Birth-death System Research Articles

Spatially inhomogeneous population dynamics: beyond the mean field approximation

We propose a novel method for numerical modeling of spatially inhomogeneous moment dynamics of populations with nonlocal dispersal and competition in continuous space. It is based on analytically solvable decompositions of the time evolution operator for a coupled set of master equations. This has allowed us—for the first time in the literature—to perform moment dynamics simulations of spatially inhomogeneous birth-death systems beyond the mean-field approach and to calculate the inhomogeneous pair correlation function using the Kirkwood superposition ansatz. As a result, we revealed a number of new subtle effects, possible in real populations. Namely, for systems with short-range dispersal and mid-range competition, strong clustering of entities at small distances followed by their deep disaggregation at larger separations are observed in the wavefront of density propagation. For populations in which the competition range is much shorter than that of dispersal, the pair correlation function exhibits long-tail asymptotics. Remarkably, the latter effect takes place only due to the spatial inhomogeneity and thus was completely unknown before. Moreover, the both effects get stronger in the direction of propagation. All these types of behaviour are interpreted as a trade-off between the dispersal and competition in the coexistence of reproductive pair correlations and the inhomogeneity of the density of the system.

Use of dirichlet distributions and orthogonal projection techniques for the fluctuation analysis of steady-state multivariate birth–death systems

Approximate weak solutions of the Fokker–Planck equation represent a useful tool to analyze the equilibrium fluctuations of birth–death systems, as they provide a quantitative knowledge lying in between numerical simulations and exact analytic arguments. In this paper, we adapt the general mathematical formalism known as the Ritz–Galerkin method for partial differential equations to the Fokker–Planck equation with time-independent polynomial drift and diffusion coefficients on the simplex. Then, we show how the method works in two examples, namely the binary and multi-state voter models with zealots.

One‐Dimensional Birth‐Death Process and Delbrück‐Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions

As a mathematical theory for the stochastic, nonlinear dynamics of individuals within a population, Delbrück‐Gillespie process (DGP) is a birth–death system with state‐dependent rates which contain the system size V as a natural parameter. For large V, it is intimately related to an autonomous, nonlinear ODE as well as a diffusion process. For nonlinear dynamical systems with multiple attractors, the quasi‐stationary and stationary behavior of such a birth–death process can be understood in terms of a separation of time scales by a T*∼eαV (α > 0): a relatively fast, intra‐basin diffusion for t≪T* and a much slower inter‐basin Markov jump process for t≫T*. In this paper for one‐dimensional systems, we study both stationary behavior (t=∞) in terms of invariant distribution , and finite time dynamics in terms of the mean first passsage time (MFPT) . We obtain an asymptotic expression of MFPT in terms of the “stochastic potential”. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the for a DGP. When n1 and n2 belong to two different basins of attraction, the MFPT yields the T*(V) in terms of Φ (x, V) ≈φ0(x) + (1/V)φ1(x). For systems with saddle‐node bifurcations and catastrophe, discontinuous “phase transition” emerges, which can be characterized by Φ (x, V) in the limit of . In terms of timescale separation, the relation between deterministic local nonlinear bifurcations, and stochastic global phase transition is discussed. The one‐dimensional theory is a pedagogic first step toward a general theory of DGP.

The Z-Transform Applied to a birth-death Process Having Varying birth and death rates

The analysis of a birth-death process using the z-transform was recently reported for processes hav-ing fixed transition probabilities between states. The current report extends that analysis to proc-esses having transition probabilities that can differ from state to state. It is then shown that the model can be used to study practical queuing and birth-death systems where the arrival, birth, ser-vice and death rates may differ from state to state.Keywords: Markov processes, birth-death models, frame synchronization, queuing systems, z-transform

Spectral Theory of Metastability and Extinction in Birth-Death Systems

We suggest a general spectral method for calculating the statistics of multistep birth-death processes and chemical reactions of the type mA-->nA (m and n are positive integers) which possess an absorbing state. The method employs the generating function formalism in conjunction with the Sturm-Liouville theory of linear differential operators. It yields accurate results for the extinction statistics and for the quasistationary probability distribution, including large deviations, of the metastable state. The power of the method is demonstrated on the example of binary annihilation and triple branching 2A--> ø, A-->3A, representative of the rather general class of dissociation-recombination reactions.

Matrix-geometric solution of discrete time MAP/PH/1 priority queue

We use the matrix-geometric method to study the discrete time MAP/PH/1 priority queue with two types of jobs. Both preemptive and non-preemptive cases are considered. We show that the structure of the R matrix obtained by Miller for the Birth-Death system can be extended to our Quasi-Birth-Death case. For both preemptive and non-preemptive cases the distributions of the number of jobs of each type in the system are obtained and their waiting times are obtained for the non-preemptive. For the preemptive case we obtain the waiting time distribution for the high priority job and the distribution of the lower priority job's wait before it becomes the leading job of its priority class. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 23–50, 1998

A stochastic multicloud model for tropical convection

A stochastic model for representing the missing variability in global climate models due to unresolved features of organized tropical convection is presented here. We use a Markov chain lattice model to represent small scale convective elements which interact with each other and with the large scale environmental variables through convective available potential energy (CAPE) and middle troposphere dryness. Each lattice site is either occupied by a cloud of a certain type (congestus, deep or stratiform) or it is a clear sky site. The lattice sites are assumed to be independent from each other so that a coarse-grained stochastic birth-death system, which can be evolved with a very low computational overhead, is obtained for the cloud area fractions alone. The stochastic multicloud model is then coupled to a simple tropical climate model consisting of a system of ODEs, mimicking the dynamics over a single GCM grid box. Physical intuition and observations are employed here to constrain the design of the models. Numerical simulations showcasing some of the dynamical features of the coupled model are presented below.