population is discussed in full detail. Pure birth and death processes are used to derive an inequality for Dirichlet polynomials.

Abstract:
Extortion strategies can dominate any opponent in an iterated prisoner's dilemma game. But if players are able to adopt the strategies performing better, extortion becomes widespread and evolutionary unstable. It may sometimes act as a catalyst for the evolution of cooperation, and it can also emerge in interactions between two populations, yet it is not the evolutionary stable outcome. Here we revisit these results in the realm of spatial games. We find that pairwise imitation and birth-death dynamics return known evolutionary outcomes. Myopic best response strategy updating, on the other hand, reveals new counterintuitive solutions. Defectors and extortioners coarsen spontaneously, which allows cooperators to prevail even at prohibitively high temptations to defect. Here extortion strategies play the role of a Trojan horse. They may emerge among defectors by chance, and once they do, cooperators become viable as well. These results are independent of the interaction topology, and they highlight the importance of coarsening, checkerboard ordering, and best response updating in evolutionary games.