Uniqueness and Extinction of Weighted Markov Branching Processes

This paper focuses on discussing some basic properties of the weighted Markov branching process which is a natural generalisation of the ordinary Markov branching process. The regularity and uniqueness criteria, which are very easy to verify, are firstly established. Some important characteristics regarding the hitting times of such structure are obtained. In particular, the closed forms for the mean extinction time and conditional mean extinction time are presented. The explosion behaviour of the process is investigated and then the mean explosion time is derived. The mean global holding time and the mean total survival time are also obtained. AMS 2000 Subject Classification Primary 60 J27; Secondary 60 J80     

General collision branching processes with two parameters

A new class of branching models, the general collision branching processes with two parameters, is considered in this paper. For such models, it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing states. Regularity and uniqueness criteria are firstly established. Explicit expressions are then obtained for the extinction probability vector, the mean extinction times and the conditional mean extinction times. The explosion behavior of these models is investigated and an explicit expression for mean explosion time is established. The mean global holding time is also obtained. It is revealed that these properties are substantially different between the super-explosive and sub-explosive cases. This work was partially supported by National Natural Science Foundation of China (Grant No. 10771216), Research Grants Council of Hong Kong (Grant No. HKU 7010/06P) and Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry of China (Grant No. [2007]1108)     

Generalized Markov interacting branching processes

We consider a very general interacting branching process which includes most of the important interacting branching models considered so far. After obtaining some key preliminary results, we first obtain some elegant conditions regarding regularity and uniqueness. Then the extinction vector is obtained which is very easy to be calculated. The mean extinction time and the conditional mean extinction time are revealed. The mean explosion time and the total mean life time of the processes are also investigated and resolved.     

Uniqueness and extinction properties of generalised Markov branching processes

This paper focuses on the basic problems regarding uniqueness and extinction properties for generalised Markov branching processes. The uniqueness criterion is firstly established and a differential-integral equation satisfied by the transition functions of such processes is derived. The extinction probability is then obtained. A closed form is presented for both the mean extinction time and the conditional mean extinction time. It turns out that these important quantities are closely related to the elementary gamma function.     

A necessary and sufficient condition for noncertain extinction of a branching process in a random environment (BPRE)

It is known that a branching process in a random environment (BPRE) which is subcritical or critical either dies with probability one or, in the trivial case, corresponds to an immortal sterile population. In the supercritical case, various conditions are known to be necessary for noncertain extinction while other conditions are known to be sufficient. In this paper, a necessary and sufficient condition for noncertain extinction of a supercritical BPRE is given. In particular, it is shown that a supercritical BPRE has noncertain extinction if and only if there exists a random truncation, depending only on the environmental sequence, such that the truncated BPRE is supercritical and such that the sequence of truncation points grows more slowly than any exponential sequence.     

A continuous-state polynomial branching process

A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The process can also be obtained from a spectrally positive Lévy process through Lamperti type transformations. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly. Some of those are also new for the classical linear branching process. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branching process arises naturally as the rescaled limit of a sequence of discrete-state processes.     

Stationary availability of a semi‐Markov system with random maintenance

We consider a reparable system with a finite state space, evolving in time according to a semi‐Markov process. The system is stopped for it to be preventively maintained at random times for a random duration. Our aim is to find the preventive maintenance policy that optimizes the stationary availability, whenever it exists. The computation of the stationary availability is based on the fact that the above maintained system evolves according to a semi‐regenerative process. As for the optimization, we observe on numerical examples that it is possible to limit the study to the maintenance actions that begin at deterministic times. We demonstrate this result in a particular case and we study the deterministic maintenance policies in that case. In particular, we show that, if the initial system has an increasing failure rate, the maintenance actions improve the stationary availability if and only if they are not too long on the average, compared to the repairs ( a bound for the mean duration of the maintenance actions is provided). On the contrary, if the initial system has a decreasing failure rate, the maintenance policy lowers the stationary availability. A few other cases are studied. Copyright © 2000 John Wiley & Sons, Ltd.     

Some results on strong solutions of SDEs with applications to interest rate models

In this work, we investigate SDEs whose coefficients may depend on the entire past of the solution process. We introduce different Lipschitz-type conditions on the coefficients. It turns out that for existence and uniqueness of a strong solution it suffices to have Lipschitz continuity in mean, in a sense to be made precise. We then investigate when it suffices to have local Lipschitz conditions. Furthermore we consider the case of drift coefficients which are locally Lipschitz in mean. Finally we show how these results can be applied to prove existence and uniqueness of solutions in interest rate term structure models.     

Markov branching processes with immigration-migration and resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are...     

带灾难的非线性马尔可夫分枝过程

本文在文献[6]的基础上，集中考虑一类带灾难的非线性马尔可夫分枝过程的基本问题-唯一性，正则性和灭绝性。文章首先给出其Q-过程唯一性的证明，然后得出该畔程的正则性与[3]非线性马尔币夫分枝过程一样，最后，我们给出该Q-过程以概1l灭绝的充要条件是Q-过程正则。     

Dynamical behavior of a stochastic food chain chemostat model with Monod response functions

This paper studies a food chain chemostat model with Monod response functions, which is perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then sufficient conditions for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. Moreover, we consider the extinction of microbes in two cases. In the first case, both the predator and prey species are extinct. In the second case, only the predator species is extinct, and the prey species survives. Finally, numerical simulations are carried out to illustrate the theoretical results.     

Quasi-stationary distributions and Yaglom limits of self-similar Markov processes

We discuss the existence and characterization of quasi-stationary distributions and Yaglom limits of self-similar Markov processes that reach 0 in finite time. By Yaglom limit, we mean the existence of a deterministic function g$g$ and a non-trivial probability measure ν$\nu$ such that the process rescaled by g$g$ and conditioned on non-extinction converges in distribution towards ν$\nu$. We will see that a Yaglom limit exists if and only if the extinction time at 0$0$ of the process is in the domain of attraction of an extreme law and we will then treat separately three cases, according to whether the extinction time is in the domain of attraction of a Gumbel, Weibull or Fréchet law. In each of these cases, necessary and sufficient conditions on the parameters of the underlying Lévy process are given for the extinction time to be in the required domain of attraction. The limit of the process conditioned to be positive is then characterized by a multiplicative equation which is connected to a factorization of the exponential distribution in the Gumbel case, a factorization of a Beta distribution in the Weibull case and a factorization of a Pareto distribution in the Fréchet case.     

Dynamics of a Stochastic Predator–Prey Model with Stage Structure for Predator and Holling Type II Functional Response

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.     

Dynamics of a rate equation describing cluster-size evolution

In this paper, we show dynamics of Smoluchowski's rate equation which has been widely applied to studies of aggregation processes (i.e., the evolution of cluster-size distribution) in physics. We introduce dissociation in the rate equation while dissociation is neglected in previous works. We prove the positiveness of solutions of the equation, which is a basic guarantee for the effectiveness of the model since the possibility that some solution may be negative is excluded. For the case of cluster coalesce without dissociation, we show both the equilibrium uniqueness and the equilibrium stability under the condition that the monomer deposition stops. For the case that clusters evolve with dissociation and there is no monomer deposition, we show the equilibrium uniqueness and prove the equilibrium stability if the maximum cluster size is not larger than three while we show the equilibrium stability by numerical simulations if the maximum size is larger than three.     

Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure

We present a predator-prey model of Beddington-DeAngelis type functional response with stage structure on prey. The constant time delay is the time taken from birth to maturity about the prey. By the uniform persistence theories and monotone dynamic theories, sharp threshold conditions which are both necessary and sufficient for the permanence and extinction of the model as well as the sufficient conditions for the global stability of the coexistence equilibria are obtained. Biologically, it is proved that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by changing the prey carrying capacity: Our results suggest that the predator coexists with prey permanently if and only if predator's recruitment rate at the peak of prey abundance is larger than its death rate; and that the predator goes extinct if and only if predator's possible highest recruitment rate is less than or equal to its death rate; furthermore, our results also show that a sufficiently large mutual interference by predators can stabilize the system.     

The critical Bienaymé-Galton-Watson process

The existence and uniqueness of invariant measures for the critical, multitype Bienayné-Galton-Watson process is established by proving corresponding results for convex solutions of Abel's functional equation. A limit theorem is obtained by conditioning future extinction. No additional moment assumptions are made.     

The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays

In this paper, we consider the effect of diffusion on the permanence and extinction of a non-autonomous nonlinear growth rate single-species dispersal model with time delays. Firstly, the sufficient conditions of the permanence and extinction of the species are established, which shows if the growth rate and dispersal coefficients is suitable, the species is permanent, on the contrary, it is extinction. Secondly, an interesting result is established, that is, if only the species in some patches even in one patch is permanent, then it is also permanent in other patches. Finally, some examples together with their numerical simulations show the feasibility of our main results.     

Competitive equilibrium and singleton cores in generalized matching problems

We study competitive equilibria in generalized matching problems. We show that, if there is a competitive matching, then it is unique and the core is a singleton consisting of the competitive matching. That is, a singleton core is necessary for the existence of competitive equilibria. We also show that a competitive matching exists if and only if the matching produced by the top trading cycles algorithm is feasible, in which case it is the unique competitive matching. Hence, we can use the top trading cycles algorithm to test whether a competitive equilibrium exists and to construct a competitive equilibrium if one exists. Lastly, in the context of bilateral matching problems, we compare the condition for the existence of competitive matchings with existing sufficient conditions for the existence or uniqueness of stable matchings and show that it is weaker than most existing conditions for uniqueness.     

On the relationship between uniqueness and stability in sum-aggregative,symmetric and general differentiable games

This article explores the relationship between uniqueness and stability in differentiable regular games, with a major focus on the important classes of sum-aggregative, two-player and symmetric games. We consider three types of popular dynamics, continuous-time gradient dynamics as well as continuous- and discrete-time best-reply dynamics, and include aggregate-taking behavior as a non-strategic behavioral variant. We show that while in general games stability conditions are only sufficient for uniqueness, they are likely to be necessary as well in models with sum-aggregative or symmetric payoff functions. In particular, a unique equilibrium always verifies the stability conditions of all dynamics if strategies are equilibrium complements, and this also holds for both continuous-time dynamics if strategies are equilibrium substitutes with bounded slopes. These findings extend to the case of aggregate-taking equilibria. We further analyze the stability relations between the various dynamics, and demonstrate that the restrictive nature of the discrete dynamics originates from simultaneity of adjustments. Asynchronous decisions or heterogeneous forward thinking may stabilize the adjustment process.     

Well-Posedness and Asymptotic Behaviors for a Predator-Prey System with Lévy Noise

In this paper, well-posedness and asymptotic behaviors for a predator-prey system with Lévy noise are studied by using stochastic analytical techniques. Firstly, the existence and uniqueness of positive global solution with positive initial value is proved. Then, stochastic permanence for the system is investigated. Finally, persistence in mean and extinction for the system are discussed and some numerical simulations are provided to support our results.