随机环境中依赖年龄分枝过程的爆炸问题

讨论了随机环境中依赖年龄分枝过程中的爆炸问题,得到了过程爆炸的判定定理以及关于两个过程爆炸的比较定理.     

Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 ? s)^1+αL(1 ? s) where α ∈ (0, 1) and L(x) varies slowly as x → 0⁺. Then we find, as t → ∞, (P{Z(t)> 0})^αL(P{Z(t)>0})～ μ/αt where μ is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u^α)^?1/α for u ?/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.     

Limit Processes for Age-Dependent Branching Particle Systems

We consider systems of spatially distributed branching particles in R ^d . The particle lifelengths are of general form, hence the time propagation of the system is typically not Markov. A natural time-space-mass scaling is applied to a sequence of particle systems and we derive limit results for the corresponding sequence of measure-valued processes. The limit is identified as the projection on R ^d of a superprocess in R ₊×R ^d . The additive functional characterizing the superprocess is the scaling limit of certain point processes, which count generations along a line of descent for the branching particles.     

Limit theorems for sums determined by branching and other exponentially growing processes

A branching process counted by a random characteristic has been defined as a process which at time t is the superposition of individual stochastic processes evaluated at the actual ages of the individuals of a branching population. Now characteristics which may depend not only on age but also on absolute time are considered. For supercritical processes a distributional limit theorem is proved, which implies that classical limit theorems for sums of characteristics evaluated at a fixed age point transfer into limit theorems for branching processes counted by these characteristics. A point is that, though characteristics of different individuals should be independent, the characteristics of an individual may well interplay with the reproduction of the latter. The result requires a sort of L^p-continuity for some 1 ? p ? 2. Its proof turns out to be valid for a wider class of processes than branching ones.For the case p = 1 a number of Poisson type limits follow and for p = 2 some normality approximations are concluded. For example results are obtained for processes for rare events, the age of the oldest individual, and the error of population predictions.This work has been supported by a grant from the Swedish Natural Science Research Council.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

On the first birth and the last death in a generation in a multi-type Markov branching process

In a multi-type continuous time Markov branching process the asymptotic distribution of the first birth in and the last death (extinction) of the kth generation can be determined from the asymptotic behavior of the probability generating function of the vector Z(^k)(t), the size of the kth generation at time t, as t tends to zero or as t tends to infinity, respectively. Apart from an appropriate transformation of the time scale, for a large initial population the generations emerge according to an independent sum of compound multi-dimensional Poisson processes and become extinct like a vector of independent reversed Poisson processes. In the first birth case the results also hold for a multi-type Bellman-Harris process if the life span distributions are differentiable at zero.     

Numerical analysis for stochastic age-dependent population equations with Poisson jumps

In this paper, stochastic age-dependent population equations with Poisson jumps are considered. In general, most of stochastic age-dependent population equations with jumps do not have explicit solutions, thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical Euler scheme and show the convergence of the numerical approximation solution to the true solution.     

HARVESTING IN POPULATION MODELS WITH DELAYED RECRUITMENT AND AGE-DEPENDENT MORTALITY

We formulate a Volterra equation model to describe the effect of harvesting on populations with delayed recruitment and age-dependent mortality. This model is more complicated mathematically than the unharvested model, and we describe some numerical simulations which indicate the possible qualitative behaviours.     

Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching

In this paper, a class of stochastic age-dependent population equations with Markovian switching is considered. The main aim of this paper is to investigate the convergence of the numerical approximation of stochastic age-dependent population equations with Markovian switching. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions. An example is given for illustration.     

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??In this paper, we introduce a class of stochastic age-dependent population equations with Poisson jumps. Existence and uniqueness of energy solutions for stochastic age-dependent population dynamic system are proved under local non-Lipschitz condition in Hilbert space.     

Numerical analysis for stochastic age-dependent population equations with fractional Brownian motion

Stochastic age-dependent population equations, one of the important classes of hybrid systems are studied. In general most equations of stochastic age-dependent population do not have explicit solutions. Thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical scheme and show the convergence of the numerical approximation solution to the analytic solution. In the last section a numerical example is given.     

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

In this paper,we form a method to calculate the probability generating function of the total progeny of multitype branching process.As examples,we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and(2-1) random walk.Consequently,we could give the probability generating functions and the distributions of the first passage time of corresponding random walks.Especially,for recurrent random walk which could stay at its position with probability 0 r 1,we show that the tail probability of the first passage time decays as 2/(π(1-r)~(1/2)) n~(1/1)= when n →∞.     

Limit theorems for flows of branching processes

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.     

The age of a Markov process

The concept of a limiting conditional age distribution of a continuous time Markov process whose state space is the set of non-negative integers and for which {0} is absorbing is defined as the weak limit as t→∞ of the last time before t an associated “return” Markov process exited from {0} conditional on the state, j, of this process at t. It is shown that this limit exists and is non-defective if the return process is ρ-recurrent and satisfies the strong ratio limit property. As a preliminary to the proof of the main results some general results are established on the representation of the ρ-invariant measure and function of a Markov process. The conditions of the main results are shown to be satisfied by the return process constructed from a Markov branching process and by birth and death processes. Finally, a number of limit theorems for the limiting age as j→∞ are given.     

Nonlinear PDEs and measure-valued branching type processes

We deal with the probabilistic approach to a nonlinear operator Λ of the form , in connection with the works of M. Nagasawa, N. Ikeda, S. Watanabe, and M.L. Silverstein on the discrete branching processes. Instead of the Laplace operator we may consider the generator of a right (Markov) process, called base process, with a general (not necessarily locally compact) state space. It turns out that solutions of the nonlinear equation Λu=0 are produced by the harmonic functions with respect to the (linear) generator of a discrete branching type process. The consideration of the general state space allows to take as base process a measure-valued superprocess (in the sense of E.B. Dynkin). The probabilistic counterpart is a Markov process which is a combination between a continuous branching process (e.g., associated with a nonlinear operator of the form Δu−uα, 1<α?2) and a discrete branching type one, on a space of configurations of finite measures. Our approach uses probabilistic and analytic potential theoretical tools, like the potential kernel of a continuous additive functional and the subordination operators.     

A nonlinear age-dependent model with spatial diffusion

The main goal of this paper is the study of the existence and uniqueness of a positive solution for a nonlinear age-dependent equation with spatial diffusion. For that, we mainly use the properties of an eigenvalue problem related to the equation and the sub-supersolution method. We justify that this method works for this equation, in which there is a potential blow-up and a nonlocal initial condition.     

Likelihood Ratio Processes for Markovian Particle Systems with Killing and Jumps

We consider Markov processes built from pasting together pieces of strong Markov processes which are killed at a position dependent rate and connected via a transition kernel. We give necessary and sufficient conditions for local absolute continuity of probability laws for such processes on a suitable path space and derive an explicit formula for the corresponding likelihood ratio process. The main tool is the consideration of the process between successive jumps – what we call ‘elementary experiments’ – and criteria for absolute continuity of laws of the process there. We apply our results to systems of branching diffusions with interactions and immigrations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Limit distributions for generalized Jiřina processes with immigration

A relationship between continuous state population-size-dependent branching (CSDB) processes with or without immigration and discrete state population-size-dependent branching (DSDB) processes with or without immigration is established via the representation of the former. Based on this relationship, some limiting distributions of CSDB processes with or without immigration are obtained.     

年龄相关的种群扩散系统广义解的正则性

证明了年龄相关的种群扩散系统广义解的正则性,为进一步讨论该系统最优控制的存在性奠定了理论基础,也为种群系统控制问题的实际研究提供了数学理论和方法.     

广义分枝过程的强遍历性

在文[1]的基础上，进一步讨论了广义分枝Q－矩阵的强遍历性及随机单调性。