Size-dependent Branching Process Research Articles

The Asymptotic Behavior for Generalized Jiřina Process

As the classic branching process, the Galton-Watson process has obtained intensive attentions in the past decades. However, this model has two idealized assumptions–discrete states and time-homogeneity. In the present paper, we consider a branching process with continuous states, and for any given n∈N, the branching law of every particle in generation n is determined by the population size of generation n. We consider the case that the process is extinct with Probability 1 since in this case the process will be substantially different from the size-dependent branching process with discrete states. We give the extinction rate in the sense of L2 and almost surely by the form of harmonic moments, that is to say, we show how fast {Zn−1} grows under a group of sufficient conditions. From the result of the present paper, we observe that the extinction rate will be determined by an asymptotic behavior of the mean of the branching law. The results obtained in this paper have the more superiority than the counterpart from the existing literature.

On the lifetime of a size-dependent branching process

We discuss lifetimes for a family of population-dependent branching processes. The attenuation factor (due to environment or competition, for example) is of Ricker type, i.e., the probability of an individual having offspring at all is of the form if the total population is n. Equivalently we can write the probability as where the carrying capacity K is the inverse of the attenuating factor. It is well known that the expected lifetime of such a process is exponential in K. If the carrying capacities vary much over time, for instance, if they are i.i.d. with a heavy-tailed distribution, the extinction scenario may change to a growth-catastrophe one with expected lifetimes much shorter. In addition to Ricker’s model, production functions of the Beverton–Holt and Hassell types are also discussed.

A Galton–Watson process with a threshold

Abstract In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

Limit theorems for generalized Jiřina processes

It is proved by a new method that continuous state size-dependent branching processes converge almost surely to 0 or ∞ under some mild conditions. In addition, these processes are almost surely extinct in the subcritical case.

Estimation of the PCR efficiency based on a size-dependent modelling of the amplification process

We propose a stochastic modelling of the PCR amplification process by a size-dependent branching process starting as a supercritical Bienaymé–Galton–Watson transient phase and then having a saturation near-critical size-dependent phase. This model based on the concept of saturation allows one to estimate the probability of replication of a DNA molecule at each cycle of a single PCR trajectory with a very good accuracy. To cite this article: N. Lalam et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Estimation of the offspring mean of a supercritical or near-critical size-dependent branching process

We consider a single-type supercritical or near-critical size-dependent branching process { N n } n such that the offspring mean converges to a limit m ⩾ 1 with a rate of convergence of order N n α as the population size N n grows to ∞ and the variance may change at the rate N n β , where α > 0 and − 1 ⩽ β < 1 . The offspring mean depends on an unknown parameter θ 0 that we estimate on the non-extinction set by using the conditional least squares method. We prove the strong consistency of the estimator of θ 0 as n → ∞ under some general conditions on the asymptotic behavior of the process. We also give its asymptotic distribution for a certain class of size-dependent branching processes. To cite this article: N. Lalam, C. Jacob, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Multitype population size-dependent branching processes with dependent offspring

In this paper, the multitype population size-dependent branching process with dependent offspring is introduced and the probability of extinction of such a process is investigated. Sufficient conditions for both almost sure extinction and survival of the process are shown to depend on the Perron–Frobenius eigenvalue of adequate limit matrices. Finally, theoretical results are illustrated with an example.

Modelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency

We propose a stochastic modelling of the PCR amplification process by a size-dependent branching process starting as a supercritical Bienaymé-Galton-Watson transient phase and then having a saturation near-critical size-dependent phase. This model allows us to estimate the probability of replication of a DNA molecule at each cycle of a single PCR trajectory with a very good accuracy.

Modelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency

We propose a stochastic modelling of the PCR amplification process by a size-dependent branching process starting as a supercritical Bienaymé-Galton-Watson transient phase and then having a saturation near-critical size-dependent phase. This model allows us to estimate the probability of replication of a DNA molecule at each cycle of a single PCR trajectory with a very good accuracy.

Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process

We consider a single-type supercritical or near-critical size-dependent branching process {N n } n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size N n grows to ∞ and the variance may vary at the rate where −1 ≤ β &amp;lt; 1. The offspring mean m(N) = m + μN -α + o(N -α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {N h ,…,N n } by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model m θ,ν̂ n (·), where ν̂ n is any estimation of the parameter of the nuisance part (O(N -α) if θ0 = m and o(N -α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

Estimation of the offspring mean in a supercritical or near-critical size-dependent branching process

We consider a single-type supercritical or near-critical size-dependent branching process {Nn}n such that the offspring mean converges to a limit m ≥ 1 with a rate of convergence of order as the population size Nn grows to ∞ and the variance may vary at the rate where −1 ≤ β &lt; 1. The offspring mean m(N) = m + μN-α + o(N-α) depends on an unknown parameter θ0 belonging either to the asymptotic model (θ0 = m) or to the transient model (θ0 = μ). We estimate θ0 on the nonextinction set from the observations {Nh,…,Nn} by using the conditional least-squares method weighted by (where γ ∈ ℝ) in the approximate model mθ,ν̂n(·), where ν̂n is any estimation of the parameter of the nuisance part (O(N-α) if θ0 = m and o(N-α) if θ0 = μ). We study the strong consistency of the estimator of θ0 as γ varies, with either h or n - h remaining constant as n → ∞. We use either a minimum-contrast method or a Taylor approximation of the first derivative of the contrast. The main condition for obtaining strong consistency concerns the asymptotic behavior of the process. We also give the asymptotic distribution of the estimator by using a central-limit theorem for random sums and we show that the best rate of convergence is attained when γ = 1 + β.

On the quasi-stationary distribution of a stochastic Ricker model

We model the evolution of a single-species population by a size-dependent branching process Z t in discrete time. Given that Z t = n the expected value of Z t+1 may be written nexp( r − γn) where r > 0 is a growth parameter and γ > 0 is an (inhibitive) environmental parameter. For small values of γ the short-term evolution of the normed process γZ t follows the deterministic Ricker model closely. As long as the parameter r remains in a region where the number of periodic points is finite and the only bifurcations are the period-doubling ones ( r in the beginning of the bifurcation sequence), the quasi-stationary distribution of γZ t is shown to converge weakly to the uniform distribution on the unique attracting or weakly attracting periodic orbit. The long-term behavior of γZ t differs from that of the Ricker model, however: γZ t has a finite lifetime a.s. The methods used rely on the central limit theorem and Markov's inequality as well as dynamical systems theory.

Population-dependent branching processes with a threshold

A branching process model where offspring distributions depend on the threshold as well as on the population size is introduced. Behaviour of such models is related to the behaviour of the corresponding deterministic models, whose behaviour is known from the chaos theory. Asymptotic behaviour of such branching processes is obtained when the population threshold is large. If the initial population size is comparable to the threshold then the size of the nth generation relative to the threshold has a normal distribution with the mean being the nth iterate of the one-step mean function. If the initial population size is negligible when compared to the threshold and the offspring distributions converge then the size of any fixed generation approaches that of a size-dependent branching process. These results are supported by a simulation study.

Geometric Growth in Near-Supercritical Population Size Dependent Multitype Galton-Watson Processes

We consider a multitype population size dependent branching process in discrete time. A process is considered to be near-supercritical if the mean matrices of offspring distributions approach the mean matrix of a supercritical process as the population size increases. We show that if the convergence of the means to the supercritical mean is fast enough and the second moments of offspring distributions do not grow too fast as the population size increases, then the process grows geometrically fast. Similarly to the classical multitype Galton-Watson process, the process grows at the geometric rate determined by the largest eigenvalue of the limiting matrix in the direction of the corresponding left eigenvector.

Stochastic Difference Equations and Generalized Gamma Distributions

We study the asymptotic growth rates of discrete-time stochastic processes $(X_n)$, where the first two conditional moments of the process depend only on the present state. Such processes satisfy a stochastic difference equation $X_{n + 1} = X_n + g(X_n) + R_{n + 1}$, where $g$ is a positive function and $(R_n)$ is a martingale difference sequence. It is known that a large class of such processes diverges with positive probability, and when properly normalized converges almost surely or converges in distribution to a normal or a lognormal distribution. Here we find a class of processes that when properly normalized converges in distribution to a generalized gamma distribution. Applications of this result to state dependent random walks and population size-dependent branching processes yield new results and reprove some of the known results.