Stochastic Interacting Population Dynamics and Related Problems

连续状态分支过程（CSBP）产生于经典离散状态Galton-Watson分支过程的时间种群尺度极限，或者产生于谱正的lsamvy过程的Lamperti变换，该过程在达到0时停止。最近在引入种群模型方面取得了进展，这些模型的繁殖机制取决于整个种群的特征。其中Li（2019）引入了一类具有非线性分支机制的csbp。它们是由具有一般速率函数的谱正lsamvy过程的广义Lamperti型随机时间变换得到的。直观地说，这种过程的分支速率取决于它当前的种群大小。结果，附加分支属性不再有效，csbp的许多标准方法都失败了。非线性分支机制允许奇异的性质，如从无穷远下降，这在Foucart等人（2020）中有详细的研究，从无穷远下降的速度确定了某些类别的速率函数。Li等人（2019a）也研究了更一般的非线性csbp的边界行为。基于同样的精神，Ren等人（2019）提出并研究了两个种群的随机Lotka-Volterra种群动态，该种群由独立lsamvy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等人（2019a）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csbp及其相互作用的种群系统进行研究。我们对非线性CSBP的熄灭行为感兴趣，并想知道，当它发生时，过程接近0的速度有多慢。我们还想证明非线性csbp的强Feller性质，我们相信这将有助于研究非线性csbp的拟平稳分布。我们将进一步研究Li et al. （2019a）和Ren et al.（2019）中的模型。对于Li et al. （2019a）的非线性csbp，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）的随机种群动态，我们计划引入并研究双向交互的种群模型。我们还建议探索将空间结构纳入研究具有平均场相交的超过程和相关spde的相似行为的可能性。本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的极端行为的影响。此外，由于一般马尔可夫过程和一般带跳跃的SDEs解的边界行为尚未得到系统的研究，因此本文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.