Stochastic Interacting Population Dynamics and Related Problems

连续态分支过程（Continuous-state branching process，简称CSBP）起源于经典离散态Galton-Watson分支过程的时间-布居标度极限，或者起源于当到达0时停止的谱正Lévy过程的Lamperti变换。最近在引入繁殖机制取决于整个种群特征的种群模型方面取得了进展。其中，Li（2019）引入了一类具有非线性分支机制的CSBPs。它们是从具有一般速率函数的谱正Lévy过程通过广义Lamperti型随机时间变换得到的。直觉上，这样一个过程的分支率取决于它当前的种群规模。结果，加性支化性质不再成立，许多标准的CSBP方法失败了。非线性分支机制允许奇异的性质，例如从无穷大下降，这在Foucart等人（2020）中进行了详细研究，其中确定了某些类速率函数从无穷大下降的速度。Li等人（2019 a）还研究了更一般的非线性CSBPs的边界行为。本着同样的精神，Ren et al.（2019）提出并研究了两个种群的随机Lotka-Volterra种群动力学，该种群动力学由独立Lévy噪声驱动的两个随机微分方程系统建模，其中两个种群都根据Li et al.（2019 a）的非线性CSBP进化，第二个种群的分支率取决于第一个种群。 我们计划继续对非线性CSBPs和相关的相互作用种群系统进行研究。我们感兴趣的是非线性CSBP的熄灭行为，并想知道，当它发生时，该过程如何缓慢地接近0。我们还想证明非线性CSBPs的强Feller性质，我们相信这将有助于研究非线性CSBPs的准平稳分布。我们将进一步研究Li等人（2019 a）的模型和Ren等人（2019）的模型。对于Li et al.（2019 a）的非线性CSBPs，一个更具挑战性的开放问题是建立边界分类的尖锐积分检验。对于Ren et al.（2019）的随机种群动力学，我们计划引入并研究具有双向相互作用的种群模型。我们还建议探索的可能性，将空间结构的超过程和相关的SPDE与平均场相交的研究类似的行为。该研究有助于更好地理解种群内部和（或）种群之间的相互作用对种群动态极端行为的影响。此外，由于一般马尔可夫过程的边界行为和一般带跳随机微分方程的解还没有得到系统的研究，所提出的研究也有望对随机过程理论做出重大贡献。

Continuous-state branching process (CSBP for short) arises either from time-population scaling limit of the classical discrete-state Galton-Watson branching processes or from the Lamperti transform of a spectrally positive Lévy process stopped whenever reaching 0. Recent progresses have been made in introducing population models whose reproduction mechanisms depend on features of the entire populations. Among them a class of CSBPs with nonlinear branching mechanisms is introduced in Li (2019). They are obtained by generalized Lamperti type random time transformations from spectrally positive Lévy processes with general rate functions. Intuitively, the branching rate for such a process depends on its current population size. As a result, the additive branching property does not hold anymore and many standard methods for CSBPs fail. The nonlinear branching mechanism allows exotic properties such as coming down from infinity, which is investigated in detail in Foucart et al. (2020) with speeds of coming down from infinity identified for certain classes of rate functions. The boundary behaviors are also investigated in Li et al. (2019a) for more general nonlinear CSBPs. In the same spirit, Ren et al. (2019) propose and study a stochastic Lotka-Volterra population dynamics of two populations modeled by a system of two stochastic differential equations driven by independent Lévy noises, where both populations evolve according to nonlinear CSBPs of Li et al. (2019a) and the branching rates of the second population depend on the first population. We plan to continue with the study on the nonlinear CSBPs and the related interacting population systems. We are interested in the extinguishing behaviors of the nonlinear CSBP and want to know, when it occurs, how slowly the process approaches to 0. We also want to prove the strong Feller property for nonlinear CSBPs, which we believe will help to investigate the quasi-stationary distributions of the nonlinear CSBPs. We are going to further study the models in Li et al. (2019a) and the models in Ren et al. (2019). For the nonlinear CSBPs of Li et al. (2019a), a more challenging open problem is to establish sharp integral tests on boundary classification. For the stochastic population dynamics of Ren et al. (2019), we plan to introduce and study population models with two-way interactions. We also propose to explore the possibility of incorporating the spatial structures to study the similar behaviors for superprocesses and related SPDEs with mean field intersections. The proposed research helps to better understand the effects of interactions within and (or) between populations on the extreme behaviors of the population dynamics. In addition, since the boundary behaviors for general Markov processes and for solutions to general SDEs with jumps have not been systematically investigated, the proposed research is also expected to make significant contributions to the theory of stochastic processes.