Stochastic Birth-death Process Research Articles

A stochastic Gompertz birth-death process

This paper shows that the stochastic Gompertz birth-death process is a special case of nonhomogeneous birth and death processes. The absolute probability, the first four cumulants as well as absorption probabilities are then derived for these processes.

Reproductive potential of Gyrodactylus bullatarudis (Monogenea) on guppies (Poecilia reticulata)

SUMMARYTwo simple experiments were undertaken using the viviparous ectoparasite, Gyrodactylus bullatarudis, and laboratory-reared guppies, Poecilia reticulata, whereby detailed records of the number and temporal sequence of all births and the age at death were obtained for flukes on isolated guppies. Gyrodactylus bullatarudis has an average fecundity of 1·68 offspring during its expected life-span of 4·20 days. The instantaneous birth rate is independent of generation but dependent on age. The first offspring is born approximately 1 day after the birth of the parent and subsequent offspring are born at 2–2·5 day intervals. The average instantancous birth rate (under given experimental conditions) is 0.43/parasite/day. The death rate increases exponentially with the age of the fluke and has an average value of 0.24/parasite/day. A simple deterministic model incorporating age structure and an age-dependent death rate was found to provide a good fit to the observed exponential increase in G. bullatarudis numbers. The variability observed in this host–parasite system was found to be largely a function of chance, as shown by Monte Carlo simulations of stochastic birth–death processes.

The Approach to Equilibrium and the Steady-State Probability Distribution of Adult Numbers in Tribolium brevicornis

The rate of population growth in adult numbers, A, for the flour beetle Tribolium was characterized by the mathematical model dA/dt = X A exp(-CA) - A D with the biological entities pupal productivity, X, adult inhibition of the immature life stages, C, and the death rate among the adults, D. A local stability analysis of the equilibrium A* = log(X/D)/C revealed that the eigenvalue λ = D log(D/X) and A* was stable if X > D. The time it takes for a perturbation to decay was evaluated using the time constant τ = 1/|λ|. The changes in adult numbers were then viewed as a stochastic birth-death process. The numbers of adults were found to asymptotically assume a constant mean value of Ā(t) = A* = log(X/D)/C and a constant variance of V(t) = V* = 1/C. Equations were established for the approach of Ā(t) and V(t) to their respective equilibrium values together with the steady-state probability distribution of adult numbers. Formulas to estimate A*, X, D, C, and λ were obtained based on the adult population size data. Experimental observations on T. brevicornis showed a good correspondence to the theoretical construct.

Orbital diamagnetism of a charged Brownian particle undergoing a birth-death process

Considers the magnetic response of a charged Brownian particle undergoing a stochastic birth-death process. The latter simulates the electron-hole pair production and recombination in semiconductors. The authors obtain non-zero, orbital diamagnetism which can be large without violating the Van Leeuwen theorem (1921).

On approximating the expected behavior of stochastic epidemiological models applicable to small populations

Many biological systems can be characterized by stochastic multivariate birth-death processes. Such models are often non-linear making the estimation of their mean behavior difficult. Several techniques for approximating the moments of such stochastic systems are proposed. Their use is illustrated by a study of the mean behavior of an epidemiological model applicable to small populations. Computer calculations using the theoretical methods are also given.

An approximate stochastic model for phage reproduction in a bacterium

The stochastic birth-death process considered in this paper provides an approximate model for phage reproduction in a bacterium. In a recent paper, Hershey [1] has discussed reproduction and recombination in phage crosses, and a deterministic model for the reproductive process has been the subject of a previous note by the author [2]. A very readable account of the process is given by Sanders [3] in his recent article, “The life of viruses”.